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+arXiv:2301.01215v1  [quant-ph]  3 Jan 2023
+Comment on ’The operational foundations of
+PT-symmetric and quasi-Hermitian quantum theory’
+Miloslav Znojil
+Nuclear Physics Institute ASCR, Hlavn´ı 130, 250 68 ˇReˇz, Czech Republic
+e-mail: [email protected]
+Abstract
+In J. Phys. A: Math. Theor. 55 (2022) 244003, Alase et al wrote that “the constraint of quasi-
+Hermiticity on observables” is not “sufficient to extend the standard quantum theory” because
+“such a system is equivalent to a standard quantum system.” Three addenda elucidating the
+current state of the art are found necessary. The first one concerns the project: In the related
+literature the original “aim of extending standard quantum theory” has already been abandoned
+shortly after its formulation. The second comment concerns the method, viz., the study in “the
+framework of general probabilistic theories” (GPT). It is noticed that a few other, mathematically
+consistent GPT-like theories are available. The authors do not mention, in particular, the progress
+achieved, under the quasi-Hermiticity constraint, in the approach using the effect algebras. We
+add that this approach already found its advanced realistic applications in the quasi-Hermitian
+models using the unbounded operators of observables acting in the infinite-dimensional Hilbert
+spaces. Thirdly, the “intriguing open question” about “what possible constraints, if any, could
+lead to such a meaningful extension” (in the future) is given an immediate tentative answer: The
+possibility is advocated that the desirable constraint could really be just the quasi-Hermiticity of
+the observables, provided only that one has in mind its recently developed non-stationary version.
+1
+
+1
+Introduction
+As a part of issue “Foundational Structures in Quantum Theory” the paper “The operational
+foundations of PT-symmetric and quasi-Hermitian quantum theory” by Abhijeet Alase, Salini
+Karuvade and Carlo Maria Scandolo [1] fitted very well the scope of the volume. In a rigorous
+mathematical style it offered the readers an interesting material confirming the compatibility
+between the three recent conceptual innovations of quantum theory. Still, we believe that the
+authors’ coverage of the subject deserves a few addenda, mainly because in loc. cit., the deeply
+satisfactory nature of the mathematical analysis seems to be accompanied by a perceivably less
+careful presentation of its implications in the context of the theoretical quantum physics.
+2
+The absence of extensions of standard quantum theory
+Our first addendum is motivated by the last sentence of the abstract in [1]. It states that “our
+results show that neither PT-symmetry nor quasi-Hermiticity constraints are sufficient to extend
+standard quantum theory consistently”.
+Indeed, it is rather unfortunate that this statement
+diverts attention from the very interesting main mathematical message of the paper (viz. from the
+rigorous confirmation of compatibility between the three alternative versions of quantum theory)
+to its much less satisfactory physical contextualization. The impression is further strengthened by
+the last paragraph of the whole text where we read that “in conclusion, neither PT-symmetry nor
+quasi-Hermiticity of observables leads to an extension of standard quantum mechanics.” Certainly,
+non-specialists could be mislead to interpret such a conclusion wrongly, as a disproof of usefulness
+of what is usually called PT-symmetric quantum theory (PTQT, an approach which is briefly
+reviewed in section 2.1 of loc. cit.) or of the so called quasi-Hermitian quantum theory (QHQT,
+cf. its compact review in the subsequent section 2.2 of loc. cit.). The misunderstanding seems
+completed by the combination of the very first sentence of the abstract with the very last sentence
+of the text: At the beginning of the Abstract we are told that “PT-symmetric quantum theory
+was originally proposed with the aim of extending standard quantum theory” (which is not too
+relevant at present), while the final question reads “what possible constraints, if any, could lead
+to such a meaningful extension” [1].
+The main weakness of such a “theory-extension” motivation and of the “physical” framing of
+paper [1] is that the original purpose of “relaxing the Hermiticity constraint on Hamiltonians”
+(as proposed, by Bender with Boettcher, in their enormously influential letter [2]) was almost
+immediately shown overambitious and unfulfilled (see, e.g., the Mostafazadeh’s 2010 very mathe-
+matical and detailed criticism and explanation “that neither PT-symmetry nor quasi-Hermiticity
+constraints are sufficient to extend standard quantum theory” [3]). Thus, the authors of [1] only
+come with their “aim to answer the question of whether a consistent physical theory with PT-
+symmetric observables extends standard quantum theory” too late. For more than twelve years
+the answer is known to be negative [4].
+3
+A comment on the method
+Naturally, nobody claims that the PTQT itself is not useful. Nobody could also deny the relevance
+and the novelty of the mathematical results presented in paper [1]. It is only a pity that its authors
+did not better emphasize how well their analysis fits the subject of the special issue, especially
+2
+
+due to their innovative turn of attention to the so called general probabilistic theories (GPT, cf.
+their compact outline in section 2.3 of [1]).
+Paradoxically, in the GPT context one immediately identifies the second weakness of the paper.
+It lies in a surprisingly short list of the GPT-approach-representing references. In the paper the
+list just incorporates the eight newer papers [5] - [12] (all of them published after the year 2000)
+plus a single older, Foulis-coauthored 1970 paper [13]. Not quite expectedly, the list of references
+does not contain any Gudder’s results – after all, paper [1] is a part of the special issue which is
+explicitly declared to honor his contribution to the field. Thus, one would expect, for example,
+a reference to his later review papers [14, 15] where he formulated one of the key GPT-related
+mathematical theses that “a physical system S under experimental investigation and governed by a
+scientific theory (which may be subject to modification in the light of new experimental evidence)
+is represented by a CB-effect algebra”. An equally unexpected gap in the references also concerns
+the absence of the Foulis’ pioneering, effect-algebras introducing 1994 paper with Bennet [16], or
+his comparatively recent review [17]. Indeed, both of these papers sought and offered operational
+foundations and gained insight into the GPT-motivating relationship between quantum theory
+and classical probability theory (this was emphasized also in [5], etc).
+What is an even worse omission is that the list of references does not contain any other subject-
+related studies like, e.g., paper [18] in which the predecessors of the present authors considered,
+explicitly, the PTQT-GPT relationship, having reconfirmed that “from the standpoint of (gen-
+eralized) effect algebra theory both representations of our quantum system coincide”. Similarly,
+the QHQT-GPT relationship may be found studied in paper [19] in which the mathematically
+fairly advanced analysis incorporated even the fairly realistic quantum models using unbounded
+operators. Indeed, the rate of the progress is striking, especially when one recalls just a few years
+younger report [14] in which the “separable complex Hilbert space” is assumed to be just “of
+dimension 1 or more”.
+4
+New and promising non-stationary constraints
+At present, it makes sense to accept the fact that in spite of the robust nature of the existing “stan-
+dard” formulations of quantum theory and, in particular, of the quantum mechanics of unitary
+systems, there still exist differences in the practical applicability of their various specific imple-
+mentations. The motivation of the diversity is that ”no [particular] formulation produces a royal
+road to quantum mechanics” [20]. In some sense this implies that the concept of the “extension”
+of the existing quantum theory is vague. The apparently minor technical differences between the
+current alternative formulations of quantum mechanics (as sampled, in [20], on elementary level)
+could happen to lead to “decisive extensions” in the future.
+A good illustrative example can be provided even within the current stationary forms of QHQT.
+Indeed, even in this framework the formalism can really be declared equivalent to its standard
+textbook form. Still, the equivalence can be confirmed only under certain fairly detailed and
+specific mathematical assumptions (cf. [21]). These assumptions are, even in the abstract context
+of functional analysis, far from trivial [22]. Paradoxically, even the popular physical quantum
+models of Bender and Boettcher [2] have been later found not to belong to the “admissible”,
+QHQT-compatible class (see, e.g., [23, 24] for the corresponding subtle details). Thus, in spite
+of their manifest and unbroken PT-symmetry, even these originally proposed benchmark models
+still wait for a “meaningful extension” of their fully consistent GPT interpretation.
+In our third, last addendum we are now prepared to reopen the vague but important question
+3
+
+of what the words of “extension” of the “standard” quantum theory could, or do, really mean.
+On one side, it is known and widely accepted that the various existing formulations of quantum
+theory “differ dramatically in mathematical and conceptual overview, yet each one makes identical
+predictions for all experimental results” [20]. On the other side, such a rigidity of the theory is
+far from satisfactory. For example, a suitable future amendment of quantum theory would be
+necessary for a still absent clarification of the concept of quantum gravity [25].
+For the sake of brevity let us skip here the discussion of the parallel questions concerning
+the PT-symmetric quantum models. This being said we believe that even the QHQT formalism
+itself did not say its last word yet. Indeed, our optimism concerning its potential “theory exten-
+sion status” is based on the recent fundamental clarifications of its scope and structure. First of
+all, it became clear that in the QHQT descriptions of unitary systems it is sufficient to distin-
+guish just between their representations in the “generalized Schr¨odinger picture” (GSP, stationary
+and best presented, by our opinion, in reviews [21, 26] and [3]) and in its non-stationary “non-
+Hermitian interaction picture” alternative (NIP, [27, 28]). Using this terminology one immediately
+reveals that the QHQT-related considerations of paper [1] just cover the GSP approach. In other
+words, the physical inner-product metric (denoted by symbol η) is perceived there as strictly
+time-independent. This means that in the GSP language one can easily identify the (stationary)
+generator G of the evolution of the wave functions with the (“observable-energy”) Hamiltonian H
+(which has real spectrum and is, by assumption, η−quasi-Hermitian).
+The situation becomes different after the extension of the QHQT approach to the non-stationary,
+NIP dynamical regime. In this case we will denote the inner-product metric by another dedicated
+symbol Θ = Θ(t) as introduced in the first description of NIP in [29]. What is important is that
+the observable-energy operator H = H(t) will get split in the sum of the two auxiliary operators
+G(t) and Σ(t). As long as they are both neither observable nor Θ-quasi-Hermitian in general,
+we will exclusively assign the name of the Hamiltonian to the instantaneous energy operator H
+(with real spectrum), adding a word of warning that a different, less consequent terminology is
+often used by some other authors (see, e.g., [30, 31]). Even though neither the spectrum of G(t)
+nor the spectrum of Σ(t) is real in general, the introduction of these operators endows the NIP
+formalism with an additional flexibility, capable, as we believe, of opening the new horizons in the
+contemporary quantum physics: In the context of relativistic quantum mechanics, for example,
+such a hypothetical “theory-extension” possibility has been discussed, in detail, in [27]. For the
+purposes of a potentially new approach to the problem of the unitary-evolution models of quantum
+phase transitions in many-body context, the formalism has slightly been adapted in [32]. Last but
+not least, our very recent paper [28] has been devoted to the possible use of the NIP evolution
+equations in a Wheeler-DeWitt-equation-based schematic model of Big Bang in the context of
+quantum gravity and cosmology. In this spirit, therefore, certain sufficiently realistic NIP-based
+models could easily happen to acquire an “extended quantum mechanics” status, perhaps, in the
+nearest future.
+5
+Conclusions
+The key subject discussed in paper [1] was the question of the possible extension of the scope of
+quantum theory in general, and of the realization of such an ambitious project, in the respective
+PTQT and QHQT theoretical frameworks, in particular. In our present commentary we reminded
+the readers, marginally, of the existence of several older, comparably sceptical conclusions as
+available in the related literature (see section 2 for details). In section 3 we then added a few
+4
+
+similar broader-context-emphasizing remarks on the mathematical, GPT-related aspects of the
+results of [1]. Still, the core of our present message (as presented in the longest section 4) concerned
+physics. We pointed out that at present, the question of the possible extension of the scope of
+the standard quantum theory should be considered open even in the narrower PTQT and QHQT
+frameworks.
+In support of the latter statement we mentioned that
+• even for the stationary and, apparently, most elementary PTQT potentials (sampled, say, by
+the most popular V (x) = ix3), the widespread initial optimism and intuitive “nothing new”
+understanding of their physical meaning and mathematical background have both recently
+been shattered by their more rigorous mathematical analysis;
+• one can hardly say “nothing new” even in a mathematically much better understood station-
+ary QHQT alias GSP framework where, typically, the use of certain stronger assumptions
+enables one to circumvent the obstacles revealed by rigorous mathematics. Indeed, even in
+the GSP framework one can search for an entirely new physics. Typically, a non-standard
+phenomenology becomes described by the QHQT models in an infinitesimally small vicinity
+of the so called exceptional points: Paper [33] offers an illustrative sample of the quantum
+systems which cannot be described by the standard quantum theory;
+• in fact, our return to optimism and expectation that the QHQT may be a “fundamentally
+innovative” theory found its most explicit formulation in section 4. Briefly we exposed there
+an enormous growth of the flexibility of the QHQT approach after its ultimate non-stationary
+NIP generalization. In some sense, the emphasis put upon the deeply promising conceptual
+nature of such a flexibility can be read as the deepest core of our present comment and
+message.
+Data availability statement
+No new data were created or analysed in this study.
+ORCID iD
+https://orcid.org/0000-0001-6076-0093
+5
+
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+7
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+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=' A: Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=' 55 (2022) 244003, Alase et al wrote that “the constraint of quasi-  Hermiticity on observables” is not “sufficient to extend the standard quantum theory” because  “such a system is equivalent to a standard quantum system.” Three addenda elucidating the  current state of the art are found necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=' The first one concerns the project: In the related  literature the original “aim of extending standard quantum theory” has already been abandoned  shortly after its formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=' The second comment concerns the method, viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=', the study in “the  framework of general probabilistic theories” (GPT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=' It is noticed that a few other, mathematically  consistent GPT-like theories are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=' The authors do not mention, in particular, the progress  achieved, under the quasi-Hermiticity constraint, in the approach using the effect algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=' We  add that this approach already found its advanced realistic applications in the quasi-Hermitian  models using the unbounded operators of observables acting in the infinite-dimensional Hilbert  spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=' Thirdly, the “intriguing open question” about “what possible constraints, if any, could  lead to such a meaningful extension” (in the future) is given an immediate tentative answer: The  possibility is advocated that the desirable constraint could really be just the quasi-Hermiticity of  the observables, provided only that one has in mind its recently developed non-stationary version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content='  1  1  Introduction  As a part of issue “Foundational Structures in Quantum Theory” the paper “The operational  foundations of PT-symmetric and quasi-Hermitian quantum theory” by Abhijeet Alase, Salini  Karuvade and Carlo Maria Scandolo [1] fitted very well the scope of the volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=' In a rigorous  mathematical style it offered the readers an interesting material confirming the compatibility  between the three recent conceptual innovations of quantum theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=' Still, we believe that the  authors’ coverage of the subject deserves a few addenda, mainly because in loc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=' cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=', the deeply  satisfactory nature of the mathematical analysis seems to be accompanied by a perceivably less  careful presentation of its implications in the context of the theoretical quantum physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content='  2  The absence of extensions of standard quantum theory  Our first addendum is motivated by the last sentence of the abstract in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=' It states that “our  results show that neither PT-symmetry nor quasi-Hermiticity constraints are sufficient to extend  standard quantum theory consistently”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content='  Indeed, it is rather unfortunate that this statement  diverts attention from the very interesting main mathematical message of the paper (viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=' from the  rigorous confirmation of compatibility between the three alternative versions of quantum theory)  to its much less satisfactory physical contextualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=' The impression is further strengthened by  the last paragraph of the whole text where we read that “in conclusion, neither PT-symmetry nor  quasi-Hermiticity of observables leads to an extension of standard quantum mechanics.” Certainly,  non-specialists could be mislead to interpret such a conclusion wrongly, as a disproof of usefulness  of what is usually called PT-symmetric quantum theory (PTQT, an approach which is briefly  reviewed in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content='1 of loc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=' cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=') or of the so called quasi-Hermitian quantum theory (QHQT,  cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=' its compact review in the subsequent section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content='2 of loc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=' cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=' The misunderstanding seems  completed by the combination of the very first sentence of the abstract with the very last sentence  of the text: At the beginning of the Abstract we are told that “PT-symmetric quantum theory  was originally proposed with the aim of extending standard quantum theory” (which is not too  relevant at present), while the final question reads “what possible constraints, if any, could lead  to such a meaningful extension” [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content='  The main weakness of such a “theory-extension” motivation and of the “physical” framing of  paper [1] is that the original purpose of “relaxing the Hermiticity constraint on Hamiltonians”  (as proposed, by Bender with Boettcher, in their enormously influential letter [2]) was almost  immediately shown overambitious and unfulfilled (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=', the Mostafazadeh’s 2010 very mathe-  matical and detailed criticism and explanation “that neither PT-symmetry nor quasi-Hermiticity  constraints are sufficient to extend standard quantum theory” [3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=' Thus, the authors of [1] only  come with their “aim to answer the question of whether a consistent physical theory with PT-  symmetric observables extends standard quantum theory” too late.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=' For more than twelve years  the answer is known to be negative [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content='  3  A comment on the method  Naturally, nobody claims that the PTQT itself is not useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=' Nobody could also deny the relevance  and the novelty of the mathematical results presented in paper [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=' It is only a pity that its authors  did not better emphasize how well their analysis fits the subject of the special issue, especially  2  due to their innovative turn of attention to the so called general probabilistic theories (GPT, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content='  their compact outline in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content='3 of [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content='  Paradoxically, in the GPT context one immediately identifies the second weakness of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content='  It lies in a surprisingly short list of the GPT-approach-representing references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=' In the paper the  list just incorporates the eight newer papers [5] - [12] (all of them published after the year 2000)  plus a single older, Foulis-coauthored 1970 paper [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=' Not quite expectedly, the list of references  does not contain any Gudder’s results – after all, paper [1] is a part of the special issue which is  explicitly declared to honor his contribution to the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=' Thus, one would expect, for example,  a reference to his later review papers [14, 15] where he formulated one of the key GPT-related  mathematical theses that “a physical system S under experimental investigation and governed by a  scientific theory (which may be subject to modification in the light of new experimental evidence)  is represented by a CB-effect algebra”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=' An equally unexpected gap in the references also concerns  the absence of the Foulis’ pioneering, effect-algebras introducing 1994 paper with Bennet [16], or  his comparatively recent review [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=' Indeed, both of these papers sought and offered operational  foundations and gained insight into the GPT-motivating relationship between quantum theory  and classical probability theory (this was emphasized also in [5], etc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content='  What is an even worse omission is that the list of references does not contain any other subject-  related studies like, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=', paper [18] in which the predecessors of the present authors considered,  explicitly, the PTQT-GPT relationship, having reconfirmed that “from the standpoint of (gen-  eralized) effect algebra theory both representations of our quantum system coincide”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=' Similarly,  the QHQT-GPT relationship may be found studied in paper [19] in which the mathematically  fairly advanced analysis incorporated even the fairly realistic quantum models using unbounded  operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=' Indeed, the rate of the progress is striking, especially when one recalls just a few years  younger report [14] in which the “separable complex Hilbert space” is assumed to be just “of  dimension 1 or more”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content='  4  New and promising non-stationary constraints  At present, it makes sense to accept the fact that in spite of the robust nature of the existing “stan-  dard” formulations of quantum theory and, in particular, of the quantum mechanics of unitary  systems, there still exist differences in the practical applicability of their various specific imple-  mentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=' The motivation of the diversity is that ”no [particular] formulation produces a royal  road to quantum mechanics” [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=' In some sense this implies that the concept of the “extension”  of the existing quantum theory is vague.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=' The apparently minor technical differences between the  current alternative formulations of quantum mechanics (as sampled, in [20], on elementary level)  could happen to lead to “decisive extensions” in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content='  A good illustrative example can be provided even within the current stationary forms of QHQT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content='  Indeed, even in this framework the formalism can really be declared equivalent to its standard  textbook form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=' Still, the equivalence can be confirmed only under certain fairly detailed and  specific mathematical assumptions (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=' [21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=' These assumptions are, even in the abstract context  of functional analysis, far from trivial [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=' Paradoxically, even the popular physical quantum  models of Bender and Boettcher [2] have been later found not to belong to the “admissible”,  QHQT-compatible class (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=', [23, 24] for the corresponding subtle details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=' Thus, in spite  of their manifest and unbroken PT-symmetry, even these originally proposed benchmark models  still wait for a “meaningful extension” of their fully consistent GPT interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content='  In our third, last addendum we are now prepared to reopen the vague but important question  3  of what the words of “extension” of the “standard” quantum theory could, or do, really mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content='  On one side, it is known and widely accepted that the various existing formulations of quantum  theory “differ dramatically in mathematical and conceptual overview, yet each one makes identical  predictions for all experimental results” [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=' On the other side, such a rigidity of the theory is  far from satisfactory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=' For example, a suitable future amendment of quantum theory would be  necessary for a still absent clarification of the concept of quantum gravity [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content='  For the sake of brevity let us skip here the discussion of the parallel questions concerning  the PT-symmetric quantum models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=' This being said we believe that even the QHQT formalism  itself did not say its last word yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=' Indeed, our optimism concerning its potential “theory exten-  sion status” is based on the recent fundamental clarifications of its scope and structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=' First of  all, it became clear that in the QHQT descriptions of unitary systems it is sufficient to distin-  guish just between their representations in the “generalized Schr¨odinger picture” (GSP, stationary  and best presented, by our opinion, in reviews [21, 26] and [3]) and in its non-stationary “non-  Hermitian interaction picture” alternative (NIP, [27, 28]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=' Using this terminology one immediately  reveals that the QHQT-related considerations of paper [1] just cover the GSP approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=' In other  words, the physical inner-product metric (denoted by symbol η) is perceived there as strictly  time-independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=' This means that in the GSP language one can easily identify the (stationary)  generator G of the evolution of the wave functions with the (“observable-energy”) Hamiltonian H  (which has real spectrum and is, by assumption, η−quasi-Hermitian).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content='  The situation becomes different after the extension of the QHQT approach to the non-stationary,  NIP dynamical regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=' In this case we will denote the inner-product metric by another dedicated  symbol Θ = Θ(t) as introduced in the first description of NIP in [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=' What is important is that  the observable-energy operator H = H(t) will get split in the sum of the two auxiliary operators  G(t) and Σ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=' As long as they are both neither observable nor Θ-quasi-Hermitian in general,  we will exclusively assign the name of the Hamiltonian to the instantaneous energy operator H  (with real spectrum), adding a word of warning that a different, less consequent terminology is  often used by some other authors (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=', [30, 31]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=' Even though neither the spectrum of G(t)  nor the spectrum of Σ(t) is real in general, the introduction of these operators endows the NIP  formalism with an additional flexibility, capable, as we believe, of opening the new horizons in the  contemporary quantum physics: In the context of relativistic quantum mechanics, for example,  such a hypothetical “theory-extension” possibility has been discussed, in detail, in [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=' For the  purposes of a potentially new approach to the problem of the unitary-evolution models of quantum  phase transitions in many-body context, the formalism has slightly been adapted in [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=' Last but  not least, our very recent paper [28] has been devoted to the possible use of the NIP evolution  equations in a Wheeler-DeWitt-equation-based schematic model of Big Bang in the context of  quantum gravity and cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=' In this spirit, therefore, certain sufficiently realistic NIP-based  models could easily happen to acquire an “extended quantum mechanics” status, perhaps, in the  nearest future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content='  5  Conclusions  The key subject discussed in paper [1] was the question of the possible extension of the scope of  quantum theory in general, and of the realization of such an ambitious project, in the respective  PTQT and QHQT theoretical frameworks, in particular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=' In our present commentary we reminded  the readers, marginally, of the existence of several older, comparably sceptical conclusions as  available in the related literature (see section 2 for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=' In section 3 we then added a few  4  similar broader-context-emphasizing remarks on the mathematical, GPT-related aspects of the  results of [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=' Still, the core of our present message (as presented in the longest section 4) concerned  physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=' We pointed out that at present, the question of the possible extension of the scope of  the standard quantum theory should be considered open even in the narrower PTQT and QHQT  frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content='  In support of the latter statement we mentioned that  even for the stationary and, apparently, most elementary PTQT potentials (sampled, say, by  the most popular V (x) = ix3), the widespread initial optimism and intuitive “nothing new”  understanding of their physical meaning and mathematical background have both recently  been shattered by their more rigorous mathematical analysis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content='  one can hardly say “nothing new” even in a mathematically much better understood station-  ary QHQT alias GSP framework where, typically, the use of certain stronger assumptions  enables one to circumvent the obstacles revealed by rigorous mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=' Indeed, even in  the GSP framework one can search for an entirely new physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=' Typically, a non-standard  phenomenology becomes described by the QHQT models in an infinitesimally small vicinity  of the so called exceptional points: Paper [33] offers an illustrative sample of the quantum  systems which cannot be described by the standard quantum theory;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content='  in fact, our return to optimism and expectation that the QHQT may be a “fundamentally  innovative” theory found its most explicit formulation in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=' Briefly we exposed there  an enormous growth of the flexibility of the QHQT approach after its ultimate non-stationary  NIP generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=' In some sense, the emphasis put upon the deeply promising conceptual  nature of such a flexibility can be read as the deepest core of our present comment and  message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content='  Data availability statement  No new data were created or analysed in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content='  ORCID iD  https://orcid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content='org/0000-0001-6076-0093  5  References  [1] Alase A, Karuvade S and Scandolo C M 2022 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=' A: Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=' 55 244003  [2] Bender C M and Boettcher S 1998 Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=' 80 5243 - 5246  [3] Mostafazadeh A 2010 Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=' Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=' Methods Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=' 07 1191 - 1306  [4] Znojil M 2015 Non-Self-Adjoint Operators in Quantum Physics: Ideas,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=' People,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=' and Trends  (New York:Wiley) ch 1 pp 7 - 58  [5] Hardy L 2001 Quantum theory from five reasonable axioms (arXiv:quant-ph/0101012)  [6] Barrett J 2007 Phys Rev A 75 032304  [7] Chiribella G,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=' D’Ariano G M and Perinotti P 2010 Phys Rev A 81 062348  [8] Hardy L 2011 Foliable Operational Structures for General Probabilistic Theories (Cambridge:  Cambridge University Press) pp 409 - 442  [9] Barnum H and Wilce A 2011 Electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=' Notes Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=' 270 3 - 15  [10] Janotta P and Hinrichsen H 2014 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=' A: Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=' 47 323001  [11] Barnum H and Wilce A 2016 Post-Classical Probability Theory (Berlin: Springer) pp 367.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content='420  [12] Scandolo C M 2018 Information-theoretic foundations of thermodynamics in general proba-  bilistic theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=' PhD Thesis,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=' University of Oxford  [13] Randall C H and Foulis D J 1970 Am Math Mon 77 363 - 374  [14] Gudder S P 2004 Rep Math Phys 54 93 - 114  [15] Gudder S P 2006 Demonstratio Math 39 43 - 54  [16] Foulis D J and Bennett M K 1994 Found Phys 24 1331 - 1352  [17] Foulis D J 2007 Rep Math Phys 60 329 - 346  [18] Paseka J 2011 Int J Theor Phys 50 1198 - 1205  [19] Paseka J,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=' Pulmannova S and Riecanova Z 2013 Int J Theor Phys 52 1994 - 2000  [20] Styer D F et al 2002 Am J Phys 70 288 - 297  [21] Scholtz F G,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=' Geyer H B and Hahne F J W 1992 Ann Phys 213 74 - 101  [22] Dieudonne J 1961 Proc Int Symp Lin Spaces (Pergamon,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=' Oxford,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=' 1961),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=' 115 - 122  [23] Siegl P and Krejˇciˇr´ık D 2012 Phys Rev D 86 121702(R)  Krejˇciˇr´ık D, Siegl P, Tater M and Viola J 2015 J Math Phys 56 103513  6  [24] Guenther U and Frank Stefani F 2019 IR-truncated PT -symmetric ix3 model and its asymp-  totic spectral scaling graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=' arXiv 1901.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content='08526  Semor´adov´a I and Siegl P 2022 SIAM J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content=' 54 5064 - 5101  [25] Rovelli C 2004 Quantum Gravity (Cambridge University Press, Cambridge, UK)  [26] Bender C M 2007 Rep Prog Phys 70 947 - 1118  [27] Znojil M 2017 Ann Phys (NY) 385 162 - 179  [28] Znojil M 2022 Universe 8 385  [29] Znojil M 2008 Phys Rev D 78 085003  Znojil M 2009 SIGMA 5 001  [30] Fring A and Moussa M H Y 2016 Phys Rev A 93 042114  [31] B´ıla H 2009 e-print arXiv: 0902.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content='0474  Gong J-B and Wang Q-H 2013 J Phys A Math Theor 46 485302  Amaouche N, Sekhri M, Zerimeche R, Maamache M and Liang J-Q 2022 eprint:  arXiv:2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}
+page_content='02477 (quant-ph)  [32] Bishop R F and Znojil M 2020 Eur Phys J Plus 135 374  [33] Znojil M 2022 Mathematics 10 3721  7' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'}

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+arXiv:2301.04190v1  [math.DG]  10 Jan 2023
+NOTES ON HARMONIC MAPS
+GEORGIOS DASKALOPOULOS AND CHIKAKO MESE
+This is set of notes prepared for the Summer School on non-Abelian Hodge theory
+in Abbaye de Saint-Jacut de la Mer June, 6-19, 2022.
+Table of Contents
+Lecture 1: Harmonic Maps Between Riemannian Manifolds
+p. 2
+Lecture 2: Existence and regularity
+p. 9
+Lecture 3: Pluriharmonic Maps and the Siu-Sampson Formula p. 16
+Lecture 4: Donaldson Corlette Theorem
+p. 30
+Date: June 2022.
+GD supported in part by NSF DMS-2105226, CM supported in part by NSF DMS-2005406. We
+would like to thank Yitong Sun for carefully reading this document and making useful suggestions to
+improve the exposition of this paper.
+1
+
+2
+GEORGIOS DASKALOPOULOS AND CHIKAKO MESE
+1. Harmonic Maps Between Riemannian manifolds
+1.1. Introduction: Basics. In this section, we define energy of maps between Rie-
+mannian manifolds, harmonic maps, and the first and second variation formulas after
+the pioneering work of Eells-Sampson [ES]. A good reference is also [J].
+1.2. The energy of maps. Let (M, g), (N, h) be Riemannian manifolds. Let f :
+M → N be a smooth map which induces a map df : TM → TN
+df
+� ∂
+∂xα
+� ����
+p
+= ∂f i
+∂xα
+∂
+∂yi
+����
+f(p)
+where (xα) (resp. (yi)) are the local coordinates of M (resp. N).
+The map f also induces a vector bundle f ∗TN over M. Let ∇ be a connection on
+f ∗TN inherited from the Levi-Civita connection on TN. Then ∇ induces an exterior
+derivative
+d∇ : C∞((ΛpT ∗M) ⊗ f ∗TN) → C∞((Λp+1T ∗M) ⊗ f ∗TN).
+We view df as a section
+df ∈ C∞(T ∗M ⊗ f ∗TN) = Ω1(f ∗TN).
+Using the notation
+∂
+∂f i := ∂
+∂yi ◦ f ∈ C∞
+loc(M, f ∗TN),
+we have
+df = ∂f i
+∂xα dxα ⊗ ∂
+∂f i .
+Let (gαβ) (resp. (hij)) be the expression of the Riemannian metric g of M (resp. h
+of N) with respect to local coordinates (xα) (resp. (yi)).
+Definition 1.1. Set
+e(f) := 1
+2|df|2 = 1
+2
+∂f i
+∂xα
+∂f j
+∂xβ gαβhij ◦ f.
+The energy of f is
+E(f) :=
+�
+M
+e(f) ⋆ 1 = 1
+2
+�
+M
+gαβ(x)hij(f(x)) ∂f i
+∂xα
+∂f j
+∂xβ
+�
+g(x) dx1 ∧ · · · ∧ dxn.
+Here, recall that the Hodge star operator ⋆ : ΛkTM → Λn−kTM, is the unique linear
+operator such that for all α, β ∈ ΛkV ,
+α ∧ ⋆β = g(α, β) ⋆ 1.
+
+NOTES ON HARMONIC MAPS
+3
+Lemma 1.2. Let f = (ft) be a smooth one-parameter family of C∞ maps
+f : M × (−ǫ, ǫ) → N,
+f(x, t) = ft(x).
+Then
+∇∂f
+∂t = ∇∂/∂tdf
+where f = ft and
+∂f
+∂t = ∂f i
+∂t
+∂
+∂f i ∈ C∞(f ∗TN).
+Proof. Both ∇ ∂f
+∂t = ∇∂/∂xα ∂f
+∂t dxα and ∇∂/∂tdf = ∇∂/∂t
+∂f
+∂xαdxα are 1-forms with values
+in f ∗TN. Here,
+∂f
+∂xα = ∂f j
+∂xα
+∂
+∂f j ∈ C∞(f ∗TN).
+Consider f as a map f : M × (−ǫ, ǫ) → N. Since
+�
+f∗
+� ∂
+∂t
+�
+, f∗
+� ∂
+∂xα
+��
+= f∗
+� ∂
+∂t,
+∂
+∂xα
+�
+= 0
+and ∇ is torsion-free,
+∇∂/∂xα ∂f
+∂t =
+�
+∇f∗(
+∂
+∂xα)f∗
+� ∂
+∂t
+��
+◦ f =
+�
+∇f∗( ∂f
+∂t )f∗
+� ∂
+∂xα
+��
+◦ f = ∇∂/∂tdf( ∂
+∂xα)
+which proves the equality.
+□
+Corollary 1.3 (First Variation Formula). For (ft) as above,
+d
+dtE(ft) =
+�
+M
+�
+∇∂f
+∂t , df
+�
+⋆ 1.
+Proof. We compute
+d
+dtE(ft) = 1
+2
+�
+M
+d
+dt ⟨df, df⟩ ⋆ 1
+=
+�
+M
+�
+∇∂/∂tdf, df
+�
+⋆ 1
+=
+�
+M
+�
+∇∂f
+∂t , df
+�
+⋆ 1.
+□
+Corollary 1.4. The critical points f of the functional E satisfy
+�
+M
+⟨∇ψ, df⟩ ⋆ 1 = 0,
+∀ψ ∈ C∞(f ∗TN).
+(1.1)
+By taking ψ compactly supported away from ∂M, we obtain the Euler Lagrange equation
+of E,
+τ(f) := −d⋆
+∇df = ⋆d∇ ⋆ df = 0.
+(1.2)
+
+4
+GEORGIOS DASKALOPOULOS AND CHIKAKO MESE
+Here, ∇ is the pullback of the Levi-Civita connection on f ∗(TN).
+Proof. Let ft be a family of maps with
+d
+dt
+����
+t=0
+ft = ψ ∈ C∞(f ∗TN).
+Taking ψ compactly supported and integrating by parts,
+d
+dt
+����
+t=0
+E(ft) =
+�
+M
+⟨ψ, −τf⟩ ⋆ 1 = 0
+which holds for every ψ ∈ C∞
+c (f ∗TM) iff τf = 0.
+□
+Definition 1.5. A smooth map f : M → N satisfying d⋆
+∇df = 0 is called a harmonic
+map.
+1.3. Harmonic map equations in local coordinates. First define
+ω
+� ∂
+∂yi
+�
+:= Γi
+jkdyk ⊗ ∂
+∂yj ,
+where Γi
+jk are Christoffel symbols on N and set
+˜ω := ω ◦ f =
+�
+Γk
+ijdyk ⊗ ∂
+∂yj
+�
+◦ f = (Γj
+ik ◦ f)∂f k
+∂xβ dxβ ⊗ ∂
+∂f j .
+Then d∇ = d + ˜ω and
+d⋆
+∇df = − ⋆ d∇ ⋆ df
+= − ⋆ (d + ˜ω)
+�
+⋆df i ⊗ ∂
+∂f i
+�
+= −(⋆d ⋆ df i) ∂
+∂f i − (−1)m−1 ⋆
+�
+⋆df i ⊗ ˜ω
+� ∂
+∂f i
+��
+= −∆f i
+∂
+∂f i − (−1)m−1 ⋆
+� ∂f i
+∂xα ⋆ dxα ∧ (Γj
+ik ◦ f)∂f k
+∂xβ dxβ ⊗ ∂
+∂f j
+�
+= −∆f i
+∂
+∂f i − (−1)m−1
+� ∂f i
+∂xα
+∂f k
+∂xβ Γj
+ik ◦ f ��� (⋆dxα ∧ dxβ) ⊗
+∂
+∂f j
+�
+= −
+�
+∆f k + gαβ ∂f i
+∂xα
+∂f j
+∂xβ Γk
+ij ◦ f
+� ∂
+∂f k .
+Thus the harmonic map equation is
+∆f k + gαβ ∂f i
+∂xα
+∂f j
+∂xβ Γk
+ij ◦ f = 0.
+(1.3)
+Examples 1.6.
+(1) Suppose N = R, then the harmonic map equation reduces to ∆f = 0; i.e. f is a
+harmonic function on M.
+
+NOTES ON HARMONIC MAPS
+5
+(2) Suppose M = S1. Then
+E(f) = 1
+2
+� 2π
+0
+| ˙f(t)|dt
+and the critical points of E(f) are geodesics. We can also see this from the harmonic
+maps equation. Since S1 is 1-dimensional we can take gαβ = δαβ. Then
+∂2f k
+∂t2 + Γk
+ij
+∂f i
+∂xα
+∂f j
+∂xβ = 0.
+This is the geodesic equation.
+(3) We’ll show later that holomorphic maps between K¨ahler manifolds are harmonic
+(cf. Remark 3.3).
+1.4. The Dirichlet and Neumann problems. If M has non-empty boundary (N
+is without boundary) there are two different boundary value problems to consider:
+• Dirichlet problem: Minimize E in a fixed homotopy class of maps from M to
+N relative to the boundary of M. This is equivalent to considering compactly
+supported variations ψ.
+• Neumann problem: Minimize E in a fixed free homotopy class of maps from
+M to N, in other words no restriction on the type of variations.
+1.5. The second variation. The Riemannian tensor
+RN : TN × TN × TN → TN
+induces and operator
+RN : f ∗TN × f ∗TN × f ∗TN → f ∗TN
+in the natural way.
+Lemma 1.7. Let ft : M → N, and let V be a vector field along ft. Then
+∇∂/∂t∇V = ∇∇∂/∂tV − RN
+�
+df, ∂f
+∂t
+�
+V.
+Proof. Both
+∇∂/∂t∇V − ∇∇∂/∂tV
+=
+�
+∇∂/∂t∇∂/∂xαV − ∇∂/∂xα∇∂/∂tV
+�
+dxα
+=
+��
+∇f∗(∂/∂t)∇f∗(∂/∂xα)V − ∇f∗(∂/∂xα)∇f∗(∂/∂t)
+�
+f∗V
+�
+◦ f dxα
+and
+RN
+�
+df, ∂f
+∂t
+�
+V =
+�
+RN
+�
+f∗
+� ∂
+∂xα
+�
+, f∗
+� ∂
+∂t
+��
+f∗(V )
+�
+◦ f dxα
+are 1−forms on M with values in f ∗TN. Thus, assertion follows from the definition
+of the Riemannian tensor RN.
+□
+
+6
+GEORGIOS DASKALOPOULOS AND CHIKAKO MESE
+Theorem 1.8 (Second Variation Formula). One has
+d2
+dt2E(ft) = ||∇∂f
+∂t ||2 −
+�
+M
+�
+RN
+�
+df, ∂f
+∂t
+� ∂f
+∂t , df
+�
+⋆ 1 +
+�
+M
+�
+∇∂/∂t
+∂f
+∂t , τf
+�
+⋆ 1.
+Proof. We compute
+d2
+dt2E(ft) =
+�
+M
+d
+dt
+�
+∇∂f
+∂t , df
+�
+⋆ 1
+=
+�
+M
+��
+∇∂/∂t∇∂f
+∂t , df
+�
++
+�
+∇∂f
+∂t , ∇∂/∂tdf
+��
+⋆ 1
+=
+�
+M
+��
+∇∇∂/∂t
+∂f
+∂t , df
+�
+−
+�
+RN
+�
+df, ∂f
+∂t
+� ∂f
+∂t , df
+�
++ ||∇∂f
+∂t ||2
+�
+⋆ 1.
+□
+2. Existence and regularity
+2.1. Introduction: Non-positive curvature. In this section, we examine the role
+of non-positive curvature of the target metric on harmonic maps. We show uniqueness
+and discuss regularity. We also study the equivariant problem and prove existence of
+equivariant harmonic maps into non-positively curved metric spaces. Some references
+are [S], [KS1], [KS2] and [GS].
+2.2. Second variation formula and non-positive curvature. The following are
+corollaries of Theorem 1.8.
+Corollary 2.1. If N has ≤ 0 sectional curvature and ft is a geodesic interpolation,
+then E(ft) is convex.
+Proof. In the second variation formula the last term vanishes and the others are ≥
+0.
+□
+Corollary 2.2. Let f, φ : M → N be homotopic with f|∂M = φ|∂M. If N has ≤ 0
+sectional curvature and f is harmonic, then
+E(f) ≤ E(φ).
+Proof. Let ft be a geodesic homotopy between f, φ, thus f0 = f, f1 = φ.
+Then
+E(t) = E(ft) is convex, and E′(0) = 0. So E(1) ≥ E(0), hence E(φ) ≥ E(f).
+□
+Corollary 2.3. If f0, f1 : M → N are homotopic harmonic maps with f0|∂M = f1|∂M
+and N has ≤ 0 sectional curvature, then:
+(1) If ∂M is nonempty, then f0 = f1.
+(2) If ∂M is empty, F is a geodesic homotopy between f0, f1 and N has sectional
+curvature < 0 at one point p in the image of F, then either f0 = f1 or the rank
+of f0 is ≤ 1.
+
+NOTES ON HARMONIC MAPS
+7
+Proof. (1) Let ft be a geodesic homotopy between f0, f1, E(t) = E(ft). Then E is
+convex. Since E′(0) = E′(1) = 0, we conclude E′(t) = 0 = E′′(t). By Theorem 1.8,
+∇∂F
+∂t = 0
+and
+�
+RN
+�
+df, ∂f
+∂t
+� ∂f
+∂t , df
+�
+= 0.
+Thus,
+∂
+∂xα||∂F
+∂t ||2 = 2
+�
+∇∂/∂xα ∂F
+∂t , ∂F
+∂t
+�
+= 0
+which implies that ||∂F/∂t|| is constant.
+But ∂F/∂t = 0 on ∂M, so ∂F/∂t = 0
+everywhere if ∂M is nonempty and hence f0 = f1.
+(2) If ||∂F/∂t|| = 0, then f0 = f1. Otherwise, ∂F/∂t ̸= 0 for every x, t. The negative
+sectional curvature at p implies df is parallel to ∂F/∂t at p and therefore everywhere.
+Thus, the image of df has dimension ≤ 1.
+□
+2.3. The Weitzenb¨ock formula.
+Theorem 2.4. Let f : M → N be a harmonic map and (eα) an orthonormal frame
+for TM. Then
+∆e(f) = |∇df|2 + 1
+2
+�
+df(RicM(eα)), df(eα)
+�
+− 1
+2
+�
+RN(df(eα), df(eβ))df(eβ), df(eα)
+�
+.
+Proof. Expanding out the Laplacian with respect to local coordinates, the harmonic
+map equation (1.3) is
+gαβf i
+/αβ − gαβ MΓη
+αβf i
+/η + gαβ NΓi
+kℓ ◦ ff k
+/αf ℓ
+/β = 0.
+We use normal coordinates at x ∈ M and f(x) = y. Thus, the metric tensors (gαβ)
+and (hij) are Euclidean up to first order at x and y respectively. Differentiating,
+f i
+/ααε = MΓη
+αα/εf i
+/η − NΓi
+kℓ/mf m
+/εf k
+/αf ℓ
+/α
+= 1
+2(gαη/αε + gαη/αε − gαα/ηε)f i
+/η
+− 1
+2(hki/ℓm + hℓi/km − hkℓ/im)f m
+/εf k
+/αf ℓ
+/α.
+Furthermore,
+gαβ
+/ǫǫ = −gαβ/ǫǫ
+and
+△hij(f(x)) = hij/klf k
+/ǫf k
+/ǫ.
+
+8
+GEORGIOS DASKALOPOULOS AND CHIKAKO MESE
+Thus,
+∆
+�1
+2gαβhij ◦ ff i
+/αf j
+/β
+�
+=
+1
+√g
+∂
+∂xσ
+�√ggστ ∂
+∂xτ
+�1
+2gαβhij ◦ ff i
+/αf j
+/β
+��
+= f i
+/ασf i
+/ασ − 1
+2(gαβ/σσ + gσσ/αβ − gσα/σβ − gσα/σβ)f i
+/αf i
+/β
++ 1
+2(hij/kℓ + hkℓ/ji − hik/jℓ − hjℓ/ik)f i
+/αf j
+/αf k
+/σf ℓ
+/σ
+= f i
+/ασf i
+/ασ + 1
+2RicM
+αβf i
+/αf j
+/β − 1
+2RN
+ikjℓf i
+/αf j
+/αf k
+/σf ℓ
+/σ.
+□
+Here,
+RicM
+αβ = gδǫRM
+αδβǫ
+is the Ricci tensor.
+2.4. Regularity. Assume N has ≤ 0 sectional curvature. Then from the Weitzenb¨ock
+formula,
+∆e(f) ≥ −Ce(f)
+(2.1)
+where C depends only on the geometry of M.
+Theorem 2.5. If f : M → N is harmonic, and N has ≤ 0 sectional curvature, then
+|f|C2+α
+loc ≤ c
+where c > 0 depends on E(f) and the geometries of M, N.
+Proof. By (2.1) and Moser iteration,
+sup
+Br(p)
+e(f) ≤ c
+�
+B2r(p)
+e(f) ⋆ 1 = E(f)
+(2.2)
+where c only depends on the geometry and r. Now the right-hand side of (2.1) is
+C0-bounded. So by elliptic regularity, fi is C1+α-bounded. But then the right-hand
+side of (2.1) is Cα-bounded, so fi is C2+α-bounded.
+□
+Corollary 2.6. If f : M → N is harmonic, and N has ≤ 0 sectional curvature, then
+f ∈ C∞(M, N).
+Proof. Keep bootstrapping with (2.1).
+□
+Theorem 2.7. If f : M → N is a harmonic map, M is compact with Ricci curvature
+≥ 0 and N has sectional curvature ≤ 0, then f is totally geodesic.
+
+NOTES ON HARMONIC MAPS
+9
+Proof. Since
+0 =
+�
+M
+△e(f) ⋆ 1 =
+�
+M
+�
+|∇df|2 + 1
+2
+�
+df(RicM(eα)), df(eα)
+�
+−1
+2
+�
+RN(df(eα), df(eβ))df(eβ), df(eα)
+��
+⋆ 1,
+and each of the terms on the right hand side is non-negative, we have
+∇df = 0.
+□
+2.5. Non-positive curvature in a metric space. A complete metric space (X, d)
+is called an NPC space if the following conditions are satisfied:
+(i) The space (X, d) is a length (or geodesic) space. That is, for any two points P
+and Q in X, there exists a rectifiable curve c so that the length of c is equal to d(P, Q)
+(which we will sometimes denote by dP Q for simplicity). We call such distance realizing
+curves geodesics.
+(ii) For any geodesic triangle with vertices P, R, Q ∈ X, let c : [0, l] → X be the
+arclength parameterized geodesic from Q to R and let Qt = c(tl). Then
+d2
+P Qt ≤ (1 − t)d2
+P Q + td2
+P R − t(1 − t)d2
+QR.
+(2.3)
+(iii) Condition (ii) implies the quadralateral comparison inequalities (cf. [KS1, Corol-
+lary 2.1.3])
+d2
+PtQt ≤ (1 − t)d2
+P Q + td2
+RS − t(1 − t)(dSP − dQR)2
+(2.4)
+d2
+QtP + d2
+Q1−tS ≤ d2
+P Q + d2
+RS − td2
+QR − 2tdSPdQR + 2t2d2
+QR
+(2.5)
+Example 2.8. The main examples we will be considering are Riemannian manifolds
+of non-positive curvature and (locally compact) Euclidean buildings.
+Example 2.9. Let (X, d) be an NPC space, P ∈ X and M be a compact Riemannian
+manifold. Let Y = L2(M, X) be a set of maps f : M → X such that
+�
+M
+d2(f, P) ⋆ 1 < ∞.
+Define a distance function dY on Y by setting
+d2
+Y (f0, f1) =
+�
+M
+d2(f0(x), f1(x)) ⋆ 1.
+Then (Y, dY ) is an NPC space (cf. [KS1, Lemma 2.1.2]) where the geodesic between
+f0 and f1 is the geodesic interpolation map ft(x) = (1 − t)f0(x) + tf1(x).
+
+10
+GEORGIOS DASKALOPOULOS AND CHIKAKO MESE
+2.6. Local existence. We solve the Dirichlet problem for a smooth Riemannian do-
+main B ⊂ M. We will motivate the construction by first considering the case X = R
+(cf. [KS1, Section 2.2]). Fix φ ∈ H1(B, X) and consider the space
+H1
+φ(B, X) = {f ∈ H1(B, X) : f − φ ∈ H1
+0(B, X)}
+Let
+E0 = inf{E(f) : f ∈ H1
+φ(B, X)}.
+By the parallelogram identity
+2
+�
+B
+|df + v
+2
+|2 ⋆ 1 + 2
+�
+B
+|df − v
+2
+|2 ⋆ 1 =
+�
+B
+|df|2 ⋆ 1 +
+�
+B
+|dv|2 ⋆ 1
+Take a minimizing sequence fi and apply the previous equality for f = fi, v = fj.
+This implies that
+2
+�
+B
+|dfi − fj
+2
+|2 ⋆ 1
+=
+�
+B
+|dfi|2 ⋆ 1 +
+�
+B
+|dfj|2 ⋆ 1 − 2
+�
+B
+|dfi + fj
+2
+|2 ⋆ 1
+≤
+2E0 + 2ǫi − 2E0 = 2ǫi.
+Hence
+lim
+�
+B
+|dfi − fj
+2
+|2 ⋆ 1 = 0.
+(2.6)
+By the Poincare inequality
+lim
+�
+B
+|fi − fj
+2
+|2 ⋆ 1 = 0
+(2.7)
+hence
+lim
+i→∞ fi = f in H1
+φ(B, X) and E(f) = E0.
+Now assume X is an NPC space. Korevaar-Schoen [KS1] showed that the energy
+density makes sense by taking difference quotients. For the purpose of these lectures,
+if X is a locally finite Euclidean building, then we can locally isometricaly embed it
+in a Euclidean space of high dimension. Then, we can define the energy density of the
+map to the building equal to the energy density of the map considered as a map to the
+Euclidean space. In fact, this was the original point of view taken in [GS]. The more
+general theory developed later in [KS1] and [KS2].
+With this, we argue as above replacing the parallelogram identity by the quadrilat-
+eral inequality. Indeed, for f, v ∈ H1
+φ(B, X), define w(x) = (1 − t)f(x) + tv(x). Then
+(2.4) with t = 1
+2 implies
+2d2(w(x), w(y)) ≤ d2(f(x), f(y)) + d2(v(x), v(y)) − 1
+2(d(f(y), v(y)) − d(f(x), v(x))2
+which then implies
+2Ew ≤ Ef + Ev − 1
+2
+�
+B
+|∇d(f, v)|2 ⋆ 1
+
+NOTES ON HARMONIC MAPS
+11
+Take a minimizing sequence fi and apply the previous inequality with f = fi and
+v = vi to conclude (cf. (2.6))
+lim
+i,j→∞
+�
+B
+|∇d(fi, fj)|2 ⋆ 1 → 0.
+By the Poincare inequality, fi is a Cauchy sequence in (Y, dY ) and converges to a map
+which is minimizing by the lower semicontinuity of energy [KS1, Theorem 1.6.1].
+2.7. Basic Regularity result of Gromov-Schoen and Korevaar-Schoen. This
+is the analogue of (2.2) without using the PDE.
+Theorem 2.10. If f ∈ H1(B, X) is a harmonic map, then f is locally Lipschitz. More
+precisely, for any B′ ⊂⊂ B, there exists a constant C only depending on the metric on
+B′ and the distance of B′ to ∂B such that
+sup
+B′ |df|2 ≤ c
+�
+B
+|df|2 ⋆ 1.
+2.8. Equivariant maps. Let ρ : π1(M) → Isom(X) be a homomorphism. A map
+v : ˜
+M → X
+is called a ρ-equivariant map, if
+v(γx) = ρ(γ)v(x).
+Since |dv|2 is π1(M)-invariant, it descends to a function on M. Define:
+E(v) =
+�
+M
+|dv|2 ⋆ 1.
+If v descends to a map to M/ρ(π1(M)) this agrees with our previous definition.
+2.9. Existence of ρ-equivariant locally Lipschitz maps. Let (M, ν) be a proba-
+bility space, X an NPC-space and f ∈ L2(M, X).
+Lemma 2.11. There exists a unique point Qf,ν that minimizes the integral
+If,ν(Q) :=
+�
+M
+d2(f(m), Q)dν(m) ∀Q ∈ X.
+We call Qf,ν the center of mass.
+Proof. Let {Qi} be a minimizing sequence and let Qij = 1
+2Qi + 1
+2Qj. By (2.3) with
+t = 1
+2,
+d2(f(x), Qij) ≤ 1
+2d2(f(x), Qi) + 1
+2d2(f(x), Qj) − 1
+4d2(Qi, Qj).
+Integrating, we obtain
+If,ν(Qij) ≤ 1
+2If,ν(Qi) + 1
+2If,ν(Qj) − 1
+2d2(Qi, Qj).
+
+12
+GEORGIOS DASKALOPOULOS AND CHIKAKO MESE
+Thus, d2(Qi, Qj) is a Cauchy sequence. We conclude that any minimizing sequence is
+a Cauchy sequence and converges to a minimizing element.
+□
+Lemma 2.12. There exists a locally Lipschitz ρ-equivariant map ˜f : ˜
+M → X. If X is
+smooth, then ˜f can be chosen to be smooth.
+Proof. Let Q0 := Qf,µ0 (resp. Q1 := Qf,µ1) be the center of mass for the function f ∈
+L2(M, X) and the probability space (M, µ0) (resp. (M, µ1)). Let Qt = (1−t)Q0+tQ1.
+By the minimizing property of Q0 and Q1,
+�
+d2(f, Q0) + d2(f, Q1) dµ0
+=
+�
+d2(f, Q0)dµ0 +
+�
+d2(f, Q1)dµ1 +
+�
+d2(f, Q1)(dµ0 − dµ1)
+≤
+2
+�
+d2(f, Q1/2) dµ0 +
+� �
+d2(f, Q1) − d2(f, Q1/2)
+�
+(dµ0 − dµ1)
+≤
+�
+d2(f, Q0) + d2(f, Q1) − 1
+4d2(Q0, Q1) dµ0
++
+� �
+d2(f, Q1) − d2(f, Q1/2)
+�
+(dµ0 − dµ1).
+The last inequality is by triangle comparison. Consequently,
+d2(Q0, Q1) ≤ 4
+� �
+d2(f, Q1) − d2(f, Q 1
+2)
+�
+(dµ0 − dµ1).
+(2.8)
+For each x ∈ ˜
+M, let
+dµx = dvol
+B1(x)
+V (x)
+,
+V (x) = vol(B1(x)).
+where vol = ⋆1 is the volume form of ˜
+M. Since µx is only dependent on the metric of
+M, x �→ µx is invariant under the isometric action of π1(M). Furthermore,
+�
+|dµx0 − dµx1| =
+� ����
+1
+V (x0)χB1(x0) −
+1
+V (x0)χB1(x0)
+���� ⋆ 1
+≤ Cρ(x0, x1)
+where ρ denotes the distance function on M.
+Let M0 be a fundamental domain. Let f(M0) = P and extend equivariantly to
+f : ˜
+M → X. For simplicity, assume M0 is compact. Then there exists a constant L
+such that
+d(f(x0), f(x1)) ≤ L whenever ρ(x0, x1) < 2.
+Thus, for ρ(x0, x1) < 1 and in the support of |dµx0 − dµx1|,
+d2(f, Q1) − d2(f, Q1/2) ≤ d2(f, Q1) + d2(f, Qt) ≤ 2L2.
+
+NOTES ON HARMONIC MAPS
+13
+Define
+˜f : ˜
+M → X,
+˜f(x) = Qf,µx
+The π1(M)-invariance of µx and the ρ-equivariance of f imply the ρ-equivariance of ˜f.
+Apply (2.8) with M = M, µ0 = µx0 and µ1 = µx1 to obtain
+d2( ˜f(x0), ˜f(x1)) ≤ 2L2
+�
+|dµx0 − dµx1| ≤ 2L2Cρ(x0, x1).
+□
+2.10. The boundary at infinity. A good reference is [BH]. Suppose X is an NPC-
+space. Two geodesic rays c, c′ : [0, ∞) → X are said to be asymptotic if there exists a
+constant K such that d(c(t), c′(t)) < K for all t > 0. The set ∂X of boundary points
+of X (which we shall also call the points at infinity) is the set of equivalence classes
+of geodesic rays, two geodesic rays being equivalent if and only if they are asymptotic.
+We denote ¯X = X ∪ ∂X. Notice that the images of two asymptotic geodesic rays
+under any isometry of X are again asymptotic geodesic rays, and hence any isometry
+extends to give a bijection of ¯X. The next proposition is [BH, Proposition 8.2].
+Proposition 2.13. If X is an NPC-space and c : [0, ∞) → X is a geodesic ray starting
+from P, then for every point P1 ∈ X there is a unique geodesic ray which starts from
+P1 and is asymptotic to c.
+The topology of ¯X is defined as follows: A sequence of points Pi converges to a point
+P ∗ ∈ ∂X if and only if the geodesics joining P0 to Pi converge (uniformly on compact
+subsets) to the geodesic ray that starts from P0 and belongs to the class of P ∗.
+Example 2.14. If X is a complete n-dimensional Riemannian manifold of non-positive
+sectional curvature, then ∂X is homeomorphic to Sn−1. Indeed, given a base point
+P0, we can obtain a homeomorphism by considering the map which associates to each
+unit vector V tangent to X at P0 the class of the geodesic ray c starting at P0 with
+velocity vector V. In particular, if X is the n-dimensional hyperbolic space, then ¯X is
+homeomorphic to the n-dimensional ball in Rn. If X is a locally compact Euclidean
+building, then ∂X is a compact spherical building (cf. [KL, Proposition 4.2.1]).
+Lemma 2.15. If Pi is a sequence in X with lim Pi = P ∗ ∈ ∂X and if Qi is another
+sequence in X with d(Pi, Qi) ≤ C independently of i, then lim Qi = P ∗.
+Proof. Fix P0 ∈ X. Let γ : [0, ∞) → ∞ be an arclength parameterized geodesic ray
+in the equivalence class P ∗ with γ(0) = P0. Let ti = d(P0, Pi) (resp. τi = d(P0, Qi))
+and let γi : [0, ti] → X (resp. ˆγi : [0, τi] → X) be the arclength parameterized geodesic
+segment connecting P0 and Pi (resp. Qi). By the triangle inequality,
+d(ˆγi(ti), ˆγi(τi)) = |ti − τi| = |d(P0, Pi) − d(P0, Qi)| ≤ d(Pi, Qi) ≤ C.
+
+14
+GEORGIOS DASKALOPOULOS AND CHIKAKO MESE
+Thus, assuming t ≤ ti ≤ τi, the NPC condition implies
+d(ˆγi(t), γi(t)) ≤ t
+ti
+d(ˆγi(ti), γi(ti)) ≤ t
+ti
+�
+d(ˆγi(ti), ˆγi(τi)) + d(ˆγi(τi), γi(ti))
+�
+≤ 2Ct
+ti
+.
+Similarly, assuming t ≤ τi < ti,
+d(ˆγi(t), γi(t)) ≤ 2Ct
+τi
+.
+Thus, for t ≤ min{ti, τi},
+d(ˆγi(t), γ(t)) ≤ d(ˆγi(t), γi(t)) + d(γi(t), γ(t)) ≤
+2Ct
+max{ti, τi} + d(γi(t), γ(t)).
+Fix T0 > ∞.
+The assumption that lim Pi = P ∗ implies that ti, τi → ∞ and the
+geodesics γi converge uniformly to γ in [0, T0]. Thus, ˆγi also converge uniformly to γ
+in [0, T0].
+□
+Lemma 2.16. The stabilizer of a point at infinity is contained in a parabolic subgroup.
+So if the image of ρ is Zariski dense it cannot fix a point at infinity.
+2.11. Global existence result. We prove existence of equivariant harmonic maps
+[GS, Theorem 7.1].
+Theorem 2.17. Let X be a locally compact NPC space. Assume that the image of ρ
+doesn’t fix a point in ∂X and that there exists a Lipschitz equivariant map v : ˜
+M → X
+with finite energy. Then there is a Lipschitz equivariant map f of least energy and the
+restriction of f to a small ball about any point is minimizing.
+Proof. Let E0 denote the infimum of the energy taken over all Lipschitz equivariant
+maps. Let vi be a sequence of Lipschitz equivariant maps with E(vi) → E0. Let B be
+a ball in ˜
+M such that γ(B) ∩ B = ∅ for all γ ∈ π1(M). We may then construct a new
+minimizing sequence ¯vi, by replacing vi with the solution to the Dirichlet problem on
+each γ(B). Clearly ¯vi is also a minimizing sequence.
+On a compact subset of B, the sequence ¯vi is uniformly Lipschitz by Theorem 2.10.
+It follows that a subsequence of ¯vi converges uniformly on compact subsets of B to
+a map into ¯X which either maps into X or maps to a single point P ∗ ∈ ∂X. We
+exclude the second possibility as follows. Let x0 ∈
+˜
+M be the center of the chosen
+ball B.
+Let C be any smooth embedded curve from x0 to γ(x0).
+An elementary
+argument using Fubini’s theorem shows that C may be chosen so that the energy of
+the restriction of each map ¯vi to C is uniformly bounded. Therefore the length of
+the curve ¯vi(C) is uniformly bounded, and in particular d(vi(x0), ρ(γ)vi((x0)) ≤ C.
+Lemma 2.15 implies lim ρ(γ)vi(x0) = P ∗, and hence ρ(γ)P ∗ = P ∗ for all γ. This is
+a contradiction. Therefore we may assume that ¯vi converges uniformly on compact
+subsets of B.
+
+NOTES ON HARMONIC MAPS
+15
+From (2.6) as before, we have�
+K
+|∇d(¯vi, ¯vj)|2 ⋆ 1 → 0
+for any compact set K ⊂ ˜
+M. Since ¯vi converges uniformly on compact subsets of B,
+the function d(¯vi, v) is uniformly bounded there. It then follows from Poincare type
+inequalities that
+�
+K
+d(¯vi, ¯vj)2 ⋆ 1 → 0.
+In particular, the sequence ¯vi → f which is a minimizer by lower semicontinuity of
+energy. The local minimizing property of f follows. This completes the proof.
+□
+
+16
+GEORGIOS DASKALOPOULOS AND CHIKAKO MESE
+3. Pluriharmonic maps and the Siu-Sampson Formula
+3.1. Introduction: Bochner methods for harmonic maps. In this section, we
+discuss the Bochner formulas of Siu [Siu] and Sampson [Sa]. Our exposition closely
+follows the approach of [LY]. We also present a variation of these formulas inspired
+by the work of Mochizuki [M]. Lastly, we sketch the existence of pluriharmonic maps
+into Euclidean buildings.
+3.2. Pluriharmonic maps from K¨ahler manifolds to Riemannian manifolds.
+Let (M, ω, J) be a K¨ahler manifold along with its K¨ahler form and complex structure.
+Let
+TM ⊗ C = T (1,0)M ⊕ T (0,1)M
+be its complexified tangent bundle decomposed into the ±√−1-eigenspaces of J. We
+can decompose v ∈ TM ⊗ C into
+v = v1,0 + v0,1 where v1,0 = 1
+2(v −
+√
+−1Jv), v0,1 = 1
+2(v +
+√
+−1Jv).
+The cotangent space T ∗M has a complex structure still denoted J defined by Jα =
+α ◦ J. Accordingly, we have an analogous decomposition
+T ∗M ⊗ C = T ∗(1,0)M ⊕ T ∗(0,1)M.
+Let (N, h) be a Riemannian manifold and TN ⊗ C its complexified tangent bundle.
+For a smooth map f : M → N, let
+E := f ∗(TN ⊗ C).
+(3.1)
+Extending complex linearly, df : TM → TN gives rise to a map df : TM ⊗ C →
+TN ⊗ C. Denote by Ωp,q(E), the space of E-valued (p, q)-forms. Define
+d′f := 1
+2(df −
+√
+−1 df ◦ J) ∈ Ω1,0(E),
+d′′f := 1
+2(df +
+√
+−1 df ◦ J) ∈ Ω0,1(E).
+We have that
+df = d′f + d′′f
+Jdf = df ◦ J = −
+√
+−1 (d′f − d′′f).
+For local coordinates (yi) of N, let
+∂
+∂fi =
+∂
+∂yi ◦ f. Then
+d′f = d′f i ∂
+∂f i
+d′′f = d′′f i ∂
+∂f i
+d′f = d′′f
+d′′f = d′f.
+Similarly, we can decompose the pullback of the Levi-Civita connection (cf. Section 1)
+as
+∇ = ∇′ + ∇′′
+where
+∇′ : C∞(E) → Ω1,0(E),
+∇′′ : C∞(E) → Ω0,1(E).
+
+NOTES ON HARMONIC MAPS
+17
+In turn, ∇′ and ∇′′ induce differential operators
+d′
+E : Ωp,q(E) → Ωp+1,q(E),
+d′′
+E : Ωp,q(E) → Ωp,q+1(E)
+where
+d′
+E(φ ⊗ s)
+=
+d′φ ⊗ s + (−1)p+qφ ⊗ ∇′
+Es
+d′′
+E(φ ⊗ s)
+=
+d′′φ ⊗ s + (−1)p+qφ ⊗ ∇′′
+Es.
+A straightforward calculation implies that
+d′
+Ed′′f = −d′′
+Ed′f,
+d′
+Ed′f = 0,
+d′′
+Ed′′f = 0.
+(3.2)
+Lemma 3.1.
+τ(f) = 2i ⋆
+� ωn−1
+(n − 1)! ∧ d′
+Ed′′f
+�
+.
+Proof. We claim
+⋆α =
+ωn−1
+(n − 1)! ∧ Jα,
+∀α ∈ Ω1(M, R).
+To check the claim, use normal coordinates (zi = xi + √−1yi) at a point x ∈ M. For
+α = dxi or α = dyi, we have
+dxi ∧
+ωn−1
+(n − 1)! ∧ Jdxi = dxi ∧ dyi ∧
+ωn−1
+(n − 1)! =
+√−1
+2
+dzi ∧ d¯zi ∧
+ωn−1
+(n − 1)! = ωn
+n!
+and
+dyi ∧
+ωn−1
+(n − 1)! ∧ Jdyi = −dyi ∧ dxi ∧
+ωn−1
+(n − 1)! =
+√−1
+2
+dzi ∧ d¯zi ∧
+ωn−1
+(n − 1)! = ωn
+n! .
+The claim follows by linearity.
+Next, note that
+Jd′f
+=
+1
+2(df ◦ J +
+√
+−1df) =
+√−1
+2
+(df −
+√
+−1df ◦ J) =
+√
+−1d′f
+Jd′′f
+=
+1
+2(df ◦ J −
+√
+−1df) =
+√−1
+2
+(df +
+√
+−1df ◦ J) = −
+√
+−1d′′f,
+
+18
+GEORGIOS DASKALOPOULOS AND CHIKAKO MESE
+which implies Jdf = Jd′f + Jd′′f = √−1(d′f − d′′f). Applying the claim for α = df,
+we use the fact that dω = 0 to obtain
+τ(f)
+=
+−d⋆
+∇df
+=
+⋆d∇(⋆df)
+=
+⋆d∇
+� ωn−1
+(n − 1)! ∧ Jdf
+�
+=
+⋆d∇
+� ωn−1
+(n − 1)! ∧ (
+√
+−1(d′f − d′′f))
+�
+=
+−
+√
+−1 ⋆
+� ωn−1
+(n − 1)! ∧ (d′
+Ed′′f − d′′
+Ed′f)
+�
+=
+−2
+√
+−1 ⋆
+� ωn−1
+(n − 1)! ∧ d′
+Ed′′f
+�
+.
+□
+Definition 3.2. f is called pluriharmonic d′
+Ed′′f = 0.
+Remark 3.3. Lemma 3.1 implies
+pluriharmonic =⇒ harmonic.
+Note that holomorphic maps between K¨ahler manifolds are pluriharmonic, and thus
+harmonic.
+3.3. Sampson’s Bochner formula.
+Theorem 3.4 (Sampson’s Bochner formula, [Sa]). For a harmonic map f : M → N
+from a K¨ahler manifold (M, g) to a Riemannian manifold (N, h),
+d′d′′{d′′f, d′′f} ∧
+ωn−2
+(n − 2)!
+=
+4
+�
+|d′
+Ed′′f|2 + Q0
+� ωn
+n!
+where {·, ·} is given in Definition 3.5 below and
+Q0 = −2gα¯δgγ ¯βRijkl
+∂f i
+∂zα
+∂f k
+∂¯zβ
+∂f j
+∂zγ
+∂f l
+∂¯zδ
+in local coordinates (zα) of M and (yi) of N.
+Proof. Combine Lemma 3.6, Lemma 3.8 and Lemma 3.20 below.
+□
+Definition 3.5. Let {si} be a local frame of E. For
+ψ = ψi ⊗ si ∈ Ωp,q(E) and ξ = ξi ⊗ si ∈ Ωp′,q′(E)
+we set
+{ψ, ξ} = ⟨si, sj⟩ψi ∧ ¯ξj ∈ Ωp+q′,q+p′
+where ⟨·, ·⟩ is the complex-linear extention of the Riemannian metric on E.
+
+NOTES ON HARMONIC MAPS
+19
+Lemma 3.6. For any smooth map f : M → N from a K¨ahler manifold to a Riemann-
+ian manifold, we have
+d′d′′{d′′f, d′′f} ∧
+ωn−2
+(n − 2)! =
+�
+−{d′
+Ed′′f, d′
+Ed′′f} + {d′′f, R(1,1)
+E
+(d′′f)}
+�
+∧
+ωn−2
+(n − 2)!
+where
+R(1,1)
+E
+= (d′
+Ed′′
+E + d′′
+Ed′
+E)
+is the (1, 1)-part of the curvature RE = d2
+E.
+Proof. Repeatedly using the fact that d′′
+Ed′′f = 0 (cf. (3.2)),
+d′d′′{d′′f, d′′f}
+=
+−{d′
+Ed′′f, d′
+Ed′′f} + {d′′f, d′′
+Ed′
+Ed′′f}
+=
+−{d′
+Ed′′f, d′
+Ed′′f} + {d′′f, (d′′
+Ed′
+E + d′
+Ed′′
+E + d′′
+E
+2)d′′f}.
+Since {d′′f, d′
+Ed′
+Ed′′f} ∧ ωn−2
+(n−2)! is an (n − 1, n + 1)-form and hence zero for dimensional
+reasons, we can complete the square to obtain
+d′d′′{d′′f, d′′f} ∧
+ωn−2
+(n − 2)!
+=
+�
+−{d′
+Ed′′f, d′
+Ed′′f} + {d′′f, (d′
+E + d′′
+E)2d′′f}
+�
+∧
+ωn−2
+(n − 2)!
+=
+�
+−{d′
+Ed′′f, d′
+Ed′′f} + {d′′f, R(1,1)
+E
+(d′′f)}
+�
+∧
+ωn−2
+(n − 2)!
+which proves the first equation.
+□
+Lemma 3.7. For any E-valued (1, 1)-form φ on M,
+−{φ, φ} ∧
+ωn−2
+(n − 2)! = 4(|φ|2 − |Traceωφ|2)ωn
+n! .
+Proof. Let (zp) be normal coordinates at x ∈ M and let φp¯qdzp ∧ d¯zq. At x,
+ωn
+n! =
+�√−1
+2
+�2 ��
+p
+dzp ∧ d¯zp
+�2
+∧
+ωn−2
+(n − 2)!.
+For p, q such that p ̸= q,
+s ̸= p or t ̸= q ⇒ dzp ∧ d¯zq ∧ d¯zs ∧ dzt ∧
+��
+j
+dzj ∧ d¯zj
+�n−2
+= 0.
+For p = q,
+s ̸= t ⇒ dzp ∧ d¯zq ∧ d¯zs ∧ dzt ∧
+��
+j
+dzj ∧ d¯zj
+�n−2
+= 0.
+
+20
+GEORGIOS DASKALOPOULOS AND CHIKAKO MESE
+Furthermore,
+φp¯qφp¯qdzp ∧ d¯zq ∧ d¯zp ∧ dzq
+=
+φp¯qφp¯qdzp ∧ d¯zp ∧ dzq ∧ d¯zq
+φp¯pφq¯qdzp ∧ d¯zp ∧ d¯zq ∧ dzq
+=
+−φp¯pφq¯qdzp ∧ d¯zp ∧ dzq ∧ d¯zq.
+Thus,
+�√−1
+2
+�2
+{φ, φ} ∧
+ωn−2
+(n − 2)!
+=
+�√−1
+2
+�2 � �
+p,q,s,t
+φp¯qφs¯tdzp ∧ d¯zq ∧ d¯zs ∧ dzt
+�
+∧
+ωn−2
+(n − 2)!
+=
+�
+p̸=q
+�
+|φp¯q|2 − φp¯pφq¯q
+� ωn
+n!
+=
+�
+p,q
+�
+|φp¯q|2 − φp¯pφq¯q
+� ωn
+n!
+=
+�
+|φ|2 − |traceωφ|2� ωn
+n! .
+□
+Lemma 3.8. For any harmonic map f : M → N from a K¨ahler manifold to a
+Riemannian manifold, we have
+−{d′
+Ed′′f, d′
+Ed′′f} ∧
+ωn−2
+(n − 2)!
+=
+4 |d′
+Ed′′f|2 ωn
+n! .
+Proof. We apply Lemma 3.7 with φ = d′
+Ed′′f. Since f is harmonic, Trωd′
+Ed′′f = 0 by
+Lemma 3.1.
+□
+Lemma 3.9. For a harmonic map f : M → N from a K¨ahler manifold to a Hermitian-
+negative Riemannian manifold, we have
+{d′′f, R(1,1)
+E
+(d′′f)} ∧
+ωn−2
+(n − 2)! = −2 Rijkld′f i ∧ d′′f k ∧ d′f j ∧ d′′f l ∧ ωn
+n!
+where R(1,1)
+E
+is defined in Lemma 3.6.
+Proof. Let (zα) (resp. (yi)) be normal coordinates at a point x ∈ M (resp. f(x) ∈ N).
+Then
+∇
+∂
+∂¯zγ ∇
+∂
+∂zβ
+∂
+∂f j
+=
+∇
+∂
+∂¯zγ
+�∂f k
+∂zβ ∇
+∂
+∂fk
+∂
+∂f j
+�
+=
+∂f k
+∂zβ
+∂f l
+∂¯zγ ∇
+∂
+∂fl ∇
+∂
+∂fk
+∂
+∂f j
+
+NOTES ON HARMONIC MAPS
+21
+and
+d′′
+Ed′
+Ed′′f
+=
+d′′
+Ed′
+E
+�∂f j
+∂¯zα d¯zα ⊗ ∂
+∂f j
+�
+=
+d′′d′
+�∂f j
+∂¯zα d¯zα
+�
+⊗ ∂
+∂f j − ∂f j
+∂¯zα d¯zα ∧ d¯zγ ∧ dzβ ⊗ ∇
+∂
+∂¯zγ ∇
+∂
+∂zβ
+∂
+∂f j
+=
+d′′d′
+�∂f j
+∂¯zα d¯zα
+�
+⊗ ∂
+∂f j + ∂f j
+∂¯zα
+∂f k
+∂zβ
+∂f l
+∂¯zγ d¯zα ∧ dzβ ∧ d¯zγ ⊗ ∇
+∂
+∂fl ∇
+∂
+∂fk
+∂
+∂f j .
+Similarly,
+d′
+Ed′′
+Ed′′f
+=
+d′d′′
+�∂f j
+∂¯zα d¯zα
+�
+⊗ ∂
+∂f j − ∂f j
+∂¯zα
+∂f k
+∂zβ
+∂f l
+∂¯zγ d¯zα ∧ dzβ ∧ d¯zγ ⊗ ∇
+∂
+∂fk ∇
+∂
+∂fl
+∂
+∂f j .
+Combining the above two equalities,
+R(1,1)
+E
+(d′′f)
+=
+∂f j
+∂¯zα
+∂f k
+∂zβ
+∂f l
+∂¯zγ d¯zα ∧ dzβ ∧ d¯zγ ⊗ Rs
+jkl
+∂
+∂f s.
+We compute
+{d′′f, R(1,1)
+E
+(d′′f)}
+=
+Rijkl
+∂f i
+∂¯zδ
+∂f j
+∂zα
+∂f k
+∂¯zβ
+∂f l
+∂zγ d¯zδ ∧ dzα ∧ d¯zβ ∧ dzγ
+=
+Rjilk
+∂f j
+∂zα
+∂f i
+∂¯zδ
+∂f l
+∂zγ
+∂f k
+∂¯zβ dzα ∧ d¯zδ ∧ dzγ ∧ d¯zβ
+=
+(−Rjlki + Rjkli)∂f j
+∂zα
+∂f i
+∂¯zδ
+∂f l
+∂zγ
+∂f k
+∂¯zβ dzα ∧ d¯zδ ∧ dzγ ∧ d¯zβ.
+Since
+Rjkli
+∂f j
+∂zα
+∂f i
+∂¯zδ
+∂f l
+∂zγ
+∂f k
+∂¯zβ dzα ∧ d¯zδ ∧ dzγ ∧ d¯zβ ∧
+ωn−2
+(n − 2)!
+=
+Rjkli
+∂f j
+∂zα
+∂f i
+∂¯zδ
+∂f l
+∂zγ
+∂f k
+∂¯zβ dzα ∧ d¯zα ∧ dzβ ∧ d¯zβ ∧
+ωn−2
+(n − 2)!
++Rjkli
+∂f j
+∂zα
+∂f i
+∂¯zδ
+∂f l
+∂zγ
+∂f k
+∂¯zβ dzα ∧ d¯zβ ∧ dzβ ∧ d¯zα ∧
+ωn−2
+(n − 2)!
+=
+0,
+
+22
+GEORGIOS DASKALOPOULOS AND CHIKAKO MESE
+we obtain
+{d′′f, R(1,1)
+E
+(d′′f)} ∧
+ωn−2
+(n − 2)!
+=
+−Rjlki
+∂f j
+∂zα
+∂f i
+∂¯zα
+∂f l
+∂zβ
+∂f k
+∂¯zβ dzα ∧ d¯zα ∧ dzβ ∧ d¯zβ ∧
+ωn−2
+(n − 2)!
+−Rjlki
+∂f j
+∂zα
+∂f i
+∂¯zβ
+∂f l
+∂zβ
+∂f k
+∂¯zα dzα ∧ d¯zβ ∧ dzβ ∧ d¯zα ∧
+ωn−2
+(n − 2)!
+=
+−2Rjlki
+∂f j
+∂zα
+∂f i
+∂¯zα
+∂f l
+∂zβ
+∂f k
+∂¯zβ dzα ∧ d¯zα ∧ dzβ ∧ d¯zβ ∧
+ωn−2
+(n − 2)!
+=
+−2Rjlkid′f j ∧ d′′f i ∧ d′f l ∧ d′′f k ∧
+ωn−2
+(n − 2)!.
+□
+Definition 3.10. A Riemannian manifold N is said to be Hermitian-negative (resp.
+strongly Hermitian negative) if
+RijklAi¯lAj¯k ≤ 0 (resp. < 0)
+for any Hermitian semi-positive matrix A =
+�
+Ai¯l�
+.
+Remark 3.11. Locally symmetric spaces whose irreducible local factors are all non-
+compact or Euclidean type are Hermitian negative (cf. [Sa, Theorem 2]).
+Theorem 3.12 (Sampson). If f : M → N is a harmonic map from a K¨ahler manifold
+into a Hermitian negative Riemannian manifold, then f is pluriharmonic.
+Proof. Integrate Sampson’s Bochner formula over M. Applying Stoke’s theorem results
+in the left hand side being 0. The two terms on the right hand side are non-negative
+pointwise, hence they must be identically equal to 0. In particular, d′
+Ed′′f = 0; i.e. f
+is is pluriharmonic.
+□
+3.4. Maps between K¨ahler manifolds. Let f : M → N be a smooth map between
+K¨ahler manifolds. By decomposing
+TN ⊗ C = T (1,0)N ⊕ T (0,1)N
+we get the decomposition of E := f −1(TN ⊗ C) as
+E = E′ ⊕ E′′ where E′ := f −1(T (1,0)N),
+E′′ := f −1(T (0,1)N).
+Denote by Ωp,q(E), Ωp,q(E′) and Ωp,q(E′′) the space of E-, E′- and E′′-valued (p, q)-
+forms respectively. If (wi) are local holomorphic coordinates in N, then { ∂
+∂fi :=
+∂
+∂wi ◦
+f,
+∂
+∂ ¯fi :=
+∂
+∂ ¯wi ◦ f} is a local frame of E. If d′f, d′f ′ are as in Section 3.2, then
+d′f = ∂f + ∂ ¯f,
+d′′f = ¯∂f + ¯∂ ¯f,
+df = d′f + d′′f = ∂f + ∂ ¯f + ¯∂f + ¯∂ ¯f.
+
+NOTES ON HARMONIC MAPS
+23
+where
+∂f = ∂f i ∂
+∂f i
+¯∂f = ¯∂f i ∂
+∂f i
+∂ ¯f = ∂ ¯f i ∂
+∂ ¯f i
+¯∂ ¯f = ¯∂ ¯f i ∂
+∂ ¯f i
+∂f = ¯∂ ¯f
+¯∂f = ∂ ¯f.
+Analogously, d∇ = d′
+E + d′′
+E is decomposed into the induced operators ∂E′, ¯∂E′, ∂E′′,
+¯∂E′′.
+A straightforward calculation yields
+∂E′ ¯∂f = −¯∂E′∂f
+∂E′′ ¯∂ ¯f = −¯∂E′′∂ ¯f
+(3.3)
+∂E′∂f = 0
+¯∂E′ ¯∂f = 0
+∂E′′∂ ¯f = 0
+¯∂E′′ ¯∂ ¯f = 0.
+(3.4)
+For any map f : M → N between K¨ahler manifolds, we have
+��∂E′′ ¯∂ ¯f
+��2 =
+��¯∂E′′∂ ¯f
+��2 =
+��∂E′ ¯∂f
+��2 .
+(3.5)
+Indeed, the left equality follows from (3.3) and the right from the fact that conjugation
+is an isometry.
+3.5. Siu’s curvature.
+Definition 3.13. Let N be a K¨ahler manifold and R its complexified curvature tensor.
+We say N has negative (resp. non-positive) complex sectional curvature, if
+R(V, ¯W, W, ¯V ) < 0 (resp. ≤ 0) ∀V, W ∈ TNC.
+In [Siu], Siu introduced the following notion of negative curvature. Recall that for
+local holomorphic coordinates (wi) of a K¨ahler manifold N, the curvature tensor is of
+type (1,1) and is given explicitly by
+Ri¯jk¯l = − ∂2hi¯j
+∂wk∂ ¯wl + hp¯q ∂hk¯q
+∂wi
+∂hp¯l
+∂ ¯wj
+where h is the K¨ahler metric on N. We say N has strongly negative (resp. strongly
+semi-negative) curvature if
+Ri¯jk¯l(AiBj − CiDj)(AlBk − ClDk) < 0 (resp. ≤ 0).
+for arbitrary complex numbers Ai, Bi, Ci, Di when AiBj − CiDj ̸= 0 for at least one
+pair of indices (i, j).
+Remark 3.14. A K¨ahler manifold N is strongly semi-negative if and only if it has
+non-positive complex sectional curvature (cf. [LSY, Theorem 4.4]).
+
+24
+GEORGIOS DASKALOPOULOS AND CHIKAKO MESE
+Lemma 3.15. Let N be a K¨ahler manifold with K¨ahler form ω and of strongly semi-
+negative curvature. Let M be another K¨ahler manifold and f : M → N be a smooth
+map. If Q : M → R is defined by setting
+Qωn
+n! = −Ri¯jk¯l ¯∂f i ∧ ∂ ¯f j ∧ ∂f k ∧ ¯∂ ¯f l ∧
+ωn−2
+(n − 2)!,
+then Q ≥ 0.
+Proof. At a point with normal coordinates in the domain
+Ri¯jk¯l ¯∂f i ∧ ∂f j ∧ ∂f k ∧ ¯∂ f l ∧
+ωn−2
+(n − 2)!
+=
+�
+α,β
+(
+√
+−1)n−2Ri¯jk¯l
+�
+−∂¯αf i∂αf j∂βf k∂¯βf l + ∂¯αf i∂βf j∂αf k∂¯βf l
++∂¯βf i∂αf j∂βf k∂¯αf l − ∂¯βf i∂βf j∂αf k∂¯αf l
+�
+∧ (∧γ(dzγ ∧ d¯zγ))
+=
+4
+�
+α,β
+Ri¯jk¯l
+�
+(∂¯αf i)(∂βf l) − (∂¯βf i)(∂αf l)
+� �
+(∂¯αf j)(∂βf k) − (∂¯βf j)(∂αf k)
+�ωn
+n!
+=
+4
+�
+α,β
+Ri¯jk¯l
+�
+(∂¯αf i)(∂βf j) − (∂¯βf i)(∂αf j)
+� �
+(∂¯αf l)(∂βf k) − (∂¯βf l)(∂αf k)
+�ωn
+n!
+≤
+0.
+The last equality is because Rı¯jk¯l = Ri¯lk¯l, and the last inequality is because of the
+assumption that N has strong semi-negative curvature.
+□
+3.6. Siu’s Bochner Formula.
+Theorem 3.16 (Siu-Bochner formula, [Siu] Proposition 2). For a harmonic map f :
+M → N between K¨ahler manifolds,
+∂ ¯∂{¯∂f, ¯∂f} ∧
+ωn−2
+(n − 2)!
+=
+�
+4
+��∂E′ ¯∂f
+��2 + Q
+� ωn
+n! .
+Proof. Combine Lemma 3.15, Lemma 3.17, Lemma 3.18 and Corollary 3.20 below.
+□
+The curvature operators of E′ and E′′ are RE′ = −(∂E′ + ¯∂E′)2 and RE′′ = −(∂E′′ +
+¯∂E′′)2 respectively.
+Lemma 3.17. For any smooth map f : M → N between K¨ahler manifolds, we have
+∂ ¯∂{¯∂f, ¯∂f} ∧
+ωn−2
+(n − 2)! =
+�
+−{∂E′ ¯∂f, ∂E′ ¯∂f} − {¯∂f, RE′(¯∂f)}
+�
+∧
+ωn−2
+(n − 2)!
+and
+∂ ¯∂{¯∂ ¯f, ¯∂ ¯f} ∧
+ωn−2
+(n − 2)! =
+�
+−{∂E′′ ¯∂ ¯f, ∂E′′ ¯∂ ¯f} − {¯∂ ¯f, RE′′(¯∂ ¯f)}
+�
+∧
+ωn−2
+(n − 2)!.
+
+NOTES ON HARMONIC MAPS
+25
+Proof. By setting ψ = ξ = ¯∂f ∈ Ω0,1(E′) in (3.5) and repeatedly using the fact that
+¯∂E′ ¯∂f = 0 (cf. (3.3)),
+∂ ¯∂{¯∂f, ¯∂f}
+=
+−{∂E′ ¯∂f, ∂E′ ¯∂f} + {¯∂f, ¯∂E′∂E′ ¯∂f}
+=
+−{∂E′ ¯∂f, ∂E′ ¯∂f} + {¯∂f, (¯∂E′∂E′ + ∂E′ ¯∂E′ + ¯∂2
+E′)¯∂f}.
+Since {¯∂f, ∂2
+E′ ¯∂f} ∧
+ωn−2
+(n−2)! is an (n − 1, n + 1)-form and hence zero for dimensional
+reasons, we can complete the square to obtain
+∂ ¯∂{¯∂f, ¯∂f} ∧
+ωn−2
+(n − 2)!
+=
+�
+−{∂E′ ¯∂f, ∂E′ ¯∂f} + {¯∂f, (∂E′ + ¯∂E′)2 ¯∂f}
+�
+∧
+ωn−2
+(n − 2)!
+=
+�
+−{∂E′ ¯∂f, ∂E′ ¯∂f} − {¯∂f, RE′(¯∂f)}
+�
+∧
+ωn−2
+(n − 2)!
+which proves the first equation. The second equation follows by setting ψ = ξ = ¯∂ ¯f ∈
+Ω0,1(E′′) in (3.5) and following exactly the same computation.
+□
+Lemma 3.18. For any harmonic map f : M → N between K¨ahler manifolds, we have
+−{∂E′ ¯∂f, ∂E′ ¯∂f} ∧
+ωn−2
+(n − 2)!
+=
+4
+��∂E′ ¯∂f
+��2 ωn
+n!
+−{∂E′′ ¯∂ ¯f, ∂E′′ ¯∂ ¯f} ∧
+ωn−2
+(n − 2)!
+=
+4
+��∂E′′ ¯∂ ¯f
+��2 ωn
+n! .
+Proof. Apply Lemma 3.7 with φ = ∂E′ ¯∂f (resp. φ = ∂E′′ ¯∂ ¯f). Since f is harmonic,
+Trω∂E′ ¯∂f = 0 and Trω∂E′′ ¯∂ ¯f = 0 by Lemma 3.1.
+□
+Lemma 3.19. For any smooth map f : M → N between K¨ahler manifolds, we have
+{¯∂f, RE′(¯∂f)} = Ri¯jk¯l ¯∂f i ∧ ∂ ¯f j ∧ ∂f k ∧ ¯∂ ¯f l = {¯∂ ¯f, RE′′(¯∂ ¯f)}.
+Proof. Using normal coordinates, we compute
+{¯∂f, RE′(¯∂f)}
+=
+{¯∂f i ∂
+∂f i , RE′(¯∂f j ∂
+∂f j )}
+=
+{¯∂f i ∂
+∂f i , ¯∂f j ∧ RE′( ∂
+∂f j )}
+=
+{¯∂f i ∂
+∂f i , ¯∂f j ∧ Rs
+jk¯l∂f k ∧ ∂f l ∂
+∂f s }
+=
+Ri¯j¯kl ¯∂f i ∧ ∂ ¯f j ∧ ¯∂ ¯f k ∧ ∂f l
+=
+Ri¯jk¯l ¯∂f i ∧ ∂ ¯f j ∧ ∂f k ∧ ¯∂ ¯f l
+
+26
+GEORGIOS DASKALOPOULOS AND CHIKAKO MESE
+which proves the first equality. The second equality is proved similarly:
+{¯∂ ¯f, RE′′(¯∂ ¯f)}
+=
+{¯∂ ¯f i ∂
+∂ ¯f i , RE′′(¯∂ ¯f j ∂
+∂ ¯f j )}
+=
+{¯∂ ¯f i ∂
+∂ ¯f i , ¯∂ ¯f j ∧ RE′′( ∂
+∂ ¯f j )}
+=
+{¯∂ ¯f i ∂
+∂ ¯f i , ¯∂ ¯f j ∧ R¯s
+¯j¯kl∂ ¯f k ∧ ¯∂f l ∂
+∂ ¯f s }
+=
+R¯ijk¯l ¯∂ ¯f i ∧ ∂f j ∧ ¯∂f k ∧ ∂ ¯f l
+=
+Rj¯ik¯l∂f j ∧ ¯∂ ¯f i ∧ ¯∂f k ∧ ∂ ¯f l
+=
+Ri¯jk¯l∂f i ∧ ¯∂ ¯f j ∧ ¯∂f k ∧ ∂ ¯f l
+=
+Ri¯jk¯l ¯∂f i ∧ ∂ ¯f j ∧ ∂f k ∧ ¯∂ ¯f l.
+□
+Corollary 3.20. For any smooth map f : M → N between K¨ahler manifolds, we have
+−{¯∂f, RE′(¯∂f)} ∧
+ωn−2
+(n − 2)! = Qωn
+n! .
+Proof. Combine Lemma 3.19 with the definition of Q given in Lemma 3.15.
+□
+Theorem 3.21. Suppose M and N are compact K¨ahler manifolds and the curvature
+of N is strongly semi-negative. If f : M → N is a harmonic map, then f is plurihar-
+monic. If, in addition, the curvature of N is strongly negative and the rankRdf ≥ 3 at
+some point of M, then f is either holomorphic or conjugate holomorphic.
+Proof. Integrate Siu’s Bochner formula over M. Applying Stoke’s theorem results in
+the left hand side being 0. The two terms on the right hand side are non-negative
+pointwise, hence they must be identically equal to 0. In particular, ∂E′ ¯∂f = 0; i.e. f
+is is pluriharmonic. If the rank is ≥ 3 at some point x, ¯∂f = 0 in some neighborhood
+of x by the definition of Q. Hence ¯∂f = 0 in all of M.
+□
+3.7. Variations of the Siu and Sampson Formulas. The following is a variation
+of the Sampson’s Bochner Formula. For harmonic metrics, this is due to Mochizuki
+(cf. [M, Proposition 21.42]).
+Theorem 3.22. For a harmonic map f : M → N from a K¨ahler manifold to a
+Riemannian manifold,
+d{d′
+Ed′f, d′′f − d′f} ∧
+ωn−2
+(n − 2)!
+=
+8
+�
+|d′
+Ed′′f|2 + Q0
+�
+∧ ωn
+n! .
+Proof. The key observation is that, since d′{d′
+Ed′′f, d′′f} ∧
+ωn−2
+(n−2)! is an (n + 1, n − 1)-
+form and d′′{d′
+Ed′′f, d′f} ∧
+ωn−2
+(n−2)! is an (n − 1, n + 1)-form, these two forms are both
+
+NOTES ON HARMONIC MAPS
+27
+identically equal to zero. Thus,
+d′{d′
+Ed′′f, d′f − d′′f} ∧
+ωn−2
+(n − 2)!
+=
+d′{d′
+Ed′′f, d′f} ∧
+ωn−2
+(n − 2)!
+=
+−d′{d′′
+Ed′f, d′f} ∧
+��n−2
+(n − 2)!
+(by (3.2)).
+=
+−d′d′′{d′f, d′f} ∧
+ωn−2
+(n − 2)!
+=
+d′d′′{d′′f, d′′f} ∧
+ωn−2
+(n − 2)!
+(3.6)
+d′′{d′
+Ed′′f, d′f − d′′f} ∧
+ωn−2
+(n − 2)!
+=
+−d′′{d′
+Ed′′f, d′′f} ∧
+ωn−2
+(n − 2)!
+=
+−d′′d′{d′′f, d′′f} ∧
+ωn−2
+(n − 2)!
+=
+d′d′′{d′′f, d′′f} ∧
+ωn−2
+(n − 2)!.
+(3.7)
+Thus,
+d{d′′
+Ed′f, d′′f − d′f} ∧
+ωn−2
+(n − 2)!
+=
+d{d′
+Ed′′f, d′f − d′′f} ∧
+ωn−2
+(n − 2)!
+(by (3.2))
+=
+(d′ + d′′){d′
+Ed′′f, d′f − d′′f} ∧
+ωn−2
+(n − 2)!
+=
+2d′d′′{d′′f, d′′f} ∧
+ωn−2
+(n − 2)! (by (3.6) and (3.7)).
+Thus, the asserted identity follows from Theorem 3.4.
+□
+By applying a similar proof as Theorem 3.23, we obtain a variation of the Siu’s
+Bochner formula.
+Theorem 3.23. For a harmonic map f : M → X between K¨ahler manifolds,
+d{¯∂E′∂f, ¯∂f − ∂f} ∧
+ωn−2
+(n − 2)!
+=
+�
+8
+��∂E′ ¯∂f
+��2 + 2Q
+�
+∧ ωn
+n! .
+Proof. As in the proof of Theorem 3.22, ∂{∂E′ ¯∂f, ¯∂f} ∧
+ωn−2
+(n−2)! = 0 = ¯∂{∂E′ ¯∂f, ∂f} ∧
+ωn−2
+(n−2)! and hence
+∂{∂E′ ¯∂f, ∂f − ¯∂f} ∧
+ωn−2
+(n − 2)!
+=
+∂ ¯∂{¯∂ ¯f, ¯∂ ¯f} ∧
+ωn−2
+(n − 2)!,
+¯∂{∂E′ ¯∂f, ∂f − ¯∂f} ∧
+ωn−2
+(n − 2)!
+=
+∂ ¯∂{¯∂f, ¯∂f} ∧
+ωn−2
+(n − 2)!.
+
+28
+GEORGIOS DASKALOPOULOS AND CHIKAKO MESE
+Consequently,
+d{¯∂E′∂f, ¯∂f − ∂f} ∧
+ωn−2
+(n − 2)!
+=
+d{∂E′ ¯∂f, ∂f − ¯∂f} ∧
+ωn−2
+(n − 2)!
+=
+(∂ + ¯∂){∂E′ ¯∂f, ∂f − ¯∂f} ∧
+ωn−2
+(n − 2)!
+=
+�
+∂ ¯∂{¯∂f, ¯∂f} + ∂ ¯∂{¯∂ ¯f, ¯∂ ¯f}
+�
+∧
+ωn−2
+(n − 2)!.
+The asserted identity follows from Theorem 3.16.
+□
+3.8. Pluriharmonic maps into Euclidean buildings.
+Theorem 3.24. Let M be a compact K¨ahler manifold and ∆(G) be the Bruhat-Tits
+building associated to a semisimple algebraic group G defined over a non-Archimedean
+local field K. For any Zariski dense representation of ρ : π1(X) → G(K), there exists
+a ρ-equivariant, locally Lipschitz pluriharmonic map f : ˜
+M → ∆(G) from the universal
+cover ˜
+M.
+Definition 3.25. A Euclidean building of dimension n is a piecewise Euclidean sim-
+plicial complex ∆ such that:
+• ∆ is the union of a collection A of subcomplexes A, called apartments, such
+that the intrinsic metric dA on A makes (A, dA) isometric to the Euclidean
+space Rn and induces the given Euclidean metric on each simplex.
+• Given two apartments A and A′ containing both simplices S and S′, there
+is a simplicial isometry from (A, dA) to (A′, dA′) which leaves both S and S′
+pointwise fixed.
+• ∆ is locally finite.
+Definition 3.26. A point x0 is said to be a regular point of a harmonic map f, if there
+exists r > 0 such that f(Br(x0)) of x is contained in an apartment of ∆. A singular
+point of f is a point of Ω that is not a regular point. The regular (resp. singular) set
+R(f) (resp. S(u)) of f is the set of all regular (resp. singular) points of f.
+Example 3.27. Consider a measured foliation defined by the quadratic differential
+zdz2 on C. The leaves of the horizontal foliation define a 3-pod T and the transverse
+measure gives T a distance function d making (T, d) into a NPC space. The projection
+along the vertical foliation u : C → T is a harmonic map.
+The leaf containing 0
+is a non-manifold point of T. Let K = u−1(0). Then K is also a 3-pod. On the
+other hand, every point of K besides 0 has a neighborhood mapping into an isometric
+copy of R and S(0) = {0}. In particular, the singular set is of Hausdorff codimension
+2. Similarly one can construct harmonic maps to other homogeneous trees by taking
+quadratic differentials of higher order.
+
+NOTES ON HARMONIC MAPS
+29
+The next two theorems are proved in [GS].
+Theorem 3.28. The singular set S(f) of a harmonic map f : M → ∆ is a closed set
+of Hausdorff codimension ≥ 2.
+Theorem 3.29. Let f : M → ∆ be as in Theorem 3.28. There exists a sequence of
+smooth functions ψi with ψi ≡ 0 in a neighborhood of S(u), 0 ≤ ψi ≤ 1 and ψi(x) → 1
+for all x ∈ S(u) such that
+lim
+i→∞
+�
+M
+|∇∇u||∇ψi| dµ = 0.
+By Theorem 3.28, Siu’s or Sampson’s Bochner formula holds at a.e. x ∈ ˜
+M. We
+now follow the proof of Theorem 3.21 where integration by parts can be justified using
+Theorem 3.28 and Theorem 3.29.
+
+30
+GEORGIOS DASKALOPOULOS AND CHIKAKO MESE
+4. Donaldson Corlette theorem
+4.1. Introduction: Higgs bundles via harmonic maps. In this lecture, we prove
+the theorem of Donaldson and Corlette relating harmonic maps to symmetric spaces
+of non-compact type and flat connections.
+We do it explicitly for SL(n, C).
+This
+correspondence is very well known and there are many excellent references to consult.
+Given the interest of the audience in this subject, we decided to give all the details of
+the proof explicitly. See also [Do], [Co] and the expositional paper [Li].
+4.2. The flat vector bundle associated to a representation. Let ρ : π1(M) →
+G = SL(n, C) be a homomorphism and
+E = ˜
+M ×ρ Cn → M
+be the associated flat vector bundle. Let H denote the space of positive definite self-
+adjoint matrices of determinant one. For g ∈ SL(n, C), define an action Ag on the
+space of (n × n)-matrices Mn×n(C) by
+Ag(h) = g−1∗hg−1.
+(4.1)
+Note that H is invariant under Ag and hence it defines an action on H.
+A ρ-equivariant map h : ˜
+M → H defines a hermitian metric on E by first defining
+H(s, t) = ¯stht
+(4.2)
+on the universal cover ˜
+M × Cn and descending to a metric on E by equivariance.
+Given the flat vector bundle (E, d) defined by ρ and the Hermitian metric H defined
+by a ρ-equivariant map, we define θ ∈ Ω1(M, End(E)) by the formula
+H(θs, t) = 1/2 (H(ds, t) + H(s, dt) − dH(s, t))
+(4.3)
+and D by the formula
+d = D + θ.
+(4.4)
+Formulas (4.3) and (4.4) immediately imply that
+H(θs, t) = H(s, θt)
+(4.5)
+and
+H(Ds, t) + H(s, Dt)
+=
+H(ds, t) − H(θs, t) + H(s, dt) − H(s, θt)
+=
+dH(s, t).
+(4.6)
+In other words, D is a Hermitian connection on (E, H).
+We claim
+θ = −1
+2h−1dh.
+(4.7)
+
+NOTES ON HARMONIC MAPS
+31
+To see (4.7), compute
+dH(s, t)
+=
+d¯sth t + ¯stdh t + ¯sth dt
+=
+H(ds, t) + ¯stdh t + H(s, dt)
+=
+H(Ds, t) + H(θs, t) + ¯stdh t + H(s, Dt) + H(s, θt)
+=
+dH(s, t) + H(θs, t) + H(s, h−1dh t) + H(s, θt).
+Thus,
+H(θs, t) = H(s, θt) = −1/2H(s, h−1dh t)
+and (4.7) follows.
+Let End0(E) denote the space of trace-less endomorphisms of E. We claim that D
+is a SL(n, C)-connection and θ ∈ End0(E). By (4.4) and since d is traceless, it suffices
+to show that θ is traceless. Indeed, since G/K is a Cartan-Hadamard space, we can
+write h = eu over a simply connected region U in M (or passing to the universal cover)
+where u(x) ∈ p for all x ∈ U. Thus,
+θ = h−1dh = du
+is traceless since u is traceless.
+As connections on End0(E),
+D = d + 1
+2
+�
+h−1dh, ·
+�
+.
+(4.8)
+We apply harmonic map theory to prove:
+Theorem 4.1. Given an irreducible representation ρ : π1(M) → SL(n, C), there exists
+a ρ-equivariant map h : ˜
+M → H such that for the Hermitian metric H, Hermitian
+connection D on End0(E) and θ ∈ Ω1(M, End0(E)) defined by (4.2), (4.3) and (4.4)
+respectively,
+d⋆
+Dθ = 0.
+(4.9)
+The proof of Theorem 4.1 is given several steps: (1) Choose h to be a harmonic map
+(cf. Section 4.3). (2) Show that the Hermitian connection D is related to the Levi-
+Civita connection on H (cf. Section 4.4). (3) Show that the harmonic map equation
+for h is equivalent to (4.9) (cf. Section 4.5).
+4.3. The equivariant map h is harmonic. The first step in the proof of Theorem 4.1
+is to choose the map h of Theorem 4.1 as a harmonic map into (H, gH) where the metric
+is given by
+gH(X, Y ) = n
+2 trace(h−1Xh−1Y ) for X, Y ∈ ThH.
+Definition 4.2. We call h or H a harmonic metric.
+
+32
+GEORGIOS DASKALOPOULOS AND CHIKAKO MESE
+For G = SL(n, C), K = SU(n), let sl(n) = k ⊕ p be the Cartan decomposition and
+B(X, Y ) = 2n trace(XY )
+be the Killing form on sl(n). The inner product is positive definite on p.
+Let Lg : G/K → G/K be left multiplication and define a metric gG/K on G/K by
+metrically identifying
+dLg−1 : TgKG/K → TeKG/K = p.
+(4.10)
+This defines (G/K, gG/K) as a symmetric space of non-compact type.
+Lemma 4.3. The map
+Ψ : G/K
+�→
+H
+gK
+�→
+g−1∗g−1 = h
+identifies (G/K, gG/K) isometrically with (H, gH) as G-spaces.
+Proof. First, Ψ is equivariant with respect to the action Lg on G/K and the action Ag
+on H. Indeed,
+Ψ ◦ Lg(g1K) = Ψ(gg1K) = (gg1)−1∗(gg1)−1 = g−1∗(g−1∗
+1
+g−1
+1 )g−1
+= g−1∗Ψ(g1K)g−1 = Ag ◦ Ψ(g1K).
+Second, Ψ is an isometry. Since the metric gG/K is defined by (4.10), we need to show
+that with h = g−1∗g−1 ∈ H,
+d(Lg−1)gK ◦ d(Ψ−1)h = d
+�
+(Ψ ◦ Lg)−1�
+h : ThH → TeKG/K = p
+is an isometry. This is a straightforward calculation: Let t �→ gt be a path in G/K with
+g0 = eK and ˙g0 ∈ TeKG/K (where dot indicates the t-derivative). For ˙g ∈ TeKG/K,
+since ˙g0 is self-adjoint,
+(dΨ)e(˙g0) = d
+dt
+���
+t=0(g−1∗
+t
+g−1
+t ) = −˙g∗
+0 − ˙g0 = −2˙g0.
+(4.11)
+For X ∈ ThH,
+d
+�
+(Ψ ◦ Lg)−1�
+h(X)
+=
+d
+�
+(Ag ◦ Ψ)−1�
+h(X) = d(Ψ−1 ◦ Ag−1)h(X)
+=
+d(Ψ−1 ◦ Ag−1)g−1∗g−1(X) = (dΨe)−1 ◦ (dAg−1)g−1∗g−1(X)
+=
+(dΨ−1)e(g∗Xg) = −1
+2g∗Xg
+=
+−1
+2Adg−1(gg∗X) = −1
+2Adg−1(h−1X).
+
+NOTES ON HARMONIC MAPS
+33
+Here we used (4.11) in the third to last equality.
+Using this formula and the Ad-
+invariance of the Killing form, we have for X, Y ∈ ThH
+B
+�
+d
+�
+(Ψ ◦ Lg)−1�
+h(X), d
+�
+(Ψ ◦ Lg)−1�
+h(Y )
+�
+= 1
+4B(h−1X, h−1Y )
+= n
+2trace(h−1Xh−1Y )
+= gH(X, Y ).
+□
+By Theorem 2.17, there exists a ρ-equivariant harmonic map f : ˜
+M → G/K. In
+view of the Lemma 4.3, we identify G/K with H and obtain a ρ-equivariant harmonic
+map
+h = f −1∗f −1 : ˜
+M → H,
+d⋆
+∇dh = 0
+where ∇ is the pullback to h∗TH of the Levi-Civita connection of (H, gH).
+4.4. The hermitian connection D and the Levi-Civita connection on (H, gH).
+Recall that H ⊂ SL(n, C) is the space of positive definite, self-adjoint matrices of
+determinant one and consider the map
+Ph : ThH → Ph(ThH) ⊂ sl(n),
+X �→ h−1X
+whose image Ph(ThH) consists of matrices self-adjoint with respect to h. Indeed,
+(h−1X)∗h = h−1(h−1X)∗h = h−1X.
+Extending this map complex linearly induces an isomorphism
+P C
+h : ThHC
+≃−→ sl(n)
+which defines a global isomorphism
+P C : THC ≃−→ H × sl(n).
+(4.12)
+The trivial connection d on H × sl(n) pulls back by the isomorphism P C to a flat
+connection ¯∇ on THC; i.e.
+¯∇XY
+���
+h = P C−1 ◦ dX ◦ P C(Y )
+���
+h.
+We next compute the formula for ¯∇ with respect to the coordinates that identify the
+space of (n × n)-matrices Mn×n(C) with Rn2. Let t �→ ht be a curve in H with h0 = h
+and ˙h0 = X(h). We have
+dX(P C(Y )) = d
+dt
+���
+t=0
+�
+h−1
+t
+Y (ht)
+�
+= h−1 d
+dt
+���
+t=0Y (ht) − h−1 ˙h0h−1 Y (h0).
+
+34
+GEORGIOS DASKALOPOULOS AND CHIKAKO MESE
+Using the embedding H ֒→ Mn×n(C), we express ht = (hij
+t ). Furthermore, we can ex-
+press X = Xij∂ij and Y = Y kl∂kl with respect to the coordinate basis (∂ij). Extending
+Y = Y kl∂kl as a vector field on Mn×n(C), we apply the chain rule to obtain
+d
+dt
+���
+t=0Y (ht) = d
+dt
+���
+t=0Y kl(ht)∂kl =
+�
+∂ijY kl���
+h
+˙hij
+0
+�
+∂kl =
+�
+Xij∂ijY kl� ���
+h ∂kl = ∂Y
+∂X
+���
+h.
+(4.13)
+Thus, the formula for the flat connection at the point h ∈ H is
+¯∇XY = ∂Y
+∂X − Xh−1Y.
+The Levi-Civita connection on TH, denoted by ∇ and extended complex linearly to
+THC, is given at h ∈ H by the formula
+∇XY = ∂Y
+∂X − 1
+2
+�
+Xh−1Y + Y h−1X
+�
+.
+Indeed:
+(i) ∇ is torsion free: First, for a function f defined near h,
+� ∂Y
+∂X − ∂X
+∂Y
+�
+f
+=
+(Xij∂ijY kl)∂klf − (Y kl∂klXij)∂ijf
+=
+(Xij∂ijY kl)∂klf + XijY kl∂ij∂klf − (Y kl∂klXij)∂ijf − Y klXij∂kl∂ijf
+=
+X(Y f) − Y (Xf) = [X, Y ]f.
+Thus,
+∇XY − ∇Y X
+=
+� ∂Y
+∂X − 1
+2(Xh−1Y + Y h−1X)
+�
+−
+�∂X
+∂Y − 1
+2(Y h−1X + Xh−1Y )
+�
+=
+∂Y
+∂X − ∂X
+∂Y = [X, Y ].
+(ii) ∇ is metric compatible: Using the path t �→ ht given above and using (4.13),
+XgH(Y, Z)
+=
+n
+2trace
+� ∂
+∂t
+���
+t=0
+�
+h−1
+t Y (ht)h−1
+t Z(ht)
+��
+=
+n
+2trace
+��
+h−1 ∂Y
+∂X − h−1Xh−1Y
+�
+h−1Z + h−1Y
+�
+h−1 ∂Z
+∂X − h−1Xh−1Z
+��
+=
+n
+2trace
+�
+h−1
+� ∂Y
+∂X − 1
+2
+�
+Xh−1Y + Y h−1X
+��
+h−1Z
+�
++n
+2trace
+�
+h−1Y h−1
+� ∂Z
+∂X − 1
+2(Xh−1Z + Zh−1X)
+��
+=
+gH(∇XY, Z) + gH(Y, ∇XZ).
+
+NOTES ON HARMONIC MAPS
+35
+The difference of the flat connection ¯∇ and the Levi-Civita connection ∇ on THC is
+� ¯∇XY − ∇XY
+�
+= 1
+2
+�
+Y h−1X − Xh−1Y
+�
+= −1
+2h
+�
+h−1X, h−1Y
+�
+.
+(4.14)
+Let ˆ∇ = P C◦∇◦P C−1 denote the pullback to H×sl(n) of the Levi-Civita connection
+∇ on THC via (4.12). Then the corresponding formula to (4.14) for the difference
+between the flat connection d and ˆ∇ on H × sl(n) → H is
+dX − ˆ∇X = −1
+2
+�
+h−1X, ·
+�
+.
+(4.15)
+The bundle H × sl(n) pulls back by h : ˜
+M → H to the trivial SL(n, C)-bundle h∗(H ×
+sl(n)) on the universal cover ˜
+M.
+h∗(H × sl(n))
+H × sl(n)
+˜
+M
+H
+h
+From (4.15), the difference of the flat connection d and the pullback h∗ ˆ∇ is given by
+the formula
+dV − (h∗ ˆ∇)V = −1
+2
+�
+h−1dh(V ), ·
+�
+.
+(4.16)
+Next, the pullback to ˜
+M of the endormorphism bundle End0(E) is isomorphic to
+the trivial bundle.
+Taking the quotient by the induced action from ρ, End0(E) ≃
+h∗(H ×ρ sl(n)) → M and the connection h∗ ˆ∇ induces a connection on End0(E) (which
+we also call ˆ∇).
+From (4.16), we have
+ˆ∇ = d + 1
+2
+�
+h−1dh, ·
+�
+.
+Hence,
+ˆ∇ = D
+by (4.8). In other words, D is the connection on End0(E) induced by the Levi-Civita
+connection on T CH.
+4.5. Completion of the proof of Theorem 4.1. The bundle isomorphism P C−1 of
+(4.12) induces a bundle isomorphism (still denoted by P C−1)
+h∗(H ×ρ sl(n))
+≃
+h∗(THC) → M
+φ
+�→
+hφ.
+Also,
+ˆ∇ = P C ◦ ∇ ◦ P C−1.
+(4.17)
+
+36
+GEORGIOS DASKALOPOULOS AND CHIKAKO MESE
+In particular, since
+θ = −1
+2h−1dh ∈ Ω1(M, End0(E)) ≃ Ω1(M, h∗(H ×ρ sl(n))),
+we have
+P C−1θ = hθ = −1
+2dh ∈ Ω1(M, h∗(THC)).
+(4.18)
+Theorem 4.1 follows from the the following implications:
+h is harmonic
+⇒
+0 = −1
+2d∗
+∇dh = d∗
+∇hθ = d∗
+∇P C−1θ
+by (4.18)
+⇒
+0 = P Cd∗
+∇P C−1θ = d∗
+ˆ∇θ = d∗
+Dθ
+by (4.4) and (4.17).
+References
+[BH]
+M. R. Bridson and A. Haefliger. Metric Spaces of Non-Positive Curvature. Springer-Verlag,
+Berlin (1999).
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+K. Corlette. Flat G-bundles with canonical metrics. J. Differential Geom. 28 (1988) 361-382.
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+J. Jost. Nonlinear Methods in Riemannian and K¨ahlerian Geometry. Birkh¨auser Verlag 1988.
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+ings. Publications mathematiques de I.H.E.S, tome 86 (1997), 115-197.
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+pact spaces. Comm. Anal. Geom. 5 (1997) 213-266.
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+N. Korevaar and R. Schoen. Global existence theorem for harmonic maps to non-locally com-
+pact spaces. Comm. Anal. Geom. 5 (1997), 333-387.
+[Li]
+Q. Li. An Introduction to Higgs Bundles via Harmonic Maps. SIGMA 15 (2019).
+[LSY] K. Liu, X. Sun, X. Yang and ST. Yau.Curvatures of moduli space of curves and applications.
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+[LY]
+K. Liu and X. Yang. Hermitian harmonic maps and non-degenerate curvatures. Mathematical
+Research Letters 21 (2014) 831-862.
+[M]
+T. Mochizuki. Asymptotic behaviour of tame harmonic bundles and an application to pure
+twistor D-modules. Memoirs of the AMS 185 (2007).
+[Sa]
+J. H. Sampson. Harmonic maps in K¨ahler geometry. Harmonic mappings and minimal im-
+mersions, 193–205, Lecture Notes in Math., 1161, Springer, Berlin, 1985.
+[S]
+R. Schoen. Analytic Aspects of the Harmonic Map Problem. Seminar on Nonlinear Partial
+Differential Equations, 1984, MSRI Publications book series, Volume 2.
+[Siu]
+Y.-T. Siu. The complex analyticity of harmonic maps and the strong rigidity of compact K¨ahler
+manifolds. Ann. of Math. 112 (1980) 73-111.
+

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+Draft version January 5, 2023
+Typeset using LATEX twocolumn style in AASTeX63
+Diagnosing limb asymmetries in hot and ultra-hot Jupiters with high-resolution transmission
+spectroscopy
+Arjun B. Savel
+,1, 2 Eliza M.-R. Kempton
+,2 Emily Rauscher
+,3 Thaddeus D. Komacek
+,2
+Jacob L. Bean
+,4 Matej Malik
+,2 and Isaac Malsky
+3
+1Center for Computational Astrophysics, Flatiron Institute, New York, NY 10010, USA
+2Astronomy Department, University of Maryland, College Park, 4296 Stadium Dr., College Park, MD 207842 USA
+3Department of Astronomy, University of Michigan, 1085 South University Avenue, Ann Arbor, MI 48109, USA
+4Department of Astronomy & Astrophysics, University of Chicago, Chicago, IL 60637, USA
+Submitted to ApJ
+Abstract
+Due to their likely tidally synchronized nature, (ultra)hot Jupiter atmospheres should experience
+strongly spatially heterogeneous instellation. The large irradiation contrast and resulting atmospheric
+circulation induce temperature and chemical gradients that can produce asymmetries across the eastern
+and western limbs of these atmospheres during transit. By observing an (ultra)hot Jupiter’s transmis-
+sion spectrum at high spectral resolution, these asymmetries can be recovered—namely through net
+Doppler shifts originating from the exoplanet’s atmosphere yielded by cross-correlation analysis. Given
+the range of mechanisms at play, identifying the underlying cause of observed asymmetry is nontrivial.
+In this work, we explore sources and diagnostics of asymmetries in high-resolution cross-correlation
+spectroscopy of hot and ultra-hot Jupiters using both parameterized and self-consistent atmospheric
+models. If an asymmetry is observed, we find that it can be difficult to attribute it to equilibrium
+chemistry gradients because many other processes can produce asymmetries. Identifying a molecule
+that is chemically stable over the temperature range of a planetary atmosphere can help establish a
+“baseline” to disentangle the various potential causes of limb asymmetries observed in other species.
+We identify CO as an ideal molecule, given its stability over nearly the entirety of the ultra-hot Jupiter
+temperature range. Furthermore, we find that if limb asymmetry is due to morning terminator clouds,
+blueshifts for a number of species should decrease during transit. Finally, by comparing our forward
+models to Kesseli et al. (2022), we demonstrate that binning high-resolution spectra into two phase
+bins provides a desirable trade-off between maintaining signal to noise and resolving asymmetries.
+Keywords: Exoplanet atmospheric composition (2021) — Radiative transfer simulations (1967) — High
+resolution spectroscopy (2096) — Hot Jupiters (753)
+1. INTRODUCTION
+Exoplanet atmospheres vary spatially.
+This is es-
+pecially the case for tidally locked exoplanets, which
+feature permanent daysides and permanent nightsides;
+such strong gradients in instellation in turn drive strong
+latitudinal and longitudinal variations in atmospheric
+temperature, dynamics, and chemistry (e.g., Showman
+& Guillot 2002; Cooper & Showman 2005; Harrington
+Corresponding author: Arjun B. Savel
+[email protected]
+et al. 2006; Cho et al. 2008; Menou & Rauscher 2009;
+Showman et al. 2009; Rauscher & Menou 2010; Perna
+et al. 2012; Mayne et al. 2014; Demory et al. 2016; Fro-
+mang et al. 2016; Kataria et al. 2016; Parmentier et al.
+2018; Zhang & Showman 2018; Kreidberg et al. 2019;
+Komacek et al. 2019; Tan & Komacek 2019; Parmentier
+et al. 2021; Roman et al. 2021).
+The spatial variations in exoplanet atmospheres have
+increasingly observable ramifications. Even with the in-
+sight gained from modeling exoplanet atmospheres as
+one-dimensional objects (e.g., Madhusudhan & Seager
+2009; Crossfield & Kreidberg 2017; Yan et al. 2019;
+arXiv:2301.01694v1  [astro-ph.EP]  4 Jan 2023
+
+ID2
+Savel et al.
+Benneke et al. 2019), a substantial and growing liter-
+ature demonstrates that upcoming JWST (Beichman
+et al. 2014) data will require consideration of 3D pro-
+cesses for accurate interpretation of exoplanet atmo-
+spheric data (Feng et al. 2016; Blecic et al. 2017; Caldas
+et al. 2019; Lacy & Burrows 2020; MacDonald et al.
+2020; Pluriel et al. 2020; Espinoza & Jones 2021; Pluriel
+et al. 2022; MacDonald & Lewis 2022; Welbanks &
+Madhusudhan 2022). Perhaps more urgently, ground-
+based high-resolution (R ≥ 15, 000) cross-correlation
+spectroscopy (HRCCS; for a review, see Birkby 2018)
+datasets already show signs of significant multidimen-
+sionality (Flowers et al. 2019; Beltz et al. 2020; Gandhi
+et al. 2022; Herman et al. 2022; van Sluijs et al. 2022).
+In transit geometry, HRCCS is similar to the more
+traditional transmission spectroscopy technique (e.g.,
+Charbonneau et al. 2002).
+Both methods leverage
+the idea that, as an exoplanet passes between its host
+star and an observer, stellar light is attenuated on a
+wavelength-dependent basis as it passes through the up-
+per layers of the planet’s atmosphere. But with HRCCS,
+the planetary absorption spectrum is buried in the stel-
+lar and telluric noise. Therefore, models of planetary ab-
+sorption often cannot be directly compared to HRCCS
+data.1
+However, by leveraging cross-correlation tech-
+niques, researchers can combine the signal from the
+many planetary absorption lines resolved at high resolu-
+tion to yield a combined, statistically significant signal
+(e.g., Snellen et al. 2010).
+The resolving of individual spectral lines allows for
+more than just binary detection/non-detection of plane-
+tary absorption: crucially, the Doppler shifts of plan-
+etary absorption lines are recoverable.
+The Doppler
+shifting of planetary lines due to the planet’s orbital
+motion is in fact central for extracting the planetary
+signal with cross-correlation techniques, as the stellar
+and telluric lines are comparatively static. Specifically,
+a template spectrum is chosen to model the planetary
+absorption signal, and it is cross-correlated against the
+combined planet, star, and telluric signal by Doppler
+shifting the template at varying velocities and multiply-
+ing the shifted template against the combined observed
+signal. The resulting cross-correlation function (CCF, a
+function of Doppler-shifted velocity) is maximized at the
+Doppler shift where the template best matches the com-
+bined observed signal—that is, at the Doppler shift of
+the planet signal in the observed combined data. Again,
+1 There are a few notable exceptions in the high-resolution spec-
+troscopy literature in which planetary absorption is strong
+enough (and data quality high enough) that individual planetary
+absorption lines can be analyzed; e.g., Tabernero et al. (2021).
+this method requires that the planet’s spectral lines
+move across a spectrograph’s pixels during observations,
+with the stellar and telluric lines largely remaining on
+the same pixel (or being easily detrended in time). With
+current instruments, this assumption is certainly justi-
+fied for tidally locked ultra-hot Jupiters, which tend to
+have high orbital velocities (e.g., Fortney et al. 2021).
+With the planetary signal identified, further Doppler
+shifting and line broadening that is not associated with
+planetary orbital motion, telluric lines, or stellar lines is
+attributable to the 3D manifestations of planetary ro-
+tation and winds (Kempton & Rauscher 2012; Show-
+man et al. 2013; Kempton et al. 2014; Brogi et al. 2016;
+Ehrenreich et al. 2020). Thus, the multidimensionality
+of exoplanetary atmospheres is imprinted on HRCCS
+data.
+Recent years have seen the intrinsic 3-dimensionality
+of these objects be uniquely constrained with transit
+HRCCS results. Observational studies such as Louden
+& Wheatley (2015), Ehrenreich et al. (2020), and Kesseli
+et al. (2022) have isolated signals from the morning
+and evening limbs of planetary atmospheres, unveiling
+Doppler shifts of multiple chemical species at multiple
+points in transit—and hence over multiple longitudi-
+nal slices. Such studies have revealed asymmetries in
+the probed Doppler velocity field (i.e., changes in the
+Doppler shift of the CCF maximum as a function of
+orbital phase), which are often attributed to physical
+asymmetries in the atmosphere.
+However, as reviewed in Section 2, an asymmetric sig-
+nal in HRCCS can arise from a combination of different
+classes of mechanisms: 1) chemistry, 2) clouds, 3) dy-
+namics, 4) orbital properties, and 5) thermal structure.
+Disentangling these effects is not a straightforward pro-
+cess.
+This may especially be the case if transmission
+spectra must be stacked together to achieve a higher
+signal-to-noise ratio (SNR), thereby smearing phase in-
+formation.
+In this work, we aim to explore the general question
+of asymmetry in exoplanet atmospheres, with particu-
+lar focus on its manifestations in high-resolution trans-
+mission spectroscopy. Section 2 examines what drives
+asymmetry in exoplanet atmospheres; we here define a
+metric that quantifies limb-to-limb asymmetry. In Sec-
+tion 3, we elaborate on diagnostics of specific mecha-
+nisms that may drive such asymmetries. This section
+additionally emphasizes how these diagnostics may be
+used to support or falsify compelling “toy models” mo-
+tivated by the drivers described in Section 2. Finally,
+we summarize our results in Section 4.
+2. SELECTED DRIVERS
+
+Diagnosing limb asymmetries in transmission
+3
+Table 1. Example drivers of phase asymmetries of ultra-hot Jupiters
+Mechanism
+Proposed diagnostic
+Selected work(s)
+Escaping atmosphere
+H Lyman-α transit duration
+Owen et al. (2023)
+Very deep, e.g., Ca II lines
+Fossati et al. (2013)
+Strong vertical winds
+Seidel et al. (2021)
+Very broadened, e.g., Na I lines
+Hoeijmakers et al. (2020)
+Large diff. between (Na) doublet lines
+Hoeijmakers et al. (2020)
+Strong H-α absorption
+Wyttenbach et al. (2020)
+Blueshifting CCF w/ phaseP
+Bourrier et al. (2020)
+Excess He absorption (10830˚A)
+Oklopˇci´c & Hirata (2018), Spake et al. (2018)
+Compare Doppler shifts of ions w/ different masses
+This work
+Scale height difference
+Blueshifting CCF w/ phase
+Kempton & Rauscher (2012)
+Strongly varying CO Doppler shift w/ phase
+Wardenier et al. (2021), this work
+H2 dissoc./recomb.
+Small phase curve amplitude
+Mansfield et al. (2020)
+High continuum/muted spectrum from H−
+Arcangeli et al. (2018)
+Weak drag state
+Blueshifted CCF
+Wardenier et al. (2021), Savel et al. (2022)
+Large phase curve offset
+May & Komacek et al. (2021)
+Cold interior
+Blueshifted CCF
+Savel et al. (2022)
+Large phase curve amplitude
+May & Komacek et al. (2021)
+Cold nightside
+May & Komacek et al. (2021)
+Superrotating jet
+CCF FWHM exceeding solid-body rotation
+Brogi et al. (2016)
+Phase curve offset
+Knutson et al. (2007)
+Day-night winds
+Blueshifted CCF
+Snellen et al. (2010)
+Equilibrium chemistry
+Limb-to-limb abundance discrepancy
+This work
+Compare chemical, dynamical timescales
+Showman et al. (2013)
+Photochemistry
+Disequilibrium abundance of species
+Tsai et al. (2017)
+Increase in product, decrease in parent w/ phaseP
+Future work
+Condensation
+Model with GCM
+Wardenier et al. (2021), Savel et al. (2022)
+Strongly blueshifting CCF w/ phase
+Ehrenreich et al. (2020)
+Eccentricity
+All lines similarly Doppler shifted
+Montalto et al. (2011), Savel et al. (2022)
+Independent orbital constraints
+Montalto et al. (2011)
+Clouds
+All species become less blueshifted w/ phase
+This work
+Blueshifted CCF
+Savel et al. (2022)
+Lines absent in low resolution present at high resolution
+Kempton et al. (2014), Hood et al. (2020)
+Comparing CCFs of water bands
+Pino et al. (2018)
+Lorentz forces
+Reduced hotspot offset
+Beltz et al. (2021)
+Increased phase curve amplitude
+Beltz et al. (2021)
+Westward hotspot offset
+Hindle et al. (2021)
+Spatially varying winds
+Variable Doppler shift over phase
+Kempton & Rauscher (2012)
+Compare ingress / egress Doppler shifts
+Kempton & Rauscher (2012)
+Compare Doppler shifts of different-strength lines
+Kempton & Rauscher (2012)
+Tidal deformation/lag
+Blueshifting CCF over phaseP
+Future work
+Light curve fitting
+Akinsanmi et al. (2019)
+T-dependent opacity
+Blueshifting CCF over phase
+Wardenier et al. (2021)
+Note—Tests with a “P” superscript have been proposed but not explicitly modeled.
+
+4
+Savel et al.
+There exist a number of potential drivers of asymme-
+try in high-resolution transmission spectroscopy.
+But
+what are the relative strengths of these drivers?
+Previous works have considered the effects of conden-
+sation, longitude-dependent winds, and orbital eccen-
+tricity in producing such asymmetries (Wardenier et al.
+2021; Savel et al. 2022). Table 1 includes these and a
+number of other potential drivers of asymmetry (along
+with potential diagnostics; Section 3).
+While many
+drivers are listed in Table 1, we consider in this work the
+relative strengths of two potentially first-order effects:
+the “scale height effect” and differences in equilibrium
+chemistry abundance across the limbs of the planet. Be-
+ing both temperature-dependent effects, the distinction
+between the two is particularly subtle from an observa-
+tional perspective, and hence interesting from a theoret-
+ical perspective.
+The scale height effect is due to the larger scale height
+in hotter regions (e.g., Miller-Ricci et al. 2008), such that
+they are “puffed up” and cover more solid angle on the
+sky. These hotter regions therefore contribute more to
+the observed net Doppler signal in HRCCS. The scale
+height effect is seen in Kempton & Rauscher (2012) as
+a slight, increasing blueshift over transit and as slight
+ingress/egress differences. The effect there is not as dra-
+matic as in planets with larger east–west limb asymme-
+try, such as WASP-76b (West et al. 2016; Wardenier
+et al. 2021; Savel et al. 2022).
+With respect to equilibrium chemistry: because of the
+strong day–night contrasts in (ultra)hot Jupiter atmo-
+spheres, there exist strong spatial variations in temper-
+ature. The day–night contrasts result in east–west con-
+trasts because the equatorial jet advects hot gas ahead
+of the substellar point to the evening limb and relatively
+cold gas from the antistellar point to the morning limb.
+Furthermore, as the planet rotates on its spin axis dur-
+ing transit, the hotter side of the planet progressively ro-
+tates into view, exacerbating these differences at egress.
+Ignoring all disequilibrium processes and scale height
+differences, there should therefore exist strong spatial
+variations in gas-phase atmospheric composition; at a
+given bulk composition, equilibrium chemistry implies
+variations in chemistry solely as a function of temper-
+ature and pressure. It is expected that asymmetries in
+transmission could hence vary as a function of temper-
+ature due to differences in chemistry alone.
+Chemical gradients are invoked to explain a number
+of observational datasets (e.g., Ehrenreich et al. 2020;
+Kesseli & Snellen 2021). However, other temperature-
+dependent effects, such as the scale height effect, may
+instead be driving observed asymmetries. With this dis-
+tinction in mind, it is prudent to consider the difference
+in strength between these two effects and whether one
+considerably outweighs the other.
+2.1. Asymmetry metric
+To quantify the asymmetry of chemical abundance in
+a planetary atmosphere, we construct a west–east asym-
+metry metric, AWE:
+AW E = 1
+C
+�
+west
+log10
+��
+nα(T, P) dl
+�
+dΩ
+− 1
+C
+�
+east
+log10
+� �
+nα(T, P) dl
+�
+dΩ,
+(1)
+where, for species α, n is the number density in an atmo-
+spheric cell, dΩ is the solid angle subtended by a given
+sky-projected radius–latitude cell, and there are C total
+cells per limb. By equilibrium chemistry, n is a func-
+tion solely of temperature T and pressure P within a
+given cell in the modeled 3D atmosphere. For each 2D
+sky-projected radius–latitude cell, dl is integrated along
+the line of sight through the planet’s 3D modeled atmo-
+sphere. This metric takes into account regions of the
+planet outside the terminator (which impacts transmis-
+sion spectra even at low resolution; e.g., Caldas et al.
+2019, Wardenier et al. 2022) by ray-striking through a
+3D atmosphere.
+AWE essentially reduces to the difference in mean (log)
+abundance between the two limbs.
+The sign of this
+quantity encodes information about the asymmetry, as
+well: positive AWE implies that the western limb is more
+abundant in a species, whereas negative AWE implies
+that the eastern limb is more abundant in a species.
+2.2. Model atmospheres
+As of yet, we have remained agnostic to the model
+that generates the temperature–pressure structure and
+defines the grid cells for an AWE calculation. Some of
+the most complex and physics-rich descriptions of 3D
+exoplanet temperature–pressure structures are given by
+general circulation models (GCMs; e.g., Showman et al.
+2009). In this study, however, we seek to gain intuition
+for the basic scaling of asymmetry with planetary tem-
+perature (which drives the scale height and equilibrium
+chemistry gradients), and the added physical complexity
+of GCMs could add “noise” to this “signal”—it would
+be difficult to isolate the effect of increasing planetary
+temperature alone. Furthermore, we here consider un-
+physical situations in order to determine the magnitude
+of the resulting difference with the correct physics. Fi-
+nally, GCMs are very computationally expensive to run
+and have a number of free parameters to tune, and we
+here aim to explore a nontrivial grid of models over a
+representative range of parameter space.
+
+Diagnosing limb asymmetries in transmission
+5
+We opt for a simple, parameterized approach instead
+of pursuing a full GCM description of our atmospheres
+for this specific experiment.
+Our model atmospheres
+have two parameters: a normalized east–west contrast
+˜∆T = (Teast − Twest)/Teast and an equilibrium temper-
+ature Teq. A normalized east–west contrast is a natural
+choice over an absolute east–west contrast for this work;
+namely, it prevents negative temperatures at low Teq,
+and it has physical meaning motivated by dynamical
+theory (e.g., Tan & Komacek 2019). In these models, the
+choice of ˜∆T also uniquely enforces the east-west tem-
+perature differences. The limb-to-limb difference cannot
+exceed the day–night difference; based on the GCMs of
+Tan & Komacek (2019) and a set of phase curve ob-
+servations (Parmentier & Crossfield 2018), we do not
+expect a day–night contrast to exceed 0.6, so we hold
+our east–west contrast below this value.
+Hence, we here sweep our parameterized atmospheric
+models in ˜∆T from 0.1–0.6, in addition to sweeping in
+Teq from 1000 K – 4000 K. Each atmosphere is charac-
+terized by two isothermal temperature–pressure profiles.
+Defining
+Teast = Teq + ∆T/2
+Twest = Teq − ∆T/2
+(2)
+and noting that ∆T = Teast−Twest, it therefore follows
+that
+Teast =
+Teq
+1 − ˜∆T/2
+Twest =
+Teq
+1 + ˜∆T/2
+.
+(3)
+With the substellar longitude at 0◦, all cells with a
+longitude φ < 180◦—the warmer evening limb—are as-
+signed temperature Teast.
+Conversely, all cells with a
+longitude φ > 180◦—the cooler morning limb—are as-
+signed temperature Twest. Pressures in the atmosphere
+run as low as 1 µbar, as one of the benefits of HRCCS
+is that it can probe low pressures such as these (e.g.,
+Kempton et al. 2014; Gandhi et al. 2020; Hood et al.
+2020). The bottom of the atmosphere is set at 0.5 bar;
+our previous 3D forward models run in Savel et al. (2022)
+across the optical and near-infrared indicate that for our
+test case of WASP-76b (West et al. 2016), this region
+is the deepest that can be probed given the expected
+continuum opacity. The parameterized modeled atmo-
+spheres in this study have no set wind fields, as in our
+models (motivated by and assuming chemical equilib-
+rium), winds do not control AWE—only the chemical
+abundance of a given cell does.
+We calculate AWE to assess the relative strength of the
+scale height and equilibrium chemistry effects. To infer
+the strength of the scale height e���ect, we construct pairs
+of model atmospheres. In each pair, one atmosphere is
+constructed self-consistently: pressure falls off per hy-
+drostatic equilibrium, with the scale height set by the
+temperature on either limb. For the models here, we
+hold composition constant across both limbs, thereby
+holding µ constant at 2.36 (appropriate for a solar-
+composition gas dominated by molecular H2; Kempton
+& Rauscher 2012). See Section 2.4 for a discussion of this
+caveat. The other atmosphere in the pair is constructed
+on the same pressure grid as the western limb at all lon-
+gitudes. That is, the eastern limb is not simulated as
+inflated compared to the western limb—removing the
+scale height effect from the projected model atmosphere
+in transmission.
+2.3. Equilibrium chemistry
+To calculate the number densities of our species in
+each modeled atmospheric cell (nα), we construct a grid
+in temperature–pressure space using the FastChem equi-
+librium chemistry code (Stock et al. 2018) and interpo-
+late the grid based on local atmospheric cell temper-
+ature and pressure.
+We initialize the code with solar
+abundances from Lodders (2003). Our chemistry code
+does not explicitly include any condensation or cloud-
+formation processes.
+Even disregarding questions of species detectability in
+HRCCS data, it is worth considering that not all species
+with FastChem thermochemical data have freely avail-
+able opacity data. With this constraint in mind, we re-
+strict our AWE molecule calculations to molecules with
+opacity data available on ExoMol,2 a popular opacity
+database for exoplanet atmosphere modeling.
+2.4. Asymmetry metric: application
+We calculate AWE for our grid of parameterized at-
+mospheres. Disregarding the scale height effect, we find
+that positive ions tend to form preferentially on the
+hotter limb of our models at an equilibrium tempera-
+ture of 2200 K (Figure 1). This is expected, as ther-
+mal ionization should increase the abundance of positive
+ions at higher temperatures. Furthermore, larger east–
+west temperature asymmetries lead to larger abundance
+asymmetries.
+Including the scale height effect increases the asymme-
+try for neutral atoms and molecules, as can be seen by
+comparing the right-hand sides of Figures 1–2. Further-
+more, there is more homogeneity across the AWE values
+2 https://www.exomol.com/
+
+6
+Savel et al.
+(a)
+(b)
+Figure 1. Asymmetry (as defined in Equation 1) of all chemical species considered in this study in our parameterized at-
+mospheres at an equilibrium temperature of 2200 K. These models do not self-consistently inflate the hotter limb of the
+parameterized model (i.e., they do not observe the “scale height effect”). The shading of each species represents the normalized
+temperature difference, ˜∆T, across the two limbs of our parameterized atmospheres; the lightest boxes have ˜∆T = 0.1, whereas
+the darkest have ˜∆T = 0.6. For illustrative purposes, we color in green tick marks for species with detections noted in Guillot
+et al. (2022) (and including the recent CO2 detection; Ahrer et al. 2022). We also draw a vertical line denoting 0 asymmetry.
+Without taking the scale height effect into account, positive ions form much more predominantly on the warmer limb (i.e., have
+negative asymmetry) than other species and reach the greatest asymmetry values.
+across positive ions, negative ions, neutral atoms, and
+neutral molecules (Figure 2). In particular, while higher
+˜∆T still implies higher absolute asymmetry in neutral
+species, the scale height effect makes it such that the
+warmer limb almost always has higher projected asym-
+metry.
+It is therefore clear that the scale height effect strongly
+tamps down genuine variation in species abundance due
+to equilibrium chemistry. However, the fact that inter-
+species variation in asymmetry remains implies that this
+variation in abundance is not completely washed out by
+the scale height effect; if the scale height effect truly and
+fully dominated, all species would have the same AWE
+value.
+When considering individual species more closely, we
+find that certain species are particularly differentially
+affected by the scale height effect.
+For example, Fig-
+ure 3 shows that there is a stark difference in whether
+the scale height effect is included for Fe. However, this
+is not as much the case for, e.g., Sr II. The meaning be-
+hind this result is evident in the equilibrium abundance
+calculations of Fe and Sr II: Fe is less sensitive to tem-
+
+= 2200K, scaled=False
+JS
+Rb 
+Ge
+Ga
+Sc -
+Ca
+Li -
+Zn
+V-
+Ti
+si -
+s
+p-
+0
+-IN
+Na
+N-
+K
+H-
+Fe
+Cu
+Cr :
+Co
+CI -
+C
+Al -
+Zr IIII
+YIII
+Sr III
+Rb IlI
+Ge llI
+Ga IlI
+Zn III
+Cu llI
+Ni III
+Co IIII
+Fe IIII
+Mn III
+Cr IIII
+VII
+Ti II
+Sc III
+S
+Ca II
+e
+KII
+Ar IIII
+Positive-ions
+CI II
+ads
+S III
+Negative ions
+P III
+Si III
+Al II
+Mg IIII
+Na III
+Nelll
+F III
+O I=I
+N III
+C III
+Li I
+Rb II
+Ge ll
+Ga lI
+Li II
+Zn II
+Si lI
+s II
+PII
+Ni lI
+Ne ll
+Na II
+NII
+Mg II
+KII
+He lI
+H3O lI
+H2 II
+HO II
+HII
+F II
+Cu II
+Co II
+CI II
+C II
+Ar II
+AIlII
+ScllI
+YII
+Sr II
+Zr II
+VII
+CrlI
+Mn II
+Call
+Ti lI
+Fel
+-102
+-101
+-100
+100
+101
+AsymmetryTeg = 2200K, scaled=False
+sis
+PS
+SO3
+SiO2
+SO2
+02
+sio
+PO
+N20
+N2
+NS
+PN
+NO
+SiH2
+H202
+TiH
+HSi
+SH
+HP
+NiH
+NaH
+HNO3
+HN
+MgH
+HKO
+FeH
+PF3
+NaF
+FMg
+HF
+CrH
+CINa
+CIK
+CIH
+Cao
+CaH
+CaF
+C2H4
+C2
+CS
+CP
+COS
+CN
+CH3
+S
+CH20
+e
+CH
+AIO
+AlH
+AIF
+AICI
+OH
+ov
+S
+O!!
+PH3
+NH3
+H2S
+H20
+CO2
+CO
+CH4
+C2 H2
+H2 
+Zn
+s
+P
+Ne
+N
+Ge
+F
+Cl
+Ar
+Co
+Al
+Z
+Sc
+Y
+Sr
+Rb
+Ga
+Cu
+IS
+0
+Li
+Mn
+Ni
+Cr
+c
+IL
+Ca
+Neutral atoms
+Mg
+Fe
+Na
+Neutral molecules
+He
+H
+-102
+-101
+-100
+100
+101
+0
+AsymmetryDiagnosing limb asymmetries in transmission
+7
+(a)
+(b)
+Figure 2. Similar to Figure 1, but now including the scale height effect (inflating the hotter limb in our parameterized models).
+Now, all species have asymmetries that favor the hotter limb (negative asymmetry)—simply because the hotter limb subtends
+more solid angle on the sky. However, there still exists inter-species variability in asymmetry, implying that the scale height
+effect does not entirely swamp genuine differences in equilibrium chemistry across limbs. Furthermore, negative ions still have
+larger asymmetries than positive ions or neutral species.
+perature variations than Sr II. This result is expected,
+as the onset of Sr II is determined by the temperature
+at which Sr I can be effectively ionized. This is gen-
+erally the case for positive ions—the temperature effect
+on chemical abundance wins out over the scale height ef-
+fect, as seen by the left-hand sides of Figures 1–2. Physi-
+cally, this behavior is because the Saha equation is more
+strongly dependent on temperature than most chemical
+equilibrium reaction rates.
+The results of this experiment indicate that the most
+temperature-sensitive species are strongly influenced by
+both abundance changes and scale height differences.
+Conversely, to isolate the scale height effect, it would be
+therefore useful to consider a species with very weakly
+temperature-dependent abundance; in this case, if a
+strong asymmetry were detected, it could be attributed
+to a scale height effect (or other non-equilibrium chem-
+istry or physics). We explore this idea further in Sec-
+tion 3.
+Note that this approach, aside from its simplified
+temperature–pressure structure, does not account for a
+variety of physics. Namely, it does not include the ef-
+fects of hydrogen dissociation and recombination that
+occurs in the ultra-hot Jupiter regime (Tan & Komacek
+2019). Inclusion of this physics would serve to decrease
+the mean molecular weight in the atmosphere, increasing
+the scale height for the hotter, eastern limb, thereby am-
+plifying the observed asymmetry. Additionally, at the
+
+= 2200K, scaled=True
+Zr -
+Sr
+Rb
+Ge
+Ga
+Sc -
+Ca
+Li -
+Zn
+V-
+Ti
+Si -
+s
+p-
+0
+-IN
+Na
+N-
+K
+H-
+Fe
+F .
+no
+Cr:
+Co
+CI -
+c
+Al -
+Zr IIII
+Y III
+Sr III!
+Rb II
+Ge llI
+Ga IlI
+Zn III
+Cu llI
+Ni III
+Co IIII
+Fe IIII
+Mn IIII
+Cr III
+VII
+Ti II
+Sc III
+S
+Ca llI
+e
+KIII
+Ar IIII
+Positive-ions
+CI IIII
+S III
+PIII
+Negative ions
+Si III
+Al III
+Mg IIII
+Na II
+Ne III
+FII
+o II
+NIII
+C II
+Li lII
+Rb II
+Ge ll
+Ga lI
+LilI
+Zn II
+Si lI
+sII
+PII
+Ni lI
+Ne lI
+Na llI
+NII
+Mg II
+KII
+He lI
+H3O II
+H2 II
+HO II
+H II
+F II
+Cu lI
+Co II
+CI II
+C II
+Ar II
+AIlII
+ScllI
+YII
+SrlI
+ZrlI
+viII
+CrlI
+Mn II
+Ca lI
+Ti II
+Fe ll
+-102
+-100
+100
+101
+-101
+0
+Asymmetry= 2200K, scaled=True
+sis
+PS
+SO3
+SiO2
+Neutralatoms
+SO2
+02
+Neutral-molecules
+sio
+ PO
+N20
+N2 
+NS
+PN
+NO
+SiH2
+H202
+TiH
+HSi
+SH
+HP
+NiH
+NaH
+HNO3
+NH
+H6W
+HKO
+FeH
+PF3
+NaF
+FMg
+HF
+CrH
+CINa
+CIK
+CIH
+Cao
+CaH
+CaF
+C2H4
+C2
+cs
+CP
+COS
+CN
+CH3
+S
+CH20
+CH
+AIO
+AlH
+Speo
+AIF
+AICI
+HO
+ov
+S
+O!!
+PH3
+NH3
+H2S
+H20
+CO2
+co
+CH4
+C2 H2
+H2 1
+Zn
+s
+P
+Ne
+N
+Ge
+F
+C1
+Co
+Al
+Z
+Sc
+Y
+Sr
+Rb
+Ga
+Cu
+Si
+0
+Li
+Mn
+Ni
+Cr
+c
+V
+IL
+Ca
+Mg
+Na
+K
+He
+H
+-102
+100
+-101
+101
+-100
+0
+Asymmetry8
+Savel et al.
+Figure 3. Asymmetry (per Equation 1) for Sr II, Fe, H2O, and CO in our parameterized atmospheres. Our grid sweeps over
+equilibrium equilibrium temperature and normalized temperature difference across limbs, and includes models that observe the
+scale height effect (circles) and do not (squares). We find that species with strong temperature-dependent abundances (e.g.,
+Sr II) are less dominated by the scale height effect than species with weaker temperature-dependent abundances.
+lower-temperature end, we did not include the effects of
+certain species being sequestered into clouds (e.g., sili-
+cate clouds). We will model the Doppler shift impact
+of optically thick clouds in Section 3.1.2. Finally, our
+approach does not include disequilibrium effects (e.g.,
+vertical / horizontal mixing) that may alter asymme-
+tries. Therefore, the results shown here motivate asym-
+metries due to equilibrium chemistry alone, which we
+expect to be a first-order driver of asymmetry; disequi-
+librium chemistry is not expected to be significant in the
+ultrahot Jupiter regime (e.g., Tsai et al. 2021).
+We further did not include the effect of temperature-
+and pressure-dependent opacities.
+At the spectrum
+level, a temperature asymmetry would be exaggerated
+by the fact that, e.g,. Fe absorbs more on the hotter
+limb than the colder limb because its opacity increases
+with temperature. This would mean that the detected
+net Doppler shift is even more strongly weighted to the
+hotter limb.
+Despite these limitations in our modeling, the trends
+listed above should hold to first order and provide intu-
+ition about the relative strengths of two potential drivers
+of asymmetry in exoplanet atmospheres.
+Broadly, it
+holds that the scale height effect appears to dominate in
+general, but relative differences in abundances of species
+as a function of temperature still matter. Given the lim-
+
+Al
+CO
+0.6
+101.
+ No scale height effect
+101.
+· Scale height effect
+least
+口
+口
+0.5
+口
+口
+0 
+0 
+口
+口
+口
+口
+C
+口
+口
+0.4
+O
+0
+-101.
+-101.
+1000
+2000
+3000
+1000
+2000
+3000
+T
+H20
+SrlII
+101
+101.
+0.3
+口
+口
+least
+口
+口
+口
+口
+口
+-0
+0
+Iwest
+口
+0.2
+口
+口
+口
+口
+口
+口
+口
+08
+口O
+0000
+:
+DO
+00
+000
+-101.
+-101.
+0.1
+1000
+2000
+3000
+1000
+2000
+3000
+ (K)
+Teq (K)
+2Diagnosing limb asymmetries in transmission
+9
+itations of simple models, we will move on to more self-
+consistent atmospheric modeling in the following sec-
+tions.
+3. SELECTED DIAGNOSTICS
+3.1. Diagnostics for specific mechanisms
+Per Section 2, even differentiating between two drivers
+of asymmetry in exoplanet atmospheres is nontrivial.
+Drivers can compete to varying degrees to produce a
+similar result: an asymmetric trend in net Doppler shifts
+in HRCCS.
+However,
+by exploiting nuances in the HRCCS
+Doppler shift signal and by independent means, it may
+be possible to disentangle even drivers that produce sim-
+ilar effects. Table 1 lists example drivers of asymmetries
+in HRCCS and how they might be diagnosed. The asso-
+ciated works listed in the table may not directly propose
+these diagnostics, but at minimum they provide founda-
+tional material for them.
+Of course, exhibiting a single diagnostic does not not
+mean that a given physical mechanism is in play. Other
+mechanisms could surely be present, and uniquely con-
+straining a single mechanism as dominant would require
+ruling out the others, as well. For instance, both day–
+night winds and morning limb condensation could result
+in a net blueshifted CCF. But if, for example, a night-
+side temperature were derived from a phase curve that
+was far too hot for any known condensate to form, then
+day–night winds would be much preferred to conden-
+sation as a physical solution. Together, collections of
+diagnostics are hence able to test the dominance of in-
+dividual mechanisms.
+In the following sections, we explore a few tests for
+specific physical mechanisms of asymmetry: using CO
+as a baseline molecule to identify the scale height effect
+and tracking the blueshifts of multiple species to identify
+the presence of clouds. We furthermore evaluate the ef-
+fectiveness of diagnostics that may be used to evaluate
+a number of different mechanisms: averaging HRCCS
+data into two phase bins and using finely phase-resolved
+HRCCS data. We additionally show how these diagnos-
+tics can further motivate or rule out “toy models” that
+at first may appear convincing.
+3.1.1. CO as a baseline molecule
+We have demonstrated (Section 2.4) that species
+with strongly temperature-dependent abundances are
+the least susceptible to the scale height effect.
+Con-
+versely, observing a species with very weak temperature-
+dependent abundance could indicate whether the scale
+height effect is in play.
+Figure 4. Volume mixing ratio of CO as a function of pres-
+sure and temperature as calculated by FastChem. Overplot-
+ted are the onset of ultra-hot Jupiters (as defined by their
+dayside temperature; Parmentier et al. 2018), the CO/CH4
+equivalency curve from Visscher (2012) as a function of pres-
+sure, the Fe condensation curve from Mbarek & Kemp-
+ton (2016), and 1D temperature–pressure profiles for a hot
+Jupiter (WASP-39b) and an ultra-hot Jupiter (WASP-18b)
+as computed with HELIOS (Malik et al. 2017). Both the con-
+densation curve and the equivalency curve are computed at
+solar metallicity. Considering the regime of ultra-hot Jupiter
+atmospheres, CO is a relatively stable chemical species.
+Consider CO. In Figure 3, its AWE values are clustered
+around 0 without the scale height effect, with relatively
+weak dependence on ˜∆T. However, CO’s AWE values
+are strongly negative when the scale height effect is in-
+cluded. We propose using CO as a tracer of the scale
+height (and other chemistry-unrelated) effects.
+As shown in Figure 4, the abundance of CO is
+relatively stable between 1000 K and 3500 K. Beltz
+et al. (2022) note that this stability holds over the
+temperature–pressure range of the observable atmo-
+sphere of the ultra-hot Jupiter WASP-76 b.
+Indeed,
+this feature remains true over the general temperature–
+pressure range of ultra-hot Jupiters. For illustrative pur-
+poses, we calculate the 1D temperature–pressure pro-
+files of a hot Jupiter (WASP-39b; Faedi et al. 2011) and
+an ultra-hot Jupiter (WASP-18b; Hellier et al. 2009).
+These profiles, calculated with the HELIOS 1D radiative-
+convective model (with full heat redistribution), indi-
+cate that the observable atmosphere for these planets is
+largely within a region of near-constant CO mixing ratio.
+The stability of CO is due to three factors: its strong
+chemical bonding, its lack of participation in gas-phase
+chemical reactions, and its lack of condensation.
+
+10-6
+-3
+Ultra-hot Jupiter
+dayside onset
+-4
+CO/CH4 equivalency
+10-5
+Fe condensation
+-5
+WASP-18 b 1D profile
+Pressure (bars)
+WASP-39 b 1D profile
+-6
+10-4
+(xp) 0x
+-7
+10-31
+-8
+-9
+10-21
+-10
+10-1
+-11
+-12
+100
+1000
+2000
+3000
+4000
+5000
+Temperature
+(K)10
+Savel et al.
+Since the strong triple bond of CO makes it diffi-
+cult to thermally dissociate, CO remains stable at tem-
+peratures that would dissociate molecules with weaker
+bonds, such as H2O (Parmentier et al. 2018), which
+has two single bonds.
+Beyond roughly 3500 K, even
+the triple bond becomes susceptible to thermal dissocia-
+tion; hence, the few exoplanets with significant portions
+of their atmosphere hotter than this temperature (e.g.,
+KELT-9b, with Teq ≈ 4050 K; Gaudi et al. 2017) would
+likely exhibit spatial variation in CO abundance. Most
+ultra-hot Jupiters, though, should fall shy of this regime.
+Furthermore, the high photoionization threshold of CO
+(relative to, e.g., H2O; Heays et al. 2017) means that
+it is not commonly photodissociated (Van Dishoeck &
+Black 1988).
+Even when it is photodissociated, recy-
+clying pathways exist in hot Jupiters that can replenish
+CO abundance, keeping it near equilibrium abundance
+even inclusive of photochemistry (Moses et al. 2011).
+Hence, the assumption of non-dissociation of CO is rea-
+sonably justified across much of the ultra-hot Jupiter
+population.
+Additionally, CO does not commonly participate in
+thermochemical reactions and is the dominant car-
+bon carrier in our temperature–pressure range of inter-
+est. While at lower temperatures the dominant carbon
+carrier becomes CH4, the ultra-hot Jupiter regime is
+squarely beyond the CO/CH4 equivalency curve (Fig-
+ure 4; Visscher 2012). Therefore, even aside from ther-
+mal dissociation, CO should not participate in gas-phase
+thermochemistry that would alter its abundance.
+Finally, CO does not form any high-temperature con-
+densates expected in ultra-hot Jupiter atmospheres.
+The condensation temperature of CO (≈80 K at 1 bar;
+Lide 2006; Fray & Schmitt 2009) is far below the
+temperature–pressure range of ultra-hot Jupiters. This
+quality makes CO a less complicated tracer of, e.g., at-
+mospheric dynamics than species that do condense in
+this region of parameter space, such as Fe, Mg, or Mn
+(Mbarek & Kempton 2016). Therefore, while the calcu-
+lations of Figure 4 do not include gas-phase condensa-
+tion, the resultant spatial constancy of CO should still
+be robust even when condensation is considered. CO
+is thus a more straightforward molecule to model than
+other, condensing species, as it does not participate in
+the complex microphysics of condensation and cloud for-
+mation (see, e.g., Gao et al. 2021).
+Beyond its spatial uniformity, there are further obser-
+vational reasons that CO is an appealing species to tar-
+get. Namely, CO has very strong spectroscopic bands
+placed across the infrared wavelength range (e.g., Li
+et al. 2015) that do not overlap with other strong ab-
+sorbers and are relatively well understood (Li et al.
+2015).
+Additionally, the high cosmic abundance of C
+and O (Lodders 2003) means that, unlike many of the
+species in the previous section, CO is readily detectable
+(and has been become a standard detection in HRCCS;
+Snellen et al. 2010; de Kok et al. 2013; Rodler et al.
+2013; Brogi et al. 2014, 2016; Flowers et al. 2019; Gia-
+cobbe et al. 2021; Line et al. 2021; Pelletier et al. 2021;
+Zhang et al. 2021; Guilluy et al. 2022; van Sluijs et al.
+2022).
+Given its stability and observational advantage, we
+propose that CO can be used as a faithful tracer of the
+atmosphere—whether it is inflated in some regions, what
+its wind profile is, whether regions are blocked by clouds,
+etc. In turn, CO may then be leveraged to better mo-
+tivate sources of asymmetry that affect other species.
+While other species with low AWE in Figure 1 (e.g., He,
+Fe, MgH, Rb II) would also appear to be good candi-
+dates for baseline species, these species are either largely
+spectroscopically inactive, have variable abundance over
+broader temperature–pressure ranges, or can condense.
+A caveat to the above is that while CO is a faithful longi-
+tudinal tracer, it is not an unbiased radial tracer (as seen
+in Figure 4). As with all chemical species, CO has its
+own balance between deep and strong lines that depends
+on the waveband considered (see, e.g., Section 3.3.1).
+Therefore, the net CO Doppler signal does not uniformly
+weight the wind profile across all altitudes. Again, this
+is a bias inherent to all chemical species.
+3.1.2. A decreasing blueshift test for clouds
+As noted in Table 1, clouds may introduce strong
+asymmetry into HRCCS data.
+Savel et al. (2022)
+demonstrated that gray, optically thick clouds produce
+stronger blueshifts in the Doppler shift signal of WASP-
+76b than the blueshifts in clear models, also changing
+the trend of Doppler shift with phase.
+But, again as
+shown in Table 1, these changes at the Doppler shift
+level are not sufficient to uniquely identify clouds as the
+driver of an observed asymmetry. Combinations of ob-
+servable quantities that would uniquely identify clouds
+as the source of observed HRCCS asymmetry are there-
+fore necessary.
+To devise such a test, we investigate in this work a
+limiting-case cloudy model. As in Savel et al. (2022),
+we construct gray, optically thick, post-processed clouds
+in our 3D ray-striking code. We here make another as-
+sumption, though: that the clouds are confined to the
+cooler, morning limb, as opposed to having a distribu-
+tion dictated by a specific species’ condensation curve.
+This distribution is based on planetary longitude (be-
+tween longitudes of 180◦ and 360◦). This approach is
+motivated by the results of Roman et al. (2021), who
+
+Diagnosing limb asymmetries in transmission
+11
+found that a subset of cloudy GCMs exhibited a cloud
+distribution strongly favoring the western limb.3
+Our
+approach benefits from providing limiting-case intuition
+for how cloudiness affects Doppler shift signals while
+avoiding the complex questions of how clouds form and
+which species contribute the most opacity (Gao et al.
+2021; Gao & Powell 2021).
+Briefly, our modeling methodology is as follows:
+1. Double-gray, two-stream GCM. GCMs such as this
+one solve the primitive equations of meteorology,
+which are a reduced form of the Navier-Stokes
+equations solved on a spherical, rotating sphere
+with a set of simplifying assumptions.4 The out-
+put of these models is temperature, pressure, and
+wind velocity as a function of latitude, longitude,
+and altitude. We use the GCM that was shown
+to best fit the Ehrenreich et al. (2020) WASP-76b
+data in Savel et al. (2022).
+2. Equilibrium chemistry with FastChem. As in Sec-
+tion 2.3, we interpolate a model grid of chem-
+istry to determine local abundances of a number
+of chemical species as determined by temperature
+and pressure conditions of the GCM output.
+3. Ray-striking radiative transfer.
+Using a code
+modified from Kempton & Rauscher (2012) (as
+detailed in Savel et al. 2022), we compute the
+high-resolution absorption spectrum of our plan-
+etary atmosphere by calculating the net absorp-
+tion of stellar light along lines of sight through
+our GCM output. This absorption is calculated
+inclusive of net motions along the lines of sight
+from atmospheric winds and rotation, inducing
+Doppler shifts relative to that of a static atmo-
+sphere’s spectrum. Limb-darkening is calculated
+with a quadratic limb-darkening law in the observ-
+able planetary atmosphere and with the batman
+code (Kreidberg 2015) for the portion of the star
+blocked by the optically thick planetary interior.
+Given its increasing utility as a benchmark planet for
+HRCCS studies (e.g., Ehrenreich et al. 2020; Kesseli &
+3 These GCMs produced clouds on a temperature–pressure basis,
+and did not model clouds as tracers.
+Therefore, they do not
+capture potential disequilibrium cloud transport (e.g., as done in
+Komacek et al. 2022), which may alter the degree of patchiness
+within the cloud deck.
+4 These assumptions are 1) local hydrostatic equilibrium, such that
+vertical motions are caused purely by the convergence and di-
+vergence of horizontal flow, 2) the “traditional approximation,”
+which removes the vertical coordinate from the Coriolis effect,
+and 3) a thin atmosphere.
+Figure 5. Atmospheric Doppler shifts, which should remain
+in the HRCCS signal after the orbital motion is subtracted,
+as a function of orbital phase for our forward models. Shown
+are representative species that span Doppler shifts and are
+noted as potentially observable by Kesseli et al. (2022): Fe,
+Sr II, and Sc. Cloud-free models are represented with solid
+lines, whereas models with fully cloudy morning limbs are
+represented with dashed lines. The first half of transit (RV1)
+and second half of transit (RV2) Doppler shifts for Fe from
+Kesseli et al. (2022) are overplotted as horizontal lines, with
+width determined by observational errors. Our cloudy mod-
+els are much more strongly blueshifted than their cloud-free
+counterparts, become less blueshifted over transit, and do
+not have significant CCF peaks at early phases.
+Snellen 2021; Landman et al. 2021; Seidel et al. 2021;
+Wardenier et al. 2021; Kesseli et al. 2022; S´anchez-L´opez
+et al. 2022), we model the ultra-hot Jupiter WASP-76b
+(West et al. 2016). We calculate 25 spectra inclusive of
+Doppler effects equally spaced in phase from the begin-
+ning to end of transit. For our cross-correlation tem-
+plate, T , we use a model that does not include Doppler
+effects.
+We then cross-correlate our template against our cal-
+culated spectrum, y:
+c(v) =
+N
+�
+i=0
+yi(λ)Ti(v, λ),
+(4)
+where the mask or template is Doppler-shifted by ve-
+locity v and interpolated onto the wavelength grid, λ,
+of y for summing. Our CCF is computed on a grid of
+velocities from −250 km s−1 and 250 km s−1 with a step
+of 1 km s−1. The final net planet-frame Doppler shift is
+calculated by fitting a Gaussian to the peak of the CCF.
+The results of our experiment are shown in Figure 5.
+When we allow clouds to extend over the entire morn-
+ing limb, note that all species become less blueshifted
+over time. Because the limb that is rotating away from
+the observer (the “receding limb”) is entirely blocked off
+by clouds, there is no wavelength-dependent absorption
+
+2
+Kesseli+22 Fe RVi
+Sc
+Kesseli+22 Fe RV2
+Sr plus
+Planet-frame RV (km/s)
+0
+Fe
+-2
+6
+8
+一10
+12
+-15
+-10
+-5
+0
+5
+10
+15
+Phase (degrees)12
+Savel et al.
+for that limb. Therefore, the contribution of redshift-
+ing from solid-body rotation on the receding limb is not
+present—the only Doppler shift contributions are from
+evening limb rotation and evening limb winds, which are
+generally in the same direction as the rotation. Hence,
+there are much stronger blueshifts at earlier phases than
+in the clear models.
+However, at later phases, the non-cloudy regions of the
+atmosphere rotate into the receding limb, thereby con-
+tributing some rotational redshift to the net Doppler
+shift signal.5
+At the earliest phases, the cloudy mod-
+els do not have enough wavelength-dependent absorp-
+tion to produce a significant CCF peak.
+Notably, all
+species follow this trend, as the blocking of clouds as
+modeled here is wavelength-independent and altitude-
+independent.
+This behavior is shown in Figure 5 for
+Fe, Sc, and Sr II—all species identified in Kesseli et al.
+(2022) has having high potential observability for ultra-
+hot Jupiters.
+From Figure 5, it is also apparent that the cloud-
+driven trend of decreasing blueshift in phase is not
+matched by the observations of Kesseli et al. (2022).
+As found in Savel et al. (2022) in comparison to the
+Ehrenreich et al. (2020) data, while the absolute magni-
+tude of the cloudy model’s Doppler shift better match
+the data than the clear model’s, the cloudy model trend
+over Doppler shift is not matched by the data. In sum,
+this limiting-case model of opaque, morning limb clouds
+does not appear to be a first-order effect driving ex-
+isting observational trends.
+This does not necessarily
+mean that clouds are not the driving factor behind limb
+asymmetries; it may simply be that a more physically
+motivated model for partial cloud coverage of the limb
+could fit the available data better.
+Also of note in Figure 5 is that the egress signatures
+of the clear and cloudy models are quite distinct. Near a
+phase of roughly 14 degrees, the clear model produces a
+sharp change in Doppler shift for all species as the lead-
+ing (rotationally redshifted) limb begins to leave the stel-
+lar disk. This sharply blueshifting behavior continues to
+the end of egress, until the last sliver of the trailing (rota-
+tionally blueshifted) limb has left the stellar disk as well.
+In the cloudy case, however, the leading limb leaving the
+stellar disk has no effect, as it is fully cloudy. While this
+effect is evident in these models, it may be less evident
+5 The degree of rotation during transit varies as a function of
+semimajor axis and host star radius, and hence from planet to
+planet. While we only model WASP-76b, other planets also have
+large (e.g., compared to the angles probed by transmission spec-
+troscopy) rotations during transit (Wardenier et al. 2022).
+in observations, which naturally cannot finely sample
+ingress and egress phases.
+3.2. Phase bins
+We have thus far examined drivers of asymmetry and
+potential diagnostics of specific mechanisms. Next, we
+will evaluate a few HRCCS data types to determine how
+robust they are and their potential ability to constrain
+a number of different physical mechanisms that give rise
+to HRCCS asymmetry.
+The first of these data types is HRCCS Doppler shifts
+that are binned in phase.
+A substantial fraction of
+HRCCS studies present detections and Doppler shifts
+integrated over the entirety of transit (e.g., Giacobbe
+et al. 2021). This approach maximizes detection SNR,
+which may be necessary for a given set of observations
+(e.g., because of a low-resolution spectrograph, small
+telescope aperture, faint star, low species abundance,
+low number of absorption lines, or weak intrinsic ab-
+sorption line strengths). While it is possible to reveal
+aspects of limb asymmetry with this approach, espe-
+cially when comparing detections of multiple species to
+one another, phase-resolving the transit (and observing
+isolated ingresses and egresses when possible) will cer-
+tainly give a more direct probe of east–west asymme-
+tries. Binning HRCCS data in phase across transit may
+provide a desirable balance between revealing asymme-
+try and maintaining high SNR.
+We seek to address this question by phase-binning
+modeled Doppler shifts to examine its biases with re-
+spect to the underlying model. We follow this experi-
+ment with a comparison to the phase-binned observa-
+tions of Kesseli et al. (2022).
+3.2.1. Theoretical phase binning
+We average our phase-resolved calculations into two
+bins: the first and second half of transit.
+Once our
+CCFs are calculated we average them in phase to ef-
+fectively reduce our data to two single bins: the first
+half of transit and the second half of transit. We make
+versions of these two half-transit bins that include or
+exclude the ingress and egress phases (when the planet
+is only partially occulting the star).
+Motivated by recent detections in the near-infrared
+(Landman et al. 2021; S´anchez-L´opez et al. 2022), we
+search for absorption from various molecules6 in our
+models, focusing on the CARMENES (Quirrenbach
+et al. 2014) wavelength range and resolution for direct
+6 We use the MoLLIST (Brooke et al. 2016), POKAZATEL
+(Polyansky et al. 2018), and Li2015 (Li et al. 2015) linelists for
+OH, H2O, and CO, respectively.
+
+Diagnosing limb asymmetries in transmission
+13
+Figure 6. Single-species (OH, CO, H2O, HCN) 3D forward-
+modeled spectra of WASP-76b. These spectra are simulated
+over the CARMENES waveband and resolution.
+Doppler
+effects are not included in these spectra, which are modeled
+at center of transit. H2O is the dominant absorber in this
+bandpass, followed by OH. HCN exhibits no spectral features
+above the continuum for WASP-76b in this bandpass.
+comparison against observational results using that in-
+strument. Of these molecules, we find that OH, H2O,
+and CO produce significant absorption over the mod-
+eled wavelength range, with OH and H2O producing the
+strongest features (Figure 6). We find that HCN does
+not produce any noticeable absorption under the as-
+sumption of chemical equilibrium and solar composition,
+implying either more exotic chemistry for WASP-76b’s
+atmosphere (i.e., photochemistry or non-solar abun-
+dances; Moses et al. 2012), or that the detection of
+HCN in this atmosphere (S´anchez-L´opez et al. 2022)
+was spurious (perhaps due to the nature of the HCN
+opacity function; Zhang et al. 2020). We furthermore
+find a moderate (≈ 4 km s−1) increase in blueshift for
+our modeled H2O.
+While this increase in blueshift is
+commensurate with the increase in blueshift described
+for H2O in S´anchez-L´opez et al. (2022), we are once
+again unable to match the high reported velocities (here
+-14.3 km s−1) with our self-consistent forward models.
+Figure 7 shows the results of this experiment. As the
+error for each phase bin, we take the average error of
+phase bins from Kesseli et al. (2022) (1.55 km s−1). We
+define the two phase bins as inconsistent if the peak of
+their respective CCFs are inconsistent at 2σ.
+We find that excluding ingress and egress phases can
+strongly reduce the difference in derived Doppler shift
+between phase bins. Furthermore, we find that, as ex-
+pected from Kempton & Rauscher (2012), differences
+between bins are maximized when just considering the
+ingress and egress phases.7
+While higher-order drivers of asymmetry are clearly
+not detectable with phase bins (e.g,. at what longitude
+condensation may begin to play a role; Wardenier et al.
+2021), certain drivers of asymmetry are accessible with
+this method. For example, ignoring for now the exact
+details of error budgets, all species in Figure 7 clearly
+blueshift over the course of transit. This provides poten-
+tial evidence for, among other things, a spatially vary-
+ing wind field, condensation, optically thick clouds, or a
+scale height effect. Furthermore, per the results of Sec-
+tion 3.1.1 the detection of CO’s blueshifting indicates
+that something besides equilibrium chemistry is driv-
+ing at least some of the asymmetry in the atmosphere.
+These underlying models are cloud-free, so these results
+imply sensitivity to, e.g., the scale height effect.
+3.2.2. Comparison to Kesseli et al. (2022)
+With our models calculated, we can now explore the
+ability of phase-resolved spectra to confront toy mod-
+els by comparing the models to observations. A prime
+observational work that made use of phase binning is
+Kesseli et al. (2022); there, the authors search for asym-
+metries in two phase bins for a wide variety of species,
+motivated by the strength of those species’ opacity func-
+tions in the data’s wavelength range.
+To consider a toy model: based on previous studies
+(Ehrenreich et al. 2020; Tabernero et al. 2021; Savel
+et al. 2022), it appears that Ca II does not follow the
+Fe-like Doppler shift trend first observed by Ehrenreich
+et al. (2020). Rather, it appears that Ca II, with its
+strong opacity and resultant deep lines, may be probing
+a non-hydrostatic region of the atmosphere (Casasayas-
+Barris et al. 2021; Deibert et al. 2021; Tabernero et al.
+2021). This region of the atmosphere cannot be cap-
+tured by the models of this work and Savel et al. (2022).
+Without a model of atmospheric escape, it seems dif-
+ficult to elevate the above picture beyond “toy model”
+status. However, by phase-resolving multiple species, a
+clearer picture can emerge.
+For our comparison with Kesseli et al. (2022), we use
+the same line lists as in that study: the National In-
+stitute of Standards and Technology (NIST; Kramida
+et al. 2019) line lists. It is crucial to use the same line
+7 We would expect that binning with fewer spectra (just includ-
+ing the ingress phases) would increase the associated error on
+Doppler shift at each bin. However, the point of this exercise is to
+illustrate the magnitude of ingress/egress Doppler shift discrep-
+ancy; observational strategies such as stacking multiple transits
+could reduce errors in practice and make these differences dis-
+cernible.
+
+H20
+CO
+1.375
+HO
+HCN
+1.350
+Transit depth (%)
+1.325
+1.300
+1.275
+1.250
+1.225
+1.200
+1.0
+1.1
+1.2
+1.3
+1.4
+1.5
+1.6
+1.7
+Wavelength (microns)14
+Savel et al.
+Figure 7. CCFs of individual species averaged over two phase bins. Each column corresponds to different species (OH, CO,
+H2O), and each row corresponds to different bin selection: without including ingress and egress, including the full transit, and
+only including ingress and egress. Central bars between the CCFs are colored blue if the difference between the CCFs is greater
+than optimal Doppler shift errors (1.55 km/s, in black; Kesseli et al. 2022); otherwise, they are colored red. In our models, CO
+only displays detectable CCF differences when only including ingress and egress. The SNR in each plot refers to the difference
+between the two phase bins’ CCF peaks relative to the optimal Doppler shift errors.
+lists for comparisons of HRCCS studies—different line
+list databases can contain vastly discrepant numbers of
+line transition, which greatly affects the resultant opac-
+ity function (see, for instance, Figure 11 of Grimm et al.
+2021).
+The results of our comparison with the species de-
+tected in Kesseli et al. (2022) are shown in Figure 8. As
+in Savel et al. (2022), these baseline models—no clouds,
+no condensation, no orbital eccentricity—cannot fully
+explain the Doppler shifts of Fe observed in WASP-
+76b.
+However, the comparison across multiple differ-
+ent species provides further constraints. Figure 8 shows
+that Fe, V, Cr, Ca II, and Sr II are strongly discrepant
+from our models for at least one half of transit, whereas
+Na, Mg, Mn, and Ni are reasonably well described by
+our models for both the first and second half of transit.
+Furthermore, Fe, V, and Cr all have stronger blueshifts
+in the second phase bin than in our models. The similar
+
+CO, no ingress / egress
+OH, no ingress / egress
+H2O, no ingress / egress
+1.4 -
+2nd half peak: i
+1st half peak:
+ 2nd half peak: i
+1st half peak:
+2nd half peak:i!
+1st half peak:
+-3.52 km s-1
+-1.3 km s-1
+-5.41 km s-1
+-3.15 km s-1
+-5.93 km s-1
+-3.73 km s-1
+1.2
+SNR: 1.4
+SNR: 1.5
+SNR: 1.4
+1.0
+0.8
+3333333333
+8 0.6
+0.4 -
+0.2
+CO, full transit
+H20, full transit
+OH, full transit
+2nd half peak:
+1st half peak:
+2nd half peak:!
+1st half peak:
+2nd half peak:
+1.4 -
+1st half peak:
+-3.73 km s-
+-0.38 km s-1
+-6.08 km s-
+-2.27 km s-1
+-6.23 km s-1
+-2.81 km s-1
+1.2 -
+0
+SNR: 2.2
+SNR: 2.5
+SNR: 2.2
+1.0 -
+0.8
+CF
+0 0.6
+0.4 -
+0.2
+CO, only ingress / egress
+OH, only ingress / egress
+H2O, only ingress / egress
+2nd half peak:i
+1.4
+1st half peak:
+2nd half peakl
+1st half peak:
+2nd half peakl
+1st half peak:
+-6.28 km s-1
+4.27 km s-1
+-8.04 km s-1T
+2.24 km s-1
+-7.98 km s-1T
+1.65 km s-1
+1.2
+■
+SNR: 6.8
+SNR: 6.6
+SNR: 6.2
+0.8
+CF
+8 0.6 -
+0.4
+0.2
+-40
+20
+0
+20
+40 
+-40 
+20
+0
+20
+40
+-40
+20 
+0
+20
+40
+Velocity (km/s)
+Velocity (km/s)
+Velocity (km/s)Diagnosing limb asymmetries in transmission
+15
+Figure 8. The net Doppler shifts of Kesseli et al. (2022) (error bars) as compared to this work’s models (crosses). The first
+phase bin is drawn thinner than the second phase bin; observed phase bins are connected by a dotted line for visibility’s sake.
+The species are ordered and colored by total observed detection SNR. Rows without crosses correspond to species that we could
+not recover via cross-correlation in our models. Our models are able to explain some species (e.g., Na), fail to explain others
+(e.g., Cr) and fail to detect yet others (e.g,. K).
+level of disagreement between Fe, V, and Cr implies that
+they share a common driver of asymmetry. This result
+in turn implies that whatever driver affects them affects
+the regions in which these species form similarly — be
+it clouds, condensation, etc.
+To bridge the toy models presented in Section 2.3 to
+our Kesseli et al. (2022) comparison, we compute a set
+of high-resolution spectra exactly as above, but with
+the same altitude grid at all latitudes and longitudes
+in an effort to effectively turn off the scale height ef-
+fect while maintaining chemical limb inhomogeneities.
+Post-processing this (self-inconsistent) model yields less
+than half the Doppler shift asymmetry as compared to
+our self-consistent models.
+This experiment confirms
+the intuition that the scale height effect is a first-order
+asymmetry effect.
+Finally, we consider the Ca II toy model previously de-
+scribed. Certain lightweight and/or ionized species may
+be entrained in an outflow, as indicated by some pre-
+vious observations (e.g., Tabernero et al. 2021) of very
+deep absorption lines in transmission that must extend
+very high up in altitude. The differential behavior of the
+Ca II and Sr II Doppler shifts lends more credence to
+this hypothesis.
+In sum, by taking advantage of phase-binned spectra,
+it is possible to better identify drivers of HRCCS asym-
+metry. Additionally, our predictions in Figure 8 indi-
+cate that most species should have roughly the same
+Doppler shift patterns. In stark contrast, observations
+reveal much larger variations in velocity across different
+species. While some interpretation may be due to spuri-
+ous detections, physics that is not included in our model
+(e.g., outflows, condensation) may be playing a driving
+role.
+3.3. Full phase-resolved spectra
+Currently, the most information-rich diagnostic avail-
+able to probe asymmetry in HRCCS is phase-resolved
+cross-correlation functions (e.g., Ehrenreich et al. 2020;
+Borsa et al. 2021)—that is, net Doppler shifts associated
+with the absorption spectrum evaluated over multiple
+points in transit. With these data, one should be able
+to directly constrain longitudinally dependent drivers of
+asymmetry, providing the best chance of disentangling
+the physical mechanisms outlined in Section 2. But how
+far can we push these data?
+3.3.1. Example: probing physics in the NIR
+To explore this question, we take as an example a
+three-species (OH, H2O, and CO) near-infrared (NIR)
+dataset over a CARMENES-like waveband as in Sec-
+tion 3.2.
+Figure 9 shows the Doppler shifts of these
+
+-14
+Sr+
+Ni
+Kesseli+22 SNR of species detection
+Co
+Model, first bin
+12
+Fe
+Model, second bin
+Kesseli+22,
+Mn
+first bin
+Kesseli+22,
+10
+cies
+Cr
+second bin
+V
+d
+S'Ca+
+8
+K
+Mg
+Na
+6
+Li
+H
+-15
+-10
+-5
+0
+5
+10
+Planet-frame Doppler shift (km/s)16
+Savel et al.
+Figure 9. Modeled phase-resolved Doppler shifts for select NIR-absorbing species, with representative error bars (Ehrenreich
+et al. 2020) drawn on. We find that OH and H2O have distinct Doppler signatures from CO; however, OH and H2O have Doppler
+shifts that are indistinguishable from one another with current best-case error bars (e.g., Ehrenreich et al. 2020). Considering
+CO as a “baseline species” here allows one to better understand how H2O and OH may change through the atmosphere.
+species as a function of phase, produced for single species
+at a time as in Section 3.2, but without any averaging.
+Without considering any data, a compelling toy model
+would be as follows: H2O is thermally dissociated on
+the hotter, approaching limb, so it preferentially exists
+on the receding limb. OH is a product of H2O photodis-
+sociation, so it forms preferentially on the approaching
+limb. CO is constant everywhere; therefore, CO should
+not experience much of a trend in Doppler shift, OH
+should be more blueshifted than CO, and H2O should
+be more redshifted than CO.
+We shall see, however, that additional, complicating
+physics is revealed by fully phase-resolved spectra. For
+our models, the relevant underlying physics is as follows:
+1. Altitude-dependent winds: H2O lines are more
+strongly blueshifted than CO lines at all phases
+because the H2O line cores over the wavelength
+range of the CARMENES bandpass more predom-
+inantly form at higher altitudes. At high altitudes,
+the atmospheric flow switches from dominantly ro-
+tational (via an eastward equatorial jet) to dom-
+inantly divergent (via day–night winds) (Ham-
+mond & Lewis 2021). This result is the opposite of
+what would be expected from the above-described
+toy model, revealing the shortcomings of simple
+models and how they can sometimes mislead us.
+2. Equilibrium chemistry: H2O and CO are less
+blueshifted than OH because OH preferentially
+forms on the approaching, blueshifted limb of the
+planet. OH being more blueshifted than the other
+molecules is in agreement with the predictions of
+the toy model.
+3. Equilibrium chemistry: The relationship be-
+tween H2O and OH changes as a function of phase
+because the ratio OH/H2O increases a function of
+temperature, and hotter regions of the planet ro-
+tate into view over transit.
+This finding is also
+qualitatively in agreement with the toy model.
+However, per Figure 9, this effect is unfortunately
+not likely to be observable given the error bars in
+current data sets.
+Now the question remains: Can we observe in real
+data the trends matching these model explanations? As
+a simple experiment, we can apply error bars representa-
+tive of the best observing nights on the best instrument
+with the most observable chemical species (roughly 2
+km/s, as drawn as vertical error bars in Figure 8; Ehren-
+reich et al. 2020) and determine whether these trends are
+still detectable. With our errorbars now applied to our
+simulated data, only the first explanation—that H2O
+forms at higher altitudes than CO—can fully be ad-
+dressed, assuming that Doppler shifts for both species
+
+CO
+OH
+- H20
+5.0
+Planet-frame RV (km/s)
+2.5
+0.0
+-2.5
+-5.0
+7.5
+-10.0
+12.5
+-15
+-10
+5
+5
+10
+15
+Phase (degrees)Diagnosing limb asymmetries in transmission
+17
+(a)
+(b)
+Figure 10. Results of an investigation into anomalous Ca II blueshift between different model runs. In panel (a), it can be
+seen that forward models that include absorption due to Sc opacity yield a larger Ca II blueshift than models that lack Sc (Fe
+Doppler shift is included for comparison). Panel (b) illustrates the cause of this anomalous blueshift: a Sc line overlapping one
+line in the optical Ca II doublet. These results imply that overlapping line profiles can subtly contaminate calculated Doppler
+shifts.
+can be obtained. The second explanation can only be
+partially addressed—we can still determine that CO is
+less blueshifted than OH.
+3.3.2. Warning: blending of Doppler shifts
+The disentangling of physics in Section 3.3.1 rests on
+a fundamental assumption: that each cross-correlation
+template directly tracks only a single species. Indeed,
+one of the promises of HRCCS is the ability to uniquely
+constrain individual species’ abundance; with individual
+line profiles resolved, different species should be readily
+identifiable from one another in cross-correlation space
+(e.g., Brogi & Line 2019). Furthermore, our noiseless
+models should be even less susceptible to degeneracies
+between different species’ spectral manifestations.
+Panel (a) of Figure 10 seems to contradict the notion
+of complete line profile independence across species. For
+models run in Savel et al. (2022), Sc was excluded. Mo-
+tivated by the search for atoms in Kesseli et al. (2022),
+however, we included Sc in this work’s models.
+Sur-
+prisingly, we found a subsequent significant difference in
+the Doppler shifts recovered from our cross-correlation
+analysis in our Sc-inclusive models.
+Panel (b) of Figure 10 reveals the source of the dis-
+crepancy. In the optical, Ca II opacity is dominated by
+a doublet; one of the lines in this doublet partially over-
+laps with a strong, narrow Sc line. When both species
+combined in a forward model, the Sc line produces ab-
+sorption just blueward of this Ca II line’s core; hence,
+the cross-correlation of the Ca II template yields a spuri-
+ous blueshift. There did exist other modeling differences
+between the two spectra (e.g., the Savel et al. (2022)
+models included TiO and VO), but none of these differ-
+ences strongly impacted the Doppler shift of Ca II.
+Because Ca II in the optical has only two strong lines,
+it is particularly susceptible to this type of error. All it
+takes is one slight overlap with another species near a
+Ca II doublet core, and the Ca II Doppler signal can be
+significantly biased. Species with forests of lines (e.g.,
+Fe in the optical) should hence be more robust to chance
+overlaps with other species’ lines.
+To guard against this error for species with few lines,
+we recommend cross-correlating templates against one
+another to get a first-order sense for the extent of species
+overlap in Doppler space.
+Furthermore, we recom-
+mend performing these analyses on HRCCS with com-
+bined species models, as opposed to single-species mod-
+els. This approach could involve a retrieval framework
+(Brogi & Line 2019; Gandhi et al. 2019; Gibson et al.
+2020), which couples a statistical sampler to an atmo-
+spheric forward model to determine the exoplanet spec-
+trum that best fits the data, inclusive of multiple chem-
+ical species at once.
+4. CONCLUSION
+The past few years have yielded asymmetric Doppler
+signals from exoplanet atmospheres as a function of
+phase.
+Compelling “toy models” notwithstanding, a
+number of physical processes can drive these asymme-
+tries, and it can be difficult to uniquely constrain the
+cause of an asymmetry.
+In this study, we determine that if an asymmetry is
+observed:
+1. It may be due to a scale height difference across
+the atmosphere, not a chemistry difference across
+
+2
+Ca+ with Sc
+Fe
+Ca+
+Planet-frame RV (km/s)
+0
+-2
+-6
+-8
+-1015
+-10
+-5
+0
+5
+10
+15
+Phase (degrees)Sc
+1.40
+Ca+ template
+(%)
+ransit depth (
+1.35
+1.30
+1.25
+1.20
+0.3931
+0.3932 0.3933 0.3934 0.3935 0.3936 0.3937 0.3938
+Wavelength (microns)18
+Savel et al.
+the atmosphere. Comparing a signal of a species in
+HRCCS to a baseline species that is guaranteed to
+be chemically stable over the atmosphere can bet-
+ter motivate whether the asymmetry could be due
+to chemistry. CO is an excellent baseline species
+for ultra-hot Jupiters, as it is stable over these
+planets’ expected temperature–pressure space, has
+many spectral lines in the near-infrared accessible
+to ground-based spectrographs, and has been de-
+tected in numerous studies.
+2. The asymmetry can be highly informative even if
+it is binned in phase, especially if multiple species
+are considered. For instance, much larger Doppler
+shifts (both blue and red) of certain species rela-
+tive to the predictions of hydrostatic GCMs can
+be used as evidence for outflowing material.
+3. The asymmetry may be boosted by including
+(and perhaps only considering) ingress and egress
+phases. Ingress and egress spectra are the the gold
+standard for asymmetric signals so long as the sig-
+nal to noise is high enough.
+4. The asymmetry may be influenced by line con-
+fusion between species, even at high resolution.
+Species with very few lines (e.g., a single doublet)
+in the observed waveband are especially suscep-
+tible to contamination by other species in cross-
+correlation analysis, and they should be carefully
+checked against theoretical models for possible
+contaminating opacity sources.
+5. If all species exhibit a similar asymmetry—
+especially if they all become less blueshifted over
+the course of transit—the asymmetry may be due
+to a large-scale effect, such as clouds blanketing
+the cooler limb.
+6. Per our comparison of near-infrared absorbers in
+the CARMENES waveband, the toy model pre-
+dictions of the H2O Doppler shift relative to CO
+was inaccurate, as it did not include information
+about the vertical coordinate. With H2O lines on
+average probing higher in the atmosphere than CO
+in this waveband, they probed a different part of
+the flow, departing from expectations of the toy
+model.
+By aiming to systematically understand even just a
+few drivers of asymmetry, this work has made it clear
+that HRCCS—already arguably abstract given its gen-
+eral inability to produce visible planetary spectra—has
+yet more nuance to uncover. As data quality continues
+to increase, it will become increasingly necessary to un-
+derstand the relationships between higher-order physical
+effects.
+ACKNOWLEDGMENTS
+A.B.S., E.M.-R.K., and E.R. acknowledge funding
+from the Heising-Simons Foundation.
+We thank Michael Zhang for a thoughtful conversa-
+tion on the cross-correlation signature of HCN. We also
+thank Anusha Pai Asnodkar for a robust discussion of
+degeneracies in HRCCS tests. Finally, we thank Serena
+Cronin for providing useful insight into applications of
+CO detections in extragalactic astronomy.
+The authors acknowledge the University of Maryland
+supercomputing resources (http://hpcc.umd.edu) made
+available for conducting the research reported in this
+paper.
+This research has made use of NASA’s Astrophysics
+Data System Bibliographic Services.
+Software:
+astropy (Price-Whelan et al. 2018),
+batman (Kreidberg 2015), FastChem (Stock et al. 2018),
+IPython (P´erez & Granger 2007), HELIOS-K (Grimm
+et al. 2021), HELIOS (Malik et al. 2017), Matplotlib
+(Hunter 2007), NumPy (Harris et al. 2020), Numba (Lam
+et al. 2015), pandas (McKinney 2010), SciPy (Virtanen
+et al. 2020), tqdm (da Costa-Luis 2019)
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+Modeling the Central Supermassive Black Holes Mass of Quasars via LSTM Approach
+Seyed Sajad Tabasi,1, 2, ∗ Reyhaneh Vojoudi Salmani,3, 2, † Pouriya Khaliliyan,3, 2, ‡ and Javad T. Firouzjaee3, 2, 4, §
+1Department of Physics, Sharif University of Technology, P. O. Box 11155-9161, Tehran, Iran
+2PDAT Laboratory, Department of Physics, K. N. Toosi University of Technology, P.O. Box 15875-4416, Tehran, Iran
+3Department of Physics, K. N. Toosi University of Technology, P. O. Box 15875-4416, Tehran, Iran
+4 School of Physics, Institute for Research in Fundamental Sciences (IPM), P. O. Box 19395-5531, Tehran, Iran
+One of the fundamental questions about quasars is related to their central supermassive black
+holes. The reason for the existence of these black holes with such a huge mass is still unclear and
+various models have been proposed to explain them. However, there is still no comprehensive ex-
+planation that is accepted by the community. The only thing we are sure of is that these black
+holes were not created by the collapse of giant stars, nor by the accretion of matter around them.
+Moreover, another important question is the mass distribution of these black holes over time. Ob-
+servations have shown that if we go back through redshift, we see black holes with more masses,
+and after passing the peak of star formation redshift, this procedure decreases. Nevertheless, the
+exact redshift of this peak is still controversial. In this paper, with the help of deep learning and the
+LSTM algorithm, we tried to find a suitable model for the mass of central black holes of quasars over
+time by considering QuasarNET data. Our model was built with these data reported from redshift
+3 to 7 and for two redshift intervals 0 to 3 and 7 to 10, it predicted the mass of the quasar’s central
+supermassive black holes. We have also tested our model for the specified intervals with observed
+data from central black holes and discussed the results.
+Keywords:
+Quasars, Supermassive Black Holes, Sloan Digital Sky Survey, QuasarNET Data Set, Deep
+Learning, and LSTM Model
+I.
+INTRODUCTION
+In recent years, the study of the high-redshift(z > 6)
+quasars was a direct probe to explore the Universe at
+the age less than 1 Gyr after the Big Bang. These early
+forming quasars are essential to studying the early growth
+of supermassive black holes (SMBHs) [1].
+By detecting the reverberation between the variations
+of broad emission lines and the continuum we can deter-
+mine SMBHs mass in quasars [2]. Until now, the time lag
+of Hβ emission lines has been confirmed and measured
+only in ∼100 quasars [3].
+The continuum and line emission from luminous
+quasars which are one of the most luminous objects, over
+a large wavelength range can be characterized by sev-
+eral leading parts.
+The broad emission line region [4]
+the optical-to-ultraviolet continuum emission, which is
+explained by a standard accretion disk extending down
+to the innermost stable circular orbit [5], X-ray emission
+with a power-law spectrum produced by inverse Compton
+scattering of photons from the accretion disk of relativis-
+tic electrons in the hot corona [6], and a soft X-ray ex-
+cess [7]. Spectroscopic observations from optical to near-
+infrared of these quasars suggest that such SMBHs are
+already established when the universe is only 700Myr
+old [8].
+To explain the existence of these SMBHs, many theo-
+∗Electronic address: [email protected]
+†Electronic address: [email protected]
+‡Electronic address: [email protected]
+§Electronic address: fi[email protected]
+retical models have been proposed like using primordial
+density seeds [9–11] and appealing a super-Eddington ac-
+cretion process [12].
+To utilize the spectroscopic observational data in phys-
+ical studies, we need an exact classification and redshift
+determination of astrophysical objects. Along the way,
+the Sloan Digital Sky Survey Catalogue 16th Data Re-
+lease Quasar Only(SDSS-DR16Q) [13], consists of two
+files, being the quasar-only main catalog of 750414 en-
+tries which contains sooner visually confirmed quasars
+SDSS-I/II/III, and a 1440615-row “superset” of SDSS-
+IV/eBOSS quasar object classifications.
+The DR16Q catalogs present multiple redshifts per ob-
+ject that are available, including the neural automated
+QuasarNET [14] redshift which is claimed > 99% ef-
+ficiency and > 99% accuracy, that rests on garnering
+deeper insights into this triumvirate connection by co-
+locating and analyzing observational data and simulated
+data. Meanwhile, the enormous increase in computing
+power over the last decades has allowed the application
+of acquired statistical methods in the analysis of big and
+complex data sets.
+Using previously-fed data has brought huge opportuni-
+ties for astronomers to develop intelligent tools and inter-
+faces, utilizing pipeline classifiers, machine learning(ML),
+and deep learning(DL) methods, to deal with data sets
+and extract novel information with possible predictions
+and estimate the relevant confidence which the behavior
+new data will have.
+In astronomy and astrophysics, ML [15, 16] and DL
+[17, 18] have been used in a broad range of subjects(e.g.
+quasars and other types of sources), such as redshift de-
+termination [19, 20], morphological classification and ref-
+erences therein [21, 22], source selection and classifica-
+arXiv:2301.01459v1  [astro-ph.GA]  4 Jan 2023
+
+2
+tion [23–25], image and spectral reconstruction [26], and
+more.
+ML methods for obtaining redshift estimation for
+quasars are becoming progressively crucial in the epoch of
+rich data astronomy. Redshift measurements of quasars
+are important as they can enable quasar population
+studies, and provide insight into the star formation
+rate(SFR), the luminosity function(LF), and the density
+rate evolution [27].
+In this work, we have used DL to model the mass of
+quasars’ central SMBH as a function of their redshift.
+Firstly, Sec.
+II is dedicated to the available observa-
+tional data and evidence on quasars. The estimation of
+a quasar’s central SMBH mass is discussed in detail in
+Sec. III. Furthermore, in Sec. IV, the mass evolution
+of these black holes(BHs) is investigated. Sec. V is the
+comparison between two newborn research platforms,
+QuasarNET and FNET, and the reasons behind using
+QuasarNET for our model are explained. Additionally,
+we use two correction methods which are explained in
+Sec.
+VI. A detailed explanation of our DL model can
+be found in Sec.
+VII to X. In Sec.
+VII we introduce
+Long short-term memory(LSTM) which is the recurrent
+neural network(RNN) that we build our model based
+on.
+We explain the chosen optimization function and
+its validation loss in Sec.
+VIII which is shown in
+multiple figures. Sec. IX presents the topology design
+of our model and finally, the comparison of the model
+predictions with other data sets is discussed in Sec. X.
+II.
+OBSERVATIONAL EVIDENCE AND DATA
+The most comprehensive observed quasi-stellar ob-
+jects(QSOs) spectra to date are cataloged in the SDSS-
+IV. SDSS has been operative since 2000 and catalogs of
+quasars have been produced and made available since
+2002. In addition to producing images, it performs spec-
+troscopic surveys across a large area of the sky. We can
+get about one million galaxies and 10,000 quasars spectra
+from the survey images of the sky, which are obtained
+by a 2.5m telescope equipped with a large format mo-
+saic Charge-coupled device(CCD) camera, and two dig-
+ital spectrographs. As part of its calibration, the SDSS
+uses observations of the US Naval Observatory’s 1m tele-
+scope to calibrate its photometry, and an array of astro-
+metric CCDs control its astrometry [28].
+The SDSS provides data necessary to study the large-
+scale structure of the universe.
+As far as the obser-
+vatory’s limit allows, the imaging survey should detect
+∼ 5 × 107 galaxies, ∼ 106 quasars, and ∼ 8 × 107 stars.
+By using photometric redshifts and angular correlation
+functions, these photometric data allow studies of large-
+scale structures that go beyond spectroscopic analysis.
+Quasars can provide information on the structure at even
+larger scales [28].
+The SDSS-DR16Q contains 750,414 quasars, with the
+automated redshift range 1 ≤ z ≤ 7.1. The number of
+sources reaches its maximum around z ≈ 2.5 and at ear-
+lier epochs i.e. higher redshifts, they are comparatively
+rare [29]. However, there is a problem with the SDSS-
+DR16Q catalog. It contains non quasar sources due to
+pipeline classification errors and incorrect redshift esti-
+mations [13].
+For example, in a search for undeclared
+quasars, the SDSS-DR16Q main quasars are found to
+contain 81 entries that are not quasars. It must there-
+fore be noted that the pipeline catalog is not an adequate
+training samples for quasars because many objects with
+z ≥ 6 as well as significant fractions of these objects at
+z ≥ 4, may not be quasars or not quasars at the given
+redshifts due to incorrect pipeline classifications [29].
+III.
+MASS ESTIMATION OF QUASARS’
+CENTRAL SMBH
+In terms of fundamental parameters of quasars, one
+can mention the central SMBH mass and structure, along
+with the ratio of the accretion rate to the Eddington
+accretion rate [30].
+The central SMBH mass can be measured via the gas
+or stellar dynamics [30] from optical or ultraviolet(UV)
+spectroscopy using empirical relations [31]. The broad
+emission line region(BLR) probably provides the best
+probe of these characteristics [32].
+The size of BLRs
+can be determined by reverberation mapping(RM) [33],
+which is a measuring technique in astrophysics. RM pro-
+vides invaluable information about the kinematic and
+ionization distribution of the gas using the time lag be-
+tween emission line and continuum variations [32].
+Assuming that gravity dominates the dynamics of the
+BLR and the virial relationship between time lag and line
+width exists, the BH mass can be estimated as [34]
+MBH = fcτv2
+G
+,
+(1)
+where τ is the mean time delay for the region of inter-
+est, v is the velocity of the gas in that region, c is the
+speed of light, G is the gravitational constant, and f is a
+scaling factor of order unity that depends on the detailed
+geometry and kinematics of the line-emitting region.
+The worth mentioning point is that the virial relation-
+ship claims a virialized system with individual clouds
+moving in their Keplerian orbits. This leads to the pro-
+portionality of mean cloud velocity and emissivity radius
+[35]
+v ∝ rBLR
+1
+2 ,
+(2)
+where rBLR is the emissivity radius.
+In the absence of RM, the quasar continuum luminosity
+is sufficient to estimate the BLR. With RM estimations,
+the best-fitting RBLR − λLλ relations were derived for
+quasars at monochromatic luminosity in both 3000 and
+5100 ˚A rest-frames as follows [37]
+
+3
+RBLR = (18.5 ± 6.6)[λL3000/1037W]
+(0.32±0.14),
+(3)
+RBLR = (26.4 ± 4.4)[λL5100/1037W]
+(0.61±0.10).
+(4)
+Here, L is the luminosity measured at a wavelength λ.
+In Eq. 1, an intrinsic Keplerian velocity of a broad-line
+gas is related to the full width at half maximum (FWHM)
+of a chosen broad emission line by the geometric factor
+f as
+VBLR = f × FWHM,
+(5)
+In other words, it is the width of a spectrum curve
+measured between those points on the y-axis which are
+half of the maximum amplitude.
+As the geometry of the BLR in radio-quiet quasars is
+currently unknown, it is generally agreed that f =
+�
+3/2,
+which is appropriate for randomly oriented orbits of the
+BLR gas.
+However, FWHM measurements for broad
+emission lines in radio-loud quasars indicate a disc-like
+geometry [38].
+Given the similarity between the opti-
+cal emission-line spectra of radio-loud and radio-quiet
+quasars, it is not unreasonable to consider the possibil-
+ity that BLRs of radio-quiet quasars that dominate the
+SDSS data can follow the same equation as well [36]
+VBLR = FWHM
+(2 sin i) .
+(6)
+Here, i represents the angle between the line of sight
+and the axis of the disc.
+Our virial BH mass estimators are derived by substi-
+tuting the calibrations of the RBLR–λLλ relations into
+Eq. 1 and determining VBLR using MgII or Hβ [37].
+Based on the L5100 which is the monochromatic lu-
+minosity at rest-frame 5100 ˚A and the Hβ line, a more
+specific expression to calculate the mass of a BH can be
+written as [39]
+MBH(Hβ) = 1.05 × 108(
+L5100
+1046ergs−1 )0.65
+(7)
+× [FWHM(Hβ)
+103kms−1
+]2M⊙,
+where MBH(Hβ) represents BH mass by considering
+Hβ line, FWHM(Hβ) is the full width at half maximum
+of Hβ line, and M⊙ is the solar mass.
+Large spectroscopic surveys like the SDSS observe
+both broad Hβ and MgII lines.
+Therefore, one can
+be calibrated against the other and based on L3000 and
+MgIIλ2798 line width, a similar expression can be de-
+rived as [35]
+MBH(MgIIλ2798) = 8.9 × 107(
+L3000
+1046ergs−1 )0.58
+(8)
+× [FWHM(MgIIλ2798)
+103kms−1
+]2M⊙,
+where MBH(MgIIλ2798) represents BH mass by con-
+sidering Hβ line, and FWHM(MgIIλ2798) is the full
+width at half maximum of MgII line.
+Based on empirical estimation of f ≃ 1.1 for the Hβ
+line, we can now write more specific expressions to calcu-
+late MBH for several emission lines like MgII as follows
+[39]
+MBH
+M⊙
+= 4.7(λL5100
+1037W )
+0.61
+[FWHM(Hβ)
+kms−1
+]
+2
+,
+(9)
+MBH
+M⊙
+= 3.2(λL3200
+1037W )
+0.62
+[FWHM(MgII)
+kms−1
+]
+2
+. (10)
+Besides, it is well-known that the relationship between
+stellar velocity dispersion and BH mass can be written
+as [39]
+log(MBH
+M⊙
+) = 4.38 × log(
+σ∗
+200kms−1 ) + 8.49,
+(11)
+where σ∗ is the stellar velocity dispersion.
+Furthermore, to estimate the mass of a BH, observa-
+tions in the local universe reveal the existence of a corre-
+lation between the central SMBH mass and the bulge of
+the host galaxies [40].
+log(MBH
+M⊙
+) = α + βlog(MBulge,∗
+1011M⊙
+),
+(12)
+where MBulge,∗ is the bulge stellar mass and the best-
+fit of α and β should be
+α = 7.93 ± 0.061; β = 1.15 ± 0.075.
+(13)
+IV.
+MASS EVOLUTION OF QUASARS’
+CENTRAL SMBH
+As studying the cosmic history of compact cosmolog-
+ical objects is so crucial to track the history line of the
+universe in a much bigger structure, we are so curious
+about the evolution of SMBHs mass. In the presence of
+a SMBH, there are obvious links between the physical
+properties and those of its host. Due to high redshifts
+that many quasars have, they are ideal to be studied to
+recognize BH evolution through time back to the early
+universe [41].
+According to the modelling of spectra from the SDSS
+first data release, the virial mass of BHs for 12698 quasars
+
+4
+in the redshift interval 0.1 ≤ z ≤ 2.1 is estimated.
+There is entirely consistent evidence to suggest that the
+BH mass of SDSS quasars lies in 107M⊙ ≤ MBH ≤
+3 × 109M⊙. The local BH mass function for early-type
+galaxies using the MBH − σ and MBH − Lbulge correla-
+tions(Eq. 11 and Eq. 12) are also estimated. In addition,
+by comparing the number density of active BHs at z ≈ 2
+with the local mass density of inactive ones, a lower limit
+is set on the lifetime of quasars, which confirms that the
+bulk of BHs with mass ≥ 108.5M⊙ are situated in place
+by z ≈ 2 [36].
+There are several different ideas on the central SMBH
+mass evolution through time in literature. Based on the
+effective flux limit along with the role of the quasar con-
+tinuum luminosity, most studies agree that the SMBH
+mass increases as a function of redshift, namely most low
+mass SMBHs can be found in the late universe(e.g. step-
+ping down from ≈ 109M⊙ at z ≈ 2.0 to ≈ 108M⊙ at
+z ≈ 0.2). Considering Eq. 9 and Eq. 10, redshift does
+not alter the mean FWHM and it can be roughly consid-
+ered to be constant. Therefore, the mean virial mass of
+the SMBH should be increased as [Lλ]0.6 [36].
+Quasars undergo important cosmic evolution accord-
+ing to optical, X-ray, and bolometric LFs. Interestingly,
+based on predictions of [42] using an extended version of
+the galaxy formation model, GALFORM code, quasars
+evolution will be influenced by different physical pro-
+cesses such as the accretion mode and the obscuration
+prescription. Observational data have also reported sim-
+ilar trends [43].
+Furthermore, SMBHs grow exponentially during a pe-
+riod in which accretion governs their mass evolution.
+When z ≳ 5, the growth of a SMBH in a quasar is as
+follows [44]
+MBH(t) = MBH(t0)etτ,
+(14)
+τ ≃ 0.4Gyr
+η
+1 − η
+1
+µ,
+(15)
+µ ≡
+L
+LEdd
+× factive,
+(16)
+where MBH(t0) is the initial mass of BH i.e. the seed’s
+mass, η is the radiative efficiency(see [45] for reported
+values of η for several objects), L is the luminosity of the
+quasar, LEdd is the luminosity at Eddington limit, factive
+is the duty cycle, and µ is a constant which is determined
+as a combination of L/LEdd and factive. Therefore, it is
+possible to calculate the growth of the BH easily as
+log MBH(z) = log MBH(z0)
+(17)
++ log[exp (R(1 − η
+η
+)zd),
+η ≡
+Lbol
+˙Mc2 ,
+(18)
+zd ≡ (1 + z)−3/2 − (1 + z0)−3/2.
+(19)
+In above equations, MBH(z0) is the mass of BHs’ seed
+and R is a constant that is defined as follows
+R ≡ 0.4Gyr
+µ
+,
+(20)
+R =
+�
+�
+�
+�
+�
+3.79322,
+µ = 0.1
+18.9661,
+µ = 0.5
+37.9322,
+µ = 1.0.
+(21)
+V.
+QUASARNET AND FNET
+To investigate the mass evolution even more precisely,
+QuasarNET and FNET are the two available research
+platforms. Using ML, QuasarNET makes deployment of
+data-driven modelling techniques possible by combining
+and co-locating large observational data sets of quasars,
+the high-redshift luminous population of accreting BHs,
+at z ≥ 3 alongside simulated data spanning the same
+cosmic epochs. The main quasar population data source
+of QuasarNET is NASA Extra-galactic Database(NED)
+which contains quasars retrieved from several indepen-
+dent optical surveys, principally the magnitude-limited
+SDSS. There is no comparison between quasars from
+SDSS and those from other surveys when it comes to
+spectra and photometry [46].
+NED contains all quasars in principle, but some are
+missing because their photometric redshifts were incor-
+rectly assigned. Photometric redshift estimation meth-
+ods suffer from degeneracy, a well-known limitation of
+current photometric redshift determination methods [47].
+QuasarNET fills in the missing sources by analyzing the
+published catalogues from all surveys. It expands to in-
+clude additional parameters used to derive BHs mass, in-
+stead of archiving only the reported masses. It contains
+136 quasars’ features, such as the position, redshift, lu-
+minosity, mass, line width, and Eddington ratio.
+Two observationally determined functions are used as
+constraints in theoretical models to describe the assembly
+history of the BHs population across time: the BH mass
+function and the Quasar Luminosity Function(QLF). As
+a statistical measurement of the combined distribution
+of BHs mass through redshifts, the BH mass function
+encodes the mass growth history. Similar to the QLF,
+which reflects their accretion history, the BH mass func-
+tion is a statistical measurement of the distribution of
+quasars’ luminosities through redshift [46].
+On the other hand, by using DL, to study quasars in
+the SDSS-DR16Q of eBOSS on a wide range of signal-
+to-noise(SNR) ratios, there is a 1-dimensional convolu-
+tional neural network(CNN) with a residual neural net-
+work(ResNet) structure, named FNet. With its 24 con-
+volutional layers and ResNet structure, which has dif-
+ferent kernel sizes of 500, 200, and 15, FNET can use
+a self-learning process to identify ”local” and ”global”
+patterns in the entire sample of spectra [29].
+
+5
+Although FNET seems to be similar to the recently
+adopted CNN-based redshift estimator and classifier, i.e.
+QuasarNET [14], their hidden layer implementations are
+distinct.
+The redshift estimation in FNET is done based on re-
+lating the hidden pattern which lies in flux to a spe-
+cific redshift, not using any information about emis-
+sion/absorption lines, while QuasarNET follows the tra-
+ditional redshift estimation procedure using the identified
+emission lines in spectra. This makes FNET to outper-
+form QuasarNET for some complex spectra(insufficient
+lines, high noise, etc.) by recognizing the global pattern.
+Moreover, FNET provides similar accuracy to Quasar-
+NET, but it is applicable for a wider range of SDSS spec-
+tra, especially for those missing the clear emission lines
+exploited by QuasarNET. In more detail, from a statis-
+tical point of view, FNET is capable to infer accurate
+redshifts even for low SNRs or incomplete spectra.
+It
+predicts the redshift of 5,190 quasars with 91.6 % accu-
+racy, while QuasarNET fails to estimate [29].
+It is important to know that the FNET vs. Quasar-
+NET comes out on top in redshift prediction, but its
+lack of quasars’ central SMBH mass information makes
+QuasarNET the preferred option for some studies like
+this work. However, if in the future SMBHs mass will be
+estimated by using redshifts from FNET approach, our
+study can be done again to achieve more accurate results.
+VI.
+FLUX AND VOLUME-LIMITED SAMPLES
+Observations are affected by flux as we move to higher
+redshifts and more distant objects.
+This is why some
+objects are not included in data sets. We suppose that
+they are not even present because their low flux makes
+them very difficult or in some cases impossible to observe.
+This will influence the results of any model that is built
+on a set of objects. To remove this bias, we must first
+correct the data set.
+Two correction methods can be put into use to build
+a corrected data set and check if the result is solid or if
+the correction can end up with a huge deviation from the
+first result.
+Using the friends-of-friends algorithm, quasars can be
+linked into systems with a specific neighbourhood radius,
+called linking length(LL). The size of the group can be
+determined based on the choice of LL or more generally
+on its scaling law. LL is parameterized upon a scaling
+law as [48]
+LL
+LL0
+= 1 + a arctan( z
+z∗
+),
+(22)
+where a = 1.00, z∗ = 0.050 and LL0 is the value of LL
+at initial redshift.
+Setting a limit for absolute magnitude is needed for
+creating volume-limited samples and all less luminous
+FIG. 1: The total number of objects available in the
+QuasarNET data set is 37648. As a result of data correction
+methods, 34403 objects were removed (red dots). The accepted
+data are the final flux and volume-limited samples, made of 3245
+Objects(blue dots).
+quasars have to be excluded from the data set.
+Flux-
+limited samples, on the other hand, are formed from
+dozens of cylinders containing quasars.
+Flux-limited
+samples can be made with both constant and varying
+LL. The constant LL0 is set as [48]
+LL0 = 250[kms−1],
+(23)
+LL0 = 0.25[h−1Mpc].
+(24)
+Following the extraction of the necessary columns and
+rejecting duplicate quasars from the data set, there is
+only one step left, which is verifying if the quasars are
+within the volume of cylinders generated by the LLs.
+To do so first we generate a cylinder, then by using the
+distance between quasars and comparing this distance
+with the volume of the cylinder, we consider a quasar
+to be an accepted object if it is located in the cylinder.
+The distance can be easily obtained from the redshift
+difference between them in the data set. This algorithm
+should be repeated as a loop for each quasar.
+As a result of applying the correction methods that are
+described, we end up with 3246 objects to work with, in-
+stead of 37648 objects that are available in QuasarNET.
+In FIG. 1 accepted and rejected quasars’ central SMBH
+of SDSS-DR16Q in terms of their redshift are illustrated.
+VII.
+LONG SHORT-TERM MEMORY
+LSTM is one of the most powerful RNN that is used in
+DL and artificial intelligence [49]. The RNN is a dynamic
+system in which there is an internal state at each step of
+the classification process [50, 51]. The circular connec-
+tions between neurons at the higher and lower layers, as
+well as the possibility of self-feedback, are responsible for
+this. These feedback connections enable RNNs to propa-
+gate data from earlier events to current processing steps.
+Thus, RNNs build a memory of time series events.
+A standard RNN is not capable of bridging more than
+5 to 10 time steps. It is because back-propagated error
+signals either grow or shrink with every time step [49]. As
+
+Removed Data
+11.0
+Accepted Data
+10.5
+10.0
+log(MBH / M。)
+9.5
+9.0
+8.5
+8.0
+7.5
+3
+5
+6
+76
+a result, the error typically blows up or disappears over a
+long period of time [52, 53]. When error signals are blown
+up, the result is oscillating weights, while vanishing er-
+rors mean learning takes too long or does not work at all.
+It is possible to solve the vanishing error problem by us-
+ing a gradient-based approach known as LSTM [53–56].
+More than 1,000 discrete time steps can be bridged us-
+ing LSTM. LSTM uses constant error carousels(CECs),
+which enforce a constant error flow within special cells.
+Cell accessibility is handled by multiplicative gate
+units, which learn when to grant access to cells [49]. Us-
+ing a multiplicative input gate unit, memory contents
+stored in j are protected from irrelevant inputs. We also
+introduce a multiplicative output gate unit that protects
+other units from being perturbed by currently irrelevant
+memory contents stored in j [57]. Considering distinct
+time steps t= 1, 2, etc., an individual step includes for-
+ward and backward passes which are the update of all
+units and calculation of error signals for all weights, re-
+spectively. The Input yin and output yout gate activation
+are computed as [54]
+netoutj(t) =
+�
+m
+ωoutjmym(t − 1), youtj(t)
+(25)
+= foutj(netoutj(t)),
+netinj(t) =
+�
+m
+ωinjmym(t − 1), yinj(t)
+(26)
+= finj(netinj(t)).
+Here, netinj and netout are the input and output gate
+activation, j indices are memory blocks, ωlm is the weight
+on the connection from unit m to l. Index m ranges over
+all source units, as specified by the network topology. For
+gates, f is a logistic sigmoid in the range of [0, 1].
+Furthermore, there are adaptive gates, which learn to
+reset memory blocks once their contents are out of date
+and therefore, useless. Like the activation of the other
+gates(Eq. 25 and Eq. 26), the forget gate activation yφ
+is calculated as
+netφj(t) =
+�
+m
+ωφjmym(t − 1), yφj(t)
+(27)
+= fφj(netφj(t)),
+where netφj is the input from the network to the forget
+gate.
+The logistic sigmoid with range [0, 1] is used as
+squashing function fφj and weighted by the hyperbolic
+tangent function which has the overall task of memory
+correction [54]. The forget gate stores all the 1 outputs
+while forgetting all the 0 outputs. Finally, LSTM can be
+written as [58]
+it = σ(Wxixt + Whiht−1 + Wcict−1),
+(28)
+ft = σ(Wxfxt + Whfht−1 + Wcfct−1),
+(29)
+ot = σ(Wxoxt + Whoht−1 + Wcoct−1),
+(30)
+ht = ot ⊙ tanh(ct).
+(31)
+Here, it , ft, and ot are input gate, forget gate and
+output gate of LSTM, ht represents LSTM output, σ is
+LSTM logistic function, ⊙ denotes element-wise product,
+W is the weight metric components, x is the input data
+in time t, and c is LSTM memory cells.
+In our application of LSTM, the forget gate and in-
+put gate share the same parameters, but are computed
+as ft = 1 − it. Note that bias terms are omitted in the
+above equations, but they are applied by default. A lin-
+ear dependence between LSTM memory cells(ct) and its
+past(ct−1) are introduced as
+ct = ft ⊙ ct−1 + it ⊙ tanh(Wxcxt + Whcxt−1).
+(32)
+VIII.
+HYPERPARAMETER SELECTION
+Hyperparameter selection in neural networks is repre-
+sented by optimization functions. Therefore, specifying
+hyperparameters such as the type of optimization func-
+tion, learning rate, number of neurons in each layer, num-
+ber of epochs, and validation are very important. Adam,
+Stochastic gradient descent(SGD), RMSProp, AdaDelta,
+and Ftrl are used as optimization functions.
+We have considered about 20% of the learning data as
+validation data. To determine the quality of the model,
+we determine the loss. The cost function that we have
+considered for the network is mean squared error(MSE).
+The number of epochs for the network learning process
+is equal to 50 and the batch size is equal to 25. Results
+of the cost function values for each learning process with
+different optimization functions and a learning rate of
+0.0005 are shown in FIG. 2.
+The results related to the loss value for learning and
+testing data with different optimization functions are
+reported in the TABLE I.
+IX.
+DATA AND NETWORK TOPOLOGY
+Using QuasarNET data we predict the SMBHs mass
+with the help of their redshift.
+We use 3245 data for
+modelling, 2596 data for the network learning process,
+and 649 data for testing the network result. Data have a
+redshift range of 3 to 7. In the first step, data are sorted
+in ascending order of their redshifts. The reason is that
+
+7
+(a)
+(b)
+(c)
+(d)
+(e)
+FIG. 2: (a) shows model evaluation for SGD optimization
+function. Optimization function loss is illustrated by the blue line
+and orange lines represent validation loss. (b) is the model
+evaluation using RMSProp whose optimization function loss and
+validation loss are shown in blue and orange. (c) illustrates the
+Adam model evaluation by comparing the Optimization function
+loss(blue line) and validation loss(orange line). The model
+evaluation for Ftel is shown in (d). loss of the optimization
+function is represented by the blue line and the validation
+function loss is shown by the orange line. (e) shows model
+evaluation for the AdaDelta optimization function. Optimization
+function loss is illustrated by the blue line and orange lines
+represent validation loss.
+Optimization functions
+Train data MSE
+Test data MSE
+SGD
+0.38
+0.39
+RMSProp
+0.37
+0.38
+Adam
+0.23
+0.23
+AdaDelta
+0.22
+0.23
+Ftrl
+0.26
+0.27
+TABLE I: This table shows the result of algorithm evaluation
+by SGD, RMSProp, Adam, Ftrl and AdaDelta optimization
+functions.
+redshift is a time series and LSTM has a recurrent archi-
+tecture which creates memory through time. Then, the
+learning and testing data are separated in chronological
+order.
+The network topology can be described by an LSTM
+layer as the dynamic layer of the network, a drop-out
+layer to prevent over-fitting, 3 dense layers as static lay-
+ers, and the output of the network which is printed by the
+last dense layer. We use the hyperbolic tangent which is
+an active function for the LSTM layer and the first dense.
+Because the hyperbolic tangent is a non-linear function
+with a symmetric range. It is a suitable option to control
+sudden changes when they are in chronological order. For
+the second dense, we use the rectified linear unit(ReLU),
+to transfer the magnitude of the positive value to the
+next layer. For the third dense, which outputs the net-
+work as a continuous number, we use a linear function.
+TABLE II shows the network structure based on the hy-
+perparameters of the network.
+Layers
+Neurons
+Computational Parameters
+Inputs
+-
+-
+LSTM
+(None,256)
+264192
+Dropout
+(None,256)
+0
+Dense
+(None,512)
+131584
+Dense
+(None,256)
+131328
+Dense
+(None,1)
+257
+Total Computational Parameters
+527361
+Trainable Computational Parameters
+527361
+Non-Trainable Computational Parameter
+0
+TABLE II: This table illustrates the network topology which
+includes each layer along with neurons and computational
+parameters.
+One of the main challenges that always exists in ML
+and DL is the issue of transparency.
+Transparency is
+a dynamic issue and solving this problem is different for
+each task. There is no specific method to solve this prob-
+lem. Many factors such as the design of an interpretable
+learning experience, the fundamental determination of
+
+SGD LoSS
+1.4 
+SGD Validation Loss
+1.2 
+1.0 
+SS0
+0.8
+0.6 
+0.4 
+0.2 
+0
+10
+20
+GE
+40
+50
+EpochsRMSProp Loss
+1.0 
+RMSProp ValidationLoss
+0.8:
+SS0
+0.6
+0.4 
+0.2 
+0
+10
+20
+GE
+40
+50
+Epochs0.27
+Adam Loss
+Adam Validation Loss
+0.26
+0.25
+SSO
+0.24 
+0.23 
+0.22
+0
+10
+20
+30
+40
+50
+EpochsFtel Loss
+70
+Ftrl Validation Loss
+ 09
+50 
+40
+30 
+20 :
+10 
+0
+10
+20
+E
+40
+50
+Epochs0.30
+AdaDelta Loss
+AdaDelta Validation Loss
+0.28
+0.26
+0.24 
+0.22
+0.20
+10
+20
+CE
+40
+50
+Epochs8
+FIG. 3: Model built using flux and volume-limited samples.
+Corrected QuasarNET data are plotted with blue dots. The black
+line represents our LSTM model best-fit. In addition, red dotted
+lines represent our models that include 95 percent of all data.
+hyperparameters by the task, the observance of the prin-
+ciples of feature selection, and the determination of the
+appropriate number of data based on characteristics can
+allow us to have a transparent model.
+Transparency in the structure of algorithms is also
+noteworthy.
+In this paper, we investigate the trans-
+parency of the model built by the designed network.
+Trained data are also based on redshifts from 3 to 7.
+With the help of the built model, SMBHs mass at 0 <
+z < 3 and 7 < z < 10 are then predicted.
+We can see the predicted changes of SMBHs mass
+through redshift in FIG. 3 based on our built model with
+its 95 percent confidence level. FIG. 4 compares the lin-
+ear best-fit with our LSTM model best-fit both before
+and after applying correction methods. It can clearly be
+seen that stated correcting methods change our model
+significantly.
+X.
+COMPARING WITH OTHER DATA
+Using corrected flux and volume-limited samples of
+QuasarNET data, we build a DL model for quasars’ cen-
+tral SMBH mass. By applying correction methods, only
+≃ 8.62% of QuasarNET data is accepted to use for mod-
+elling. FIG. 1 shows the accepted data along with the
+removed data.
+Moreover, FIG. 3 illustrated our model whose best-
+fit contains 95 percent of corrected data samples. The
+model shows that SMBHs mass increases in 0 < z < 4.72
+and reaches its peak at z ≃ 4.72. The mass then falls
+exponentially with increasing redshift at z > 4.72.
+It
+should be noted that our model yields a different result
+than what is shown in other recent works like [44], where
+the peak is z < 4. Nevertheless, in some studies which
+attempt to show quasars’ central SMBHs mass evolution,
+Eq. 14 is used that does not include any peaks(e.g. see
+[59]).
+The model is then evaluated by using different data
+sets which are available in multiple tables. We use the
+results of the long-term spectroscopic monitoring of 15
+PG quasars that have relatively strong Fe II emission to
+generate TABLE III [60]. Moreover, TABLE IV shows
+(a)
+(b)
+FIG. 4: (a) compares our model with the linear best-fit. Blue
+dots indicate train data, the LSTM model prediction is showed
+the colour red, and the orange line is the linear best-fit. (b)
+illustrates our model and compares it with linear best-fit based on
+flux and volume-limited samples. Train data is shown as blue
+dots, LSTM model prediction as red, and linear best-fit as an
+orange line.
+(a)
+(b)
+FIG. 5: (a) shows the examination of our model using multiple
+data sets in the redshift range of 0 < z < 7. An overview of the
+utilized data can be found in TABLE III and IV. (b) is also the
+model examination at 3 < z < 10 whose data is available in
+TABLE V and VI.
+
+11
+LSTM Model Best-fit
+95% CI
+QuasarNET Data
+10
++
+C. Hu et al. (2021)
+M. Vestegaard et al. (2005)
+9
+log(MBH / Mo)
+7
+6 
+0
+1
+2
+3
+4
+5
+611
+LSTM Model Best-fit
+95% CI
+QuasarNETData
+Y.Aggarwal (2022)
+10
+J.Yang et al. (2021)
+9
+g(MBH / M。)
+60
+8
+7
+6
+3
+4
+5
+6
+8
+9
+1011
+LSTM Model Best-fit
+95% CI
+QuasarNET Data
+10-
+log(
+8 -
+7 -
+6 
+0
+1
+2
+3
+5
+6
+711.0:
+TrainData
+LSTM Model Prediction
+Linear Best-Fit
+10.5
+10.0
+log(M_bh)
+9.5
+0'6
+B.5
+7.5
+3.0
+3.5
+4.0
+4.5
+5.0
+5.5
+6.0
+6.5
+7.0Trained Data
+LSTM Model Best-fit
+10.5
+Linear Best-fit
+.
+10.0
+log(MBH / Mo)
+9.5
+9.0
+8.5
+8.0 
+3.0
+3.5
+4.0
+4.5
+5.0
+5.5
+6.0
+6.59
+relatively nearby quasars with redshifts obtained from
+the NED and central SMBHs mass determined through
+multi-epoch spectrophotometry and RM [61]. A list of
+69 high-redshift quasars is also available in TABLE V
+and TABLE VI. For each quasar, the most accurate es-
+timation of its central SMBH mass using Mg II emis-
+sion lines along with its uncertainty is shown [62, 63].
+While the model matches observational data quite well at
+3 < z < 10, there is a minor deviation at lower redshifts,
+i.e. 0 < z < 3. The comparison between our model pre-
+dictions and the observational data for both low-redshift
+and high-redshift quasars can be seen in FIG. 5.
+In addition to gas being sucked into SMBHs, there is
+an alternative process that turns them into stars. There
+has been a comparison of SMBH accretion rate and SFR
+on a galactic scale in several observational studies [64–
+67].
+In our next work, we will address the SFR and its
+effects on the model.
+Thus, it is possible to fix the
+minor deviation between the model and observations.
+Further, there are more data available for lower redshift
+quasars, compared to higher ones, whose reasons should
+be studied and may have an impact on the final results
+of our model.
+XI.
+CONCLUSIONS
+The question of how the SMBHs that have been ob-
+served in the universe came into being is one of the
+biggest questions in cosmology. In recent years, it has
+been established that stellar BHs cannot accrete mass,
+resulting in such BHs. If we want to consider these BHs
+as stellar BHs that have reached such incredible mass due
+to accretion, the age of the universe should have been
+much longer than it is. On the other hand, it is impossi-
+ble for a star to form a SMBH as a result of its collapse.
+In addition, there is another idea that states that these
+BHs are actually primordial BHs. Although this idea is
+very controversial, it has not been rejected yet. There
+are even hopes to prove such a thing.
+One of the most interesting surveys available for
+quasars is the SDSS. In this paper, we have used SDSS-
+DR16Q. In particular, we have taken advantage of the
+QuasarNET research platform. QuasarNET specifically
+has focused on the study of SMBHs.
+Although 37648
+data in redshifts between 3 and 7 have been reported in
+it, these data need accurate corrections to be used. These
+corrections are flux and volume-limited, which makes the
+right conditions to work on SMBHs over time for training
+the machine. After applying these corrections, 3246 data
+remained and 34403 data were removed. In FIG. 1 we
+have plotted accepted and removed data after correcting
+them.
+Considering the remaining 3246 data of the mass of
+BHs in the center of quasars at redshifts between 3 and
+7, we have modeled them over time with the help of the
+LSTM RNN. We have elaborated details of our used DL
+approach in several sections.
+The model we have pre-
+sented with the help of QuasarNET data tries to predict
+the mass of the central massive BHs of quasars at red-
+shifts between 0 and 10.
+Firstly, in FIG. 4, we have compared our prediction
+with the linear best-fit of QuasarNET data before and
+after correcting data. Then, we illustrated the best-fit
+and a band that 95 percent of the QuasarNET data is
+within 2 standard deviations of the mean for our model
+in redshifts 0 to 10.
+Eventually, we should have compared our model with
+other observational data at redshifts between 0 and 3
+and also 7 and 10. This will enable us to see whether our
+model works or not. We have used four data sets for this
+comparison. Two of them are related to redshifts 0 to 3
+and the other two are related to redshifts 7 to 10. FIG. 5
+demonstrates two redshift ranges, 0 to 7 and 3 to 10. As
+it is evident, at redshifts higher than 7, our model has a
+very good description of the data and can make a reliable
+prediction, but at redshifts below 3, it seems that there
+is a slight deviation.
+This deviation can be due to not considering other pa-
+rameters describing quasars. We have only used the esti-
+mation of the mass of the central SMBHs of quasars and
+their redshift in QuasarNET data. However, data such as
+the Eddington ratio and bolometric luminosity are also
+available and can be used for subsequent modeling.
+Another thing that can improve the model is to con-
+sider star formation with the help of other observational
+data sets. Accurately obtaining the time of star forma-
+tion causes the redshift of the peak of the model we ob-
+tained to change to lower redshifts. This issue makes our
+model predict more massive central SMBHs at redshifts
+below 3, and as a result, it fits better with other data.
+Finally, we must state that this effort to model SMBHs
+at high redshifts will help us to find out when and how
+they have been formed and their role in the formation
+of the structures.
+Furthermore, if the process of their
+growth through the accretion and merger of primordial
+BHs is also studied in future works, it will probably yield
+interesting results. Because by going back through time,
+the initial masses of these central SMBHs can be exam-
+ined.
+Acknowledgement
+Authors thank Shant Baghram for the great discus-
+sions that helped us to model and correct the Quasar-
+NET data and Rahim Moradi for helpful discussion.
+Data availability
+The catalogue underlying this paper is available in
+the Sloan Digital Sky Survey Quasar catalogue: 16th
+
+10
+data release (DR16Q) at https://www.sdss.org/dr16/
+algorithms/qsocatalog/ [13].
+The data that support the findings of this study
+are
+openly
+available
+at
+https://www.kaggle.com/
+datasets/quasarnet/quasarnet,
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+Accretion Rates in Active Galactic Nuclei. XII. Reverber-
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+lution study (QUEST). II. The spectral energy distribu-
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+trophysics 545 (2012): A45.
+[68] QuasarNet
+(2022).
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+[Dataset].
+https://www.kaggle.com/datasets/quasarnet/quasarnet
+[69] Wang, Feige, et al. ”A luminous quasar at redshift 7.642.”
+The Astrophysical Journal Letters 907.1 (2021): L1.
+a redshift of 7.5, Nature 553 473 (2017)
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+12
+[70] Mortlock, Daniel J., et al. ”A luminous quasar at a red-
+shift of z= 7.085.” Nature 474.7353 (2011): 616-619.
+[71] Matsuoka, Yoshiki, et al. ”Discovery of the First Low-
+luminosity Quasar at z¿ 7.” The Astrophysical Journal
+Letters 872.1 (2019): L2.
+[72] Wang, Feige, et al. ”The discovery of a luminous broad
+absorption line quasar at a redshift of 7.02.” The Astro-
+physical Journal Letters 869.1 (2018): L9.
+[73] Wang, Feige, et al. ”A Significantly Neutral Intergalactic
+Medium Around the Luminous z 7 Quasar J0252–0503.”
+The Astrophysical Journal 896.1 (2020): 23.
+[74] B.P. Venemans et al., Discovery of Three z 6.5 Quasars
+in the VISTA Kilo-Degree Infrared Galaxy (VIKING)
+Survey, ApJ 779 24 (2013)
+[75] Reed, Sophie L., et al. ”Three new VHS–DES quasars at
+6.7¡ z¡ 6.9 and emission line properties at z¿ 6.5.” Monthly
+Notices of the Royal Astronomical Society 487.2 (2019):
+1874-1885.
+[76] Matsuoka, Yoshiki, et al. ”SUBARU HIGH-z EXPLO-
+RATION OF LOW-LUMINOSITY QUASARS (SHEL-
+LQs). I. DISCOVERY OF 15 QUASARS AND BRIGHT
+GALAXIES.” The Astrophysical Journal 828.1 (2016):
+26.
+[77] Mazzucchelli, C., et al. ”Physical properties of 15 quasars
+.” The Astrophysical Journal 849.2 (2017): 91.
+[78] Eilers, Anna-Christina, et al. ”Detecting and character-
+izing young quasars. I. Systemic redshifts and proximity
+zone measurements.” The Astrophysical Journal 900.1
+(2020): 37.
+[79] Onoue, Masafusa, et al. ”No Redshift Evolution in the
+Broad-line-region Metallicity up to z= 7.54: Deep Near-
+infrared Spectroscopy of ULAS J1342+ 0928.” The As-
+trophysical Journal 898.2 (2020): 105.
+[80] Mortlock, D. J., et al. ”Discovery of a redshift 6.13 quasar
+in the UKIRT infrared deep sky survey.” Astronomy and
+Astrophysics 505.1 (2009): 97-104.
+
+13
+Object
+Redshift
+MBH(×107M⊙)
+PG 0003+199
+0.0259
+0.50+0.18
+−0.18
+PG 0804+761
+0.1005
+4.14+0.91
+−0.98
+PG 0838+770
+0.1316
+2.89+1.01
+−1.13
+PG 1115+407
+0.1542
+7.76+2.23
+−1.95
+PG 1322+659
+0.1678
+3.35+1.92
+−0.71
+PG 1402+261
+0.1643
+3.41+1.28
+−1.51
+PG 1404+226
+0.0972
+0.68+0.14
+−0.23
+PG 1415+451
+0.1132
+1.75+0.36
+−0.32
+PG 1440+356
+0.0770
+1.49+0.49
+−0.55
+PG 1448+273
+0.0646
+1.01+0.38
+−0.23
+PG 1519+226
+0.1351
+4.87+0.49
+−0.86
+PG 1535+547
+0.0385
+1.55+0.84
+−0.82
+PG 1552+085
+0.1187
+1.30+0.68
+−0.65
+PG 1617+175
+0.1144
+4.79+2.94
+−2.83
+PG 1626+554
+0.1316
+19.17+2.98
+−2.73
+TABLE III: This table contains 15 low redshift quasars at z < 1 with their central SMBH mass reported in [60].
+Object
+Redshift
+log(M/M⊙) (Hβ, rms)
+Mrk 335
+0.02578
+7.152+0.101
+−0.131
+PG 0026+129
+0.14200
+8.594+0.095
+−0.122
+PG 0052+251
+0.15500
+8.567+0.081
+−0.100
+Fairall 9
+0.04702
+8.407+0.086
+−0.108
+Mrk 590
+0.02638
+7.677+0.063
+−0.074
+3C 120
+0.03301
+7.744+0.195
+−0.226
+Ark 120
+0.03230
+8.176+0.052
+−0.059
+PG 0804+761
+0.10000
+8.841+0.049
+−0.055
+PG 0844+349
+0.06400
+7.966+0.150
+−0.231
+Mrk 110
+0.03529
+7.400+0.094
+−0.121
+PG 0953+414
+0.23410
+8.441+0.084
+−0.104
+NGC 3783
+0.00973
+7.474+0.072
+−0.087
+NGC 4151
+0.00332
+7.124+0.129
+−0.184
+PG 1226+023
+0.15830
+8.947+0.083
+−0.103
+PG 1229+204
+0.06301
+7.865+0.171
+−0.285
+PG 1307+085
+0.15500
+8.643+0.107
+−0.142
+Mrk 279
+0.03045
+7.543+0.102
+−0.133
+PG 1411+442
+0.08960
+8.646+0.124
+−0.174
+NGC 5548
+0.01717
+7.827+0.017
+−0.017
+PG 1426+015
+0.08647
+9.113+0.113
+−0.153
+Mrk 817
+0.03145
+7.694+0.063
+−0.074
+PG 1613+658
+0.12900
+8.446+0.165
+−0.270
+PG 1617+175
+0.11240
+8.774+0.019
+−0.115
+PG 1700+518
+0.29200
+8.893+0.091
+−0.103
+3C 390.3
+0.05610
+8.458+0.087
+−0.110
+Mrk 509
+0.03440
+8.115+0.035
+−0.038
+PG 2130+099
+0.06298
+8.660+0.049
+−0.056
+NGC 7469
+0.01632
+7.086+0.047
+−0.053
+TABLE IV: This table contains 28 low redshift quasars at z < 1 with their central SMBH mass from [61].
+
+14
+Object
+Redshift
+MBH(×109M⊙)
+Refence
+J0313-1806
+7.64
+0.16+0.4
+−0.4
+[69]
+ULAS J1342+0928
+7.541
+0.91+0.13
+−0.14
+[59]
+J100758.264+211529.207
+7.52
+1.5+0.2
+−0.2
+[8]
+ULAS J1120+0641
+7.085
+2.0+1.5
+−0.7
+[70]
+J124353.93+010038.5
+7.07
+0.33+0.2
+−0.2
+[71]
+J0038-1527
+7.021
+1.33+0.25
+−0.25
+[72]
+DES J025216.64–050331.8
+7
+1.39+0.16
+−0.16
+[73]
+ULAS J2348-3054
+6.886
+2.1+0.5
+−0.5
+[74]
+VDES J0020-3653
+6.834
+1.67+0.32
+−0.32
+[75]
+PSO J172.3556+18.7734
+6.823
+3.7+1.3
+−1.0
+[74]
+ULAS J0109-3047
+6.745
+1.0+0.1
+−0.1
+[74]
+HSC J1205-0000
+6.73
+1.15+0.39
+−0.39
+[75]
+VDES J0244-5008
+6.724
+3.7+1.3
+−1.0
+[74]
+PSO J338.2298
+6.658
+3.7+1.3
+−1.0
+[74]
+ULAS J0305-3150
+6.604
+1.0+0.1
+−0.1
+[74]
+PSO J323.1382
+6.592
+1.39+0.32
+−0.51
+[77]
+PSO J231.6575
+6.587
+3.05+0.44
+−2.24
+[77]
+PSO J036.5078
+6.527
+3+0.92
+−0.77
+[77]
+V DESJ0224 − 4711
+6.526
+2.12+0.42
+−0.42
+[75]
+PSOJ167.6415
+6.508
+0.3+0.008
+−0.012
+[74]
+PSOJ261 + 19
+6.483
+0.67+0.21
+−0.21
+[78]
+PSOJ247.2970
+6.476
+5.2+0.22
+−0.25
+[77]
+PSOJ011 + 09
+6.458
+1.2+0.51
+−0.51
+[78]
+CFHQSJ0210 − 0456
+6.438
+0.08+0.055
+−0.04
+[41]
+CFHQSJ2329 − 0301
+6.417
+2.5+0.4
+−0.4
+[41]
+HSCJ0859 + 0022
+6.388
+0.038+0.001
+−0.0018
+[76]
+HSCJ2239 + 0207
+6.245
+1.1+3
+−2
+[77]
+V DESJ0330–4025
+6.239
+5.87+0.89
+−0.89
+[78]
+V DESJ0323–4701
+6.238
+0.55+0.126
+−0.126
+[78]
+PSOJ359–06
+6.164
+1.66+0.21
+−0.21
+[78]
+CFHQSJ0221 − 0802
+6.161
+0.7+0.75
+−0.47
+[41]
+HSCJ1208 − 0200
+6.144
+0.71+0.24
+−0.52
+[79]
+ULASJ1319 + 0950
+6.13
+2.7+0.6
+−0.6
+[80]
+CFHQSJ1509 − 1749
+6.121
+3.0+0.3
+−0.3
+[41]
+PSOJ239–07
+6.114
+3.63+0.2
+−0.2
+[78]
+HSCJ2216 − 0016
+6.109
+0.7+0.14
+−0.23
+[79]
+CFHQSJ2100 − 1715
+6.087
+3.37+0.64
+−0.64
+[41]
+PSOJ158–14
+6.057
+2.15+0.25
+−0.25
+[78]
+CFHQSJ1641 + 3755
+6.047
+0.24+0.1
+−0.8
+[41]
+CFHQSJ0055 + 0146
+5.983
+0.24+0.9
+−0.7
+[41]
+PSOJ056–16
+5.975
+0.75+0.007
+−0.007
+[78]
+TABLE V: This table contains 41 high-redshift quasars at z > 5 with their central SMBH mass from different
+references which are identified in the fourth column.
+
+15
+Object
+Redshift
+MBH(×109M⊙)
+J002429.77+391319.0
+6.620 ± 0.004
+0.27 ± 0.02
+J003836.10-152723.6
+6.999 ± 0.001
+1.36 ± 0.05
+J004533.57+090156.9
+6.441 ± 0.004
+0.63 ± 0.02
+J021847.04+000715.2
+6.766 ± 0.004
+0.61 ± 0.07
+J024655.90-521949.9
+6.86 ± 0.02
+1.05 ± 0.37
+J025216.64-050331.8
+6.99 ± 0.02
+1.28 ± 0.09
+J031343.84-180636.4
+7.611 ± 0.004
+1.61 ± 0.40
+J031941.66-100846.0
+6.816 ± 0.004
+0.40 ± 0.03
+J041128.63-090749.8
+6.827 ± 0.006
+0.95 ± 0.09
+J043947.08+163415.7
+6.519 ± 0.003
+0.63 ± 0.02
+J052559.68-240623.0
+6.543 ± 002
+0.002 0.29±
+J070626.39+292105.5
+6.5925 ± 0.0004
+2.11 ± 0.04
+J082931.97+411740.4
+6.384 ± 0.004
+1.40 ± 0.16
+J083737.84+492900.4
+6.773 ± 0.007
+0.71 ± 0.18
+J083946.88+390011.5
+6.702 ± 0.001
+0.81 ± 0.02
+J091054.53-041406.8
+6.9046 �� 0.0003
+0.671 ± 0.003
+J092120.56+000722.9
+6.610 ± 0.003
+0.41 ± 0.03
+J092347.12+040254.4
+6.719 ± 0.005
+0.26 ± 0.01
+J092359.00+075349.1
+6.5654 ± 0.0002
+1.77 ± 0.02
+J100758.26+211529.2
+6.682 ± 0.002
+0.49 ± 0.15
+J105807.72+293041.7
+7.48 ± 0.01
+1.43 ± 0.22
+J110421.59+213428.8
+6.585 ± 0.005
+0.54 ± 0.03
+J112001.48+064124.3
+6.766 ± 0.005
+1.69 ± 0.15
+J112925.34+184624.2
+7.070 ± 0.003
+1.35 ± 0.04
+J113508.93+501133.0
+6.824 ± 0.001
+0.29 ± 0.02
+J121627.58+451910.7
+6.579 ± 0.001
+1.49 ± 0.05
+J131608.14+102832.8
+6.648 ± 0.003
+0.61 ± 0.20
+J134208.10+092838.6
+7.51 ± 0.01
+0.81 ± 0.18
+J153532.87+194320.1
+6.370 ± 0.001
+3.53 ± 0.33
+J172408.74+190143.0
+6.480 ± 0.001
+0.67 ± 0.08
+J200241.59-301321.7
+6.673 ± 0.001
+1.62 ± 0.27
+J210219.22-145854.0
+6.652 ± 0.003
+0.74 ± 0.11
+J221100.60-632055.8
+6.83 ± 0.01
+0.55 ± 0.24
+J223255.15+293032.0
+6.655 ± 0.003
+3.06 ± 0.36
+J233807.03+214358.2
+6.565 ± 0.009
+0.56 ± 0.03
+TABLE VI: This table contains 35 high-redshift quasars at z > 6 with their central SMBH mass from [63].
+

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+Semi-Supervised Learning with Pseudo-Negative Labels for Image
+Classification
+Hao Xua, Hui Xiaoa, Huazheng Haoa, Li Donga, Xiaojie Qiub and Chengbin Penga,∗
+aNingbo University, Ningbo, China
+bZhejiang Keyongtai Automation Technology Co., Ltd., Ningbo, China
+A R T I C L E I N F O
+Keywords:
+Semi-Supervised Learning
+Image Classification
+Mutual Learning
+A B S T R A C T
+Semi-supervised learning frameworks usually adopt mutual learning approaches with multiple
+submodels to learn from different perspectives. To avoid transferring erroneous pseudo labels between
+these submodels, a high threshold is usually used to filter out a large number of low-confidence
+predictions for unlabeled data. However, such filtering can not fully exploit unlabeled data with
+low prediction confidence. To overcome this problem, in this work, we propose a mutual learning
+framework based on pseudo-negative labels. Negative labels are those that a corresponding data item
+does not belong. In each iteration, one submodel generates pseudo-negative labels for each data item,
+and the other submodel learns from these labels. The role of the two submodels exchanges after
+each iteration until convergence. By reducing the prediction probability on pseudo-negative labels,
+the dual model can improve its prediction ability. We also propose a mechanism to select a few
+pseudo-negative labels to feed into submodels. In the experiments, our framework achieves state-of-
+the-art results on several main benchmarks. Specifically, with our framework, the error rates of the
+13-layer CNN model are 9.35% and 7.94% for CIFAR-10 with 1000 and 4000 labels, respectively.
+In addition, for the non-augmented MNIST with only 20 labels, the error rate is 0.81% by our
+framework, which is much smaller than that of other approaches. Our approach also demonstrates
+a significant performance improvement in domain adaptation.
+1. Introduction
+Deep learning is widely used in many areas, and the
+performance of deep learning models [10] heavily relies on
+the amount of training data. However, in many real-world
+scenarios [16, 24, 5, 37], labeled data are often limited, and
+the annotation for unlabeled data can usually be expensive.
+In such cases, a semi-supervised learning framework can be
+adopted.
+Semi-supervised learning frameworks include generative-
+based models [18], graph-based models [25], consistency-
+based regularization [20, 33, 27, 2, 35, 15, 4], self-training
+with pseudo-labels [8, 32], and so on.
+Among them, self-training methods can expand the
+training set by producing pseudo labels for unlabeled data
+to improve the model performance. Nevertheless, single
+models are not robust to noisy data. Inspired by DML [40],
+a natural idea is to simultaneously train two independently
+initialized models, and predictions of one submodel can be
+used as the learning target for the other submodel.
+To avoid transferring erroneous predictions to each other
+and alleviate parameter coupling between submodels in the
+early stages of training, a dual student framework [15] is
+proposed.
+It prevents the mutual transfer of erroneous knowledge
+by only passing high-confidence predictions to the other
+learning model. However, such a mechanism can waste a
+large amount of unlabeled data during training.
+∗Corresponding author
+[email protected] (C. Peng)
+ORCID(s):
+Figure 1: The dual model on the left side represents general
+mutual learning, i.e., the models pass strong information
+to each other such as information about the category with
+the highest prediction probability. The dual model near the
+right side exchanges weak information between each other,
+indicating which category the data does not belong to.
+To address these problems, we propose a new semi-
+supervised classification framework based on dual pseudo-
+negative label learning. This framework comprises two
+submodels, and each submodel generates pseudo-negative
+labels as learning targets for the other submodel. Each sub-
+model also provides pseudo-negative labels on augmented
+data for self-training. The difference between our frame-
+work and general mutual learning is shown in Figure 1. We
+also propose a selection mechanism to identify the most
+representative pseudo-negative labels for the other model.
+The main contributions can be summarized as follows:
+• We propose a Dual Negative Label Learning (DNLL)
+framework, which not only improves the utiliza-
+tion of unlabeled data but also significantly reduces
+model parameter coupling compared to general mu-
+tual learning methods.
+Xu.eal: Preprint submitted to Elsevier
+Page 1 of 10
+arXiv:2301.03976v1  [cs.CV]  10 Jan 2023
+
+aM1
+M1'
+Dog
+Not Cat
+Not Bird
+M2
+Unlabeled Data
+M2'• We propose a selection mechanism to help select
+representative pseudo-negative labels and prove the
+effectiveness of this approach theoretically.
+• We demonstrate the effectiveness of the proposed
+method experimentally on different benchmarks.
+2. Related Work
+2.1. Data Augmentation
+Data augmentation plays a key role in model training,
+which is widely used in classification or segmentation. Data
+augmentation is used to expand the training set by applying
+random perturbations to improve algorithm performance
+and robustness. Simple augmentation methods include ran-
+dom flips, horizontal or vertical transitions, geometric trans-
+formations, changing the contrast of images, and so on.
+There are also complex operations. Mixup randomly selects
+two images and mixes them by a random proportion to
+expand the data set. The Cutout method replaces randomly
+selected image pixel values with zeros while leaving the
+labels unchanged [7]. In order to maximize the effect of data
+augmentation, strategies combining a range of augmentation
+techniques are proposed, such as AutoAugmentation [38],
+RandAugmentation [6], etc. We also employ data augmen-
+tation methods similar to other semi-supervised learning
+frameworks [2, 1].
+2.2. Semi-Supervised Learning
+Semi-supervised learning has received a lot of attention
+in recent years. The main task of semi-supervised learn-
+ing is to utilize labeled and unlabeled data to train algo-
+rithms. Many approaches based on consistency regularity,
+Pi-Model, Temporal Ensembling Model [20], Mean Teacher
+[33], Dual Student [15], and so on. Later, a series of holistic
+analysis methods, such as MixMatch [2], ReMixMatch
+[1], FixMatch [32], have been proposed. Alternatively, in
+DMT, inconsistency between two models has also been
+used to exploit the correctness of pseudo-labels [9]. In this
+work, we propose an efficient semi-supervised classification
+framework with dual negative label learning.
+2.3. Learning with Noisy Labels
+In this case, models are trained with correctly labeled
+data and mistakenly labeled data. For example, based on
+the recent memory effect of a neural network, co-teaching
+[11] trains two models simultaneously, and each model can
+help the other one to filter out samples with large losses.
+Kim et al. [17] proposes a negative learning method for
+training convolutional neural networks with noisy data. This
+method provides feedback for input images about classes
+to that they do not belong. In this work, we propose to
+use low-confidence pseudo-labels as noisy labels for further
+learning.
+2.4. Learning from Complementary Labels
+A category corresponding to the complementary label is
+that a data item does not belong. Due to difficulties in col-
+lecting labeled data, complementary-label learning is used
+in fully supervised learning methods [14] and noisy-label
+learning methods [17]. Complementary labels can be gen-
+erated based on noisy labels [14, 17]. In our method, com-
+plementary labels are generated based on model-generated
+pseudo labels.
+3. Methodology
+3.1. Problem Definition
+In traditional multi-model frameworks, learning models
+under-fitted in the early stage of training are likely to pass
+erroneous pseudo-labels to other models. Such errors can
+be accumulated and need to be filtered out. In addition,
+consistency loss on the same erroneous pseudo-labels can
+also lead the multi-model framework to degenerate into a
+self-training model.
+Therefore, in this section, we propose a multi-model
+semi-supervised learning framework to improve the utiliza-
+tion of unlabeled data and alleviate degeneration. We first
+describe the novel mutual learning framework called Dual
+Negative Label Learning. That detailed framework is shown
+in Figure 2, and then proposes an effective selection mech-
+anism for choosing representative pseudo-negative labels.
+In semi-supervised learning (SSL), the goal is to train a
+model by utilizing a small amount of labeled data and a large
+amount of unlabeled data. Formally, we define a training set
+퐷 consisting of labeled data 퐷푙={(푋푖, 푌푖
+) ; 푖 ∈ (1, ..., 푁)
+}
+and unlabeled data 퐷푢={(푋푗
+) ; 푗 ∈ (1, ..., 푀)
+}, and we use
+a dual model to allow each submodel learning from the
+other. The label 푌푖 of the 푖-th data item is a one-hot vector.
+3.2. Supervised Learning
+In supervised learning, labeled data are augmented by
+different weak augmentations for different submodels.
+푋(1)
+푖
+=퐴(1)
+푤 (푋푖),
+(1)
+푋(2)
+푖
+=퐴(2)
+푤 (푋푖).
+(2)
+where 퐴(1)
+푤 , 퐴(2)
+푤 denote different weak augmentation opera-
+tions and 푋(1)
+푖 , 푋(2)
+푖
+denote weakly augmented data sets.
+We use the cross-entropy (CE) function for the super-
+vised loss. In classification tasks, the image-level CE loss is
+as follows:
+퐻(푌 , ̂푌 ) = −
+∑
+푖
+푌푖푙표푔( ̂푌푖)
+(3)
+where ̂푌 is the predicted label, and 푌 is the ground truth.
+The supervised losses of the two submodels are as
+follows:
+퓁(1)
+푠푢푝 = 퐻(푓휃(푋(1)
+푖 ), 푌푖),
+(4)
+퓁(2)
+푠푢푝 = 퐻(푓휑(푋(2)
+푖 ), 푌푖).
+(5)
+where 푓휃 and 푓휑 represent the operations of two submodels
+respectively, and 휃 and 휑 represent parameters correspond-
+ing submodels.
+Xu.eal: Preprint submitted to Elsevier
+Page 2 of 10
+
+Figure 2: Overview of the DNLL framework. We use a small amount of labeled data and a large amount of unlabeled data
+to train a dual model. Each submodel within the dual model has the same structure and is initialized independently. For each
+labeled data, weak augmentations such as random cropping and random flipping are applied. A cross-entropy function is
+used to calculate the supervised loss. For each unlabeled data, besides weak augmentations, strong augmentations such as
+color jittering are applied. Each submodel generates pseudo-negative labels based on predictions of weakly augmented data,
+and these labels are used to teach the other submodels when predicting strongly augmented data.
+3.3. Unsupervised Learning
+3.3.1. Dual pseudo-negative label Learning
+Most unsupervised learning parts in semi-supervised
+learning frameworks are realized by allowing each sub-
+model to learn with pseudo-positive labels from other sub-
+models. To avoid model degeneration and error accumula-
+tion in this process, we propose a novel dual negative label
+learning approach. In this approach, each submodel teaches
+the other that a given data item should not belong to a
+certain category. It allows model diversity and can reduce
+transferring of erroneous information.
+Pseudo-negative labels, namely, the labels that a corre-
+sponding data item does not belong to, are generated by
+taking complementary labels of the predicted label by a
+submodel. In our approach, we also select a few pseudo-
+negative labels as representative pseudo-negative labels.
+For data item 푗, its pseudo label ̂푌푗 and its representative
+pseudo-negative label 푌 푐
+푗 are randomly selected from all the
+candidates with equal probability (EP) as follows:
+̂푌푗 = 푓(푋푗),
+(6)
+푌 푐
+푗 ∈ 푧(푓(푋푗), 푚),
+(7)
+where 푚 is one by default, and 푧 is defined as follows:
+푧(푓(푋푗), 푚) ={푣|푣 ∈ {0, 1}퐾,
+∑
+푖
+푣푖 = 푚,
+and 푣[arg max ̂푌푗] ≠ 1}.
+(8)
+Here, 퐾 is the number of categories, and {0, 1}퐾 represents
+a vector of length 퐾 with elements equal to zero or one. To
+increase the convergence rate, we can allow each submodel
+to generate multiple representative pseudo-negative labels
+for each weakly augmented data item for the other submodel
+to learn. Thus, 푚 can also be positive integers larger than one
+and less than 퐾.
+By teaching each other with pseudo-negative labels
+only, we reduce the coupling between submodels. The loss
+function can be written as follows:
+퐿( ̂푌 , 푌 푐) = −
+∑
+푖
+푌 푐
+푖 log(1 − ̂푌푖)
+(9)
+where ̂푌 denotes the predictions from one submodel and 푌 푐
+is the representative pseudo-negative labels from the other
+submodel.
+We also use weak and strong data augmentations for un-
+labeled data to improve the generalization ability of the dual
+model. The weak augmentations can be random cropping,
+random flipping, or simply outputting the original images.
+The strong augmentation operations can be color dithering
+or noise perturbations. Usually, predictions for weakly aug-
+mented data by a submodel will be more accurate than that
+for strongly augmented data. Thus, in our framework, the
+predictions of weakly augmented data by one submodel are
+used for generating pseudo-negative labels. We use these
+labels as learning targets for the other submodel feed by
+strongly augmented images. The augmentation process can
+be written as follows:
+푋(푤)
+푗
+=퐴푤(푋푗),
+(10)
+푋(푠)
+푗
+=퐴푠(푋푗),
+(11)
+where 퐴푤 and 퐴푠 denote the weak and strong augmentation
+operations, respectively. 푋(푤)
+푗
+and 푋(푠)
+푗
+denote the weakly
+Xu.eal: Preprint submitted to Elsevier
+Page 3 of 10
+
+Weak Augmentations
+Labeled Prediction
+Supervised Loss
+Labeled Data
+Unlabeled Prediction
+Pseudo Label
+Pseudo-Negative Labe
+Net A
+.....
+Unsupervised Loss
+GT
+Unlabeled Data
+NetB
+Pseudo Label
+Pseudo-Negative Labe
+Supervised Loss
+Strong Augmentations
+Labeled Predictionand strongly augmented data items. Consequently, we have
+푌 푐1 ∈ 푧(푓휃(푋(푤)
+푗
+), 푚),
+(12)
+푌 푐2 ∈ 푧(푓휑(푋(푤)
+푗
+), 푚).
+(13)
+Therefore, the loss of learning between submodels is as
+follows:
+퓁(1)
+푐푟표푠푠 = 퐿(푓휃(푋(푠)
+푗 ), 푌 푐2),
+(14)
+퓁(2)
+푐푟표푠푠 = 퐿(푓휑(푋(푠)
+푗 ), 푌 푐1).
+(15)
+To further utilize the augmented data, we also developed
+a self-learning approach. In this approach, the generated
+pseudo-negative labels with weakly augmented data are also
+used by the same submodel to feed strong augmented data.
+The loss function can be written as follows:
+퓁(1)
+푠푒푙푓 = 퐿(푓휃(푋(푠)
+푗 ), 푌 푐1),
+(16)
+퓁(2)
+푠푒푙푓 = 퐿(푓휑(푋(푠)
+푗 ), 푌 푐2).
+(17)
+The unsupervised loss of the dual model is a combina-
+tion of the previous loss functions:
+퓁(1)
+푢푛푠푢푝 = 퓁(1)
+푐표푟푠푠 + 퓁(1)
+푠푒푙푓,
+(18)
+퓁(2)
+푢푛푠푢푝 = 퓁(2)
+푐표푟푠푠 + 퓁(2)
+푠푒푙푓.
+(19)
+The final total loss of the dual model in the DNLL is a
+combination of the supervised loss and the unsupervised
+one, as follows:
+퓁(1) = 퓁(1)
+푠푢푝 + 휆퓁(1)
+푢푛푠푢푝,
+(20)
+퓁(2) = 퓁(2)
+푠푢푝 + 휆퓁(2)
+푢푛푠푢푝,
+(21)
+where 휆 is a hyperparameter to balance the supervised
+loss item and the unsupervised loss item. The complete
+algorithm is shown in Algorithm 1.
+From this pseudo code, we can see that the running
+time is proportional to the size of the input data. If the
+size of unlabeled data, 푀, is much larger than that of the
+labeled data, 푁, which usually happens in semi-supervised
+learning, the running time is approximately proportional to
+the size of the unlabeled data. Thus, the time complexity is
+푂(푀).
+3.3.2. Error Perception Mechanism for Selecting
+Pseudo-Negative Labels
+In the above section, for an unlabeled data item, a rep-
+resentative pseudo-negative label is randomly selected from
+all the candidates with equal probability. To incorporate the
+performance of each submodel in different categories, we
+propose an Error Perception Mechanism (EPM).
+In this approach, for a given data item, if a submodel is
+prone to misclassify it into the other category, the pseudo-
+negative label generated by the other submodel should
+include that misclassified category. Therefore, we compute
+the probability of misclassification for each category of each
+Figure 3: The generating process of pseudo-negative labels.
+For an unlabeled data item, a submodel makes a prediction
+to generate a pseudo label (3 in this example) and then
+randomly selects two pseudo-negative labels according to
+푅 of the other submodel.
+submodel using labeled data. Formally, for a submodel, we
+define a vector 푃푟푘 for category 푘 with its 푖-th element
+defined as follows:
+푃푟푘[푖] =
+{∑푁푘
+푗=1 푝푖푗,
+푖 ≠ 푘
+0
+푖 = 푘
+(22)
+where 푁푘 denotes the total number of data with category
+푘 being misclassified into category 푖, and 푝푖푗 represents the
+confidence that the 푗-th misclassified sample belongs to the
+푖-th category. We may also use EMA to update 푃푟푘 for
+stability.
+It is then normalized with a softmax function.
+푅푘 = 푆표푓푡푚푎푥(푃푟푘).
+(23)
+We use superscripts to represent submodels, so 푅(1)
+푘 and 푅(2)
+푘
+are misclassification probabilities for the first and the second
+submodels. An example of the 푅푘-based pseudo-negative
+label generation process is shown in Figure 3.
+Therefore, when computing 퓁2
+푐푟표푠푠, we sample 푌 푐1 from
+푧(푓휃(푋(푤)
+푗
+), 푚) such that the probability that 푌 푐2
+푗 [푘] = 1
+is proportional to 푅(2)
+푘 . A similar approach applies when
+computing 퓁1
+푐푟표푠푠.
+3.4. Theoretical Analysis
+First, we demonstrate that in the mutual learning frame-
+work based on a dual model, passing pseudo-negative labels
+between submodels is less likely to have error accumulation
+than that of passing pseudo labels, especially at the early
+stages of training.
+Theorem 3.1. The error rate (ER) for transferring pseudo-
+negative labels from one submodel to the other is expected
+to be
+푚
+퐾−1 of the ER when transferring pseudo labels, where
+푚 is the number of selected pseudo-negative labels and 퐾 is
+the number of categories for each data item.
+Proof. Without loss of generality, we define that the pre-
+diction accuracy of one submodel is 푞 for unlabeled data.
+Xu.eal: Preprint submitted to Elsevier
+Page 4 of 10
+
+0.26
+0.12
+0.10
+0.17
+0.07
+0.05
+0.03
+0.09
+0.11
+2
+3
+4
+5
+6
+7
+8
+0
+9
+R for category 3 of Submodel B
+2
+Pseudo label by SubmodeI A
+Pseudo-negative labels for Submodel BAlgorithm 1 Pseudo code for the training process of DNLL.
+Input:The labeled dataset 퐷푙={(푋푖, 푌푖
+) ; 푖 ∈ (1, ..., 푁)
+}
+and the unlabeled dataset 퐷푢={(푋푗
+) ; 푗 ∈ (1, ..., 푀)
+}.
+The two submodels are 푓휃 and 푓휑.
+1: for each epoch do
+2:
+for each batch do
+3:
+(휒푙, 푌푙) ∶ select a batch of data from 퐷푙
+4:
+(휒푢) ∶ select a batch of data from 퐷푢
+5:
+휒(1)
+푙
+= 퐴(1)
+푤 (휒푙)
+6:
+휒(2)
+푙
+= 퐴(2)
+푤 (휒푙)
+7:
+휒(푤)
+푢
+= 퐴푤(휒푢)
+8:
+휒(푠)
+푢
+= 퐴푠(휒푢)
+9:
+퓁(1)
+푠푢푝 = 퐻(푓휃(휒(1)
+푙 ), 푌푙)
+10:
+퓁(2)
+푠푢푝 = 퐻(푓휑(휒(2)
+푙 ), 푌푙)
+11:
+푌 푐1 ∈ 푧(푓휃(휒(푤)
+푢
+), 푚)
+12:
+푌 푐2 ∈ 푧(푓휑(휒(푤)
+푢
+), 푚)
+13:
+퓁(1)
+푢푛푠푢푝 = 퐿(푓휃(휒(푠)
+푢 ), 푌 푐2)
+14:
+퓁(2)
+푢푛푠푢푝 = 퐿(푓휑(휒(푠)
+푢 ), 푌 푐1)
+15:
+푓휃 = arg min푓휃(퓁(1)
+푠푢푝 + 휆퓁(1)
+푢푛푠푢푝)
+16:
+푓휑 = arg min푓휑(퓁(2)
+푠푢푝 + 휆퓁(2)
+푢푛푠푢푝)
+17:
+end for
+18: end for
+return 푓휃, 푓휑
+Therefore, when transferring pseudo labels, the probability
+that that submodel provides correct learning targets to the
+other is 푞.
+When transferring 푚 pseudo-negative labels, if the sub-
+model predicts correctly, it transfers correct negative labels.
+If the submodel predicts mistakenly, the chance of providing
+correct negative labels is
+퐶푚
+퐾−2
+퐶푚
+퐾−1
+,
+(24)
+where 퐶푚
+퐾−1 denotes the total number of combinations of
+selecting 푚 pseudo-negative labels from all the 퐾 − 1
+pseudo-negative labels, and 퐶푚
+퐾−2 denotes the number of
+combinations of selecting 푚 pseudo-negative labels from
+퐾 − 2 truly negative labels. 퐾 − 2 is obtained by taking
+all the 퐾 categories except two categories corresponding to
+one pseudo label and one ground-truth label. Therefore, the
+probability of providing the correct learning target is
+푞 + (1 − 푞)
+퐶푚
+퐾−2
+퐶푚
+퐾−1
+= 1 − (1 − 푞)푚
+퐾 − 1 .
+(25)
+Therefore, the error rate of transferring pseudo-negative
+labels is
+1 − (1 − (1 − 푞)푚
+퐾 − 1 ) = (1 − 푞)
+푚
+퐾 − 1.
+(26)
+As the error rate of transferring pseudo labels is 1 − 푞, the
+error rate of transferring pseudo-negative labels is
+푚
+퐾−1 of
+that of transferring pseudo labels. Therefore, transferring
+pseudo labels can provide a better learning target, and a
+smaller 푚 and a larger 퐾 can further reduce the error
+accumulation.
+For two submodels with the same structure, when they
+are converged to be the same, they can no longer be used for
+semi-supervised learning. We need to avoid such scenarios,
+especially in the early training stages. In the unsupervised
+learning part, we demonstrate that when transferring knowl-
+edge with pseudo-negative labels, it is unlikely to have two
+submodels degenerate into the same.
+Theorem 3.2. When transferring representative pseudo-
+negative labels randomly, the probability that two submod-
+els are optimized for different objectives is 1 −
+√
+2휋푚
+푒퐾 ( 푚
+퐾 )푚
+approximately, where 푚 is the number of representative
+pseudo-negative labels and 퐾 is the number of categories.
+Proof. Without loss of generality, we assume that two sub-
+models produce the same prediction with probability 푞 and
+when they produce the same pseudo labels, the probability
+that the two submodels can produce the same representative
+pseudo-negative labels is
+1
+퐶푚
+퐾−1
+.
+(27)
+Similarly, the probability that two submodels produce dif-
+ferent predictions is 1 − 푞, and when they produce different
+predictions, the probability that they produce the same
+pseudo labels is
+1
+퐶푚
+퐾−2
+.
+(28)
+Thus, the probability that the two submodels transfer-
+ring the same representative pseudo-negative label is
+푞
+퐶푚
+퐾−1
++ 1 − 푞
+퐶푚
+퐾−2
+(29)
+=푚!(퐾 − 2 − 푚)!(퐾 − 1 − 푞푚)
+(퐾 − 1)!
+(30)
+≈(퐾 − 1 − 푞푚)×
+√
+2휋푚( 푚
+푒 )푚√
+2휋(퐾 − 2 − 푚)( 퐾−2−푚
+푒
+)퐾−2−푚
+√
+2휋(퐾 − 1)( 퐾−1
+푒 )퐾−1
+(31)
+≈
+√
+2휋푚
+푒퐾
+( 푚
+퐾 )푚
+(32)
+where the approximation in Eq. (31) is obtained by the Stir-
+ling’s approximation, and that in Eq. (32) is by considering
+퐾 >> 푚.
+4. Experiments
+In this section, we first introduce benchmarks used
+in experiments and briefly describe the details of the ex-
+periments. Then we compare DNLL with other methods.
+Xu.eal: Preprint submitted to Elsevier
+Page 5 of 10
+
+Finally, we evaluate the efficiency of DNLL from different
+perspectives.
+4.1. Benchmark datasets
+In the classification task, we use the public bench-
+mark datasets CIFAR-10 [19], SVHN [28], and MNIST as
+many others. The CIFAR-10 dataset includes 50000 training
+images and 10000 test images, and the total number of
+categories is ten. We randomly select 500 images for each
+category as the validation set. The total number of categories
+of SVHN Dataset is ten, in which the training set contains
+73257 images and the test set contains 26032 images. We
+also randomly select 500 images for each category as the
+validation set. The MNIST dataset includes 60000 training
+images and 10000 test images, and the total number of
+categories also is ten. We randomly select 50 images for
+each category as the validation set.
+4.2. Implementation Details
+Our approach is implemented on Pytorch. For the train-
+ing stage, the following configurations are used. The learn-
+ing rate is 0.03, and the weight decay is 5 × 10−4. The
+momentum is 0.9. We use the cosine annealing technique
+with batch size 256. We report performances on the test
+set averaged from three runnings. For dual models, we use
+WideResNet-28-2 (WRN-28-2)[39] and 13-layer CNN as
+other approaches [2, 15].
+We use data augmentation techniques in our experi-
+ments. The data augmentation operation for each data set
+is performed exactly following its corresponding literature
+for fairness. Specifically, for the MNIST dataset, we do
+not change the input data [25]. For the CIFAR-10 dataset,
+when using the 13-layer CNN as the model [15], we make
+the original image as a weakly augmented version and the
+noise-processed image as a strongly augmented version.
+When using WideResNet-28-2 as the model [9], the weak
+augmentation operations we used include random cropping
+and random flipping, and the strong augmentation operation
+is random color jittering. For the SVHN dataset [20], we
+only use the horizontal translation as the strong augmenta-
+tion operation and the original image as the weakly aug-
+mented version.
+4.3. Comparison on Benchmarks
+In experiments with the CIFAR-10 dataset, we randomly
+select 1K, 2K, and 4K data items, respectively, as labeled
+data and the rest as unlabeled data.
+We compare our method with others: Π model, Tempo-
+ral Ensembling [20], VAT[27] and Mean Teacher [33] based
+on consistency regularization; Π+STNG [25], LP+SSDL
+and LP-SSDL-MT [13] based on graph methods; Filtering
+CCL, Temperature CCL [23], TSSDL, TSSDL-MT [31] and
+TNAR-VAE [36] based on mean-teacher frameworks; Cur-
+riculum Labeling (CL) [3] based self-training; MixMatch
+[2] based on strong hybrid method. We also compare our ap-
+proach with others based on dual models: Deep Co-Training
+(DCT) [29], Dual student(DS) [15], Mutual Learning of
+Complementary Networks(CCN) [34] and Dynamic Mutual
+Table 1
+Accuracy on the Test Set of CIFAR-10 with the 13-layer CNN
+as the backbone.
+Method
+1K
+2K
+4K
+Π model†
+68.35
+82.43
+87.64
+Temporal ensembling†
+76.69
+84.36
+87.84
+Mean Teacher
+81.78
+85.67
+88.59
+Π+SNTG†
+78.77
+85.35
+88.64
+LP-SSDL†
+77.98
+84.34
+87.31
+LP-SSDL-MT†
+83.07
+86.78
+89.39
+Filtering CCL†
+81.78
+85.67
+88.59
+Temperature CCL†
+83.01
+87.43
+89.37
+TSSDL†
+78.87
+85.35
+89.10
+TSSDL-MT†
+81.59
+86.46
+90.70
+TNAR-VAE†
+-
+-
+91.15
+DCT
+-
+-
+90.97
+Dual Student
+85.83
+89.28
+91.11
+CCN
+87.95
+89.63
+91.2
+DNLL (Ours)
+87.87
+90.65
+92.06
+Table 2
+Accuracy on the Test Set of CIFAR-10 with the WRN-28-2 as
+the backbone.
+Method
+1K
+4K
+VAT†
+81.36
+88.95
+Mean Teacher†
+82.68
+89.64
+CL
+90.61
+94.02
+MixMatch
+92.25
+93.76
+DMT
+91.51
+94.21
+DNLL (Ours)
+92.03
+94.29
+Training (DMT) [9]. The symbol † indicates that the results
+are reported in [4] and [12]. The symbol ’-’ indicates that
+the corresponding results have not been reported in this
+literature.
+From Table 1 and Table 2, we can find that our method
+performs relatively well with 1k labels and outperforms all
+the other methods in other cases. From Table 1, the accuracy
+of our approach ranges between 87.87% and 92.06%, which
+outperforms most of the other methods using the dual
+model, i.e., DCT, Dual Student, and CCN. From Table 2,
+the MixMatch is 0.53% lower than our approach at the
+accuracy with 4K labels. The DMT is 0.41% and 0.08%
+lower than our approach at the accuracy with 1K and 4K
+labels, respectively. Figure 4 demonstrates the performance
+of DNLL during the training process on the test set. As the
+epoch number increases, the training accuracy increases.
+In the SVHN dataset, 1K and 4K items are also ran-
+domly selected as labeled data. We compare our method
+with others as follows: Π model [20], Pseudo-Labeling [21],
+VAT [27] and Mean Teacher [33]. The symbol † indicates
+that the results are reported in [12]. All the approaches
+use WideResNet-28-2 as the backbone model. As shown in
+Table 3, our method outperforms all the other approaches.
+Xu.eal: Preprint submitted to Elsevier
+Page 6 of 10
+
+Table 3
+Accuracy on the Test Set of SVHN with the WRN-28-2 as the
+backbone.
+Method
+1K
+4K
+Pseudo-Labeling
+90.06
+-
+Π model
+92.46
+-
+VAT†
+94.02
+95.80
+Mean Teacher†
+96.25
+96.61
+DNLL (Ours)
+96.41
+96.84
+Table 4
+Accuracy on the Test Set of MNIST with the 13-layer CNN
+as the backbone.
+Method
+20
+50
+100
+ImprovedGAN†
+83.23
+97.79
+99.07
+Triple GAN†
+95.19
+98.44
+99.09
+Π model†
+93.68
+98.98
+99.11
+Π + SNTG†
+98.64
+99.06
+93.34
+DNLL (Ours)
+99.19
+99.32
+99.54
+Figure 4: Performance of DNLL on the test set during
+training with the CIFAR-10 dataset of 1000 and 4000 labeled
+data.
+For the MNIST dataset, 20, 50, and 100 data items are
+randomly selected as labeled data. We compare the DNLL
+with other semi-supervised methods, i.e., ImprovedGAN
+[30], Triple GAN [22], Π model [20] and Π + STNG [25].
+The symbol † indicates that the results are reported in [25].
+All the above methods use the 13-layer CNN as the model.
+As shown in Table 4, the DNLL outperforms the other
+approaches.
+4.4. Sensitivity Analysis
+We conduct a sensitivity analysis on the CIFAR-10
+dataset with 4K labeled data items to analyze the relation-
+ship between representative pseudo-negative label number
+푚 and the accuracy of the model under different selection
+mechanisms that were introduced in the methodology sec-
+tion: Equal Probability (EP) vs. Error Perception Mech-
+anism (EPM). As the number of representative pseudo-
+negative labels tends to be less than half of the total number
+Table 5
+Accuracy under different choices of 푚 and different selection
+mechanisms for representative pseudo-negative labels.
+Selection Method
+푚 = 1
+푚 = 2
+푚 = 3
+푚 = 4
+EP
+92.9
+93.76
+94.01
+93.78
+EPM
+93.12
+93.84
+94.29
+93.77
+Table 6
+Comparison of the performance of mutual learning (ML) and
+self-learning (SL) with DNLL.
+Method
+4k labels
+SL w/o EPM
+92.78
+SL
+93.03
+ML w/o EPM
+94.01
+ML
+94.29
+of categories, here we compare with 푚 ≤ 4. From Table. 5,
+we can find that generally, the error perception mechanism
+performs better than selecting with equal probability, and
+moderately increasing 푚 is helpful to increase the perfor-
+mance. When 푚 is too large, for example, close to half of
+the total number of categories, pseudo labels are likely to be
+selected, and the performance can be undermined.
+4.5. Comparison with variants of DNLL
+In this part, we demonstrate that using mutual learning
+framework in DNLL is more efficient compared to a self-
+learning framework. We compare the performance of these
+two learning frameworks. We can see from Table. 6 that the
+mutual learning framework under the dual model is better.
+This is mainly because erroneous information can be filtered
+out by each other with different capabilities, avoiding the
+accumulation of errors.
+4.6. Visualization of embeddings
+We conduct experiments on MNIST with 20 labels
+without augmentation [25]. We visualize the embeddings of
+DNLL and a fully supervised learning method, respectively,
+on testing data under the same settings. We use t-SNE [26]
+to project the representations of the last hidden layer into
+two dimensions. Figure 5 shows the results. Each point
+corresponds to an item in the testing set, and different
+ground-truth classes are encoded with different colors. It
+demonstrates that the representations obtained from DNLL
+can better identify each class in the embedding space.
+4.7. Generalizability of DNLL
+To verify the generalizability of DNLL, we combine the
+ideology of DNLL method with the Dual Student method
+and the Mean Teacher method. For Dual Student, we use
+DNLL on the unstable samples discarded by the Dual
+Student. As can be observed from the left side of Figure 6,
+our approach can take advantage of the discarded unlabeled
+data, which in turn improves the overall performance. In
+addition, we combine DNLL with Mean Teacher to use all
+Xu.eal: Preprint submitted to Elsevier
+Page 7 of 10
+
+Training process of CIFAl-10 with 1k/4k labels
+90
+80
+Accurary(%)
+70
+60
+ModelA with 1klabels
+Model B with 1k labels
+Model A with 4k labels
+Model B with 4k labels
+50
+0
+100
+200
+300
+400
+500
+600
+700
+EpochFigure 5: The t-SNE plot of the last hidden layer on the test
+data of MNIST with 20 labels: the baseline model (left) and
+our model (right). Our model can learn more discriminative
+representation.
+the unlabeled data together. From the right side of Figure 6,
+we can see that DNLL contributes significantly to the overall
+performance improvement. These experiments demonstrate
+that DNLL can be used in combination with other semi-
+supervised methods to jointly improve model performance.
+Figure 6: The left side of the above figure shows the iteration
+process of combining DNLL and Dual Student. The right side
+shows the training process of combining DNLL and Mean
+Teacher.
+4.8. Domain Adaptation using DNLL
+Figure 7: Test curves of domain adaptation from USPS to
+MNIST versus the number of epochs. The DNLL avoids
+overfitting and improves the result remarkably.
+Domain adaptation is the closely related to semi-supervised
+learning. It aims at knowledge transfer from the source
+Table 7
+The execution time (seconds) of DNLL and other competitive
+methods such as Mean Teacher (MT) and Dual Student
+(DS).
+MT
+DS
+DNLL
+Train iteration time
+0.072
+0.145
+0.143
+Inference iteration time
+0.0183
+0.0189
+0.0184
+domain to the target domain. Zhan et al. [15] propose Dual
+Student method to overcome the shortcomings of Mean
+Teacher and demonstrate the effectiveness of a dual model
+in domain adaptation tasks. In this section, we use DNLL for
+adapting digital pattern recognition from USPS to MNIST.
+We use USPS as the source domain and MNIST as the target
+domain and show that the DNLL has advantages over the
+Dual Student and Mean Teacher.
+USPS and MNIST are both grayscale hand-written digi-
+tal datasets, the difference is that the image size is 16x16 for
+USPS and 28x28 for MNIST. The training set of USPS con-
+tains 7291 images, and the training set of MNIST contains
+60,000 images. And the test set for the experiments uses
+the MNIST test set containing 10,000 images. We compare
+DNLL with Dual Student, Mean Teacher, fully supervised
+learning for the source domain and fully supervised learning
+for the target domain with 7k balanced labels. Following
+experiment settings in Dual Student [15], we use cubic
+spline interpolation to match the resolution between the
+two dataset images and employ a 3-layer CNN [15] as the
+backbone, with random noise for data augmentation.
+Figure 7 shows the test accuracy versus the number
+of epochs. We can see that as the number of epochs in-
+creases, overfitting occurs in both Mean Teacher and the
+fully supervised learning for the source domain. From this
+figure, we can see that DNLL not only avoids the overfitting
+phenomenon but also is superior to Dual Student, and its
+performance is very close to that of the target domain
+supervision.
+4.9. Execution time of DNLL
+In this section, we conduct experiments to investigate
+the execution time of DNLL. We report the average time
+for each iteration during training and testing. We evaluate
+the execution time with the CIFAR-10 dataset using 4000
+randomly selected training samples as labeled data. The
+batch size is set to 100. The number of both labeled and
+unlabeled data in a batch is 50. We compare DNLL with
+Mean Teacher and Dual Student in the same settings in
+terms of execution time. The experiment is performed on a
+GTX 3060 GPU with Pytorch-1.10.2 software toolkit. The
+system memory is 64 GB, and the CPU is Intel Core i5-
+11400F. The experimental results are shown in Table 7 and
+Figure 8.
+From Table 7 and Figure 8, we can see that Mean
+Teacher takes the shortest training time but produces the
+lowest testing accuracy on the testing set. As both DNLL
+Xu.eal: Preprint submitted to Elsevier
+Page 8 of 10
+
+100
+90
+80
+Accurary(%)
+70
+60
+50
+MNIST Supervised
+40
+DNLL
+DS
+MT
+30
+USPS Supervised
+20
+40
+60
+80
+100
+0
+Epoch80
+DS
+DS+DNLL
+75
+70
+Accurary(%)
+65
+60
+55
+0
+20
+40
+60
+80
+100
+EpochMT
+MT+DNLL
+75
+70
+Accurary(%)
+65
+60
+55
+0
+20
+40
+60
+80
+100
+Epochand Dual Student use a dual model structure, the train
+time for each iteration is approximately twice that of Mean
+Teacher, but both have higher accuracy. The training time
+of DNLL and Dual Student are similar, but the performance
+of DNLL is higher than that of Dual Student. The average
+testing time of each iteration is shown in Table 7. Due to
+the similarity in model architectures, the testing time of all
+methods is similar.
+Figure 8: The training time (seconds) for each iteration and
+the testing accuracies of DNLL, Mean Teacher and Dual
+Student.
+5. Discussions
+Our approach has several advantages over existing semi-
+supervised algorithms. Firstly, in semi-supervised learning,
+our approach outperforms state-of-the-art approaches on
+benchmarks. Secondly, the unsupervised learning part of
+our methods can easily be used as add-ons for other semi-
+supervised learning methods to improve their performance.
+Finally, our approach fits domain adaptation tasks as well.
+We discuss the differences between DNLL and other meth-
+ods that use a dual model.
+Mean Teacher (MT): MT [33] has been proposed to
+improve the temporal-ensembling model [20]. The frame-
+work of MT consists of a student model and a teacher
+model. The student model is trained by perturbing the input
+data. The output of the student model is trained to be
+consistent with the output of the teacher model. Different
+from DNLL, in MT, the teacher model is only updated
+by EMA. Thus, the predictions between the teacher model
+and the student model converge to be the same relatively
+fast during training. In addition, submodels in DNLL can
+generate pseudo-negative labels to help each other filter
+out erroneous information, while the student model and the
+teacher model in MT cannot.
+Dual Student (DS): DS [15] has been proposed to im-
+prove MT. DS trains two submodels online simultaneously
+with different initialization parameters in order to avoid
+coupling between the two models in the early training
+stages. To transfer reliable knowledge, submodels in DS fil-
+ter unlabeled data with low prediction confidences or inter-
+submodel consistency. This can lead to an underutilization
+of a significant amount of unlabeled data. On the other
+hand, in DNLL, most of the unlabeled data can be used
+in the training process, and the transferring of erroneous
+information is also reduced by using pseudo-negative labels.
+Mutual Learning of Complementary Networks: This
+method proposes a complementary correction network
+(CCN) [34] based on Deep Mutual Learning (DML) [40].
+This method simultaneously trains three submodels, in-
+cluding two submodels with the same structure and one
+CCN. The CCN takes the output from one submodel and
+the intermediate features extracted by another submodel as
+input and is trained with labeled data only. This network
+is then used to correct predictions by submodels. The
+prediction is then used as pseudo-labels for one of the
+submodels. The performance of the CCN can significantly
+determine the quality of the pseudo label, which in turn
+affects the training of the underlying submodel. On the other
+hand, DNLL is trained in a much simpler and more effective
+way.
+Dynamic Mutual Training (DMT): DMT [9] uses a
+weighted loss to control the selection of unlabeled data
+items so that data items with inconsistent predictions by
+submodels are filtered in the loss calculation. In addition,
+this method uses a course learning strategy in which unla-
+beled data are gradually used in the training process rather
+than used as a whole from the beginning. Compared with
+DNLL, this method also suffers from the underutilization
+of unlabeled data, and it is also time-consuming to train
+repetitively during course learning.
+6. Conclusion
+The paper analyzes submodel degeneration and under-
+utilization problems suffered from traditional mutual learn-
+ing approaches. To address these problems, we propose a
+novel mutual learning method for semi-supervised learning.
+Submodels in this approach provide each other with pseudo-
+negative labels instead of traditional pseudo labels. It can
+reduce error accumulation and promote unlabeled data uti-
+lization and is justified theoretically and experimentally. We
+also propose the error perception mechanism to help select
+efficient pseudo-negative labels. This framework can also be
+useful in different tasks.
+Acknowledgements
+This work was supported by the Natural Science Foun-
+dation of Zhejiang Province (NO. LGG20F020011), Ningbo
+Science and Technology Innovation Project (No. 2022Z075),
+and Open Fund by Ningbo Institute of Materials Technology
+& Engineering, the Chinese Academy of Sciences.
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+JOURNAL OF LATEX CLASS FILES, 2019
+1
+Edge Preserving Implicit Surface
+Representation of Point Clouds
+Xiaogang Wang, Yuhang Cheng, Liang Wang, Jiangbo Lu, Senior Member, IEEE , Kai Xu, Senior
+Member, IEEE , Guoqiang Xiao
+Abstract— Learning implicit surface directly from raw data recently has become a very attractive representation method for 3D
+reconstruction tasks due to its excellent performance. However, as the raw data quality deteriorates, the implicit functions often lead
+to unsatisfactory reconstruction results. To this end, we propose a novel edge-preserving implicit surface reconstruction method, which
+mainly consists of a differentiable Laplican regularizer and a dynamic edge sampling strategy. Among them, the differential Laplican
+regularizer can effectively alleviate the implicit surface unsmoothness caused by the point cloud quality deteriorates; Meanwhile, in order
+to reduce the excessive smoothing at the edge regions of implicit suface, we proposed a dynamic edge extract strategy for sampling near
+the sharp edge of point cloud, which can effectively avoid the Laplacian regularizer from smoothing all regions. Finally, we combine them
+with a simple regularization term for robust implicit surface reconstruction. Compared with the state-of-the-art methods, experimental
+results show that our method significantly improves the quality of 3D reconstruction results. Moreover, we demonstrate through several
+experiments that our method can be conveniently and effectively applied to some point cloud analysis tasks, including point cloud edge
+feature extraction, normal estimation,etc.
+Index Terms—Implicit surface representation, Differential Laplacian regularizer, Dynamic edge sampling, Point cloud, Geometric
+modeling, Shape analysis.
+!
+1
+INTRODUCTION
+Recently, Implicit Neural Representations (INRs) has gained made
+great strides in the field of 3D reconstruction [1]–[8]. In contrast
+to traditional explicit representations such as point clouds
+[9],
+voxels [10], [11] and mesh
+[12]–[15], implicit neural repre-
+sentations represent surface function primarily through neural
+networks, providing higher quality, flexibility, and fidelity without
+discretization errors, and significantly save amounts of storage
+space to store high-quality results.
+However, most of these methods need ground truth data as
+supervision [1]–[3], which have difficulty in generalizing well to
+unseen shapes that are dissimilar to the training samples. Recently,
+some methods [16]–[20] have been proposed to reconstruct im-
+plicit neural representations directly from raw data (point clouds,
+triangle soups, unoriented meshes, etc.). Compared to data-driven
+approaches, building implicit neural representations directly from
+raw data is obviously more appealing. Generally speaking, the
+core idea of such methods is to impose explicit/implicit regularity
+constraints to reduce reliance on dataset. SAL [18] proposed a
+unsigned regression loss to a given unsigned distance function
+to raw data, which can produce signed solutions of implicit
+functions. Specifically, starting from raw data (e.g., point clouds,
+real scanned grids, etc.), implicit neural representations learn in a
+self-supervised manner and can be trained reliably relying only
+on raw input data by minimizing unsigned regression. Subse-
+quently, SALD
+[17], a generalized version of SAL
+[18] was
+proposed, which can obtain higher quality reconstruction results
+•
+Xiaogang Wang, Yuhang Cheng, Liang Wang and Guoqiang Xiao are
+with College of Computer and Information Science, Southwest University,
+China.
+•
+Jiangbo Lu is with the SmartMore Co., Ltd.
+•
+Kai Xu is with the National University of Defense Technology, China.
+Fig. 1: Effect of edge preserving differential Laplacian regularizer.
+(b) are the optimization results of the edge preserving differential
+Laplacian regularizer which is incorporated into the state-of-the-
+art methods IGR [19] . (a) and (c) are the results of IGR and
+Ground truth respectively.
+arXiv:2301.04860v1  [cs.CV]  12 Jan 2023
+
+(a) IGR
+(b) Our
+(c) GTJOURNAL OF LATEX CLASS FILES, 2019
+2
+by incorporating an explicit gradient constraint on SAL. Gropp et
+al. [19] proposed a novel implicit geometric regularization (IGR)
+method to directly learn an implicit neural representation from
+raw data and achieved surprising results. Different from SAL [18]
+and SALD [17], IGR only relies on implicit regularization con-
+straints, without the need for a unsigned distance function. More
+specifically, IGR proposes an implicit geometric regularization,
+which amounts to solving a particular Eikonal boundary value
+problem that constrains the norm of spatial gradients to be 1
+almost everywhere. Yet, when the normal information cannot be
+available and the number of input points is not dense enough,
+the above algorithms often lead to unsatisfactory reconstruction
+results (See Figure 1(a)).
+We observed that the main reason for the unsatisfactory re-
+construction results is that the implicit function needs to fit the
+input point cloud as much as possible, and the noise information
+in the point cloud tends to cause the implicit surface to be very
+unsmooth. In other words, the main reason for this phenomenon is
+the inconsistency of normal in the local region of the reconstructed
+surface. Therefore, it is an intuitive idea to keep the local normal
+of the surface consistent as much as possible; Meanwhile, it
+should be noted that not all regions are restricted in their normal
+consistency, for example, obviously sharp edges often exist in the
+surface (as shown in Figure 1). In the reconstruction process,
+we hope that this part of the area will not be overly smoothed.
+Therefore, The edge preserving local normal consistency is more
+accurate for implicit surface representation .
+In view of the above problem, it can be visually viewed
+as a standard Laplacian minimization problem; Meanwhile, we
+can also use the Laplacian operator to identify the edge region
+effectively, which has achieved good results in many image
+processing tasks. Therefore, in other words, we can design an
+intuitive Laplacian regularization, which can effectively improve
+the quality of reconstruction results.
+However, in this task, the raw data type we consider is point
+cloud data, and the difference method cannot be directly used to
+approximate high-order derivatives, mainly because point cloud
+data does not have a clear topological relationship like mesh
+or image. If the algorithm similar to KNN is used, the nearest
+neighbor points searched cannot guarantee the correct topology
+structure (as shown in Figure 3), especially when the point cloud
+is not dense and the normal are not available , such wrong nearest
+neighbor results will easily lead to the anti-optimization results (as
+shown in Figure 4(a)).
+Recently, there is growing interest in differentiable optimiza-
+tion of implicit neural representations that enable differential
+nature as supervision in learning frameworks
+[3], [19], [21]–
+[25]. The advantage of differentiable implicit neural represen-
+tations is that it can directly solve the higher derivative of the
+input signal instead of discretization approximation, which greatly
+improves its optimization performance and application range.
+Thanks to the analytically-differentiable nature of implicit neural
+representation, we can easily design a differentiable Laplacian
+regularizer. Meanwhile, the differentiable Laplacian regularizer
+can be easily and intuitively incorporated into implicit neural
+surface representations (as shown in Figure 1). We show that it
+significantly improve the quality of 3D reconstruction. Meanwhile,
+in order to facilitate qualitative and quantitative comparisons in
+this paper, unless otherwise stated, in this paper, all experimental
+results are obtained by incorporating them into IGR
+[19]. We
+carefully evaluate its performance through a series of ablation
+studies. Meanwhile, we demonstrate through several experiments
+that our method can be conveniently and effectively applied to
+some point cloud analysis tasks, including point cloud edge feature
+extraction, normal estimation, etc.
+In summary, we make the following contributions: In this pa-
+per, we use the infinite differentiability property of implicit neural
+representation to propose a novel edge-preserving implicit surface
+reconstruction method, which mainly consists of a differentiable
+Laplican regularizer and a dynamic edge sampling strategy. 1),
+Among them, the differential Laplican regularizer can effectively
+alleviate the implicit surface unsmoothness caused by the point
+cloud quality deteriorates; 2), Meanwhile, in order to reduce
+the excessive smoothing at the edge regions of implicit suface,
+we proposed a dynamic edge extract strategy for sampling near
+the sharp edge of point cloud, which can effectively avoid the
+Laplacian regularizer from smoothing all regions.
+2
+RELATED WORK
+2.1
+Data-driven based Implicit surface reconstruction
+3D surface reconstruction from raw data has gained significant
+research progress in recent year, benefiting from the advances in
+machine learning techniques
+[1]–[8]. Early studies
+[26]–[28]
+most utilize predefined geometric priors (such as local linearity
+and smoothness) towards specific tasks. These geometric priors
+often encode statistical properties of raw data and are designed
+to be optimized, such as poisson equation
+[28], [29], radius
+basis function [26], moving least squares [27]. Recently, implicit
+neural representation has gained significant research progress for
+geometry reconstruction
+[1]–[3], [6], [7], [16], [30]–[34] and
+object representation [3], [23], [35]–[45] due to their simplicity
+and excellent performance, which learn an approximate implicit
+function with multi-layer perceptron (MLP). Compared to the
+traditional continuous and discrete representations (grid, point
+cloud and voxel), implicit neural representations have many poten-
+tial benefits, which can provide higher modeling quality without
+discretization errors, flexibility and fidelity, and save storage
+space. However, most of these methods need ground truth data
+as supervision [1]–[3], which have difficulty in generalizing well
+to unseen shapes that are dissimilar to the training samples.
+In addition, there are hybridization-based methods [46]–[50]
+that combine data-driven priors with optimization strategy that can
+achieve state-of-the-art performance. However, the above methods
+also require additional ground truth data as supervision, which
+seriously limits their applicability.
+2.2
+Sign Agnostic Implicit surface reconstruction
+Recently, some methods [17]–[20] have been proposed to re-
+construct implicit neural representations directly from raw data.
+Compared to big data-driven approaches, building implicit neural
+representations directly from raw data is obviously more appeal-
+ing. These methods can avoid the need for a large number of
+ground truth signed distance representation of training data as
+supervision. SAL
+[18] introduces a sign agnostic regression
+loss to a given unsigned distance function to raw data, which
+is the signed version of unsigned distance function. Meanwhile,
+that avoids the use of surface normals by properly initializing
+implicit decoder networks so that they can only produce signed
+solutions of implicit functions using unsigned distance function.
+Subsequently, SALD
+[17], a generalized version of SAL
+[18]
+
+JOURNAL OF LATEX CLASS FILES, 2019
+3
+was proposed, which can obtain higher quality reconstruction
+results by incorporating an explicit gradient constraint on SAL.
+Similarly, in this paper, our approach also uses implicit neural
+representation to estimate level set functions directly from raw
+data. The major difference is that our proposed regularization
+terms are directly based on differentiable implicit optimization,
+and does not explicitly enforce some regularization on the zero
+level set, such constraints, when the normal information cannot be
+available and the number of input point cloud is not dense enough,
+the implicit neural representation often lead to unsatisfactory
+reconstruction results.
+2.3
+Differentiable implicit neural representation
+Compared with general implicit neural representation, differen-
+tiable implicit neural representation has the advantage that it
+can directly use various properties of differential geometry in-
+stead of discretization approximation, which can lead to more
+stable solutions in many optimization problems. Recently, there is
+growing interest in differentiable optimization of implicit neural
+representation that enable differential nature as supervision in
+learning frameworks [3], [19], [21], [21]–[25]. General numerical
+optimization often uses the discrete approximation of differential
+geometry, for example, finite difference method is often used to
+enhance the smoothness between adjacent samples in space. But
+thanks to the analytically-differentiable nature of implicit neural
+representation, differentiable implicit neural representations can
+make direct use of many properties in differential geometry, such
+as gradients [19], [21], [23], curvatures [24], and the solution of
+partial differential equations [22], [25]. Recently, Gropp et al. [19]
+proposed to use the differentiable implicit neural representation
+to directly reconstruct surface from raw data. More specifically,
+it proposes an implicit regularization constraint, which amounts
+to solving a particular Eikonal boundary value problem that
+constrains the norm of spatial gradients to be 1 almost everywhere.
+Similarly, Sitzmann et al. [21] uses the proposed a differentiable
+periodic activation functions to represent signed distance fields
+in a fully-differentiable manner. Both of these works [19], [21]
+, however, when the normal information cannot be available and
+the number of input points is not dense enough, often lead to
+unsatisfactory reconstruction results. In this paper, our work is also
+based on the differentiability of implicit neural representations
+to optimize implicit level set function estimated directly from
+the input point cloud. Specifically, we designed an implicit dif-
+ferentiable Laplacian regularizer, which effectively alleviated the
+problem of unsatisfactory reconstruction results caused by direct
+fitting of input point cloud by implicit neural function.
+3
+METHOD
+We present a differentiable laplacian regularizer for neural implicit
+representation directly from input point cloud without normal
+supervision. Note that our differential Laplacian regularizer can
+be incorporated into any implicit neural representation, such as
+IGR [19],SAL [18],SALD [17]. In this paper, unless otherwise
+noted, we incorporate it in the IGR, which use level sets of neural
+network to represent 3D shape (Sec. 3.1). More specifically, IGR
+proposes an implicit geometric regularization, which amounts to
+solving a particular Eikonal boundary value problem that con-
+strains the norm of spatial gradients to be 1 almost everywhere.
+Yet, when the normal information cannot be available and the
+number of input points is not dense enough, IGR often lead
+Fig. 2: Illustrations of the local normal consistency.
+to unsatisfactory reconstruction results (See Figure 1(a)). We
+observed that the main reason for the unsatisfactory reconstruction
+results is that the implicit function needs to fit the input point cloud
+as much as possible, and the noise information in the point cloud
+tends to cause the implicit surface to be very unsmooth.
+To
+overcome
+this
+problem,
+we
+use
+the
+analytically-
+differentiable nature of implicit neural representation, to propose
+a differential Laplacian regularizer, which can effectively alleviate
+the unsatisfactory reconstruction results (Sec. 3.2). Meanwhile, in
+order to reduce the excessive smoothing at the edge regions of
+3D shape (such as man-made shapes), a dynamic edge extraction
+strategy (Sec. 3.2) is introduced for sampling near the sharp edge
+of input point cloud, which can effectively avoid the Laplacian
+regularizer from smoothing all regions, so as to effectively im-
+prove the quality of reconstruction results while maintaining the
+edge.
+3.1
+Background
+A neural implicit representations is a continuous function that
+approximate the signed distance function. The underlying surface
+of 3D shape is implicitly represented by the zero level set of this
+function,
+fθ(x) = 0, ∀x ∈ X.
+(1)
+where θ indicates the parameters to be learned and X indicates the
+set of input point cloud. In general, one parameterize this function
+using a multi-layer perceptron (MLP). Meanwhile, in order to
+conveniently use the analytically-differentiable (such as, gradi-
+ents,etc.) nature of implicit neural representation, recent works
+[19], [21] usually replace the commonly used ReLU activation
+function with a non-linear differentiable activation functions, thus
+transforming MLP into a continuous and infinitely differentiable
+function.
+In IGR, the training is done by minimizing the loss that
+encourages f to vanish on X:
+Lvanish =
+1
+N(X)
+�
+x∈X
+|fθ(x)|
+(2)
+where N(X) is the number of point set X, | • | indicates abso-
+lute value. if the input point cloud includes normal information
+ngt(x), the corresponding loss function can be designed to make
+the predicted normal (the differentiable gradient ▽fθ(x) of the
+implicit function) as close as possible to the ground truth normal
+ngt(x):
+Lnormal =
+1
+N(X)
+�
+x∈X
+||▽fθ(x) − ngt(x)||2
+(3)
+In addition to the above two intuitive fitting loss terms, IGR
+[19] based on the Eikonal partial differential equation presents
+an additional loss (Eikonal loss), which is equivalent to solve
+
+(b)
+a
+CJOURNAL OF LATEX CLASS FILES, 2019
+4
+Fig. 3: Illustrations of two different N nearest neighbors of non-
+topological preservation (b) and topological preservation (c) for
+geometric structure (a).
+boundary value problems of a particular Eikonal that constrains
+the norm of spatial gradients ▽fθ(x) to be 1 almost everywhere:
+Leikonal =
+1
+N(X)
+�
+x∈X
+(||▽fθ(x)||2 − 1)2
+(4)
+Note that, in our approach, we do not consider normal infor-
+mation as supervision, so we will not consider Lnormal term in
+all subsequent experiments. More specifically, our approach builds
+upon the above two items Lvanish and Leikonal.
+3.2
+Differentiable laplace regularization
+Neighborhood normal consistency. A high-quality result can be
+generated based on the above two terms (Lvanish and Leikonal)
+when the input point data is large enough, however, when the
+normal information cannot be available and the number of input
+points is not dense enough, often lead to unsatisfactory reconstruc-
+tion results (See Figure 1(a)).
+We observed that the main reason for the unsatisfactory re-
+construction results is that the implicit function needs to fit the
+input point cloud as much as possible, and the noise information
+in the point cloud tends to cause the implicit surface to be
+very unsmooth. More specifically, the optimization results are
+not guaranteed to provide a high-quality reconstruction result,
+which is intuitively reflected by the possibility that the normal
+of reconstruction result is inconsistent in the neighborhood.
+From another perspective, it is well known that 3D shapes
+tend to be piecewise smooth, that is, flat surfaces are more
+likely than high-frequency structures [51]. For this purpose, we
+incorporate this prior into implicit neural function by encouraging
+the geometric smoothness of the reconstructed results. Therefore,
+an intuitive solution is to constrain the consistency of the neighbor-
+hood normal of the reconstruction results (as shown in Figure 2):
+Lneibor =
+�
+x∈X
+�
+xi∈nei(x)
+||▽fθ(x) − ▽fθ(xi)||2
+(5)
+where nei(x) indicates the neighbor point set of point x.
+However, in this paper, the raw data type we consider is point
+cloud data, which does not have a clear topological structure
+like mesh or voxels. If the algorithm similar to KNN is used,
+the nearest neighbor points searched cannot guarantee that they
+maintain the correct topology structure, especially when the point
+cloud is not dense and the normal are not available, as shown in
+Figure 3(b) where the three points P4, P5 and P6 do not meet the
+nearest neighbor result of N = 5 under the maintenance of the
+topology structure, and the correct set of nearest neighbor points
+Fig. 4: The comparison of Lneibor (a) and Llaplacian (b).
+should be {P1, P2, P3, P8, P9}. Moreover, it is difficult to get a
+reasonable value for this parameter nei(x) in practice. As shown
+in Figure 4, we can easily see that the wrong reconstructed results,
+which is mainly caused by the above reasons.
+Differentiable Laplacian regularizer. In fact, the above con-
+straint Lneibor is mainly used to constrain the normal consistency
+in the local domain, which can be easily interpreted as a discrete
+Laplace operator. The Laplacian operator △f is a second-order
+differential operator in n-dimensional euclidean space, defined as
+the divergence (▽ · f) of the gradient (▽f). Thanks to the infinite
+differentiability of implicit neural representation, we can design a
+simple but effective differentiable Laplacian regularizer:
+Llaplacian =
+�
+x∈X
+△fθ(x)2
+(6)
+where △fθ(x) indicates the differentiable Laplace operator of
+point x.
+As shown in Figure 4(b), compared with the explicit regular-
+ization constraint Lneibor based on the nearest neighbor normal
+consistency, the differentiable Laplacian regularizer can obtain
+more stable results without introducing hyperparameter nearest
+neighbors N.
+3.3
+Dynamic edge sampling
+However, while the differentiable Laplacian regularizer restricts
+the normal consistency, it also brings a new problem: It imposes
+undifferentiated constraints on all 3D regions, even in the sharp-
+edge regions, as shown in Figure 6. As we know, complex 3D
+shapes are generally constructed by multiple piecewise smooth
+surfaces, which may not be differentiable at the joints, and are
+more likely to form sharp edges. Therefore, in essence, a complex
+3D shape (piecewise smooth model with sharp edges) cannot
+be accurately represented by an implicit function, because it is
+obviously not differentiable at sharp edges, so if it is forced to be
+represented by an implicit function, especially only sparse point
+sets without normal information are used as supervision, it is easy
+to form an overly smooth reconstruction at the sharp edges (as
+shown in Figure 6).
+The most intuitive solution is to implicitly represent each
+piecewise smooth surface separately, but this is difficult to do in
+practice because it first requires the segmentation of the input point
+set, which is difficult to do accurately in unsupervised conditions.
+Therefore, we propose a novel dynamic edge sampling
+strategy to effectively extract sharp edge regions in the training
+
+P
+p
+P
+3
+2
+p
+p
+4
+4
+P
+p
+D
+P
+5
+5
+P
+P
+9
+6
+P8
+p,
+P
+P,
+D
+8
+(a)
+(b)
+(c)(a) KNN=9
+(b) Our
+(c) GTJOURNAL OF LATEX CLASS FILES, 2019
+5
+Fig. 5: Statistics of Laplacian operators |△fθ(x)| and edge thresh-
+old τ selection.
+process. In theory, the remaining regions not only satisfy the
+differentiable property, but also conform to the normal consistency
+constraint, which can effectively avoid the indifference smoothing
+of all regions, including the edge regions, of the laplace regular-
+izer.
+Specifically, for each point p in the input point set, we may
+quickly determine whether it is an edge point according to its
+differentiable Laplacian operator △fθ(x). Essentially, Laplacian
+is mainly used to describe the rate of change of gradient, and
+is often used for edge detection in image processing. From the
+perspective of differential geometry, it is used to describe the
+change rate of spatial position normal. Therefore, the larger the
+laplacian of the point, the stronger the possibility that the point is
+an edge point. We threshold the Laplacian |△fθ(x)| < τ to obtain
+a corresponding set of non-edge points X′. According to statistics
+(as shown in Figure 5), we set the parameter τ = 20 throughout
+our experiments. This operation is performed before the back-
+propagation of each iteration, therefore, we call it dynamic edge
+sampling.
+Llaplacian =
+�
+x∈X′
+△fθ(x)2
+(7)
+where X′ indicates the non-edge subset of the input point cloud
+X. Finally, we optimize the total loss:
+Ltotal = Lvanish + λ1Leikonal + λ2Llaplacian
+(8)
+In which, we set λ1 = 0.1 and λ2 = 0.001 throughout our
+experiments.
+4
+DETAILS, RESULTS AND EVALUATIONS
+4.1
+Implementation details
+Data preparation. To facilitate quantitative evaluation of our
+method on multiple tasks, including reconstruction , edge ex-
+traction and normal estimation, we selected 100 3D shapes with
+rich geometric topologies to construct the evaluation dataset (See
+Figure 8) from ABC dataset [52], which provides more than 1
+million standard 3D CAD models with multiple types of standard
+CAD format files. In addition to 3D geometry and normal informa-
+tion, the geometric edges information mentioned above does not
+provide us explicitly. To this end, we have developed a tool that,
+Fig. 6: The comparison of with (b) and without Dynamic Edge
+Sampling (DES) (a).
+for each 3D shape, can quickly and easily extract the geometric
+edge information from the multiple CAD files, thus fully meeting
+the needs of our method for multi-task quantitative evaluation.
+Point sampling. For each model, we sample it into a point
+cloud containing 16, 384 points by uniform point sampling.
+Meanwhile, in order to simulate the real point cloud noise, we
+added Gaussian noise with mean µ = 0 and standard devia-
+tion δ = 0.005 to each sampling point. In each case, except
+where otherwise stated, the network is trained on the noisy data
+throughout our experiments. A few metrics on point cloud multi-
+tasks accuracy are defined to support quantitative evaluation of our
+approach; see the following subsections for details.
+4.2
+Metrics
+In our experiments, both qualitative and quantitative evaluations
+are provided. We evaluate our approach via ablation studies
+(Section 4.6), comparisons to state-of-the-art methods for 3D
+reconstruction (Section 4.3) , edge detection (Section 4.4) and
+normal estimation (Section 4.5). For the quantitative assessment of
+the 3D reconstruction results, we used the two-sided Chamfer dC
+and Hausdorff distances dH introduced by [19]. For the evaluation
+of the normal estimation, we use the angle dangle between the
+predicted normal and the groudtruth normal as the metric. To
+evaluate edge detection, we measure precision/recall and the
+IoU between predictions and ground truth, while to evaluate the
+geometric accuracy of the reconstructed edges, we employ the
+Edge Chamfer Distance (ECD) introduced by [1].
+4.3
+Reconstruction
+Comparison with IGR [19]. To facilitate a fair comparison with
+IGR [19], our network architecture is consistent with IGR [19]. In
+all experiments, we used the default training procedure specified
+in IGR to train our network, except that we did not use normal
+information in the training and set iterations to 10000. We set the
+loss parameters (see equation (8)) λ2 = 0.1 and λ3 = 0.001
+throughout our experiments. Qualitative and quantitative experi-
+ments are reported in Table 1 and Figure 7 we can also see that
+the performance of our method is significantly better.
+Comparison with state-of-the-art methods SAL [18] and
+SALD [17]. In addition to IGR [19], our method is also compared
+with SAL [18] and SALD [17], two state-of-the-art sign agnostic
+learning based methods from raw data. The results shown in
+Table 1(row 1 and 2) are inferior to those of our method. As shown
+in Figure 7, the results demonstrate the significant advantage of
+
+4096
+8192
+300
+16384
+250
+abs(Laplacian)
+200
+150
+100
+50
+0
+0
+2500
+5000
+7500
+10000
+12500
+15000
+Points(a) Our (w/o DES
+(b) Our
+(c) GTJOURNAL OF LATEX CLASS FILES, 2019
+6
+Fig. 7: Qualitative comparison with state-of-the-art methods IGR [19], SAL [18] and SALD [17].
+Fig. 8: An overview of multi-task evaluation dataset.
+dC
+dH
+Mean
+Median
+Mean
+Median
+SAL [18]
+0.019
+0.016
+0.094
+0.050
+SALD [17]
+0.016
+0.015
+0.053
+0.042
+IGR [19]
+0.028
+0.011
+0.111
+0.034
+Our (Llaplace)
+0.017
+0.009
+0.068
+0.026
+Our (Llaplace + DES)
+0.007
+0.007
+0.021
+0.021
+TABLE 1: A quantitative comparison of our method and ablation
+against IGR [19], SAL [18] and SALD
+[17] on multi-task
+evaluation dataset.
+our approach, due to the fact that differential Laplacian regularizer
+can effectively alleviate the unsatisfactory reconstruction results.
+4.4
+Edge recognition
+Specifically, for each point p in the input point set, we may quickly
+determine whether it is an edge point according to its differentiable
+laplace operator △fθ(x) . Essentially, laplace operator is mainly
+used to describe the rate of change of gradient, and is often used
+for edge detection in image processing. From the perspective of
+differential geometry, it is used to describe the change rate of
+spatial position normal. Therefore, the larger the laplace operator
+of the point, the stronger the possibility that the point is an edge
+point. We threshold the laplace operator |△fθ(x)| > τ to obtain a
+corresponding set of non-edge points Xedge. We set the parameter
+τ = 20 throughout our experiments, as shown in Figure 10.
+In addition to IGR [19], we also choose two representative
+classical non-learning based methods: Voronoi Covariance Mea-
+sure (VCM) [53], and Edge-Aware Resampling (EAR) [54], as
+both have been adopted in the point-set processing routines of the
+well known CGAL library. As reported in Table 4, our method
+completely outperforms these classical methods, This is mainly
+because we use the differentiable Laplacian operator of each
+sampling point as the metric, which can be approximate to the
+average curvature in the implicit surface representation. Note that,
+there are a large number of high-quality edge detection methods
+based on data-driven. We do not use these methods as references
+here, mainly because ours is a self-supervised learning approach.
+4.5
+Normal estimation
+Essentially, an implicitly represented MLP with softplus activation
+funtion represents a differentiable Signed Distance Functions d =
+fθ(x). According to the properties of differential geometry, the
+gradient operator of each point on the implicit surface fθ(x) = 0
+can be regarded as the normal vector of the current point x.
+Therefore, after the training, for each point in the input point
+cloud, we can directly calculate the gradient operator ▽fθ(x) of
+the differentiable function fθ(x) at the current point x, that is, the
+normal vector of the current point x. The experimental results are
+reported in Table 1. The comparison results demonstrate how our
+method achieves significantly better performance; as immediately
+quantified by the fact that dangle is larger than the one reported
+for our method.
+
+(b) IGR
+(c) SAL
+(a) Input
+(d) SALD
+(e) Our
+(f) GT(a) Point cloud
+(b) Normal
+(c) EdgeJOURNAL OF LATEX CLASS FILES, 2019
+7
+Fig. 9: Visualization normal estimation of differential Laplacian regularizer (c) and dynamic edge sampling strategy (d).
+Fig. 10: Visualization edge recognition of differential Laplacian regularizer (c) and dynamic edge sampling strategy (d).
+dC
+dH
+Mean
+Median
+Mean
+Median
+X = 0.010
+0.0102
+0.0108
+0.0509
+0.0543
+X = 0.005
+0.0069
+0.0069
+0.0206
+0.0209
+X = 0.000
+0.0055
+0.0057
+0.0148
+0.0153
+D = 4, 096
+0.0075
+0.0075
+0.0350
+0.0328
+D = 8, 192
+0.0071
+0.0072
+0.0352
+0.0269
+D = 16, 384
+0.0069
+0.0069
+0.0206
+0.0209
+TABLE 2: Algorithm performance with respect to noise X and
+sampling density D.
+4.6
+Analysis of parameters and networks
+Effect of noise. We stress test Laplacian regularizer by increasing
+the level of noise. Specifically, we randomly add a Gaussian noise
+whose mean is 0 and variance is X to each sampling point on
+the surface of the 3D shape, where we tested four values of
+X = {0, 0.005, 0.01, 0.02}. In each case, the implicit neural
+surface was trained with the noise-added data. Table 2 shows
+the quantitative results. As we can observe that, the Laplacian
+regularizer, even when trained with noisy data, can still out-
+perform these state-of-the-art methods [17]–[19] when they are
+tested on point cloud with 0.005 noise.
+Effect of density. We also train our method on point clouds
+at a reduced density. Specifically, for each 3D shape, we sam-
+pled a different number D of points to verify whether our
+network could handle the sparser point clouds, where D =
+
+(c) +Liaplacian
+(a) Input
+(b) IGR
+(e) GT(c) +Liaplacian
+(a) Input
+(b) IGR
+(e) GTJOURNAL OF LATEX CLASS FILES, 2019
+8
+Fig. 11: Effect of edge preserving differential Laplacian regu-
+larizer. (b) are the optimization results of the edge preserving
+differential Laplacian regularizer which is incorporated into the
+state-of-the-art method SALD [17]. (a) and (c) are the results of
+SALD and Ground truth respectively.
+dC
+dH
+dangle
+IGR [19]
+0.028
+0.111
+0.514
+Our (+Llaplacian)
+0.017
+0.068
+0.274
+Our (+Llaplacian + DES)
+0.009
+0.036
+0.133
+TABLE 3: Ablation studies – We evaluate the quantitative per-
+formance of our method with/without components Llaplacian and
+dynamic edge sampling (DES).
+{4, 096, 8, 192, 16, 384}. (Results in Table 2 reveal a similar
+trend as from the previous stress test. Namely, our network, when
+trained on sparser point clouds, can still outperform these state-of-
+the-art methods [17]–[19] when they are tested on or trained on
+data at full resolution (16,384 points).
+Effect of Llaplacian. To evaluate the effectiveness of loss
+Llaplacian, We incorporate this into another state-of-the-art
+method, SALD [17], This qualitative result is shown in Figure 11,
+we can find that, compared with the original algorithm, the re-
+construction quality can be effectively improved by incorporating
+Laplacian. This is mainly because the differentiable Laplacian reg-
+ularizer can effectively alleviate the unsatisfactory reconstruction
+results.
+Dynamic edge sampling. We evaluate the effect of dynamic
+edge sampling strategy on reconstruction quality. We experiment
+with the dynamic edge sampling, while keeping all other
+parameters the same. From Table 1 and Figure 7 and 12 , we can
+see that at the sharp edges, we can effectively improve the quality
+of modeling compared with state-of-the-art methods (Table 1
+(rows 1 3)) and the baseline method without dynamic edge
+sampling, this is largely due to thedynamic edge sampling
+strategy for sampling near the sharp edge of input point cloud,
+which can effectively avoid the regularizer from smoothing all
+regions.
+ECD
+IoU
+Precision
+Recall
+VCM [53]
+0.0017
+0.1925
+0.2238
+0.5998
+EAR [54]
+0.0071
+0.1146
+0.2399
+0.1933
+IGR [19]
+0.0063
+0.0880
+0.0958
+0.5620
+Our
+0.0015
+0.2375
+0.2665
+0.6934
+TABLE 4: Comparison state-of-the-art edge recognition tech-
+niques - VCM [53], EAR [54], and IGR [19].
+5
+CONCLUSION AND LIMITATION
+We present a differential Laplacian regularizer for neural implicit
+representation directly from input point cloud without normal
+supervision. More specifically, we use the infinite differentiability
+property of implicit neural representation to propose a differen-
+tiable Laplacian regularizer, which can effectively alleviate the
+unsatisfactory reconstruction results. Meanwhile, we propose a
+dynamic edge sampling strategy for sampling near the sharp
+edge of input point cloud, which can effectively avoid the Lapla-
+cian regularizer from smoothing all regions, so as to effectively
+improve the quality of reconstruction results while maintaining
+the edge. Moreover, the differentiable Laplacian regularizer can
+be easily and intuitively incorporated into implicit neural sur-
+face representations. We carefully evaluate its generation quality
+through a series of ablation studies, which show that our method
+significantly improve the quality of 3D reconstruction. In addition
+to 3D reconstruction, our method can also be conveniently applied
+to other point cloud analysis tasks, including edge extraction and
+normal estimation, etc.
+Limitation. Our approach has a few limitations, which point
+out the directions of future study. Some representative failure cases
+are shown in Figure 13. First, our method is prone to problems
+in the reconstruction of ultra-thin geometric structures, probably
+because the point cloud data is noisy, resulting in the geometric
+structure has been completely destroyed. Second, Our method
+for extremely detailed structure may be overlooked, resulting in
+incorrect reconstruction results.
+ACKNOWLEDGEMENT
+We thank the anonymous reviewers for their valuable comments.
+This work was supported in part by Natural Science Foundation
+of China (62102328), and Fundamental Research Funds for the
+Central Universities (SWU120076).
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@@ -0,0 +1,2401 @@
+Multiplicative topological semimetals
+Adipta Pal,1, 2 Joe H. Winter,1, 2, 3 and Ashley M. Cook1, 2
+1Max Planck Institute for Chemical Physics of Solids, N¨othnitzer Strasse 40, 01187 Dresden, Germany
+2Max Planck Institute for the Physics of Complex Systems, N¨othnitzer Strasse 38, 01187 Dresden, Germany
+3SUPA, School of Physics and Astronomy, University of St. Andrews, North Haugh, St. Andrews KY16 9SS, UK
+Exhaustive study of topological semimetal phases of matter in equilibriated electonic systems and myriad
+extensions has built upon the foundations laid by earlier introduction and study of the Weyl semimetal, with
+broad applications in topologically-protected quantum computing, spintronics, and optical devices. We extend
+recent introduction of multiplicative topological phases to find previously-overlooked topological semimetal
+phases of electronic systems in equilibrium, with minimal symmetry-protection. We show these multiplicative
+topological semimetal phases exhibit rich and distinctive bulk-boundary correspondence and response signatures
+that greatly expand understanding of consequences of topology in condensed matter settings, such as the limits
+on Fermi arc connectivity and structure, and transport signatures such as the chiral anomaly. Our work therefore
+lays the foundation for extensive future study of multiplicative topological semimetal phases.
+I
+Introduction
+Topological semimetals are a vast family1,2 of topological
+phases of matter studied in great depth experimentally3–11 in
+the search for table-top, quasiparticle realizations of high-
+energy physics12. At the simplest level, the topological de-
+generacies of band structures in these topological semimetal
+phases are realized quite generically if either time-reversal
+symmetry13 or inversion symmetry14 are broken.
+This is
+the requirement for two-fold topological degeneracies char-
+acteristic of the Weyl semimetal phase, although it is desire-
+able to realize such degeneracies in the vicinity of the Fermi
+level15,16, with minimal contributions to the Fermi surface
+from other electronic states. In such cases, the key signa-
+tures of Weyl semimetals are especially prominent, includ-
+ing the distinguishing Fermi arc surface states17–20, and trans-
+port signatures associated with the chiral anomaly21–27. Such
+isolation of Weyl nodes in the vicinity of the Fermi level is
+also facilitated—and the physics of topological semimetals
+enriched—by systematic study of these topological phases in
+compounds with wide-ranging phenomena, including super-
+conductivity, strong spin-orbit coupling, and strong correla-
+tions28–34. Much progress has also been made in identifying
+other topological semimetals with more complex topological
+degeneracies2,35–37 in electronic band structures, protected by
+a large set of crystalline point group symmetries in combi-
+nation with additional anti-unitary symmetries such as time-
+reversal.
+The present work returns to the foundations of topologi-
+cal semimetal studies by introducing previously-unidentified
+topological semimetal phases of matter of electronic systems
+in equilibrium, which may then be generalized in the same
+manner as outlined above. We do so by studying the first topo-
+logical semimetal realizations of multiplicative topological
+phases, a recently-identified set of topological phases of mat-
+ter described by Bloch Hamiltonians in an infinitely-large, pe-
+riodic bulk, which are symmetry-protected tensor products of
+“parent” Bloch Hamiltonians. These multiplicative topolog-
+ical semimetal (MTSM) phases are therefore straightforward
+constructions described by tensor products of Weyl semimetal
+Bloch Hamiltonians, yet exhibit rich phenomena distinct from
+all other known topological semimetals.
+We first review the Weyl semimetal phase and its canonical
+models. We then construct the first examples of multiplicative
+topological semimetal phases using these past results. The
+multiplicative topological semimetals are then first character-
+ized in the bulk, and their bulk-boundary correspondence es-
+tablished.
+II
+Review of the Weyl semimetal phase and suitable
+models for constructing multiplicative phases
+The Weyl semimetal is a topologically non-trivial phase
+of matter characterized by topologically-protected, doubly-
+degenerate and linearly-dispersing band crossings in the Bril-
+louin zone38. That is, these band-crossings, known as Weyl
+points or nodes, cannot be removed from the electronic struc-
+ture through smooth deformations of the Hamiltonian, but
+rather only through mutual annihilation of the Weyl nodes, by
+bringing two nodes of opposite topological charge to the same
+point in the Brillouin zone to gap out these band-touchings.
+When the Fermi level intersects only the Weyl nodes of this
+semimetal phase, their low-energy physics dominates, yield-
+ing a variety of intensely-studied exotic phenomena of interest
+for applications. At the simplest level, the Weyl nodes serve
+as quasiparticle, table-top realizations of Weyl fermions pre-
+dicted in high-energy physics. However, they are also a start-
+ing point in going well beyond high-energy physics, by tilting
+the Weyl cone to realize a type-II Weyl semimetal phase39,
+in which the low-energy physics of the Weyl nodes is not
+Lorentz-invariant.
+Weyl semimetal phases can be realized in effectively non-
+interacting systems where certain discrete symmetries are bro-
+ken rather than respected, in contrast to many other effectively
+non-interacting topological phases.
+They may be derived
+through symmetry-breaking starting from the Dirac semimetal
+state40,41, for instance, (which could be topologically-robust
+or fine-tuned) by breaking either time-reversal symmetry T or
+inversion symmetry I, which pulls the two Weyl nodes com-
+prising the Dirac node away from one another in momentum-
+space42. This phase, characterized by Weyl nodes in the Bril-
+louin zone, is topologically stable so long as Weyl nodes of
+opposite topological charge do not annihilate one another43.
+I-breaking Weyl semimetal phases are of tremendous ex-
+arXiv:2301.02404v1  [cond-mat.mes-hall]  6 Jan 2023
+
+2
+perimental interest, but are described by Bloch Hamiltonian
+models with four bands at minimum. A more natural starting
+point in deriving multiplicative topological semimetal phases
+is instead to use the minimal Weyl semimetal Bloch Hamil-
+tonian achieved by breaking T , which possesses only two
+bands. Such two-band models for the Weyl semimetal cor-
+respond to the non-trivial homotopy group π3(S2) and, sim-
+ilarly to the two-band Chern and Hopf insulators44 and the
+two-band Kitaev chain model 45, may be combined using
+known constructions to form a multiplicative counterpart of
+the Weyl semimetal phase, the multiplicative Weyl semimetal
+phase (MWSM).
+We therefore consider a well-established two-band Bloch
+Hamiltonian previously used in study of Weyl nodes, with
+various instances of this model serving as the parents of the
+MWSM.
+HW SM(k) =t1 sin kxτ x + t2 sin kyτ y
++ t3(2 + γ − cos kx − cos ky − cos kz)τ z.
+(1)
+where the τ j (j = x, y, z) are the Pauli matrices in the orbital
+basis. The two band spectrum,
+E(k) = ±
+�
+t2
+1 sin2 kx + t2
+2 sin2 ky + ϵ(k)2,
+ϵ(k) = t3(2 + γ − cos kx − cos ky − cos kz),
+(2)
+has two gapless nodes at k = (0, 0, ±k0), for cos k0 = γ. We
+refer to these as the Weyl nodes. The equation of motion for
+Bloch electrons in the k-space in the presence of Berry curva-
+ture is represented by ˙r = vk + ˙k×F(k). For the equation of
+motion to remain invariant under T -symmetry, one must have
+the equality, F(k) = −F(−k). The breaking of T -symmetry,
+then involves a minimum of two Weyl nodes with opposite
+Berry curvature at opposite momenta. Therefore, close to the
+Weyl nodes, we have,
+H±(k) = ±t1kxτ x + t2kyτ y ± t3 sin k0kzτ z,
+(3)
+which in turn corresponds to the Berry curvatures,
+F±(k)|0,0,±k0 = ±
+t1t2t3 sin k0
+2[t1k2x + t2k2y + (t3 sin k0)2k2z]3/2 (kx, ky, kz).
+(4)
+The Chern number of the lower-energy band for the range,
+kx = 0, ky = 0 and kz ∈ (−k0, k0) is C = ±1 depend-
+ing on the direction of the magnetic field corresponding to the
+monopoles at the two Weyl points. The Weyl nodes are in-
+volved with exotic boundary states at surfaces perpendicular
+to the z-axis, called the Fermi Arc surface states. For the case
+where the surfaces are open in the x-direction, the surface dis-
+persion is given by,
+E(ky) = ±t2 sin ky,
+(5)
+and the arc-states,
+Ψ(x, ky, kz) = e+ikyy+ikzz(e−λ1x − e−λ2x) 1
+√
+2
+�
+1
+±i
+�
+.
+In the k-space, this includes all contours cos ky + cos kz >
+1 + cos k0.
+III
+Multiplicative Weyl Semimetal (MWSM) in the bulk
+A protocol for constructing the child Hamiltonian for the
+MWSM, Hc derived from Hp1 and Hp2 as first reported in
+Cook and Moore46, is given as follows. Given two two-band
+Bloch Hamiltonians Hp1 and Hp2 written in a general form,
+with momentum-dependence suppressed, as
+Hp1 =
+�
+a b
+c d
+�
+;
+Hp2 =
+�
+α β
+γ δ
+�
+,
+(6)
+the multiplicative child Bloch Hamiltonian constructed
+from these two parents can be written as Hc
+12, where
+Hc
+12 =
+�
+�
+�
+aδ
+−aγ
+bδ
+−bγ
+−aβ
+aα
+−bβ
+bα
+cδ
+−cγ
+dδ
+−dγ
+−cβ
+cα
+−dβ
+dα
+�
+�
+� .
+(7)
+Expressing the two-band parent Bloch Hamiltonians
+Hp1(k) and Hp2(k) more compactly as the following,
+Hp1(k) = d1(k) · τ;
+Hp2(k) = d2(k) · σ,
+(8)
+where d1(k) and d2(k) are momentum-dependent, three-
+component vectors of scalar functions, and each of σ and τ is
+the vector of Pauli matrices, the multiplicative child Hamilto-
+nian may more compactly be written as,
+Hc
+12(k) = (d11, d21, d31) · τ ⊗ (−d12, d22, −d32) · σ, (9)
+to highlight the tensor product structure of the child Hamil-
+tonian, which can be symmetry-protected as discussed in ear-
+lier work by Cook and Moore on multiplicative topological
+phases, and therefore can describe phases of matter, even in
+the presence of additional bands46.
+The tensor-product structure guarantees that the energy
+spectrum of the child Hamiltonian is a product of the energy
+spectrum of Hp1(k), Ep1(k), and of Hp2(k), Ep2(k), respec-
+tively,
+Ec
+12(k) = ±Ep1(k)Ep2(k).
+(10)
+This implies that bands of the child Hamiltonian dispersion
+are at least doubly degenerate everywhere in the bulk Bril-
+louin zone.
+We will consider two cases in this work: (1) the Weyl
+node separation of each parent is along one axis in the Bril-
+louin zone, and (2) the axis along which Weyl nodes are
+separated in one parent is perpendicular to the axis along
+which Weyl nodes are separated in the other parent. Spectral
+and magneto-transport properties differ significantly between
+these two cases, as we will show, demonstrating the richness
+of MTSM phases of matter.
+A
+Multiplicative Weyl Semimetal - parallel axis par-
+ents
+The construction of the MWSM for both parents along the
+same axis is derived from two parent WSMs. As an example,
+
+3
+we consider the following parents and the resulting child:
+Hp1(k) =t11 sin kxτ x + t21 sin kyτ y
++ t31(2 + γ1 − cos kx − cos ky − cos kz)τ z,
+(11a)
+Hp2(k) =t12 sin kxσx + t22 sin kyσy
++ t32(2 + γ2 − cos kx − cos ky − cos kz)σz,
+(11b)
+Hc(k) =[t11 sin kxτ x + t21 sin kyτ y
++ t31(2 + γ1 − cos kx − cos ky − cos kz)τ z]
+⊗ [−t12 sin kxσx + t22 sin kyσy
+− t32(2 + γ2 − cos kx − cos ky − cos kz)σz].
+(11c)
+Each parent Hamiltonian realizes Weyl nodes at k
+=
+�
+0, 0, cos−1 γi
+�
+when −1 < γi < 1, (i = 1, 2). Examples of
+such topologically non-trivial dispersion are shown in Fig. 1
+(a) and (b), respectively.
+/2 0
+/2
+1
+0
+1
+E(k)
+(a) WSM parent 1
+/2 0
+/2
+momentum kz
+1
+0
+1
+E(k)
+(b) WSM parent 2
+/2
+0
+/2
+momentum kz
+1.0
+0.5
+0.0
+0.5
+1.0
+E(k)
+(c) MWSM parallel child
+band 1
+band 2
+band 3
+band 4
+FIG. 1: Dispersion E(k) for (a) WSM Parent Hamiltonian
+with γ1 = 0.5 along kz and t11 = t21 = t31 = 1, (b) WSM
+Parent Hamiltonian with γ2 = −0.5 along kz and
+t12 = t22 = t32 = 1, and (c) the resulting MWSM parallel
+Child Hamiltonian along kz.
+From these parent Hamiltonian dispersions, we can find the
+dispersion of the child. As given in Eq. 10, the bulk spectrum
+is doubly degenerate and determined by the spectra of the par-
+ent 1, Ep1(k), and parent 2, Ep2(k), respectively, which take
+the following forms:
+Ep1(k) = [t2
+11 sin2 kx + t2
+21 sin2 ky + ϵ1(k)2]1/2,
+Ep2(k) = [t2
+12 sin2 kx + t2
+22 sin2 ky + ϵ2(k)2]1/2,
+(12)
+where ϵ1/2(k) = t31/2(2 + γ1/2 − cos kx − cos ky − cos kz).
+For the sake of convenience, we refer to the MWSM with
+Weyl node separation for each parent along the same axis in
+the Brillouin zone (as in the case of parents given by Eq. 22a
+and Eq. 22b) as MWSM||. For the MWSM|| bulk spectrum
+given by Eq. 10 and Eq. 12, gapless points occur at the po-
+sitions in the Brillouin zone where gapless points are present
+for the parent systems. As γ1 and γ2 control separation of the
+Weyl nodes in the Brillouin zone for the parents, they play a
+major role in determining the number of nodes, the location
+of the nodes, and the polynomial order of the nodes in the
+Brillouin zone for the child. When γ1 = γ2, for instance, we
+have two gapless points but the dispersion near the nodes is
+quadratic. In contrast, for γ1 ̸= γ2 as for parents depicted
+in Fig. 1 (a) and (b), the child MWSM|| has four nodes, and
+bands disperse linearly in the vicinity of the nodes, as shown
+in Fig. 1 (c). Each node is four-fold degenerate.
+While such degeneracy naively suggests Dirac nodes or
+Weyl nodes of higher charge, the multiplicative nodes are dis-
+tinct in a number of ways. To examine this difference, we look
+at the child Hamiltonian in the vicinity of each multiplicative
+node for the case −1 < γ1 ̸= γ2 < 1. From the tensor product
+structure, it easy to check that ∂E±
+∂ki = const. which implies
+that the dispersion is linear at each of the gapless nodes of the
+MWSM. Therefore the possibility of a higher order Weyl node
+is nullified. The position of each of the multiplicative nodes
+are determined by the nodes in the respective parents. We re-
+fer to (0, 0, ±k01) as the Weyl node positions derived from
+the first parent, and (0, 0, ±k02) as the Weyl node positions
+derived from the second parent. Here γi = cos k0i, (i = 1, 2).
+If the gapless point is (0, 0, k02), then we define MWSM|| in
+the vicinity as Hc
+||,2, and,
+Hc
+||,2 = t31(γ1−γ2)τ z(−t12kxσx+t22kyσy−t32 sin k02¯kz,2σz),
+(13)
+where ¯kz,2 = (kz − k02). Surprisingly, this looks like a Dirac
+semimetal Hamiltonian, whose Dirac node has been shifted in
+k-space. Since it is no longer at the origin, the time-reversal
+symmetry is broken. For the other node, γ1 = cos k01 for
+(0, 0, k01), we define the multiplicative Hamiltonian in the
+vicinity as Hc
+||,1, so that,
+Hc
+||,1 = (t11kxτ x+t21kyτ y+t31 sin k01¯kz,1τ z)t32(γ1−γ2)σz,
+(14)
+where ¯kz,1d = (kz − k01) and contains off-diagonal terms for
+the block Hamiltonian. But again, it is possible to perform
+a similarity transformation on this Hamiltonian, in the form
+U = R−1
+τ (θ, φ) ⊗ Rσ(θ, φ), so that we get another ‘shifted’
+Dirac semimetal type Hamiltonian,
+¯Hc
+||,1 = t32(γ1−γ2)τ z(t11kxσx+t21kyσy+t31 sin k01¯kz,1σz).
+(15)
+Again, the shift from the origin breaks the time-reversal sym-
+metry of the original Dirac semimetal. It is therefore appropri-
+ate to refer to the MWSM|| as possessing degeneracies con-
+sisting of Weyl nodes, rather than possessing Dirac nodes, and
+exhibit strikingly different physics as a result.
+B
+MWSM - perpendicular axis parents
+Before characterizing bulk-boundary correspondence and
+transport signatures of MTSMs, we explore further richness
+of multiplicative constructions by considering cases where
+
+4
+parent Weyl nodes are separated along orthogonal axes in k-
+space. As a specific case, we choose parent Hamiltonians such
+that the first parent has Weyl node separation along the y-axis,
+while the second one has Weyl node separation along the z-
+axis,
+Hp1(k) =t11 sin kxτ x + t21 sin kzτ y
++ t31(2 + γ1 −
+�
+i
+cos ki)τ z,
+(16a)
+Hp2(k) =t12 sin kxσx + t22 sin kyσy
++ t32(2 + γ2 −
+�
+i
+cos ki)σz.
+(16b)
+Again the bulk spectrum is derived from the tensor product
+structure,
+Ep1(k) = [t2
+11 sin2 kx + t2
+21 sin2 kz + ϵ2
+1(k)]1/2,
+Ep2(k) = [t2
+12 sin2 kx + t2
+22 sin2 ky + ϵ2
+2(k)]1/2,
+Ec
+⊥k = ±Ep1(k)Ep2(k),
+(17)
+where ϵ1/2(k) = t31/32(2+γ1/2 −cos kx −cos ky −cos kz).
+Examples of parent and child dispersion in this case are shown
+in Fig. 2 for the values, γ1 = 0.5 and γ2 = −0.5.
+We gain greater understanding of the multiplicative struc-
+ture in this case by examining the low-energy expansion of
+the Child Hamiltonian in the vicinity of its nodes. Taylor ex-
+panding up to linear order around the point, (0, k0,1, 0) for
+γ1 = cos k0,1, one gets,
+Hc
+⊥,1(k) =(t11kxτ x + t21kzτ y + t31 sin k0,1¯ky,1τ z)
+⊗ (t22 sin k0,1σy − t32(γ2 − γ1)σz).
+(18)
+Similarly, expanding around (0, 0, k0,2) for γ2 = cos k0,2, we
+get,
+Hc
+⊥,2(k) =(t21 sin k0,2τ y + t31(γ1 − γ2)τ z)
+⊗ (−t12kxσx + t22kyσy − t32 sin k0,2¯kz,2σz).
+(19)
+One notices that Hc
+⊥,2(k) is equivalent to a DSM when
+γ1 = γ2.
+C
+Discrete Symmetries of the MWSM
+The discrete symmetries satisfied by the parent WSMs include
+invariance under particle-hole conjugation given by P = σxκ,
+such that the Hamiltonian satisfies,
+σxH∗
+1/2(k)σx = −H1/2(−k),
+and invariance under spatial inversion given by I = σz, such
+that the Hamiltonian satisfies,
+σzH1/2(k)σz = H1/2(−k).
+The MWSM|| or ⊥ child systems are instead invariant un-
+der time reversal given by T = iτ xσxκ corresponding to the
+transformation,
+τ xσxH∗
+c(k)τ xσx = Hc(−k).
+ky
+1
+0
+1
+E
+(a)
+kz
+1
+0
+1
+E
+(b)
+(0, 0, 0)
+(0, 0, )
+(0, , )
+(0, , 0)
+(0, 0, 0)
+(0, , )
+k
+7.5
+5.0
+2.5
+0.0
+2.5
+5.0
+7.5
+E
+(f)
+ky
+0
+kz
+0
+E
+5
+0
+5
+(e)
+kz
+0.5
+0.0
+0.5
+E
+(c)
+/20
+/2
+ky
+0.5
+0.0
+0.5
+E
+(d)
+FIG. 2: Dispersion E(k) (t11 = t12 = 1, t21 = t22 = 1,
+t31 = t32 = 1) for (a) WSM Parent Hamiltonian with
+γ1 = 0.5 along ky, (b) WSM Parent Hamiltonian with
+γ2 = −0.5 along kz and the resulting MWSM perpendicular
+Child Hamiltonian along (c) kz and (d) kz. The energy
+dispersion plotted along both ky and kz is shown in (e) and
+the dispersion along a high-symmetry path in the first
+quadrant of the two-dimensional (2d) BZ is shown in (f).
+Inversion symmetry relates the nodes in the first quadrant to
+those in the other quadrants, giving rise to four gapless nodes
+in the 2d BZ.
+They are also invariant under spatial inversion given by I =
+τ zσz, corresponding to the transformation,
+τ zσzHc(k)τ zσz = Hc(−k).
+The MWSM should then satisfy the symmetry, T I, which
+may also protect the Dirac semi-metal phase. Indeed, in some
+cases, the Dirac Hamiltonian for the MWSM near the nodes
+is reminiscent of the corresponding low-energy Hamiltonian
+for a Dirac semi-metal. This invariance of the multiplicative
+bulk Hamiltonian under products of transformations, which
+leave each parent Hamiltonian invariant, is expected given the
+multiplicative dependence of the child on the parents.
+
+5
+D
+Bulk characterization of topology with Wilson loops
+As calculated in Supplementary section S1, the Berry connec-
+tion for the MWSM is given as
+A = (A1,kx − A2,kx, A1,ky − A2,ky, A1,kz − A2,kz), (20)
+where Aj,l = (i ⟨+j| ∂l |+j⟩ , i ⟨−j| ∂l |−j⟩).
+Using this
+expression for the Berry connection, we compute Wilson
+loops and associated Wannier spectra by integrating over kx
+for a given ky, as detailed in Alexandradinata et al47.
+In
+the parallel case illustrated in Fig. 3(a), the Wannier spectra
+derived from Wilson loop calculations show that only in
+regions where only one of the parent phases is non-trivial do
+we get non-trivial Wannier spectra distinguished by π values
+for Wannier charge centers. However, the Wannier spectra in
+the region where each parent is topological appears trivial,
+given the dependence of child Wannier spectra on parent
+Wannier spectra distinctive of multiplicative topological
+phases. We have referred to a pair of Weyl nodes of equal
+and opposite topological charge as a ‘dipole’. We observe,
+that the orientation of this dipole due to the two constituent
+parents is important, as anti-parallel dipoles, as depicted
+in Fig.
+3(b), show non-trivial Wilson loop eigenvalues in
+a region in the 2d BZ where neither of the parent systems
+have non-trivial topological character. Analogous results for
+the MWSM⊥ are shown in Fig. 4, although the Wannier
+spectrum structure is far richer than in the parallel case.
+/2 0
+/2
+kz
+/2
+0
+/2
+ky
+(a)WSM 1 x
+/2 0
+/2
+kz
+/2
+0
+/2
+ky
+(b)WSM 2 x
+/2 0
+/2
+kz
+/2
+0
+/2
+ky
+(c)MWSM pll
+/2 0
+/2
+kz
+/2
+0
+/2
+ky
+(d)MWSM pll
+0.25
+0.00
+0.25
+x
+0.25
+0.00
+0.25
+x
+0.50
+0.25
+0.00
+0.25
+0.50
+x, 1
+0.50
+0.25
+0.00
+0.25
+0.50
+x, 2
+(a) Each parent has Weyl node ‘dipole’ oriented in +ˆz direction.
+/2
+0
+/2
+kz
+/2
+0
+/2
+ky
+(a)WSM 1 x
+/2
+0
+/2
+kz
+/2
+0
+/2
+ky
+(b)WSM 2 x
+/2
+0
+/2
+kz
+/2
+0
+/2
+ky
+(c)MWSM pll
+/2
+0
+/2
+kz
+/2
+0
+/2
+ky
+(d)MWSM pll
+0.25
+0.00
+0.25
+x
+0.25
+0.00
+0.25
+x
+0.50
+0.25
+0.00
+0.25
+0.50
+x, 1
+0.50
+0.25
+0.00
+0.25
+0.50
+x, 2
+(b) Parent 1 with dipole oriented along +ˆz direction and parent 2
+with dipole oriented in −ˆz direction.
+FIG. 3: Wannier spectra in MWSM parallel for two filled
+bands derived from Wilson loop around kx for parent 1 with
+γ1 = −0.5 and parent 2 with γ2 = 0.5. The upper row (a)
+and the lower row (b) have opposite orientation of the Weyl
+node ’dipole’ for parent 2. Corresponding Wannier spectra of
+the MWSM for the lowest-energy and second-lowest in
+energy occupied bands is shown in (c) and (d), respectively.
+
+6
+/2 0
+/2
+kz
+/2
+0
+/2
+ky
+(a)WSM 1 x
+/2 0
+/2
+kz
+/2
+0
+/2
+ky
+(b)WSM 2 x
+/2 0
+/2
+kz
+/2
+0
+/2
+ky
+(c)MWSM perp
+/2 0
+/2
+kz
+/2
+0
+/2
+ky
+(d)MWSM perp
+0.25
+0.00
+0.25
+x
+0.25
+0.00
+0.25
+x
+0.50
+0.25
+0.00
+0.25
+0.50
+x, 1
+0.50
+0.25
+0.00
+0.25
+0.50
+x, 2
+(a) Parent 1 has Weyl node ‘dipole’ oriented along +ˆy direction and
+parent 2 along −ˆz direction.
+/2 0
+/2
+kz
+/2
+0
+/2
+ky
+(a)WSM 1 x
+/2 0
+/2
+kz
+/2
+0
+/2
+ky
+(b)WSM 2 x
+/2 0
+/2
+kz
+/2
+0
+/2
+ky
+(c)MWSM perp
+/2 0
+/2
+kz
+/2
+0
+/2
+ky
+(d)MWSM perp
+0.25
+0.00
+0.25
+x
+0.25
+0.00
+0.25
+x
+0.50
+0.25
+0.00
+0.25
+0.50
+x, 1
+0.50
+0.25
+0.00
+0.25
+0.50
+x, 2
+(b) Parent 1 has Weyl node ’dipole’ oriented in +haty direction and
+parent 2 Weyl node dipole oriented in −ˆz direction.
+FIG. 4: Wannier spectra in MWSM perpendicular for two
+filled bands derived from Wilson loop around kx for parent 1
+with γ1 = −0.5 and parent 2 with γ2 = 0.5. The upper row
+(a) and the lower row (b) have opposite orientation of the
+Weyl node ’dipole’ for parent 2. Corresponding Wannier
+spectra of the MWSM for the lowest-energy and
+second-lowest in energy occupied bands is shown in (c) and
+(d), respectively.
+IV
+MWSM with open-boundary conditions–
+1
+Slab spectra of MWSM:
+An important aspect of WSM physics is its distinctive
+bulk-boundary correspondence:
+Weyl nodes in the three-
+dimensional bulk Brillouin zone serve as termination points of
+topologically-protected boundary states known as Fermi arcs
+when projected to a slab Brillouin zone corresponding to open
+boundary conditions in one direction. We expect analogous
+topologically-protected surface states in MTSMs and explore
+possible realizations of these Fermi arc states in this section.
+One might expect that the tensor product structure of the
+multiplicative phases is visible in the surface spectrum of the
+MWSM. Numerical simulations show that this is the case. For
+the parent WSMs, the surface spectra is given as, E(ky) ∼
+sin(ky)(Fig. 5(a) and (b) and Fig.
+6 2nd row) for nodes
+along the z-axis and open boundaries along the x-direction
+and E(kz) ∼ sin(kz)(Fig. 6 1st row) for nodes along the
+y-axis and open boundaries along the x-direction.
+Indeed,
+corresponding surface spectra of child Hamiltonians depend
+on these surface spectra in a multiplicative way.
+Numeri-
+cal simulation from Fig. 5 (c) shows that, for MWSM||, the
+slab spectra disperses as E(ky) ∼ sin2(ky) for two parents
+each with surface spectrum E(ky) ∼ sin(ky). In contrast,
+Fig. 6 (c) shows that the surface spectrum instead disperses
+as E(ky, kz) ∼ sin(ky) sin(kz) for MWSM⊥ when one par-
+ent has the former surface spectrum and the other has the lat-
+ter. We also show, for the case of each parent surface spec-
+trum along kz, which exhibits flat bands between the two
+Weyl nodes (Fig. 5(a) and (b)) corresponds to flat bands be-
+tween all four gapless points in the MWSM parallel system
+(Fig. 5(c)). However, fitting sin2(ky) curves to each of the
+parallel and perpendicular MWSM spectra reveals that, except
+in special cases when γ1 = γ2 where the fit is exact, the slab
+spectra does not disperse as sin2(ky) and instead exhibits kz-
+dependence. One can check this by comparing E vs. ky slab
+spectra in the range −min(k0,1, k0,2) < kz < min(k0,1, k0,2)
+and min(k0,1, k0,2) < kx < max(k0,1, k0,2). The spectra ap-
+pears linear near zero in the latter case.
+2
+Stability of surface states of MWSM
+For the MWSM|| system, the low-energy expansion about
+a node is reminiscent of a Dirac node, and it is therefore
+possible to break apart the four-fold degeneracy at each
+of the nodes by introducing an external magnetic field.
+We introduce minimal coupling, ky → ky − eBx for the
+MWSM|| to simulate the effect of applied magnetic field
+on the spectral density of the Fermi arc surface states. We
+observe that the Fermi arcs split but are not destroyed by the
+applied field as in the case of the DSM.
+3
+Fermi Arcs for the MWSM as a stack of MCIs:
+WSMs can be interpreted as a set of Chern insula-
+tors(CIs),
+each
+defined
+in
+a
+2d
+submanifold
+of
+the
+3d BZ of the WSM (e.g.,
+each kx-ky
+plane) for a
+given value of kz, yielding a stack of CIs in the kz-
+direction.
+The Weyl nodes then correspond to topological
+
+7
+/2
+0
+/2
+ky
+2
+1
+0
+1
+2
+E
+(e)MWSM ||
+/2
+0
+/2
+ky
+2
+1
+0
+1
+2
+E
+(f)MWSM ||
+/2
+0
+/2
+kz
+2
+1
+0
+1
+2
+E
+(f)MWSM ||
+/2
+0
+/2
+ky
+2
+0
+2
+E
+(a)Parent 1
+/2
+0
+/2
+kz
+2
+0
+2
+E
+(b)Parent 1
+/2
+0
+/2
+ky
+2
+0
+2
+E
+(c)Parent 2
+/2
+0
+/2
+kz
+2
+0
+2
+E
+(d)Parent 2
+FIG. 5: Finite slab spectra(in x-direction, Lx = 80) along ky
+(kz = 0) and kz(ky = 0) respectively for (a,b) WSM with
+γ1 = −0.5, (c,d) WSM with γ2 = 0.5. In (e,f) the slab
+spectra(Lx = 80) E vs. ky for the MWSM|| child created
+from the above two parents for kz = 0 and kz = π
+2
+respectively. (g) shows the slab spectra E vs. kz at ky = 0
+for the same MWSM|| child system.
+phase
+transitions—corresponding
+to
+gap-closings—in
+the stack between intervals in kz
+with topologically-
+distinct CIs.
+Specifically,
+we use the SCZ model48,
+/2 0
+/2
+kz
+2.5
+0.0
+2.5
+E
+(a) WSM parent 1   x
+/2 0
+/2
+kz
+2
+0
+2
+E
+(b) WSM parent 2
+/2 0
+/2
+kz
+2
+0
+2
+E
+=   (c) MWSM perp. child
+/2 0
+/2
+ky
+2.5
+0.0
+2.5
+E
+/2 0
+/2
+ky
+2
+0
+2
+E
+/2 0
+/2
+ky
+2
+0
+2
+E
+/2 0
+/2
+kz
+2.5
+0.0
+2.5
+E
+/2 0
+/2
+ky
+2
+0
+2
+E
+/2 0
+/2
+(kz + ky)
+2
+0
+2
+E
+FIG. 6: Finite slab spectra (in x direction, Nx = 80) with the
+constituent parent Hamiltonians - WSM parent Hamiltonian
+1 with γ1 = 0.5 and Weyl nodes along the ky-direction is
+shown along column (a), WSM parent Hamiltonian 2 with
+γ = −0.5 with Weyl nodes along the kz-direction along
+column (b) and the MWSM perpendicular child Hamiltonian
+along column (c). It is apparent how the surface spectra along
+kz(for ky = 0) and ky(for kz = 0) combine multiplicatively
+to create the surface spectra for the MWSM perpendicular
+system. The lowest diagram along column (c) especially
+shows the spectra along the diagonal kz + ky direction where
+the component spectra sin(kz) and sin(ky) have combined to
+produce sin(kz) sin(ky) as the leading term.
+HCI = B(2+M −cos kx −cos ky)σz +sin kxσx +sin kyσy
+in particle-hole space. In the WSM, the mass term is given
+as M = γ − cos kz. Here, for the range, −1 < γ < 1,
+kz ∈ [− cos−1 γ, cos−1 γ]. The Fermi arcs we observe in the
+2d BZ defined in the ky − kz for open boundary conditions in
+the x-direction are projections of the chiral edge states of the
+slices of the corresponding CIs in the stack.
+The multiplicative counterpart of a Chern insulator was
+introduced recently by Cook and Moore46 as Multiplicative
+Chern Insulators(MCIs). Here, one must notice that the MCI
+has two mass terms derived from each of the parent systems,
+one from each of the parent systems. Hence, there exists more
+than one way to stack the MCIs in the kz direction. Either
+parent mass term can be kz-dependent, for instance, or both
+can be. Here, we have attached the momentum dependence
+to both the mass terms, so that the difference in parent mass
+parameters remains constant. We then characterize the multi-
+plicative Fermi arc states by opening boundary conditions in
+the x- and y-directions, and plotting the probability density for
+
+8
+the sum of 40 eigenstates nearest in energy to zero in Fig. 7
+for kz = 0 (a 2D submanifold of the BZ realizing an MCI) and
+kz = π
+2 (a 2D submanifold of the BZ that is topologically triv-
+ial). For the former case shown in Fig. 7(a) and (b), the proba-
+bility density in the corresponding child is localized at sites at
+the boundary, but also at the sites adjacent to these sites. For
+the latter case, parent 1 has edge states and parent 2 does not
+as shown in Fig. 7(d) and (e). The resultant child probability
+density shows low-energy states localize only at the boundary
+sites as shown in Fig. 7(f). This localization behavior is sim-
+ilar to that of the multiplicative Kitaev chain presented in a
+second work by the present authors, where, if each parent is
+topological, edge states are localized at lattice sites right at the
+edge, but also at sites adjacent to these sites. We expect such
+localization to protect the edge states from backscattering to
+some extent, which we will explore in future work.
+4
+Boundary states disconnected from bulk states—
+The MCI introduced by Cook and Moore46 can exhibit topo-
+logically robust yet floating edge states, which are separated
+from the bulk by a finite energy gap. MTSMs constructed
+from MCIs can inherit this exotic boundary state connectivity,
+displaying boundary states disconnected from the bulk band
+structure.
+To realize such a MWSM, we first note when edge states
+are disconnected from bulk states for the case of the MCI:
+HCI,p1(k) =B1(2 + M1 − cos kx − cos ky)τ z
++ sin kxτ x + sin kyτ y,
+(21a)
+HCI,p2(k) =B2(2 + M2 − cos kx − cos ky)σz
++ sin kxσx + sin kyσy,
+(21b)
+HMCI,c(k) =[B1(2 + M1 − cos kx − cos ky)τ z
++ sin kxτ x + sin kyτ y]
+⊗ [−B2(2 + M2 − cos kx − cos ky)σz
+− sin kxσx + sin kyσy],
+(21c)
+the range of parameters over which this is possible is M1 ∈
+[−4, −2] and M2 ∈ [−2, 0] which corresponds to Chern num-
+bers C = +1 and C = −1 respectively. We therefore con-
+struct a MWSM for which the Weyl nodes of one parent WSM
+are separated in k-space by a stack of Chern insulators, each
+with total Chern number C = +1, and the Weyl nodes of the
+other parent are separated by a stack of Chern insulators, each
+with total Chern number C = −1. Comparing Eqn. (22a)
+with (21a) and Eqn. (22b) with (21b), it is clear that, for each
+Chern insulator in the stack, the following mapping holds,
+Mi = γi − cos kz, i ∈ {1, 2}, and i labeling the parent. From
+this mapping, it is not possible to have M2 ∈ (−4, −2) while
+γi ∈ (−1, 1), i ∈ {1, 2}. We therefore generalize the map-
+ping to the following form, Mi = γi − ri cos kz, i ∈ {1, 2},
+so that the parents and the child Hamiltonian for the MWSM
+parallel are,
+Hp1(k) =t11 sin kxτ x + t21 sin kyτ y
++ t31(2 + γ1 − cos kx − cos ky − r1 cos kz)τ z,
+(22a)
+Hp2(k) =t12 sin kxσx + t22 sin kyσy
++ t32(2 + γ2 − cos kx − cos ky − r2 cos kz)σz,
+(22b)
+Hc(k) =[t11 sin kxτ x + t21 sin kyτ y
++ t31(2 + γ1 − cos kx − cos ky − r1 cos kz)τ z]
+⊗ [−t12 sin kxσx + t22 sin kyσy
+− t32(2 + γ2 − cos kx − cos ky − r2 cos kz)σz].
+(22c)
+To construct one parent with Chern number of this stack
+non-trivial and opposite in sign to the Chern number of the
+stack in the other parent, we first introduce some terminology.
+We refer to the region between Weyl nodes including kz = 0
+as regular Weyl region (RWR) and the region including kz =
+±π as the irregular Weyl region (IWR). The existence of Weyl
+nodes requires |r1,2| ≥ 1 for |γ1,2| < 1. It is then possible to
+realize a RWR with negative Chern number by varying r1,2,
+so that γ1,2 − r1,2 cos kz ∈ (−4, −2). These RWRs—one of
+each parent system—must then occur over the same interval in
+kz, however, to realize topological floating surface states. We
+set γ2 = 0 and r2 = 3, which means we have C = −1 for the
+range [− cos−1( 2
+3), cos−1( 2
+3)] when M2 = γ2 − r2 cos kz ∈
+[−3, −2]. Then we must have γ1 = cos π
+3 = 0.5 and r1 = 1
+so that in the region kz ∈ [− cos−1( 2
+3), cos−1( 2
+3)], we have
+the same kind of MCI with edge states gapped from the bulk
+as described in Cook and Moore46. These results are shown in
+Fig. 8.
+The MWSM⊥ case of topologically robust yet floating
+Fermi arc surface states is constructed similarly, and we de-
+fer thorough investigation of this case to later work.
+V
+Effect of Magnetic field on MWSM and Chiral
+anomaly
+We now investigate response signatures of MTSMs.
+As
+we consider MWSMs here, which may be constructed from
+WSM parent systems, we focus in particular on the question
+of whether there is a multiplicative generalization of the chi-
+ral anomaly, one of the most important signatures of Weyl
+semimetals: application of non-orthogonal electric and mag-
+netic fields can pump electrons between Weyl nodes of oppo-
+site chirality49. More specifically, applying an external mag-
+netic field parallel to the axis along which Weyl nodes are
+separated in k-space yields a single chiral Landau level near
+each of the Weyl nodes. In Weyl semimetals, this suppresses
+backscattering of electrons with opposite chirality, manifest-
+ing as a negative magnetoresistance (MR). Weyl semimetals
+
+9
+FIG. 7: Probability densities of superposition of 40 edge state eigenvectors in a 30 × 30(Lx × Ly) square lattice at kz = 0 and
+kz = π
+2 for (a, d) Parent WSM 1 (γ1 = −0.5), (b, e) Parent WSM 2(γ2 = 0.5) and (c, f) MWSM || child (γ1 = −0.5 and
+γ2 = 0.5) respectively. At kz = 0, both the parent systems are topological as seen from a visible edge state which results in
+localization at both the edge and second last edge sites in the MWSM || child system. When kz = π
+2 , the parent 1 is still
+topological but the parent 2 is trivial as seen from the absence of edge states which results in localization only at the edge sites
+of the MWSM || child system.
+therefore serve as condensed matter platforms for study of the
+chiral anomaly, also known as Adler-Bell-Jackiw anomaly, as-
+sociated with the Standard Model of particle physics50. When
+the external magnetic field is instead oriented perpendicular to
+the k-space axis along which Weyl nodes are separated, semi-
+classical calculations indicate the presence of quantum oscil-
+lations in the density of states51, observable in magnetization,
+magnetic torque, and MR measurements50.
+To study the effects of external fields on the MWSM,
+we first derive the Landau level structure for the the Weyl
+semimetal in the cases of external magnetic fields applied par-
+allel and perpendicular to the Weyl node axis. We can then
+draw parallels between these results and their generalizations
+in the case of the MWSM.
+A
+Chiral anomaly in WSM
+To study the chiral anomaly in a WSM, we consider a par-
+ticular Bloch Hamiltonian HW SM(k) characterizing a Weyl
+semimetal phase and its expansion around the kz-axis, i.e.
+k → (0, 0, kz) (up to 2nd order in kx and ky),
+HW SM(k) =t(2 + γ − cos kx − cos ky − cos kz)σz
++ t′ sin kyσy + t′ sin kxσx,
+≈t(Q + 1
+2(k2
+x + k2
+y))σz + t′kyσy + t′kxσx,
+(23)
+where Q = γ − cos kz. Applying the magnetic field, B =
+Bˆz along the Weyl node axis, Peierls substitution changes the
+momenta in the following way, kx → k′
+x = kx, ky → k′
+y =
+ky + eBx, and kz → k′
+z = kz. The position-momentum
+commutator, implies, [k′
+y, k′
+x] = ieB, so that, it is possible to
+define bosonic ladder operators,
+a = k′
+x − ik′
+y
+√
+2eB
+;
+a† = k′
+x + ik′
+y
+√
+2eB
+;
+[a, a†] = 1.
+(24)
+
+10
+/2
+0
+/2
+ky
+4
+2
+0
+2
+4
+E
+(e)MWSM ||
+/2
+0
+/2
+kz
+4
+2
+0
+2
+4
+E
+(f)MWSM ||
+/2
+0
+/2
+ky
+2.5
+0.0
+2.5
+E
+(a)Parent 1
+/2
+0
+/2
+kz
+2.5
+0.0
+2.5
+E
+(b)Parent 1
+/2
+0
+/2
+ky
+2.5
+0.0
+2.5
+E
+(c)Parent 2
+/2
+0
+/2
+kz
+2.5
+0.0
+2.5
+E
+(d)Parent 2
+FIG. 8: Slab spectra along ky (subfigure a) and kz (subfigure
+b) for WSM parent 1 with γ1 = 0, r1 = 3, and slab spectra
+along ky (subfigure c) and kz (subfigure d) for WSM parent 2
+with γ2 = 2/3, r2 = 1, respectively. Corresponding slab
+spectra for the MWSM|| with t11 = t12 = 1, t21 = t22 = 1,
+t31 = t32 = 1 along (e) ky and (f) kz, respectively, with
+edges separate from the bulk slab spectra along ky.
+Applying Eqn.S18, after substituting k → k′, we get the fol-
+lowing system which looks similar to the polariton conserving
+Jaynes-Cummings Hamiltonian,
+HW SM(k′) ≈ t(Q+eB(a†a+1
+2))σz+t′√
+2eB(aσ++a†σ−),
+(25)
+where σ± =
+1
+2(σx ± iσy) are the spin ladder operators in
+the basis {|+⟩ , |−⟩} of σz (σz |±⟩ = ± |±⟩). The ground
+state from the above Hamiltonian is given by the eigenvec-
+tor, |ψLLL⟩ = |0; −⟩ (states denoted as |n; s⟩ where n is the
+bosonic number and s is the spin direction), which leads to the
+lowest Landau level energy,
+ELLL = −t(Q + 1
+2eB).
+(26)
+Near each of the Weyl nodes, it is easy to observe that |ψLLL⟩
+is chiral as shown in Fig. 9. The other Landau levels can be
+derived by restricting to the two dimensional disjoint spaces,
+{|n, −⟩ , |n − 1, +⟩}, parametrized by the bosonic number, n
+so that in each such basis, the Hamiltonian is,
+H(kz, n) = −teB
+2 σ0 −t(Q+eBn)σz +t′√
+2eBnσx. (27)
+The energy for the other Landau levels parametrized by n =
+1, 2, ... is given by the eigenvalues of Eqn. 27,
+EnLL = −teB
+2
+±
+�
+t2(Q + eBn)2 + 2t′2eBn.
+(28)
+We have illustrated the analytically calculated Landau levels
+in Fig. 9 and compared them to numerical calculations of Lan-
+dau levels. The numerical computation involves plotting the
+bands for the Peierls substituted Weyl semimetal with periodic
+boundary conditions, say in the x-direction, and subjected to
+magnetic field in integer multiples of 2π
+L , where L is the size of
+the lattice in the x-direction. We observe that the chiral Lan-
+dau level from both analytical and numerical methods overlap,
+with an approximate overlap of the other Landau levels since
+we only considered till second order in kx and ky.
+Next we consider the case when the magnetic field is di-
+rected perpendicular to the Weyl node axis, say B = Bˆy.
+Expanding the first line of Eqn. 23 around the Weyl node,
+k = (0, 0, k0 = cos−1 γ) of positive chirality, and setting
+t = t′ = 1, we get,
+HW SM(k) ≈ sin k0(kz − k0)σz + kyσy + kxσx,
+=⇒ H′
+W SM(k) ≈ − kyσz + kxσx + sin k0(kz − k0)σy,
+(29)
+where in the second line we have rotated the Hamiltonian to
+a new basis via, σx → σz and σx → −σx. In the presence
+of mentioned magnetic field perpendicular to the Weyl node
+axis, the Peierls substitution is applied as kx → k′
+x = kx,
+ky → k′
+y = ky and kz → k′
+z = kz − eBx. The commuta-
+tion relation, [kx, sin k0(kz − k0 − eBx)] = ieB sin k0, then
+constructs the bosonic ladder operators,
+b = kz − k0 − eBx − ikx
+√2eB sin k0
+;
+b† = kz − k0 − eBx + ikx
+√2eB sin k0
+.
+(30)
+
+11
+2
+0
+2
+kz
+4
+2
+0
+2
+4
+E
+Numerical
+Analytical
+Chiral LLL(numerical)
+Chiral LLL(analytical)
+2
+0
+2
+ky
+4
+2
+0
+2
+4
+E
+Numerical
+Analytical(near WN)
+Chiral LLL(numerical)
+Chiral LLL(analytical near WN)
+FIG. 9: Landau Levels for the two-band Weyl Semimetal
+calculated analytically from Eqn. 25 and numerically, with
+t = 1 = t′, γ = 0 and B = 2π
+51 ˆz(upper) and
+B = 2π
+51 ˆy(lower). The (black) bands indicate the numerically
+calculated Landau levels and the (red) bands for the
+analytically calculated Landau levels for n = 1, 2, ..., 19. The
+(blue) band and the dotted (magenta) band is the Lowest
+Landau Level(LLL) calculated numerically and analytically,
+and is responsible for the Chiral anomaly in the upper figure
+and Weyl orbits in the lower figure.
+The system in Eqn. 29 then changes to,
+HW SM(k′) ≈ −kyσz +
+�
+2eB sin k0(bσ+ + b†σ−). (31)
+Similar
+to
+the
+previous
+case,
+it
+is
+possible
+to
+re-
+solve the Hamiltonian into the subspaces spanned by
+{|n, −⟩ , |n − 1, +⟩}, where n is the eigenvalue of the num-
+ber operator, b†b. We get two chiral lowest Landau levels with
+energies, E = ±ky in the bulk, which are responsible for the
+chiral anomaly50.
+B
+Chiral anomaly in the MWSM
+We now study the response of the MWSM to external fields
+for comparison with the signatures of the chiral anomaly in the
+WSM reviewed in the previous section. We treat the MWSM
+parallel and perpendicular cases separately, given the expected
+sensitivity of the response to orientation of the axes of node
+separation relative to the orientation of the external fields.
+1
+Landau levels in the MWSM parallel system:
+In Sec. III A we have derived the Dirac Hamiltonian for the
+MWSM|| in the vicinity of each of its two nodes, (0, 0, k01)
+and (0, 0, k02) derived respectively from each of its two par-
+ents.
+Hc
+||,1(k) =(t′
+1kxτ x + t′
+1kyτ y
++ t1 sin k01¯kz,1τ z)t2(γ1 − γ2)σz,
+Hc
+||,2(k) =t1(γ1 − γ2)τ z(−t′
+2kxσx + t′
+2kyσy
+− t2 sin k02¯kz,2σz)
+In this section, we will only consider cases where γ1 ̸= γ2. To
+investigate the response to external fields for the MWSM||, we
+consider the effect of magnetic field along the Weyl node axis,
+i.e., B = Bˆz. We use the exact Peierls substitution in Eqn.
+S18, so that the two expressions above transform as follows,
+Hc
+||,1(k′) =t2(γ1 − γ2)(t1 sin k01¯kz,1τ z
++ t′
+1
+√
+2eB(aτ + + a†τ −))σz,
+Hc
+||,2(k′) =t1(γ1 − γ2)τ z(−t2 sin k01¯kz,2σz
+− t′
+2
+√
+2eB(aσ− + a†σ+)).
+(32)
+Here τ ± =
+1
+2(τ x ± iτ y) and σ± =
+1
+2(σx ± iσy) are the
+pseudo-spin ladder operators in the τ and σ spaces. The low-
+est Landau levels from the above two expressions are given
+below,
+Hc
+||,1 →E1,LLL = ±(γ1 − γ2)t1t2 sin k01¯kz,1,
+|ψ1,LLL⟩ = |0; −, ±⟩ ,
+Hc
+||,2 →E2,LLL = ∓(γ1 − γ2)t2t2 sin k02¯kz,2,
+|ψ2,LLL⟩ = |0; ±, +⟩ .
+(33)
+One may notice that the eigenvector |0; −, +⟩ occurs in the
+vicinity of each node.
+Therefore, we calculate its energy
+eigenvalue if one expands the MWSM parallel system in the
+vicinity of the kz axis. The details of the calculation can be
+found in the Supplementary Materials S3. We find the energy
+is given as,
+E|0;−,+⟩ = (Q1Q2 + 1
+2eB(Q1 + Q2)).
+(34)
+We show that this expression is consistent with the numer-
+ically calculated Landau levels in Fig. 10.
+The other chi-
+ral Landau level consistent with the other two eigenvectors,
+|0; −, −⟩ and |0; +, +⟩ near their respective Weyl nodes ap-
+pears distinct from |0; −, +⟩ away from the Weyl nodes.
+
+12
+/2
+0
+/2
+kz
+0.6
+0.4
+0.2
+0.0
+0.2
+0.4
+0.6
+E
+(a)Landau Levels for B = Bz in MWSM pll
+/2
+0
+/2
+ky
+0.6
+0.4
+0.2
+0.0
+0.2
+0.4
+0.6
+E
+(b)Landau Levels for B = By in MWSM pll
+FIG. 10: The Landau Levels for the MWSM parallel
+Hamiltonian with γ1 = −0.5, γ2 = 0.5,
+t1 = t′
+1 = t2 = t′
+2 = 1 and B = 2π
+80 . (a) and (b) show the
+Landau levels for the magnetic field along the Weyl axis and
+perpendicular to the Weyl axis (at Weyl node (0, 0, π
+3 )
+respectively. The (red) bands refer to the lowest Landau
+levels and the (black) bands form the bulk Landau levels.
+In Fig. 10, for certain values of γ1 and γ2, it appears, at
+first glance, as if there are two separate, chiral Landau lev-
+els corresponding to |0; −, −⟩ and |1; , −, −⟩ respectively. All
+four Weyl nodes are connected by each of these LLLs, how-
+ever, and the two LLLs in combination furthermore account
+for each chirality at each node. Although this is reminiscent of
+the Dirac semimetal, there is potentially a distinction in char-
+acter between the chiralities at each node. If each parent cor-
+responds to a particular degree of freedom, for instance, and
+these dofs are physically distinct from one another in some
+sense, such as one parent corresponding to a two-fold valley
+dof, and the other corresponding to a two-fold layer dof, the
+chiral anomalies are inequivalent and do not compensate one
+another as they would for a Dirac semimetal.
+The two apparently ’separated’ LLLs seem to only scatter
+between the Weyl nodes derived from their respective parents,
+i.e.
+intra-parent scattering.
+Upon closer inspection, how-
+ever, we see the intersection point between two apparently
+separated Landau levels is actually a very small gap.
+We
+have verified in Supplementary Sec. S3, that the gap is fi-
+nite in analytical calculations performed to second order in
+momenta. The gap is an emergent feature of the multiplica-
+tive chiral anomaly, with the single LLL reducing to |0; −, −⟩
+and |0; +, +⟩ at nodes associated with a particular parent. We
+therefore interpret the multiplicative chiral anomaly as ex-
+hibiting parent-graded features as well as emergent features
+not associated with either individual parent. This is reminis-
+cent of the topologically robust floating bands of the multi-
+plicative Chern insulator46.
+2
+Landau levels in the MWSM perpendicular system:
+In Sec. III B, we had shown the linear expansion of the
+MWSM⊥ Bloch Hamiltonian near each of the nodes corre-
+sponding to one parent with Weyl nodes separated along the
+ky axis and the other parent with Weyl nodes separated along
+the kz axis in Eqn. 18 and 19. Without loss of generality, we
+consider, t31 = t32 = t21 = t22 = 1 = t11 = t12. There
+exists three separate cases one needs to check - (i) magnetic
+field along the Weyl axis of the first parent, B = Bˆy, (ii) mag-
+netic field along the Weyl axis of the second parent, B = Bˆz,
+and (iii) magnetic field perpendicular to the Weyl axis of both
+parents, B = Bˆx.
+• Case 1 (B = Bˆy) : Substituting, kx → k′
+x = kx+eBz,
+and using the bosonic ladder operators, a⊥,y = kz−ik′
+x
+√
+2eB ,
+a†
+⊥,y = kz+ik′
+x
+√
+2eB , we have, from Eqn. 18,
+H⊥,1(k′) =(sin k0,1(ky − k0,1)τ z +
+√
+2eB(a⊥,yτ + + a†
+⊥,yτ −)
+⊗ (sin k0,1σy + (γ1 − γ2)σz).
+(35)
+For the expression from Eqn.
+19, we instead con-
+sider the following bosonic ladder operators, ˜a⊥,y =
+˜
+kz−ik′
+x
+√
+2eB sin k0,2 and ˜a†
+⊥,y =
+˜
+kz+ik′
+x
+√
+2eB sin k0,2 , which gives us,
+H⊥,2(k′) =(sin k0,2τ y + (γ1 − γ2)τ z)
+⊗ (kyσy −
+�
+2eB sin k0,2(˜a⊥,yσ+
+y + ˜a†
+⊥,yσ−
+y )).
+(36)
+It is easy to find the lowest Landau level energies in
+the vicinity of each node. From Eqn. 35 and 36, we
+respectively have the LLL energies,
+Ey,1,LLL = ±
+�
+sin2 k0,1 + (γ1 − γ2)2 sin k0,1(ky − k0,1),
+Ey,2,LLL = ±
+�
+sin2 k0,2 + (γ1 − γ2)2ky.
+(37)
+We then find two ky-dependent chiral LLLs connecting
+the nodes of the first parent, while we have two chiral
+LLLs at ky = 0 due to the second parent, as shown in
+Fig. 11 (a). The following result was expected if one
+considers the Landau levels for the parents for different
+
+13
+directions of the magnetic field discussed in the previ-
+ous subsection. For the MWSM perpendicular case, the
+incident magnetic field in this case is both parallel to
+the Weyl axis of parent 1 and perpendicular to the Weyl
+axis of parent 2, so that we get both kinds of Landau
+levels simultaneously.
+• Case 2 (B = Bˆz): This produces results similar to Case
+1, as shown in Fig. 11 (b). A similar calculation gives
+us the lowest Landau level energies,
+Ez,1,LLL = ±
+�
+sin2 k0,1 + (γ1 − γ2)2kz,
+Ez,2,LLL = ±
+�
+sin2 k0,2 + (γ1 − γ2)2 sin k0,2(kz − k0,2).
+(38)
+VI
+Discussion and Conclusion
+In this work, we have introduced the previously-unidentified
+multiplicative topological semimetal phases of matter, distin-
+guished by Bloch Hamiltonians with a symmetry-protected
+tensor product structure. Parent Bloch Hamiltonians, with ei-
+ther one or both of the parents being topologically non-trivial,
+may then be combined in the tensor product to realize mul-
+tiplicative topological semimetal phases inheriting topology
+from the parent states.
+We consider foundational examples of multiplicative topo-
+logical semimetals, with Bloch Hamiltonians constructed as
+tensor products of two-band Bloch Hamiltonians, each char-
+acterizing a Weyl semimetal phase. These multiplicative topo-
+logical semimetal phases are protected by a combination of
+symmetries of class DIII at the level of the child, and each
+parent Bloch Hamiltonian in class D. Given the great vari-
+ety of exotic crystalline point group symmetries considered to
+protect most recently-identified topological semimetal phases,
+it is remarkable that the symmetry-protection of these multi-
+plicative semimetal phases is relatively simple, and suggests
+many additional multiplicative semimetal phases may be iden-
+tified by enforcing these many other symmetries on parent
+Bloch Hamiltonians.
+We first characterize multiplicative topological semimetal
+phases in the bulk, showing the bulk spectrum of the child
+Bloch Hamiltonian depends in a multiplicative way on the
+spectra of the parent Bloch Hamiltonians: each eigenvalue of
+the child, at a given point in k-space, is a product of eigen-
+values, one from each parent. We furthermore consider two
+different constructions of the multiplicative Weyl semimetal,
+either for the case of each parent having a pair of Weyl nodes
+separated along the same axis in k-space (parallel construc-
+tion), or along perpendicular axes in k-space (perpendicu-
+lar construction). For either construction, the multiplicative
+symmetry-protected structure can then naturally yield nodal
+degeneracies reminiscent of Dirac nodes or higher-charge
+Weyl nodes. However, the multiplicative degeneracies are dis-
+tinguished from these more familiar quasiparticles by distinc-
+tive Wannier spectra signatures in the bulk, and exotic bulk-
+boundary correspondence. Importantly, bulk characterization
+/2
+0
+/2
+ky
+0.6
+0.4
+0.2
+0.0
+0.2
+0.4
+0.6
+E
+(a)Landau Levels for B = By in MWSM perp
+/2
+0
+/2
+kz
+0.6
+0.4
+0.2
+0.0
+0.2
+0.4
+0.6
+E
+(b)Landau Levels for B = Bz in MWSM perp
+FIG. 11: Landau levels for the MWSM perpendicular system
+with γ1 = −0.5 and γ2 = 0.5 representing separation of
+Weyl nodes along the ky and kz direction respectively. We
+show two cases, (a) when magnetic field is along the
+y-direction and (b) when magnetic field is along the
+z-direction. Red lines indicate the chiral Landau levels. since
+the magnetic field is paralle to one Weyl node separation and
+perpendicular to another Weyl node separation, the above
+behaviour is expected.
+by Wannier spectra reveals a complex dependence of Berry
+connection in the child Bloch Hamiltonian on Berry connec-
+tion of each parent Bloch Hamiltonian, depending on whether
+the parents are constructed with Weyl nodes separated along
+the same axis in momentum-space (parallel) or not (perpen-
+dicular). Additionally, the connectivity of Fermi arc surface
+states for the multiplicative Weyl semimetal is far more com-
+plex than in standard Dirac or Weyl semimetals, reflecting the
+underlying dependence of the child topology on the topology
+of the parents. An especially interesting example is the re-
+alization of topologically-protected—yet floating—boundary
+states.
+Response signatures of the multiplicative Weyl semimetal
+
+14
+also inherit response signatures of the parents, with the po-
+tential for emergent phenomena beyond that of either parent
+individually. Here, we consider the multiplicative analog of
+one of the defining response signatures of the Weyl semimetal,
+the chiral anomaly, finding instead multiple co-existing chiral
+anomalies graded by the parent degrees of freedom, as well as
+emergent features in the Landau level structure not inherited
+from a particular parent. In the case of parents correspond-
+ing to effectively the same degree of freedom, the response
+reduces to a signature reminiscent of a Dirac semimetal. This
+brings up the possibility of controlled manipulation of partic-
+ular properties of an electronic system similar to spintronics.
+Future work will characterize other signatures of multi-
+plicative topological semimetals anticipated given the exten-
+sive characterization of Weyl and Dirac semimetals, particu-
+larly optical and non-linear responses given the tremendous
+interest in the bulk photovoltaic effect in Weyl semimetals, as
+well as symmetry-protection of more exotic topological quasi-
+particles, such as multiplicative generalizations of multifold
+fermions or nodal lines. Given the immense body of work
+on topological semimetals and the surprising consequences
+of multiplicative topology for bulk-boundary correspondence,
+nodal band structure, and Berry phase structure, our intro-
+duction of previously-unidentified multiplicative topological
+semimetals into the literature lays the foundation for consid-
+erable future theoretical and experimental study, which will
+greatly expand and deepen our understanding of topological
+semimetal phases.
+Acknowledgements - We gratefully acknowledge help-
+ful discussions with J. E. Moore, I. A. Day, D. Varjas and
+R. Calderon.
+Correspondence
+-
+Correspondence
+and
+requests
+for materials should be addressed to A.M.C. (email:
+[email protected]).
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+17
+Supplemental material for “Multiplicative topological semimetals”
+Adipta Pal1,2, Joe H. Winter1,2,3, and Ashley M. Cook1,2,∗
+1Max Planck Institute for Chemical Physics of Solids, N¨othnitzer Strasse 40, 01187 Dresden, Germany
+2Max Planck Institute for the Physics of Complex Systems, N¨othnitzer Strasse 38, 01187 Dresden, Germany
+3SUPA, School of Physics and Astronomy, University of St. Andrews, North Haugh, St. Andrews KY16 9SS, UK
+∗Electronic address: [email protected]
+S1
+Wilson loops for multiplicative Weyl semi-metal:
+Labelling the parent Hamiltonians as Hp1 = (d1x, d1y, d1z) · τ and Hp2 = (d2x, d2y, d2z) · σ, with eigenvectors, {|+1⟩ |−1⟩}
+and {|+2⟩ , |−2⟩} respectively, the child Hamiltonian is given by Hc = Hp1 ⊗ H′
+p2, where H′
+p2 = (−d2x, d2y, −d2z) · σ. The
+ground state subspace of the child Hamiltonian is then spanned by, {|+1⟩ |−2⟩′ , |−1⟩ |+2⟩′} = {|+1⟩ |+2⟩, |−1⟩ |−2⟩}, where
+|ψ⟩ denotes complex conjugation and ’ denotes an eigenstate of H′
+p2. The non-abelian Berry connection is then given as follows:
+Aµ =i
+�
+⟨+1, +2| ∂µ |+1, +2⟩
+⟨−1, −2| ∂µ |−1, −2⟩
+�
+= i
+�
+⟨+1| ∂µ |+1⟩
+⟨−1| ∂µ |−1⟩
+�
++ i
+�
+⟨+2|∂µ|+2⟩
+⟨−2|∂µ|−2⟩
+�
+,
+=
+�A+
+1,µ − A+
+2,µ
+A−
+1,µ − A−
+2,µ
+�
+,
+(S1)
+where Al
+j,µ = i ⟨lj| ∂µ |lj⟩. For Berry connection around a loop in the Brillouin zone, the values of µ are {kx, ky, kz} for a 3d
+Brillouin Zone. This clearly shows the difference between the parallel multiplicative phases and the perpendicular multiplicative
+phases. For a 1d BZ, as shown in past work46, the connection for parallel MKC is A = (A1,kx − A2,kx, 0, 0), while for
+the perpendicular MKC it is, A = (A1,kx, −A2,ky, 0). For 2d or 3d parent systems, it then becomes very straightforward to
+extrapolate this trend such that the Berry connection looks qualitatively like the combination of the parallel and perpendicular
+MKC connections based on which directions the parents have in common. This is particularly interesting for the case of parallel
+and perpendicular Multiplicative Chern Insulators(MCIs), where parent CIs are each defined over a 2d BZ, and the parents can
+share one or two axes. We illustrate the MCI parallel with two parent CIs on the x-y plane. The resultant Berry connection is
+then A = (A1,kx − A2,kx, A1,ky − A2,ky, 0). The MCI perpendicular on the other hand is constructed with one parent in the x-y
+plane and another in the x-z plane. The resulting Berry connection is then, A = (A1,kx − A2,kx, A1,ky, −A2,kz). The MWSM,
+on the other hand, is a 3d system, so we instead consider parent Weyl nodes separated along parallel or perpendicular axes in
+k-space. As explained in the main text, the parallel MWSM has parent Weyl nodes separated along the same axis in k-space
+(the kz axis) while the perpendicular MWSM has parent 1 and parent 2 Weyl nodes separated along the ky-axis and kz-axis,
+respectively. The resultant Berry connection is then, A = (A1,kx − A2,kx, A1,ky − A2,ky, A1,kz − A2,kz).
+S2
+Calculation for the surface state spectrum of MWSM:
+We write down here the derivation for the surface state energy for the MWSM parallel and MWSM perpendicular Hamiltonians,
+for the case of open boundary conditions in the ˆx direction and periodic boundary conditions in the ˆy and ˆz directions. First, we
+briefly specify how such a calculation should be done for the two band Weyl semi-metal.
+A
+Slab spectra for WSM:
+We start by writing down the WSM Hamiltonian used,
+HW SM(k) =t3(2 + γ − cos kx − cos ky − cos kz)σz + t2 sin kyσy + t1 sin kxσx,
+=t3(f − cos kx)σz + t2 sin kyσy + t1 sin kxσx,
+(S2)
+where f = 2 + γ − cos ky − cos kz. Surface states decay into the bulk, so for open boundaries in the x-direction, we carry out
+the transformation, kx → iq for edge states on the left side (x = 0), so that,
+HW SM(iq, ky, kz) = t3(f − cosh q)σz + t2 sin kyσy + it1 sinh qσx.
+(S3)
+We claim that the determinant derived from the matrix due to the following limit must be zero,
+lim
+q1→q2
+H(iq1) − H(iq2)
+2 sinh q−
+= −t3 sinh q+σz + it1 cosh q+σx,
+(S4)
+where q± = 1
+2(q1 ± q2). Carrying out the determinant, we get the following two conditions,
+t3 sinh q+ = ±t1 cosh q+.
+(S5)
+
+18
+Choosing the + sign, the RHS in Eqn. S4 becomes, −t1 cosh q+(σz − iσx), so that the null eigenvector derived from it is one
+of the eigenvectors for the surface spectra,
+|ψ+⟩ =
+1
+√
+2
+�
+1
+i
+�
+.
+(S6)
+The energy corresponding to this eigenvector can be found by solving the eigenvalue for the RHS in Eqn. S3 with the above
+eigenvector. This gives us the eigen-energy, E, and the equation to determine the eigen-function, for the left boundary
+E = t2 sin ky,
+(S7a)
+(t3 + t1)e−2q + 2fe−q + (t3 − t1) = 0,
+=⇒ e−q± = −f ±
+�
+f 2 − (t2
+3 − t2
+1)
+(t3 + t1)
+,
+Ψ+(x, y, z) ∼ (e−q+x − e−q−x)eikyy+ikzz |ψ+⟩ .
+(S7b)
+The eigen-function in the last line has the following form based on the boundary condition on the left edge, Ψ(x = 0) = 0. The
+other edge can be derived similarly by shifting x → L + 1 − x where L is the length of the system along the x-direction.
+B
+Slab spectra for MWSM parallel:
+We use the same method as in section S2A of the supplementary materials to derive surface states and spectra for the MWSM
+parallel system. The Hamiltonian is given as follows,
+HMW SM||(k) = [t31(f1 − cos kx)τ z + t21 sin kyτ y + t11 sin kxτ x] ⊗ [−t32(f2 − cos kx)σz + t22 sin kyσy − t12 sin kxσx],
+(S8)
+where f1/2 = 2 + γ1/2 − cos ky − cos kz. To ease our calculations, we carry out the following rotation on the four band basis,
+τ z → τ y, τ y → −τ z and σz → −σy, σy → −σz. The Hamiltonian then becomes,
+HMW SM||(k) =[t31(f1 − cos kx)τ y − t21 sin kyτ z + t11 sin kxτ x] ⊗ [−t32(f2 − cos kx)σy − t22 sin kyσz − t12 sin kxσx],
+=[t31(f1 − cos kx)τ y + t11 sin kxτ x][−t32(f2 − cos kx)σy − t12 sin kxσx]
+− t21 sin kyτ z[−t32(f2 − cos kx)σy − t12 sin kxσx] − t22 sin ky[t31(f1 − cos kx)τ y + t11 sin kxτ x]σz
++ t21t22 sin2 kyτ zσz.
+(S9)
+Again, without loss of generality, we set t11 = t21 = t31 = 1 = t32 = t22 = t12. The edge modes on the left edge (x = 0),
+require we carry out the substitution, kx → iq, and the Hamiltonian is now,
+HMW SM||(iq, ky, kz) =[(f1 − cosh q)τ y + i sinh qτ x][−(f2 − cosh q)σy − i sinh qσx]
+− sin kyτ z[−(f2 − cosh q)σy − i sinh qσx] − sin ky[(f1 − cosh q)τ y + i sinh qτ x]σz
++ sin2 kyτ zσz.
+(S10)
+Carrying out our previous limit on the rotated Hamiltonian above, we get the following matrix,
+lim
+q1→q2
+HMW SM||(iq1) − HMW SM||(iq2)
+2 sinh q−
+=
+�
+�
+�
+0
+i sin kyS+
+−i sin kyS+
+S+(−(f1 + f2) + 2S+)
+i sin kyS−
+0
+−f1S− + f2S+
+i sin kyS+
+−i sin kyS−
+f1S+ − f2S−
+0
+−i sin kyS+
+S−((f1 + f2) − 2S−)
+i sin kyS−
+−i sin kyS−
+0
+�
+�
+� ,
+(S11)
+where S± = cosh q+ ± sinh q+. The determinant of the RHS of Eqn. S11 must be zero, i.e., we have the condition,
+S−S+[sin2 kyS−(f1 + f2 − 2S+)(f1 − f2)(S− + S+) − sin2 kyS+(f1 + f2 − 2S−)(f1 + f2)(S+ − S−)
+− (f1 + f2 − 2S+)(f1 + f2 − 2S−)(f1S− − f2S+)(−f2S− + f1S+)] = 0
+(S12)
+Let us start with the first condition, S− = 0. The RHS of Eqn. S11 then becomes,
+lim
+q1→q2
+HMW SM||(iq1) − HMW SM||(iq2)
+2 sinh q−
+=S+(−(f1 + f2) + 2S+)τ +σ+ + f2S+τ +σ− + f1S+τ −σ+
++ i sin kyS+τ zσ+ − i sin kyS+τ +σz,
+(S13)
+
+19
+where τ ± = 1
+2(τ x ±iτ y) and σ± = 1
+2(σx ±iσy) are the two level ladder operators. Here, if {|+⟩ , |−⟩} are eigen-vectors of τ z,
+then τ + |−⟩ = |+⟩, τ + |+⟩ = 0, τ − |−⟩ = 0 and τ − |+⟩ = |−⟩. Similar relations exist for the σ counterpart. |ψ1⟩ = |+⟩⊗|+⟩ is
+a null eigen-vector to the above expression on the RHS. We solve for the energy eigenvalue first for the special case γ1 = γ2.Then
+from HMW SM||(iq, ky, kz) in Eqn. S10 due to the eigen-vector |ψ1⟩, we have the energy and the condition,
+E = sin2 ky;
+(S14a)
+(f1 − cosh q + sinh q)(f2 − cosh q + sinh q) = 0.
+(S14b)
+S3
+Landau Level repulsion in the MWSM parallel system:
+We start with the MWSM parallel case,
+HMW SM,||(k) =[t1(2 + γ1 − cos kx − cos ky − cos kz)τ z + t′
+1 sin kyτ y + t′
+1 sin kxτ x]
+⊗ [−t2(2 + γ2 − cos kx cos ky − cos kz)σz + t′
+2 sin kyσy − t′
+2 sin kxσx].
+(S15)
+We expand the Bloch Hamiltonian near the z-axis i.e. k → (0, 0, kz),
+HMW SM,||(k) ≈[t1(Q1 + 1
+2(k2
+x + k2
+y))τ z + t′
+1kyτ y + t′
+1kxτ x]
+⊗ [−t2(Q2 + 1
+2(k2
+x + k2
+y))σz + t′
+2kyσy − t′
+2kxσx],
+(S16)
+where Qi = γi − cos kz (i=1,2). Expanding only up to second order in momenta, we have,
+HMW SM,||(k) ≈ − t1t2(Q1Q2 + (Q1 + Q2)1
+2(k2
+x + k2
+y))τ zσz
+− t1t′
+2Q1τ z(kxσx − kyσx) − t′
+1t2Q2(kxτ x + kyτ y)σz
+− t′
+1t′
+2(k2
+xτ xσx − k2
+yτ yσy − 1
+2(kxky + kykx)τ xσy + 1
+2(kxky + kykx)τ yσx).
+(S17)
+/2
+0
+/2
+kz
+0.4
+0.2
+0.0
+0.2
+0.4
+E
+numerical
+1st order
+2nd order
+FIG. S12: Comparison of the numerically calculated Landau Levels of the MWSM||. system with the analytically calculated
+lower Landau levels for first order(blue dashed) and second order(red) expansion in momenta along the direction perpendicular
+to kz. Level repulsion between two parent graded lowest Landau levels are only observed if one expands to second order in
+momenta.
+We consider B = Bˆz. After Peierls substitution, kx → k′
+x = kx, ky → k′
+y = ky + eBx, and kz → k′
+z = kz. The position-
+momenta commutator leads to the commutator, [k′
+y, k′
+x] = ieB. Here, e is the charge of the particle in consideration. One can
+therefore construct bosonic ladder operators of the form,
+a = k′
+x − ik′
+x
+√
+2eB
+;
+a† = k′
+x + ik′
+y
+√
+2eB
+;
+[a, a†] = 1.
+(S18)
+
+20
+We calculate some important identities via Eqn.S18 which we will be using in the next few lines,
+1
+2(k′
+x
+2 + k′
+y
+2) = eB(a†a + 1
+2);
+k′
+xσx + k′
+yσy =
+√
+2eB(aσ+ + a†σ−);
+k′
+xσx − k′
+yσy =
+√
+2eB(aσ− + a†σ+),
+k′
+x
+2 − k′
+y
+2 = eB(a2 + a†2);
+i[k′
+xk′
+y + k′
+yk′
+x] = eB(a†2 − a2),
+(S19)
+where we have used τ ± = 1
+2(τ x ± iτ y) and σ± = 1
+2(σx ± iσy), which are spin ladder operators in the basis {|+⟩ , |−⟩} in
+both the τ and σ spaces. Now, substituting k for k′ in Eqn. S17 and then transforming them via Eqn. S19, we get the following
+expression,
+HMW SM,||(k′) ≈ − t1t2(Q1Q2 + (Q1 + Q2)eB(a†a + 1
+2))τ zσz − t1t′
+2Q1
+√
+2eBτ z(aσ− + a†σ+) − t′
+1t2Q2
+√
+2eB(aτ + + a†τ −)σz
+− t′
+1t′
+2(2eB)(a†a + 1
+2)(τ +σ+ + τ −σ−) − t′
+1t′
+2(2eB)(a2τ +σ− + a†2τ −σ+).
+(S20)
+Let us ignore the second order perturbations not in the mass term (i.e. τ zσz) and simplify the Hamiltonian,
+HMW SM,||(k′) ≈ −(Q1Q2 +(Q1 +Q2)eB(a†a+ 1
+2))τ zσz −Q1
+√
+2eBτ z(aσ− +a†σ+)−Q2
+√
+2eB(aτ + +a†τ −)σz. (S21)
+We obtain one of the lowest Landau levels, |ψ⟩1,LLL = |0; −, +⟩ with energy E1,LLL = (Q1Q2 + eB
+2 (Q1 + Q2)) which match
+exactly both numerically and analytically in first and second order expansions. For the other lowest Landau level, we observe an
+amalgamation of chiral Landau levels obtained from each parent which cause level repulsion at the intersection point.
+S4
+Euler space topology calculation
+In the main text, we have already reported that the MWSM system possesses both time reversal, T and inversion symmetry, I and
+hence the combined symmetry, T ′ denoted by τ yσyκ, where κ refers to complex conjugation. However, here T ′2 = 1, so that
+a Z2 invariant is not possible. Instead, it is possible to find a basis, where T ′ = κ. Here we provide the unitary transformation
+which makes this possible,
+V = 1
+2[(1 + i)τ 0σ0 + (1 − i)τ yσy].
+(S22)
+Based on the method provided in the appendix in a previous work52, the above unitary transformation satisfies, V τ yσyV T = 1,
+so that we get a Hamiltonian, ˜H(k) = V H(k)V † which satisfies, ˜H(k) = ˜H∗(k), and is real and symmetric. Denoting the
+MWSM in a condensed notation,
+H = (M1τ z + Q1τ x + R1τ y) ⊗ (−M2σz − Q2σx + R2σy),
+(S23)
+we obtain after the transformation,
+˜H =
+M1(−M2τ zσz − Q2τ zσx + R2τ xσ0)
+− Q1(M2τ xσz + Q2τ xσx + R2τ zσ0)
+− R1(M2τ 0σx − Q2τ 0σz − R2τ yσy).
+(S24)
+Comparing with the method introduced in52, it is possible to view the real Hamiltonian as an element of a Real oriented Grass-
+mannian, ˜GR
+2,4 which is diffeomorphic to S2×S2. For a given kz, then it is possible to define a mapping from the 2d BZ spanned
+by kx and ky (for MWSM ||) into (n1, n2) ∈ S2 × S2 and the topology of ˜H is then determined by the two skyrmion numbers,
+Q[n1] = q1 and Q[n2] = q2 of parent 1 and parent 2, respectively. The Euler class topology is then found from these skyrmion
+numbers as follows,
+EI = q2 − q1;
+EII = q2 + q1.
+(S25)
+The Euler numbers are unique up to the mapping (EI, EII) → (−EI, −EII).
+

8dAzT4oBgHgl3EQfE_q4/content/tmp_files/2301.01004v1.pdf.txt

@@ -0,0 +1,307 @@
+arXiv:2301.01004v1  [math.RT]  3 Jan 2023
+SPIN NORM AND LAMBDA NORM
+CHAO-PING DONG AND DU CHENGYU
+Abstract. Given a K-type π, it is known that its spin norm (due to first-named author)
+is lower bounded by its lambda norm (due to Vogan). That is, ∥π∥spin ≥ ∥π∥lambda. This
+note aims to describe for which π one can actually have equality. We apply the result to
+tempered Dirac series. In the case of real groups, we obtain that the tempered Dirac series
+are divided into #W 1 parts among all tempered modules with real infinitesimal characters.
+1. Introduction
+Let G be a linear real reductive Lie group which is in the Harish-Chandra class [5]. That
+is,
+• G has only a finite number of connected components;
+• The derived group [G, G] has finite center;
+• The adjoint action Ad(g) of any g ∈ G is an inner automorphism of g = (g0)C, where
+g0 is the Lie algebra of G.
+Let θ be a Cartan involution of G. We assume the subgroup K = Gθ of fixed points
+of θ is a maximal compact subgroup of G. Let g0 = k0 ⊕ s0 be the corresponding Cartan
+decomposition of g0. We drop the subscript for the complexification.
+Let ˆGtemp,o denote the set of irreducible tempered representations with real infinitesimal
+character (up to equivalence). Let ˆK denote the set of K-types. The following bijection was
+noted by Trapa [10], after Vogan’s paper [12].
+Theorem 1.1. Let X be any irreducible tempered (g, K)-module with real infinitesimal char-
+acter. Then X has a unique lowest K-type which occurs with multiplicity one. Moreover,
+the map
+φ : ˆGtemp,o → ˆK
+defined by taking the lowest K-type, is a well-defined bijection.
+Motivated by the lambda norm introduced by Vogan [11], the first-named author intro-
+duced spin norm [4] for the classification of Dirac series (i.e., irreducible unitary representa-
+tions of G with non-vanishing Dirac cohomology). We refer the reader to [6] and references
+therein for the notion of Dirac cohomology. It was proven in [4] that the spin norm of a
+K-type π is bounded below by its lambda norm. That is,
+(1)
+∥π∥spin ⩾ ∥π∥lambda.
+2010 Mathematics Subject Classification. Primary 22E46.
+Key words and phrases. lambda norm, spin norm, tempered representations.
+1
+
+2
+CHAO-PING DONG AND DU CHENGYU
+This inequality turns out to have a nice interpretation in the setting of Theorem 1.1. Indeed,
+let ˆGtemp,d collect the members of ˆGtemp,o with non-zero Dirac cohomology. Put
+ˆKe := {π ∈ ˆK| ∥π∥spin = ∥π∥lambda}.
+Theorem 1.2. ([2]) The map φ restricts to ˆGtemp,d is a bijection onto ˆKe. More precisely,
+any member π ∈ ˆGtemp,o is a Dirac series if and only if the inequality (1) becomes an equality
+on its unique lowest K-type.
+Given an arbitrary K-type π, it is not easy to compute neither ∥π∥lambda nor ∥π∥spin.
+Thus, it is subtle to detect whether the inequality (1) is strict or not. This note aims to
+give a criterion on this aspect. Our main result is Theorem 3.4. The main idea is to insert
+an intermediate value between ∥π∥lambda and ∥π∥spin. As an application, our result suggests
+that the tempered Dirac series should be separated into #W 1 parts. See (5) for the definition
+of W 1.
+The note is outlined as follows: In Section 2, we recall lambda norm and spin norm. Then
+we deduce our main result in Section 3. The last section considers tempered Dirac series.
+2. Preliminaries
+In this section, we briefly recall the definitions of the spin norm and the lambda norm.
+2.1. The lambda norm. We keep the notations K, G, k, s, θ, etc as in the previous section.
+Let T be a maximal torus of K and t0 be the Lie algebra of T. Recall that the analytic
+Weyl group is defined by
+W(k, t) = NK(T)/AK(T).
+It acts on the root system ∆(k, t). Fix a choice of positive roots ∆+(k, t), and define
+R(G) := {r ∈ W(k, t)|r∆+(k, t) = ∆+(k, t)}.
+Given a K-type π, by Lemma 0.1 of [9], the collection of highest weights of π as k-module
+is a single orbit of R(G) on ˆT ∈ it∗
+0, where ˆT is the abelian group of characters of T.
+Now given any K-type π, take a highest weight µ of it. Then µ ∈ it∗
+0 is dominant integral
+for ∆+(k, t). Denote by ρc the half sum of all roots in ∆+(k, t). Choose a positive root system
+∆+(g, t) making µ + 2ρc dominant. Denote by ρ the half sum of all roots in ∆+(g, t). Let
+P be the projection map to the dominant chamber C(g) corresponding to ∆+(g, t). Then
+∥P(µ+2ρc −ρ)∥ is independent of the choices of µ and ∆+(g, t), cf. Section 1 and Corrollary
+2.4 of [9]. Now we are ready to talk about the lambda norm.
+Definition 2.1. ([11, 1]) For any π ∈ ˆK, the lambda norm of π is defined to be
+(2)
+∥π∥lambda := ∥P(µ + 2ρc − ρ)∥,
+where µ is any highest weight of π. For any irreducible admissible (g, K)-module X, the
+lambda norm of X is defined to be
+(3)
+∥X∥lambda := min
+π ∥π∥lambda,
+where π runs over all the K-types occurring in X. A K-type π is called a lowest K-type of
+X if it occurs in X and ∥π∥lambda = ∥X∥lambda.
+
+SPIN NORM AND LAMBDA NORM
+3
+2.2. The spin norm. Although the original definition of the spin norm involves the spin
+module SG of the Clifford algebra C(s), our discussion here does not need a deep under-
+standing of it. Our tool is mainly the root systems and their Weyl groups.
+Definition 2.2. ([4]) For any π ∈ ˆK, its spin norm is defined to be
+∥π∥spin := min ∥γ + ρc∥,
+where γ runs over all the highest weights of the ˜K-types in π ⊗ SG. For any irreducible
+admissible (g, K)-module X, its spin norm is defined to be
+∥X∥spin := min
+π ∥π∥spin,
+where π runs over all the K-types occurring in X. We call π a spin lowest K-type of X if
+it occurs in X and ∥π∥spin = ∥X∥spin.
+3. When is the inequality (1) strict?
+We fix a positive root system ∆+(k, t), and denote the half sum of roots in it by ρc. Let
+W(g, t) (resp., W(k, t))) be the Weyl group of ∆(g, t) (resp., ∆(k, t)). Let C(k) be the closed
+dominant Weyl chamber for ∆+(k, t). For any µ ∈ t∗, we use {µ} to denote the unique weight
+in C(k) to which µ is conjugate under the action of W(k, t). Let ∆+(g, t) be a positive root
+system of ∆(g, t) containing ∆+(k, t).
+Lemma 3.1. ([4, Lemma 3.5]) For any K-type π with a highest weight µ ∈ t∗, we have
+(4)
+∥µ∥spin = min
+w∈W 1 ∥{µ − wρ + ρc} + ρc∥,
+where
+(5)
+W 1 := {w ∈ W(g, t)|wC(g) ⊆ C(k)}.
+Lemma 3.2. ([7, §13.3, Lemma B]) Let λ ∈ C(k). Then
+∥λ + ρc∥ ⩾ ∥wλ + ρc∥
+for any w ∈ W(k, t). Moreover, the equality holds if and only if λ = wλ.
+Lemma 3.3. Let ∆ be a root system with Weyl group W. Fix a positive set ∆+ of roots
+and denote by ρ the half sum of all positive roots. For any dominant weight λ, we have the
+following inequality
+(6)
+∥λ − ρ∥ ⩽ ∥λ − wρ∥, ∀w ∈ W.
+Moreover, if λ is dominant with respect to w∆+, we have
+∥λ − ρ∥ = ∥λ − wρ∥
+Otherwise, the inequality (6) is strict.
+Proof. We first prove the inequality. Compute the following difference
+(∗)
+∥λ − ρ∥2 − ∥λ − wρ∥2 = −2(λ, ρ − wρ).
+A widely known fact is that ρ − wρ is a sum of positive roots. The weight λ is dominant by
+assumption. Thus the pairing (λ, ρ − wρ) is non-negative, and (6) follows.
+
+4
+CHAO-PING DONG AND DU CHENGYU
+Now suppose λ is dominant with respect to w∆+. Notice that the half sum of positive
+roots with respect to w∆+ is wρ. Applying (6) to λ, w∆+ and wρ gives
+∥λ − wρ∥ ⩽ ∥λ − w−1(wρ)∥ = ∥λ − ρ∥.
+Therefore, (6) becomes an equality in the current setting.
+Now suppose λ is not dominant with respect to the new positive set w∆+. Define
+Dw := {γ ∈ ∆−|γ ∈ w∆+},
+where ∆− = −∆+. It is well-known that
+ρ − wρ =
+�
+γ∈Dw
+(−γ).
+By assumption, λ is not dominant with respect to w∆+. There must exist β ∈ w∆+ such
+that (λ, β) < 0. But it cannot live in ∆+, because λ is dominant with respect to ∆+. As a
+consequence, β ∈ Dw. Continuing with (∗), we have that
+−(λ, ρ − wρ) = −
+
+λ,
+�
+γ∈Dw
+(−γ)
+
+ =
+
+λ,
+�
+γ∈Dw
+γ
+
+ ⩽ (λ, β) < 0.
+Thus (6) is strict in this case.
+□
+Let us state the main result of this section.
+Theorem 3.4. Let π be an irreducible representation of K with be a highest weight µ.
+Choose a positive root system ∆+(g, t) making µ + 2ρc dominant. Let C(g) be the closed
+dominant Weyl chamber corresponding to ∆+(g, t). Then the inequality (1) is strict if and
+only if one of the following conditions holds:
+(a) µ + 2ρc is irregular for ∆(g, t).
+(b) µ − wρ + ρc /∈ C(k) for all w ∈ W(g, t) such that µ + 2ρc ∈ wC(g).
+Proof. Let P(·) be the projection map to the cone C(g). It suffices to show that
+(7)
+∥π∥lambda = ∥P(µ + 2ρc − ρ)∥ ≤ ∥µ + 2ρc − ρ∥ ≤ ∥π∥spin,
+that the first equality happens if and only if (a) holds, and that the second equality happens
+if and only if (b) holds.
+By the Pythagorean theorem, the first inequality in (7) holds, and it becomes an equality
+if and only if P(µ + 2ρc − ρ) = µ + 2ρc − ρ, which is equivalent to µ + 2ρc − ρ ∈ C(g). The
+latter is equivalent to (a) since µ + 2ρc ∈ C(g) and µ + 2ρc is integral.
+Now let us consider the second inequality in (7). We collect all w ∈ W(g, t) such that
+µ + 2ρc ∈ wC(g) as W 1(µ). Since µ + 2ρc ∈ C(k), it follows that W 1(µ) ⊆ W 1. Moreover,
+the identity element e ∈ W 1(µ) due to µ + 2ρc ∈ C(g).
+Using Lemma 3.1 and 3.2, we have that
+(8)
+∥π∥spin = min
+w∈W 1 ∥{µ − wρ + ρc} + ρc∥ ⩾ min
+w∈W 1 ∥µ − wρ + 2ρc∥.
+Take λ = µ + 2ρc and ∆+ = ∆+(g, t) in Lemma 3.3. We have
+(9)
+∥µ − wρ + 2ρc∥ ≥ ∥µ − ρ + 2ρc∥.
+
+SPIN NORM AND LAMBDA NORM
+5
+Furthermore, the inequality (9) is strict when w /∈ W 1(µ); yet it is an equality when w ∈
+W 1(µ). Now the second inequality in (7) follows from (8) and (9).
+Assume (b) holds. For any w ∈ W 1 \ W 1(µ), one has that
+∥{µ − wρ + ρc} + ρc∥ ⩾ ∥µ − wρ + 2ρc∥ > ∥µ − ρ + 2ρc∥.
+On the other hand, for all w ∈ W 1(µ), one has that
+∥{µ − wρ + ρc} + ρc∥ > ∥µ − wρ + 2ρc∥ = ∥µ − ρ + 2ρc∥.
+The first strict inequality is due to the assumption that µ − wρ + ρc /∈ C(k) and Lemma 3.2.
+Assume (b) does not hold. Then there exists some w0 ∈ W 1(µ) such that µ − w0ρ + ρc ∈
+C(k). Therefore,
+∥{µ − w0ρ + ρc} + ρc∥ = ∥µ − w0ρ + 2ρc∥ = ∥µ − ρ + 2ρc∥.
+Since we have proven that
+min
+w∈W 1 ∥{µ − wρ + ρc} + ρc∥ ≥ ∥µ − ρ + 2ρc∥,
+we must have ∥π∥spin = ∥µ − ρ + 2ρc∥.
+To sum up, the two inequalities in (7) are controlled by (a) and (b), respectively. Thus
+∥π∥spin > ∥π∥lambda happens if and only if at least one of (a) and (b) holds.
+□
+We record an interesting corollary from the above proof.
+Corollary 3.5. If µ + ρc − wρ ∈ C(k) for some w ∈ W 1(µ). Then
+(10)
+∥µ∥spin = ∥µ + 2ρc − ρ∥.
+4. Application to tempered Dirac series
+We call an irreducible tempered representations with non-zero Dirac cohomology a tem-
+pered Dirac series. Combining Theorems 3.4 and 1.2, we have the following.
+Theorem 4.1. Let X be a tempered (g, K)-module with real infinitesimal character. Let π
+be the unique lowest K-type of X which has a highest weight µ. Then HD(X) = 0 if and
+only if
+(a) µ + 2ρc is irregular for ∆(g, t); or
+(b) µ − wρ + ρc is not dominant for ∆+(k, t) for any w ∈ W 1(µ).
+Example 4.2. In the special case that W 1 = {e}, which is met for complex Lie groups,
+SL(2n + 1, R), SL(n, H) and the linear E6(−26), we always have that µ + 2ρc is regular for
+∆(g, t). Thus HD(X) = 0 if and only if µ − ρ + ρc is dominant for ∆+(k, t).
+When #W 1 > 1, pick up two distinct elements w1, w2 from W 1 such that w1C(g)∩w2C(g)
+is a codimension one facet of w1C(g). Then condition (a) holds for any µ such that µ+2ρc ∈
+w1C(g) ∩ w2C(g). This suggests that the tempered Dirac series of G should be divided into
+#W 1 parts by those irreducible tempered X such that HD(X) vanishes.
+From now on, we shall use a circle to stand for a K-type, and paint it if and only if (1)
+is an equality. Let us see some concrete examples.
+
+6
+CHAO-PING DONG AND DU CHENGYU
+-4
+0
+4
+Figure 1. Some K-types of SL(2, R)
+(0,0)
+(4,4)
+(-4,-4)
+(4,-4)
+Figure 2. Some K-types of Sp(4, R)
+Example 4.3. Consider SL(2, R), where ∆(g, t) = ∆(s, t) = {±2}. Then #W 1 = 2 and
+C(g) ∩ sC(g) = {0}, where s is the non-trivial element in W 1. Condition (b) does not take
+effect here since ∆(k, t) is empty. Thus µ = 0 is the unique K-type such that ∥µ∥spin >
+∥µ∥lambda, and the tempered Dirac series of SL(2, R) are separated into two parts.
+See
+Figure 1.
+Example 4.4. Consider G = Sp(4, R). Let K = U(2) and T = U(1) × U(1). Thus k has a
+one-dimensional center. Fix
+∆+(k, t) = {(1, −1)},
+∆+(g, t) = {(1, −1), (2, 0), (0, 2), (1, 1)}.
+The corresponding simple roots are α1 = (1, −1) = 2ρc, and α2 = (0, 2). The highest weight
+of a K-type is represented by a pair of integers (x, y) such that x ≥ y.
+Condition (a) of says that the K-types on the three lines y = 1, x = −1 and y = −x
+should not be painted. These lines intersect at the point (−1, 1), which is −2ρc. Condition
+(b) further says that (1, 0) and (0, −1) should not be painted. Now Figure 2 suggests that
+the tempered Dirac series of Sp(4, R) are separated into four parts.
+Example 4.5. Let G be G2(2), the linear split G2, which is centerless, connected, but not
+simply connected. We adopt the simple roots of ∆+(g, t) and ∆+(k, t) as in Knapp [8]. Let
+
+SPIN NORM AND LAMBDA NORM
+7
+[0,0]
+[0,4]
+[22,0]
+Figure 3. Some K-types of the linear split G2
+α1 be the short simple root and α2 be the long one. In this case, ∆(g, t) is of type G2, while
+∆(k, t) is of type A1 × A1. We fix ∆+(k, t) = {γ1, γ2}, where γ1 := α1 and γ2 := 3α1 + 2α2.
+Let ω1, ω2 be the fundamental weights for ∆(k, t) such that (ωi, α∨
+j ) = δij. The K-types are
+parameterized via the highest weight theorem by [a, b] := aω1 + bω2, a, b ∈ Z⩾0 such that
+a + b is even.
+We show some of the K-types in Figure 3, where the a-coordinates of the bottom line are
+0, 2, 4, 6, 8, . . . , and so are the b-coordinates of the left-most column.
+Now condition (a) says that K-types on the two lines a = b and a = 3b + 4 should not
+be painted. These two lines intersect at [−2, −2] = −2ρc. From Figure 3, one sees that the
+tempered Dirac series are divided into three parts by the two lines. Condition (b) further
+says that [2, 0] should not be painted.
+To sum up, we have recovered Corollary 8.4 of [3].
+Funding
+Dong is supported by the National Natural Science Foundation of China (grant 12171344).
+References
+[1] J. Carmona, Sur la classification des modules admissibles irr´eductibles, pp.11–34 in Noncommutative
+Harmonic Analysis and Lie Groups, J. Carmona and M. Vergne, eds., Lecture Notes in Mathematics
+1020, Springer-Verlag, New York, 1983.
+[2] J. Ding and C.-P. Dong, Spin Norm, K-Types, and Tempered Representations, J. Lie Theory 26 (2016),
+651–658.
+[3] J. Ding, C.-P. Dong, and L. Yang, Dirac series for some real exceptional Lie groups, J. Algebra 559
+(2020) 379–407.
+[4] C.-P. Dong, On the Dirac cohomology of complex Lie group representations, Transform. Groups 18 (2013),
+61-79. [Erratum: Transform. Groups 18 (2013), 595–597.]
+[5] Harish-Chandra, Harmonic analysis on real reductive Lie groups. I. The theory of the constant term J.
+Funct. Anal. 19 (1975), 104–204.
+[6] J.-S. Huang and P. Pandˇzi´c, Dirac cohomology, unitary representations and a proof of a conjecture of
+Vogan, J. Amer. Math. Soc. 15 (2002), 185–202.
+
+8
+CHAO-PING DONG AND DU CHENGYU
+[7] J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer-Verlag, New York,
+1972.
+[8] A. Knapp, Lie Groups, Beyond an Introduction, 2nd edition, Birkh¨auser, 2002.
+[9] S. Salamanca-Riba and D. Vogan, On the classification of unitary representations of reductive Lie groups,
+Ann. of Math. 148 (1998), 1067–1133.
+[10] P. Trapa, A parametrization of ˆK (after Vogan), Notes from an AIM workshop, July 2004.
+[11] D. Vogan, Representations of Real Reductive Groups, Birkh¨auser, 1981.
+[12] D. Vogan, Unitarizability of certain series of representations, Ann. of Math. 120 (1984), 141–187.
+(Dong) School of Mathematical Sciences, Soochow University, Suzhou 215006, P. R. China
+Email address: [email protected]
+(Du) School of Mathematical Sciences, Soochow University, Suzhou 215006, P. R. China
+Email address: [email protected]
+

8dAzT4oBgHgl3EQfE_q4/content/tmp_files/load_file.txt

@@ -0,0 +1,303 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf,len=302
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='01004v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='RT]  3 Jan 2023  SPIN NORM AND LAMBDA NORM  CHAO-PING DONG AND DU CHENGYU  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' Given a K-type π, it is known that its spin norm (due to first-named author)  is lower bounded by its lambda norm (due to Vogan).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' That is, ∥π∥spin ≥ ∥π∥lambda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' This  note aims to describe for which π one can actually have equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' We apply the result to  tempered Dirac series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' In the case of real groups, we obtain that the tempered Dirac series  are divided into #W 1 parts among all tempered modules with real infinitesimal characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' Introduction  Let G be a linear real reductive Lie group which is in the Harish-Chandra class [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' That  is,  G has only a finite number of connected components;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='  The derived group [G, G] has finite center;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='  The adjoint action Ad(g) of any g ∈ G is an inner automorphism of g = (g0)C, where  g0 is the Lie algebra of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='  Let θ be a Cartan involution of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' We assume the subgroup K = Gθ of fixed points  of θ is a maximal compact subgroup of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' Let g0 = k0 ⊕ s0 be the corresponding Cartan  decomposition of g0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' We drop the subscript for the complexification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='  Let ˆGtemp,o denote the set of irreducible tempered representations with real infinitesimal  character (up to equivalence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' Let ˆK denote the set of K-types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' The following bijection was  noted by Trapa [10], after Vogan’s paper [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' Let X be any irreducible tempered (g, K)-module with real infinitesimal char-  acter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' Then X has a unique lowest K-type which occurs with multiplicity one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' Moreover,  the map  φ : ˆGtemp,o → ˆK  defined by taking the lowest K-type, is a well-defined bijection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='  Motivated by the lambda norm introduced by Vogan [11], the first-named author intro-  duced spin norm [4] for the classification of Dirac series (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=', irreducible unitary representa-  tions of G with non-vanishing Dirac cohomology).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' We refer the reader to [6] and references  therein for the notion of Dirac cohomology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' It was proven in [4] that the spin norm of a  K-type π is bounded below by its lambda norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' That is,  (1)  ∥π∥spin ⩾ ∥π∥lambda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='  2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' Primary 22E46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='  Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' lambda norm, spin norm, tempered representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='  1  2  CHAO-PING DONG AND DU CHENGYU  This inequality turns out to have a nice interpretation in the setting of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' Indeed,  let ˆGtemp,d collect the members of ˆGtemp,o with non-zero Dirac cohomology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' Put  ˆKe := {π ∈ ˆK| ∥π∥spin = ∥π∥lambda}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' ([2]) The map φ restricts to ˆGtemp,d is a bijection onto ˆKe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' More precisely,  any member π ∈ ˆGtemp,o is a Dirac series if and only if the inequality (1) becomes an equality  on its unique lowest K-type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='  Given an arbitrary K-type π, it is not easy to compute neither ∥π∥lambda nor ∥π∥spin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='  Thus, it is subtle to detect whether the inequality (1) is strict or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' This note aims to  give a criterion on this aspect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' Our main result is Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' The main idea is to insert  an intermediate value between ∥π∥lambda and ∥π∥spin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' As an application, our result suggests  that the tempered Dirac series should be separated into #W 1 parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' See (5) for the definition  of W 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='  The note is outlined as follows: In Section 2, we recall lambda norm and spin norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' Then  we deduce our main result in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' The last section considers tempered Dirac series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' Preliminaries  In this section, we briefly recall the definitions of the spin norm and the lambda norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' The lambda norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' We keep the notations K, G, k, s, θ, etc as in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='  Let T be a maximal torus of K and t0 be the Lie algebra of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' Recall that the analytic  Weyl group is defined by  W(k, t) = NK(T)/AK(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='  It acts on the root system ∆(k, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' Fix a choice of positive roots ∆+(k, t), and define  R(G) := {r ∈ W(k, t)|r∆+(k, t) = ∆+(k, t)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='  Given a K-type π, by Lemma 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='1 of [9], the collection of highest weights of π as k-module  is a single orbit of R(G) on ˆT ∈ it∗  0, where ˆT is the abelian group of characters of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='  Now given any K-type π, take a highest weight µ of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' Then µ ∈ it∗  0 is dominant integral  for ∆+(k, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' Denote by ρc the half sum of all roots in ∆+(k, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' Choose a positive root system  ∆+(g, t) making µ + 2ρc dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' Denote by ρ the half sum of all roots in ∆+(g, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' Let  P be the projection map to the dominant chamber C(g) corresponding to ∆+(g, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' Then  ∥P(µ+2ρc −ρ)∥ is independent of the choices of µ and ∆+(g, t), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' Section 1 and Corrollary  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='4 of [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' Now we are ready to talk about the lambda norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' ([11, 1]) For any π ∈ ˆK, the lambda norm of π is defined to be  (2)  ∥��∥lambda := ∥P(µ + 2ρc − ρ)∥,  where µ is any highest weight of π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' For any irreducible admissible (g, K)-module X, the  lambda norm of X is defined to be  (3)  ∥X∥lambda := min  π ∥π∥lambda,  where π runs over all the K-types occurring in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' A K-type π is called a lowest K-type of  X if it occurs in X and ∥π∥lambda = ∥X∥lambda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='  SPIN NORM AND LAMBDA NORM  3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' The spin norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' Although the original definition of the spin norm involves the spin  module SG of the Clifford algebra C(s), our discussion here does not need a deep under-  standing of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' Our tool is mainly the root systems and their Weyl groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' ([4]) For any π ∈ ˆK, its spin norm is defined to be  ∥π∥spin := min ∥γ + ρc∥,  where γ runs over all the highest weights of the ˜K-types in π ⊗ SG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' For any irreducible  admissible (g, K)-module X, its spin norm is defined to be  ∥X∥spin := min  π ∥π∥spin,  where π runs over all the K-types occurring in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' We call π a spin lowest K-type of X if  it occurs in X and ∥π∥spin = ∥X∥spin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' When is the inequality (1) strict?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='  We fix a positive root system ∆+(k, t), and denote the half sum of roots in it by ρc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' Let  W(g, t) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=', W(k, t))) be the Weyl group of ∆(g, t) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=', ∆(k, t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' Let C(k) be the closed  dominant Weyl chamber for ∆+(k, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' For any µ ∈ t∗, we use {µ} to denote the unique weight  in C(k) to which µ is conjugate under the action of W(k, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' Let ∆+(g, t) be a positive root  system of ∆(g, t) containing ∆+(k, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' ([4, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='5]) For any K-type π with a highest weight µ ∈ t∗, we have  (4)  ∥µ∥spin = min  w∈W 1 ∥{µ − wρ + ρc} + ρc∥,  where  (5)  W 1 := {w ∈ W(g, t)|wC(g) ⊆ C(k)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' ([7, §13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='3, Lemma B]) Let λ ∈ C(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' Then  ∥λ + ρc∥ ⩾ ∥wλ + ρc∥  for any w ∈ W(k, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' Moreover, the equality holds if and only if λ = wλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' Let ∆ be a root system with Weyl group W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' Fix a positive set ∆+ of roots  and denote by ρ the half sum of all positive roots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' For any dominant weight λ, we have the  following inequality  (6)  ∥λ − ρ∥ ⩽ ∥λ − wρ∥, ∀w ∈ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='  Moreover, if λ is dominant with respect to w∆+, we have  ∥λ − ρ∥ = ∥λ − wρ∥  Otherwise, the inequality (6) is strict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' We first prove the inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' Compute the following difference  (∗)  ∥λ − ρ∥2 − ∥λ − wρ∥2 = −2(λ, ρ − wρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='  A widely known fact is that ρ − wρ is a sum of positive roots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' The weight λ is dominant by  assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' Thus the pairing (λ, ρ − wρ) is non-negative, and (6) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='  4  CHAO-PING DONG AND DU CHENGYU  Now suppose λ is dominant with respect to w∆+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' Notice that the half sum of positive  roots with respect to w∆+ is wρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' Applying (6) to λ, w∆+ and wρ gives  ∥λ − wρ∥ ⩽ ∥λ − w−1(wρ)∥ = ∥λ − ρ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='  Therefore, (6) becomes an equality in the current setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='  Now suppose λ is not dominant with respect to the new positive set w∆+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' Define  Dw := {γ ∈ ∆−|γ ∈ w∆+},  where ∆− = −∆+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' It is well-known that  ρ − wρ =  �  γ∈Dw  (−γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='  By assumption, λ is not dominant with respect to w∆+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' There must exist β ∈ w∆+ such  that (λ, β) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' But it cannot live in ∆+, because λ is dominant with respect to ∆+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' As a  consequence, β ∈ Dw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' Continuing with (∗), we have that  −(λ, ρ − wρ) = −  \uf8eb  \uf8edλ,  �  γ∈Dw  (−γ)  \uf8f6  \uf8f8 =  \uf8eb  \uf8edλ,  �  γ∈Dw  γ  \uf8f6  \uf8f8 ⩽ (λ, β) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='  Thus (6) is strict in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='  □  Let us state the main result of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' Let π be an irreducible representation of K with be a highest weight µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='  Choose a positive root system ∆+(g, t) making µ + 2ρc dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' Let C(g) be the closed  dominant Weyl chamber corresponding to ∆+(g, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' Then the inequality (1) is strict if and  only if one of the following conditions holds:  (a) µ + 2ρc is irregular for ∆(g, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='  (b) µ − wρ + ρc /∈ C(k) for all w ∈ W(g, t) such that µ + 2ρc ∈ wC(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' Let P(·) be the projection map to the cone C(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' It suffices to show that  (7)  ∥π∥lambda = ∥P(µ + 2ρc − ρ)∥ ≤ ∥µ + 2ρc − ρ∥ ≤ ∥π∥spin,  that the first equality happens if and only if (a) holds, and that the second equality happens  if and only if (b) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='  By the Pythagorean theorem, the first inequality in (7) holds, and it becomes an equality  if and only if P(µ + 2ρc − ρ) = µ + 2ρc − ρ, which is equivalent to µ + 2ρc − ρ ∈ C(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' The  latter is equivalent to (a) since µ + 2ρc ∈ C(g) and µ + 2ρc is integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='  Now let us consider the second inequality in (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' We collect all w ∈ W(g, t) such that  µ + 2ρc ∈ wC(g) as W 1(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' Since µ + 2ρc ∈ C(k), it follows that W 1(µ) ⊆ W 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' Moreover,  the identity element e ∈ W 1(µ) due to µ + 2ρc ∈ C(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='  Using Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='2, we have that  (8)  ∥π∥spin = min  w∈W 1 ∥{µ − wρ + ρc} + ρc∥ ⩾ min  w∈W 1 ∥µ − wρ + 2ρc∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='  Take λ = µ + 2ρc and ∆+ = ∆+(g, t) in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' We have  (9)  ∥µ − wρ + 2ρc∥ ≥ ∥µ − ρ + 2ρc∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='  SPIN NORM AND LAMBDA NORM  5  Furthermore, the inequality (9) is strict when w /∈ W 1(µ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' yet it is an equality when w ∈  W 1(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' Now the second inequality in (7) follows from (8) and (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='  Assume (b) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' For any w ∈ W 1 \\ W 1(µ), one has that  ∥{µ − wρ + ρc} + ρc∥ ⩾ ∥µ − wρ + 2ρc∥ > ∥µ − ρ + 2ρc∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='  On the other hand, for all w ∈ W 1(µ), one has that  ∥{µ − wρ + ρc} + ρc∥ > ∥µ − wρ + 2ρc∥ = ∥µ − ρ + 2ρc∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='  The first strict inequality is due to the assumption that µ − wρ + ρc /∈ C(k) and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='  Assume (b) does not hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' Then there exists some w0 ∈ W 1(µ) such that µ − w0ρ + ρc ∈  C(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' Therefore,  ∥{µ − w0ρ + ρc} + ρc∥ = ∥µ − w0ρ + 2ρc∥ = ∥µ − ρ + 2ρc∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='  Since we have proven that  min  w∈W 1 ∥{µ − wρ + ρc} + ρc∥ ≥ ∥µ − ρ + 2ρc∥,  we must have ∥π∥spin = ∥µ − ρ + 2ρc∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='  To sum up, the two inequalities in (7) are controlled by (a) and (b), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' Thus  ∥π∥spin > ∥π∥lambda happens if and only if at least one of (a) and (b) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='  □  We record an interesting corollary from the above proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' If µ + ρc − wρ ∈ C(k) for some w ∈ W 1(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' Then  (10)  ∥µ∥spin = ∥µ + 2ρc − ρ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' Application to tempered Dirac series  We call an irreducible tempered representations with non-zero Dirac cohomology a tem-  pered Dirac series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' Combining Theorems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='4 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='2, we have the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' Let X be a tempered (g, K)-module with real infinitesimal character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' Let π  be the unique lowest K-type of X which has a highest weight µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' Then HD(X) = 0 if and  only if  (a) µ + 2ρc is irregular for ∆(g, t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' or  (b) µ − wρ + ρc is not dominant for ∆+(k, t) for any w ∈ W 1(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='  Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' In the special case that W 1 = {e}, which is met for complex Lie groups,  SL(2n + 1, R), SL(n, H) and the linear E6(−26), we always have that µ + 2ρc is regular for  ∆(g, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' Thus HD(X) = 0 if and only if µ − ρ + ρc is dominant for ∆+(k, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='  When #W 1 > 1, pick up two distinct elements w1, w2 from W 1 such that w1C(g)∩w2C(g)  is a codimension one facet of w1C(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' Then condition (a) holds for any µ such that µ+2ρc ∈  w1C(g) ∩ w2C(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' This suggests that the tempered Dirac series of G should be divided into  #W 1 parts by those irreducible tempered X such that HD(X) vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='  From now on, we shall use a circle to stand for a K-type, and paint it if and only if (1)  is an equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' Let us see some concrete examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='  6  CHAO-PING DONG AND DU CHENGYU  4  0  4  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' Some K-types of SL(2, R)  (0,0)  (4,4)  (-4,-4)  (4,-4)  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' Some K-types of Sp(4, R)  Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' Consider SL(2, R), where ∆(g, t) = ∆(s, t) = {±2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' Then #W 1 = 2 and  C(g) ∩ sC(g) = {0}, where s is the non-trivial element in W 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' Condition (b) does not take  effect here since ∆(k, t) is empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' Thus µ = 0 is the unique K-type such that ∥µ∥spin >  ∥µ∥lambda, and the tempered Dirac series of SL(2, R) are separated into two parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='  See  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='  Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' Consider G = Sp(4, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' Let K = U(2) and T = U(1) × U(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' Thus k has a  one-dimensional center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' Fix  ∆+(k, t) = {(1, −1)},  ∆+(g, t) = {(1, −1), (2, 0), (0, 2), (1, 1)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='  The corresponding simple roots are α1 = (1, −1) = 2ρc, and α2 = (0, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' The highest weight  of a K-type is represented by a pair of integers (x, y) such that x ≥ y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='  Condition (a) of says that the K-types on the three lines y = 1, x = −1 and y = −x  should not be painted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' These lines intersect at the point (−1, 1), which is −2ρc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' Condition  (b) further says that (1, 0) and (0, −1) should not be painted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' Now Figure 2 suggests that  the tempered Dirac series of Sp(4, R) are separated into four parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='  Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' Let G be G2(2), the linear split G2, which is centerless, connected, but not  simply connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' We adopt the simple roots of ∆+(g, t) and ∆+(k, t) as in Knapp [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' Let  SPIN NORM AND LAMBDA NORM  7  [0,0]  [0,4]  [22,0]  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' Some K-types of the linear split G2  α1 be the short simple root and α2 be the long one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' In this case, ∆(g, t) is of type G2, while  ∆(k, t) is of type A1 × A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' We fix ∆+(k, t) = {γ1, γ2}, where γ1 := α1 and γ2 := 3α1 + 2α2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='  Let ω1, ω2 be the fundamental weights for ∆(k, t) such that (ωi, α∨  j ) = δij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' The K-types are  parameterized via the highest weight theorem by [a, b] := aω1 + bω2, a, b ∈ Z⩾0 such that  a + b is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='  We show some of the K-types in Figure 3, where the a-coordinates of the bottom line are  0, 2, 4, 6, 8, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' , and so are the b-coordinates of the left-most column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='  Now condition (a) says that K-types on the two lines a = b and a = 3b + 4 should not  be painted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' These two lines intersect at [−2, −2] = −2ρc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' From Figure 3, one sees that the  tempered Dirac series are divided into three parts by the two lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' Condition (b) further  says that [2, 0] should not be painted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='  To sum up, we have recovered Corollary 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='4 of [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='  Funding  Dong is supported by the National Natural Science Foundation of China (grant 12171344).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='  References  [1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' Carmona, Sur la classification des modules admissibles irr´eductibles, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='11–34 in Noncommutative  Harmonic Analysis and Lie Groups, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' Carmona and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' Vergne, eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=', Lecture Notes in Mathematics  1020, Springer-Verlag, New York, 1983.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='  [2] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' Ding and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' Dong, Spin Norm, K-Types, and Tempered Representations, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content=' Lie Theory 26 (2016),  651–658.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
+page_content='  [3] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'}
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+Astronomy & Astrophysics manuscript no. TCrA_Rigliaco
+©ESO 2023
+January 5, 2023
+Disk Evolution Study Through Imaging of Nearby Young Stars
+(DESTINYS): Characterization of the young star T CrA and its
+circumstellar environment ⋆
+E. Rigliaco1, R. Gratton1, S. Ceppi2, C. Ginski.3, 4, M. Hogerheijde3, 4, M. Benisty5, 6, T. Birnstiel7, 8, M. Dima1, S.
+Facchini2, A. Garufi9, J. Bae10, M. Langlois11, G. Lodato2, E. Mamajek12, C.F. Manara13, F. Ménard14, Á. Ribas15, and
+A. Zurlo16, 17, 18
+1 INAF/Osservatorio Astronomico di Padova, Vicolo dell’osservatorio 5, 35122 Padova e-mail: [email protected]
+2 Dipartimento di Fisica, Università Degli Studi di Milano, Via Celoria, 16, Milano, 20133, Italy
+3 Anton Pannekoek Institute for Astronomy, University of Amsterdam, Science Park 904, 1098XH Amsterdam, The Netherlands
+4 Leiden Observatory, Leiden University, PO Box 9513, 2300 RA, Leiden, The Netherlands
+5 Unidad Mixta Internacional Franco-Chilena de Astronomía, CNRS/INSU UMI 3386, Departamento de Astronomía, Universidad
+de Chile, Camino El Observatorio 1515, Las Condes, Santiago, Chile
+6 Univ. Grenoble Alpes, CNRS, IPAG, 38000 Grenoble, France
+7 University Observatory, Faculty of Physics, Ludwig-Maximilians-Universität München, Scheinerstr. 1, 81679 Munich, Germany
+8 Exzellenzcluster ORIGINS, Boltzmannstr. 2, D-85748 Garching, Germany
+9 INAF, Osservatorio Astrofisico di Arcetri, Largo Enrico Fermi 5, 50125, Firenze, Italy
+10 Department of Astronomy, University of Florida, Gainesville, FL 32611, United States of America
+11 CRAL, UMR 5574, CNRS, Université Lyon 1, 9 avenue Charles André, 69561 Saint-Genis-Laval Cedex, France
+12 Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, CA 91109, USA
+13 European Southern Observatory, Karl-Schwarzschild-Strasse 2, 85748 Garching bei München, Germany
+14 Univ. Grenoble Alpes, CNRS, IPAG, 38000 Grenoble, France
+15 Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge, CB3 0HA, UK
+16 Núcleo de Astronomía, Facultad de Ingeniería y Ciencias, Universidad Diego Portales, Av. Ejercito 441, Santiago, Chile
+17 Escuela de Ingeniería Industrial, Facultad de Ingeniería y Ciencias, Universidad Diego Portales, Av. Ejercito 441, Santiago, Chile
+18 Aix Marseille Univ, CNRS, CNES, LAM, Marseille, France
+Received 12 October 2022; accepted 22 December 2022
+ABSTRACT
+Context. In recent years it is emerging a new hot-topic in the star and planet formation field: the interaction between circumstellar
+disk and its birth cloud. Birth environments of young stars have strong imprints on the star itself and their surroundings. In this context
+we present a detailed analysis of the wealthy circumstellar environment around the young Herbig Ae/Be star T CrA.
+Aims. Our aim is to understand the nature of the stellar system and the extended circumstellar structures as seen in scattered light
+images.
+Methods. We conduct our analysis combining archival data, and new adaptive optics high-contrast and high-resolution images.
+Results. The scattered light images reveal the presence of a complex environment around T CrA composed of a bright forward
+scattering rim of the disk’s surface that is seen at very high inclination, a dark lane of the disk midplane, bipolar outflows, and streamer
+features likely tracing infalling material from the surrounding birth cloud onto the disk. The analysis of the light curve suggests that
+the star is a binary with a period of 29.6 years, confirming previous assertions based on spectro-astrometry. The comparison of the
+scattered light images with ALMA continuum and 12CO (2–1) line emission shows that the disk is in keplerian rotation, and the
+northern side of the outflowing material is receding, while the southern side is approaching to the observer. The overall system lays
+on different geometrical planes. The orbit of the binary star is perpendicular to the outflows and is seen edge on. The disk is itself seen
+edge-on, with a position angle of ∼7◦. The direction of the outflows seen in scattered light is in agreement with the direction of the
+more distant molecular hydrogen emission-line objects (MHOs) associated to the star. Modeling of the spectral energy distribution
+(SED) using a radiative transfer scheme well agrees with the proposed configuration, as well as the hydrodynamical simulation
+performed using a Smoothed Particle Hydrodynamics (SPH) code.
+Conclusions. We find evidence of streamers of accreting material around T CrA. These streamers connect the filament along which
+T CrA is forming with the outer parts of the disk, suggesting that the strong misalignment between the inner and outer disk is due to
+a change in the direction of the angular momentum of the material accreting on the disk during the late phase of star formation. This
+impacts the accretion on the components of the binary, favoring the growth of the primary with respect the secondary, as opposite to
+the case of aligned disks.
+Key words. stars: pre-main sequence, circumstellar matter – protoplanetary disks – ISM: individual object: T CrA – ISM: jets and
+outflows
+Article number, page 1 of 17
+arXiv:2301.01486v1  [astro-ph.SR]  4 Jan 2023
+
+A&A proofs: manuscript no. TCrA_Rigliaco
+1. Introduction
+Herbig Ae/Be stars (Herbig 1960) are pre-main sequence stars
+with intermediate mass covering the range between low-mass T
+Tauri stars (TTSs) and the embedded massive young stellar ob-
+jects. The formation of stars in the low and intermediate-mass
+regimes involves accreting disks formed during the collapse of
+the protostar, and fast collimated outflows and jets. The circum-
+stellar environment of these objects is highly dynamic and multi-
+wavelengths observations show large photometric and spectro-
+scopic variability (e.g., Pikhartova et al. 2021; Mendigutía et al.
+2011) that can be used as a tool to understand the physics of ac-
+cretion and ejection related to the interaction between the star
+and its circumstellar environment.
+T CrA (RA=19:01:58.79 DEC=-36:57:50.33) is an Herbig
+Ae/Be star member of the Coronet Cluster, belonging to the
+Corona Australis star-forming region, which is one of the near-
+est (149.4±0.4 pc, Galli et al. 2020) and most active regions of
+ongoing star formation. The Coronet Cluster is centered on the
+Herbig Ae/Be stars R CrA and T CrA. It is very active in star
+formation (e.g. Lindberg & Jørgensen 2012), harboring many
+Herbig-Haro (HHs) objects and Molecular Hydrogen emission-
+line Objects (MHOs). It has been target of many surveys, and
+all studies agree in assigning the Coronet an age <3 Myr (e.g.
+Meyer & Wilking 2009; Sicilia-Aguilar et al. 2011). In this pa-
+per we investigate the variable star T CrA. T CrA is classified
+as F0 by Joy (1945) with effective temperature Teff=7200 K,
+and according to Cazzoletti et al. (2019) and Herczeg & Hil-
+lenbrand (2014) this corresponds to L∗ ∼29 L⊙, and stellar mass
+∼2.25 M⊙ using the evolutionary tracks by Siess et al. 2000, and
+adopting the average distance of 154 pc calculated by Dzib et al.
+(2018). The Gaia-DR2 and DR3 catalogs (Gaia Collaboration
+et al. 2016, 2021) do not provide proper motion or parallax for
+T CrA. This star was not observed by the Hipparcos satellite and
+it is also not listed in the UCAC5 catalog. The former UCAC4
+catalog (Zacharias et al. 2012) provides a proper motion result
+(µα cos δ = 2.0 ± 3.8 mas yr−1, µδ=-22.6±3.8 mas yr−1), which
+is consistent with membership in Corona-Australis (within the
+large uncertainties of that solution). Galli et al. (2020) provided
+an updated census of the stellar population in the Corona Aus-
+tralis deriving an average distance of 149.4±0.4 pc. This is the
+distance we will use throughout the paper. A deep H2 v=1–0
+S(1) 2.12 µm narrow-band imaging survey of the northern part
+of the Corona Australis cloud conducted by Kumar et al. (2011)
+identified many new MHOs (Davis et al. 2010). Among these
+objects, two are considered unambiguously associated to T CrA:
+MHO2013 and MHO2015, see Figure 3 in Kumar et al. (2011).
+MHO 2015 is a clear bow-shock feature, lying to the south of
+T CrA, and it marks the southern lobe of the bipolar outflow
+originating from T CrA. MHO 2013 marks the northern lobe.
+The hypothetical line connecting the two MHOs crosses the po-
+sition of T CrA. This is the only unambiguously detected bipolar
+outflow traced by two complementing bow-shock features in the
+entire Coronet region (Kumar et al. 2011). We reproduce the im-
+age shown in Kumar et al. (2011) in the left panel of Fig. 1.
+T CrA was suggested to be a binary system by Bailey (1998)
+and Takami et al. (2003) who adopted spectro-astrometry in the
+Hα line suggesting that the system is a binary with a compan-
+ion at >0.14′′. However, no companion has been detected us-
+ing spectro-astrometry in the fundamental rovibrational band of
+CO at 4.6µm (Pontoppidan et al. 2011) nor with K-band speckle
+⋆ Based on observations collected at the European Organisation for
+Astronomical Research in the Southern Hemisphere under ESO pro-
+gramme 1104.C-0415(H).
+imaging (Ghez et al. 1997; Köhler et al. 2008). In the same
+years, infrared speckle observations performed by Ghez et al.
+(1997) did not show the presence of a stellar companion. The
+non-detection of the companion by Ghez et al. (1997) implies
+that the possible companion has a contrast in the K-band larger
+than 3 mag (that is a K-magnitude fainter than 10.5) or a sep-
+aration smaller than 0.1 arcsec at the epoch of the observation
+(April 26, 1994; see also Takami et al. 2003).
+Recently, the circumstellar environment of T CrA has been
+investigated. SOFIA/FORCAST (Faint Object infraRed CAm-
+era for the SOFIA Telescope, Herter et al. 2018) observations
+show very strong excess in the far-IR. T CrA was also de-
+tected in all Herschel/PACS (Photodetector Array Camera and
+Spectrometer) bands (Sandell et al. 2021), highlighting the pres-
+ence of warm or hot dust. Mid-infrared interferometric data
+obtained with VLT/MIDI (MID-infrared Interferometric instru-
+ment) show the presence of disk emission from the inner regions,
+where the temperature is sufficiently high (Varga et al. 2018).
+The presence of the inner disk is also given by the spectral en-
+ergy distribution (SED) which shows near-IR excess emission
+(Sicilia-Aguilar et al. 2013). Optical and IR spectra covering
+the [OI] λ6300 and [NeII] 12.81 µm lines (Pascucci et al. 2020)
+show emission attributed to a jet nearly in the plane of the sky.
+Moreover, continuum ALMA observations of T CrA at 1.3 mm
+(230 GHz) were conducted as part of the survey of protoplan-
+etary disks in Corona Australis (Cazzoletti et al. 2019) and the
+data show a ∼22σ detection at 1.34′′ from the nominal Spitzer
+position that is considered as detection. The 1.3 mm continuum
+flux is then converted into a dust mass (Mdust) under the assump-
+tion of optically thin and isothermal sub-millimeter emission,
+yielding Mdust=3.64±0.27 M⊕. No information on the 12CO(2-
+1) gas content in the disk are provided. The average disk mass
+in CrA is 6±3 M⊕, and it is significantly lower than that of disks
+in other young (1–3 Myr) star forming regions (Lupus, Taurus,
+Chamaeleon I, and Ophiuchus) and appears to be consistent with
+the average disk mass of the 5–10 Myr-old Upper Sco (Cazzo-
+letti et al. 2019).
+In this paper we analyze images of T CrA acquired with the
+Very Large Telescope at ESO’s Paranal Observatory in Chile.
+We employ polarimetric differential imaging (PDI) observations
+obtained with SPHERE (Spectro-Polarimetric High-contrast Ex-
+oplanet REsearch, Beuzit et al. 2019) in the H band to explore
+the circumstellar environment by tracing light scattered by the
+small (µm-sized) dust grains. Moreover, we use archival pho-
+tometric and imaging data to investigate the multiplicity of the
+system. The paper is organized as follows. In Sect. 2 we describe
+the data collected from the archive and newly acquired. In Sect. 3
+we describe the data analysis. First we discuss the multiplicity of
+the system as suggested by the photometric data, the analysis of
+the proper motion and the analysis of the PSF subtracted images.
+Second we analyze the geometry of the system with the analy-
+sis of the disk and the extended emission seen in scattered light.
+In Sect. 4 we propose a scenario that reconciles all the findings,
+showing a model of the system, and discussing a modeling of
+the spectral energy distribution and hydrodynamical simulation.
+In Sect. 5 we summarize and conclude.
+2. Observations
+2.1. SPHERE data
+T CrA was observed on 2021 June 6th with SPHERE/IRDIS
+(InfraRed Dual-band Imager and Spectrograph (IRDIS; Dohlen
+et al. 2008) in dual-beam polarimetric imaging mode (DPI; de
+Article number, page 2 of 17
+
+Rigliaco et al.: DESTINYS–TCrA
+Boer et al. 2020; van Holstein et al. 2020) in the broadband H
+filter with pupil tracking setting, as part of the DESTINYS pro-
+gram (Disk Evolution Study Through Imaging of Nearby Young
+Stars, Ginski et al. (2020, 2021)). An apodized Lyot coronagraph
+with an inner working angle of 92.5 mas was used to mask the
+central star. The individual frame exposure time was set to 32 s,
+and a total of 136 frames were taken separately in 34 polari-
+metric cycles of the half-wave plate. The total integration time
+was 72.5 minutes. Observing conditions were excellent with an
+average seeing of 0.8′′ and an atmosphere coherence time of
+6.4 ms. In addition to the science images, flux calibration images
+were obtained by offsetting the star position by about 0.5 arcsec
+with respect to the coronagraph using the SPHERE tip/tilt mir-
+ror, and inserting a suitable neutral density filter to avoid image
+saturation. Two flux calibration sequences were acquired, before
+and after the science observation. We used the public IRDAP
+pipeline (IRDIS Data reduction for Accurate Polarimetry; van
+Holstein et al. 2020) to reduce the data. The images were astro-
+metrically calibrated using the pixel scale and true north offset
+given in Maire et al. (2016). Because the data were taken in pupil
+tracking mode, we were able to perform an angular differential
+imaging (ADI; Marois et al. 2006) reduction in addition to the
+polarimetric reduction, resulting in a total intensity image and
+a polarized intensity image. We show the initial combined and
+flux calibrated Stokes Q and U images as well as the QΦ and UΦ
+images in Appendix A.
+Additional SPHERE observations of T CrA were acquired
+in 2016 and 2018 with the ESO programs 097.C-0591(A) and
+0101.C-0686(A) (P.I. Schmidt) in classical imaging mode, using
+a classical Lyot coronagraph and the broadband H filter (BB_H).
+The data were reduced through the SPHERE Data Center (De-
+lorme et al. 2017). The 2016 data have very low S/N ratio and
+they are not usable for this work. The 2018 IRDIS data are in-
+stead of good quality and are used to confirm the features de-
+tected in the 2021 images.
+2.2. NACO data
+To perform our analysis we also employed archival NACO data.
+Adaptive optics corrected near-infrared imaging of T CrA was
+obtained with NAOS-CONICA (NACO; Lenzen et al. 2003;
+Rousset et al. 2003) at the VLT in July 12th 2007 (program ID
+079.C-0103(A)), March 29th 2016 (program ID 097.C-0085(A))
+and May 21st 2017 (program ID 099.C-0563(A)). In all cases
+images were obtained in Ks band (λc=2.18 µm) using the S13
+camera, with a 13.72 mas/pixel scale. In 2007, 3000 frames of
+0.6 seconds were taken with an average seeing of 0.8. In 2016,
+540 frames of 0.5 seconds each were taken with average see-
+ing of 1.5. In 2017, 756 frames of 0.35 seconds each were taken
+with average seeing of 1.4. The final images are obtained as the
+median of all the exposures for each year, after re-centering and
+rotating the single-exposure images.
+2.3. Photometric data
+We collected long-term optical photometry of T CrA from the
+AAVSO Database1 (American Association of Variable Star Ob-
+servers: Kafka 2020) in order to investigate its secular evolution.
+We also considered data acquired within the ASAS (Pojman-
+1 https://www.aavso.org/data-access
+ski 1997)2 and ASAS-SN surveys (Shappee et al. 2014)3. While
+more accurate than the AAVSO data, they have a much more
+limited temporal coverage. Results are fully consistent with the
+long-term light curve obtained from the AAVSO data, but no fur-
+ther insight could be obtained. So we will not discuss the ASAS
+data further.
+2.4. ALMA data
+T CrA was observed by ALMA on 2016 August 1–2 (project
+2015.1.01058.S). Details of the observations and calibration are
+described in Cazzoletti et al. (2019). These authors also present
+an analysis of the continuum emission. For the current paper,
+the continuum emission was imaged using Hogböm CLEANing
+with Brigss weighting, a robust parameter of 0.5, and a manu-
+ally drawn CLEAN mask. The resulting beam size is 0.36×0.27
+arcsec (PA +78◦). The noise level is 0.12 mJy, and a continuum
+flux of 3.1 mJy is detected. These values are not corrected for
+the primary beam response, which can be expected to affect the
+results since the observations was not centered on the target. A
+2D Gaussian fit to the continuum emission shows that the con-
+tinuum emission is slightly resolved, with a size of 0.54 × 0.37
+and a PA of +23◦.
+The 12CO line emission was imaged using natural weight-
+ing and 0.5 km s−1 channels, from VLSR = −5 to +15 km s−1;
+no emission was detected outside this range. We used hand
+drawn masks for each individual channel and applied multi-scale
+CLEAN with scales of 0,5,15,25 pixels. A pixel scale of 12.251
+mas was used, coincident with the SPHERE pixel scale. Because
+the CrA region contains extended CO emission around the sys-
+temic velocity of T CrA (e.g., Cazzoletti et al. 2019), we re-
+moved all baselines shorter than 55 kλ. This removed most, but
+not all, of the extended line flux but also limits the recovered
+spatial scales to ∼ 3.75 arcsec.
+3. Data Analysis
+The new and archival data described in the previous section al-
+low us to investigate the nature of T CrA as young stellar object.
+In this section we will analyze the observational evidences we
+have for the stellar system, its environment, and the geometry of
+the extended structures visible in scattered light. In Sect. 3.1 we
+analyze the clues related to the binarity of the system. In Sect. 3.2
+we show the newly acquired polarized light image in H-band of
+T CrA, describing all the features that we see in the image.
+3.1. T CrA as binary system
+The light curve (Fig. 2) shows alternate and periodic maxima
+and minima. The photometric time series analyzed in this study
+consists of more than 5100 V-band data points collected from the
+AAVSO Database and taken in a period of over 100 years, be-
+tween 1910 and 2010. Each point in Figure 2 is the mean value
+over each year. The secular evolution of the light curve is well
+reproduced by a sinusoidal function with a period of 29.6 years.
+Sinusoidal light curves, like the one observed in T CrA, can be
+due to different reasons such as rotation, pulsation, the presence
+of eclipsing binaries, or occulting binaries. In the case of oc-
+culting binaries, the period is generally longer than in the other
+cases, and the occultation is not only due to the stars, but also
+2 http://www.astrouw.edu.pl/asas/?page=aasc&catsrc=
+asas3
+3 https://asas-sn.osu.edu/variables
+Article number, page 3 of 17
+
+A&A proofs: manuscript no. TCrA_Rigliaco
+Fig. 1: SPHERE/IRDIS polarized light image in H-band of T CrA. Left panel: H2 image of the Coronet sub-region. The image is
+adapted from Kumar et al. 2011. The red line shows the line connecting the two MHOs associated to T CrA. The orange box shows
+the IRDIS field of view. Middle panel: Field of view (∼12.5′′) of the SPHERE/IRDIS polarized light image in H-band of T CrA.
+The extended emission features analyzed in the manuscript are labeled. The orange box shows the innermost region of the system.
+Right panel: Zoom-in of the innermost 2′′ around the central system. The disk and the shielded disk mid-plane seen as dark lane are
+labeled.
+Fig. 2: Secular light curve of T CrA with the photometry col-
+lected from the AAVSO archive. Each point is the mean value
+for each year; error bar is the standard deviation of the mean.The
+horizontal dashed lines show the ∆V-mag variation. The period
+of the light curve, measured as the mean between the difference
+of the first and third maxima and minima, is labeled.
+to the circumstellar disks surrounding one or both the stars. The
+light curve of T CrA is suggestive of the motion of an occulting
+binary star. The variation (∆V) in V-magnitude is of the order of
+∼1.4±0.2 mag (see Fig. 2).
+Evidence of the presence of a binary star is also provided by
+the peculiar proper motion of T CrA. Indeed T CrA shows a rela-
+tive average motion of 7.5±3.8 mas yr−1 with respect to the clus-
+ter in the direction (PAPM)=156±30◦ over the period 1998 (mean
+epoch of UCAC4 and PPMXL observation) and 2016 (epoch of
+Gaia DR3). These values are given by the difference between
+the proper motion of T CrA, µα cos δ = 4.2 ± 2.5 mas yr−1 in
+RA and µδ=-6.2±2.9 mas yr−1 in DEC (see Appendix B), and
+the average proper motion of the on-cloud Coronet cluster mem-
+bers (µα cos δ = 4.3 mas yr−1 and µδ=-27.3 mas yr−1, Galli et al.
+2020). This result might indicate a peculiar (large) motion of
+T CrA with respect to the Coronet cluster. However the position
+of T CrA is also constrained and defined by the position of the
+two associated MHOs (Kumar et al. 2011). We measured the po-
+sition angle of the straight line connecting MHO 2013 and 2015,
+that are thought to be connected to the star (Kumar et al. 2011),
+and crossing T CrA, finding the position angle of the bipolar out-
+flow (PAMHO) to be PAMHO ≃33◦. This represents the direction
+of the large scale bipolar outflows. We notice that the minimum
+distance between T CrA and the line connecting the two MHOs
+is only 0.44′′. While this small offset is within the errors in the
+MHO positions, it can be used to set an upper limit to the relative
+proper motion of T CrA with the Coronet cloud in the direction
+perpendicular to this straight line, that is roughly along the di-
+rection where we found an offset between the proper motion of
+T CrA measured above and that of the Coronet cluster. The exact
+value depends on the time elapsed between the expulsion of the
+material responsible for the MHO and the observation by Ku-
+mar et al. (2011). Given the projected distances from the star
+of the MHO’s are 217′′ (MHO 2013) and 64′′ (MHO 2015),
+considering the distance of the Coronet cluster and assuming
+the collimated fast outflowing gas has a speed of approximately
+200 km/s as typical for jets from young stars (e.g., Frank et al.
+2014), we obtain that the material was expelled 765 year ago (for
+MHO2013) and 224 years ago (for MHO 2015). The upper limit
+on the proper motion of T CrA with respect to the cloud is then
+obtained by dividing the measured offset between the barycenter
+of the system that includes T Cra and the line connecting the two
+MHOs: the result is about 1 mas/yr, an order of magnitude less
+than the offset in proper motions considered above and consis-
+tent with the typical scatter of stars in the Coronet cluster. We
+conclude that this offset is not due to a real peculiar motion of
+Article number, page 4 of 17
+
+36:54:00.0
+F103
+N
+MHO2013
+PAdisk
+E
+dark lane
+ 102
+36:56:00.0
+Extended
+Extended emission
+DEC (J2000)
+emission
+(feature 1)
+(feature 2)
+101
+tail
+36:58:00.0
+disk
+TCrA
+H2image of
+the Coronet
+subregion
+MHO2015
+10°
+2"
+0.5"-75AU
+PA
+-37:00:00.0
+02:10.019:02:00.001:50.0
+RA (J2000)Rigliaco et al.: DESTINYS–TCrA
+T CrA, that moves as the Coronet cluster, and should then be
+an apparent or transient effect, that might be due to the orbital
+motion of the central binary star.
+Additional evidence of T CrA as a binary system can also be
+found in the images acquired with IRDIS in 2018 and 2021 and
+NACO in 2007, 2016 and 2017. We subtracted a median PSF,
+obtained by rotating and averaging the PSF image in steps of 1
+degree, to the raw NACO images taken in 2007 and 2016, 2017.
+For IRDIS, we used the flux calibration images that are acquired
+before and after the science sequence. The technique, described
+by Bonavita et al. (2021), allows to make a differential image
+that cancels static aberrations. The output of the procedure is
+a contrast map that allows to spot stellar companions. Due to
+the contrast limit and to the limits imposed by the diffraction
+patterns, none of the images obtained allows us to clearly and
+uniquely detect the presence of a companion star. However, The
+PSF of the NACO 2016 and 2017 data set clearly show an exten-
+sion in the same direction (see Fig. 3), namely NW–SE, but in
+the NACO 2007 data set we do not see this extension. A slight
+extension can be seen in the SPHERE 2018 data set, while no ex-
+tension in the SPHERE 2021 data set. The observed extensions,
+all in the same direction, are very unlikely to be caused by adap-
+tive optic effect, but might indicate a distortion of the PSF due to
+an unresolved companion.
+3.2. The geometry of the system
+Figure 1 shows the polarized light image in H-band of T CrA.
+The image shows several structures, as annotated. In the right
+panel the brightly illuminated top-side of the outer disk is clearly
+visible, as well as the shielded disk mid-plane, seen as a stark
+dark lane in approximately the N-S direction. On larger scale,
+in the middle panel, we can identify two different extended
+emissions. The extended emission labeled as "feature 1" is two-
+lobed and extends in the NE–SW direction, up to 2′′ from the
+central source. The extended emission labeled as "feature 2"
+appears two lobed as well, it is approximately oriented along
+the N-S direction. The South lobe extends out to the edge of
+SPHERE/IRDIS field of view, while the North lobe extends up
+to ∼5′′ from the central source. In the following section we will
+analyze these different structures.
+3.2.1. Outer Disk
+Figure 1 in the right panel shows a very prominent morpholog-
+ical feature composed by a dark lane and a bright region that
+represents the disk surface. This outer disk appears highly in-
+clined, and oriented almost edge-on with respect to the observer,
+and extends almost to the edge of the coronagraph. The dark
+lane has a maximum width of ∼0.2′′ along the E–W direction,
+corresponding to ∼30 au if it were seen exactly edge-on. More-
+over, the disk seen as a dark lane shows an offset with respect
+to the center of the image that corresponds to ∼10 pixels in the
+West direction (∼122 mas) that is about four times the FWHM
+of the point spread function. The disk surface is instead shown
+by the bright regions that extend further out. The PA of the disk
+measures PAdisk=7±2◦, shown as green line in Fig. 1. The disk
+appears highly inclined and seen as a dark lane, as in the case
+for DoAr25 (Garufi et al. 2020), MY Lup and IM Lup (Aven-
+haus et al. 2018). From the images we cannot provide a precise
+estimate of the disk inclination, but we can make a few con-
+siderations. The brightness asymmetry between the bright disk
+top-side, and the diffuse disk bottom-side, indicates that the disk
+is not exactly seen edge-on, indeed in that case we should expect
+top- and bottom-side of the disk to be equally bright. Moreover,
+the offset between the dark-lane and the center of the image pro-
+vides another hint of a non-exactly edge-on disk. From simple
+trigonometric consideration, we can measure the inclination of
+the disk from the angle between the center of the image and
+the center of the dark lane and dividing for half the lengths of
+the dark lane, finding an inclination of ∼87◦. We can conserva-
+tively assume that the T CrA disk, identified as a dark lane in
+the SPHERE image has an inclination between 85-90◦. Another
+possible interpretation for the dark lane could be that it is due to
+a shadow cast by a highly inclined inner disk close to the cen-
+ter, as in the case of SU Aur (Ginski et al. 2021). However, in
+this scenario, we can not reconcile the brightness asymmetry be-
+tween the bright top-side and the diffuse bottom-side of the disk.
+Moreover, we should expect the shadow to cross the center of
+the image, while it appears shifted to the west by ∼10 pixels.
+In order to investigate the innermost region of the outer disk,
+we have plotted the radial profile of the flux seen in QΦ scat-
+tered light along a slice oriented as the disk, seven pixels wide
+and 2.5′′ long. The radial profile, normalized to the brightness
+peak of the disk, is shown in Fig. 4 as a black line. The grey
+area shows the coronagraph. The disk has a gap that extends up
+to ∼25 au and is quite symmetric in the innermost region. As
+far as 60 au the disk start to look asymmetric, and extends up to
+∼100 au. The observed asymmetry might be due to the outflow-
+ing material that overlaps with the disk itself in the north side (as
+discussed in the next section). From this analysis we consider for
+the outer disk an inner rim with radius rin=0.17′′ (∼25 au) and
+an outer rim rout=0.67′′ (∼100 au). We performed the same anal-
+ysis of the radial profile in the direction orthogonal to the disk,
+and shown as blue-dotted line in Fig. 4. In the East side there is
+emission from the scattered light down to the border of the coro-
+nagraph (rin−east ≲14 au), and inside the disk rim measured along
+the disk direction. As expected, in the West-side the emission
+starts further out, due to the presence of the disk’s dark silhou-
+ette (rin−west ∼30 au). We notice that in the West direction at ra-
+dial distances >50 au there is contamination with the outflowing
+material. We will discuss the presence of scattered light emission
+inside the outer disk gap in the following section, showing that it
+may suggest the presence of an intermediate circumbinary disk
+surrounding the central binary system.
+3.2.2. Extended emission
+The structure seen in scattered light in the NE–SW direction,
+identified as feature 1, is consistent with an outflow in the di-
+rection of the line connecting the two MHOs (MHO2013 and
+MHO2015) that are unambiguously associated to T CrA (show
+in the left panel of Fig. 1), which are however at a projected sepa-
+ration of ∼35,000 au and ∼10,000 au, respectively. The presence
+of the MHOs is a clear sign that the source has in the past al-
+ready experienced outflowing phenomena, hence it is consistent
+to consider the emission seen in scattered light in the same direc-
+tions as associated to outflowing material close to the star. From
+a geometrical point of view, the dust seen in scattered light in
+the direction of the outflow has a position angle PAoutflow ∼35◦
+with semi-aperture of ∼25◦, consistent with the PAMHO previ-
+ously discussed.
+The extended emission that elongates approximately in the
+N-S direction, and identified as feature 2, is two lobed as well.
+In the North it extends up to 4.5′′ from the center, and appears
+bent toward the West direction. The Southern feature 2 extends
+up to the edge of the field of view and appears brighter than
+Article number, page 5 of 17
+
+A&A proofs: manuscript no. TCrA_Rigliaco
+0.2
+0.1
+0.0
+0.1
+0.2
+0.2
+0.1
+0.0
+0.1
+0.2
+∆Dec (arcsec)
+NACO/2007
+0.2
+0.1
+0.0
+0.1
+0.2
+NACO/2016
+0.2
+0.1
+0.0
+0.1
+0.2
+∆RA (arcsec)
+NACO/2017
+0.2
+0.1
+0.0
+0.1
+0.2
+SPHERE/2018
+0.2
+0.1
+0.0
+0.1
+0.2
+0.2
+0.1
+0.0
+0.1
+0.2
+SPHERE/2021
+Fig. 3: PSF for all the epochs T CrA was observed. The size of the PSF for every single epochs is shown in the bottom-right corner.
+For NACO 2016, 2017 data sets we can notice an elongation of the PSF in the NW–SW direction.
+Fig. 4: Radial profile of the Qφ image. The black profile shows
+the radial profile obtained along a 2.5′′ long slice centered on
+the star in the N-S direction, with PA=7◦ and extending along
+the disk (black-dashed box in the insert). The blue-dotted profile
+shows the radial profile obtained in the orthogonal direction (E-
+W, blue-dashed box in the insert). All profiles are normalized to
+the brightness peak of the disk. The gray area shows the radius
+of the coronagraph.
+the North feature. We can also detect a faint dust tail extend-
+ing from the main disk toward SE. As it happens in the case of
+SU Aur, where several tails are detected (Ginski et al. 2021),
+we can trace the tail structure until it merges with the disk. Fea-
+ture 2 is most likely showing the presence of accretion streamers
+that bring material from the forming cloud filament to the outer
+disk. From the polarized (Fig. 1) and total intensity images of
+T CrA we can see that in both cases the northern streamer is
+fainter than the southern streamer, indicating that we overall re-
+ceive more photons from the South than from the North side of
+the extended structure. Moreover, the ratio between the polarized
+and total intensity image shows that the overall degree of polar-
+ization is similar on both sides. This indicates that light from the
+South streamer is scattered with angles smaller than 90◦, favor-
+ing the forward scattering. Because the Northern streamer shows
+a similar degree of polarization, but overall fainter signal, we
+conclude that the light is scattered with angles larger than 90◦.
+Hence, the South streamer is angled toward the observed and the
+North streamer is angled away from the observer.
+4. Discussion
+The environment around T CrA is very complex and the analysis
+of new and archival data shows several features. In the following
+we will discuss each of the evidences presented in the previous
+sections, and we will provide a global picture of its circumstel-
+lar environment. A cartoon of the proposed model, showing all
+the observational evidences analyzed in the previous section, is
+shown in Fig. 5.
+Fig. 5: Not-to-scale cartoon of the proposed model for the T CrA
+system. All the features seen in the scattered light images are
+labeled. Moreover, the central binary system, and the size of the
+coronagraph is shown.
+4.1. Modeling of the light curve
+Motivated by the light curve, the peculiar proper motion and the
+PSF distortion, we conducted a detailed analysis of the pho-
+tometric and proper motion data, to be compared to the new
+information on the system’s geometry gathered thanks to the
+Article number, page 6 of 17
+
+[argsec]
+-0.5
+0.5
+1
+N
+Rin(N-s) = 0.17"
+Rout(N-s) = 0.70"
+1
+E
+W
+intensity
+Arbitrary
+0.5
+South-side
+North-side
+0
+East-side
+West-side
+-100
+0
+100
+Radial distance (AU)Accretion
+streamer
+Outflow
+Flows from
+outer to inner
+disk
+Intermediate
+(circumbinary)
+Coronagraph
+disk
+edge
+Outer disk
+dark lane
+Outer disk
+surface
+tailRigliaco et al.: DESTINYS–TCrA
+Parameters
+Value
+log(q)
+-0.27±0.17 M⊙
+T0
+2006.06±0.4 years
+AV0
+6.7±1.1 mag
+Disk Thickness
+54.7±20.2 mas
+Disk Offset
+90.7±19.2 mas
+Table 1:
+Stellar parameters obtained from the modeling of
+T CrA as a binary star. The primary mass star is assumed to be
+1.7M⊙, the orbit to be circular, and period 29.6 years.
+SPHERE’s images. In the attempt to reproduce the observed
+light curve and the H-band magnitude collected from 2MASS,
+we develop a Monte Carlo (MC) model that accounts for the
+light emitted from a binary system and partially absorbed by a
+disk seen edge-on, modeled as a slab with an exponential pro-
+file, and inclined with respect to the binary’s orbit by 35◦, corre-
+sponding to an orbit perpendicular to the outflow’s direction. For
+this simplified model we assume for the binary system a circular
+orbit seen itself edge-on. While the circular orbit is an assump-
+tion made to reduce the number of free parameters, and hence
+avoid degeneracy in the models, the high-inclination of the bi-
+nary orbit is supported by the observation. Indeed, as discussed
+in Pascucci et al. (2020), evidence from the small blueshift of
+the [OI] and [NeII] forbidden lines of T CrA suggests that the
+inner disk is itself close to edge-on, with the microjets close to
+the plane of the sky. We assume for the F0 star a mass of 1.7M⊙
+for the primary star, corresponding to 2 Myrs from the BHAC
+evolutionary tracks (Baraffe et al. 2015), circular orbit, and a
+period of 29.6 years as found from the light curve. The model
+provides the mass ratio (q) between the primary and secondary
+component of the binary system, the epoch of the minimum dis-
+tance between the two components (T0, in years), the offset of
+the center of mass with respect to the absorbing slab (disk offset,
+in mas), the disk thickness (in mas) and the maximum absorption
+at the disk center (AV0, in mag). The proper motion between the
+1998 and 2016 is also measured to be compared to the apparent
+proper motion of T CrA.A corner plot of the derived quantities
+is shown in Appendix C. The MC model computes one million
+random sampling of the priors, and provides solutions with re-
+duced χ2 <2.3. Figure 6 shows the comparison between the ob-
+served secular evolution of T CrA and the light curve obtained
+from the model. There is a very good agreement between the
+observed and modeled light curve. The best fit parameters for
+each of the computed values, obtained as the median value of all
+the solutions with χ2 <2.3, are reported in Table 1. According
+to this model T CrA is a binary system, whose primary star is
+a 1.7M⊙ star, and the secondary is a ∼0.9M⊙, and it is orbiting
+with a 29.6 years period. The corresponding semi-major axis of
+the orbit is ∼12 au, seen edge-on, and with the line of nodes of
+the orbit almost perpendicular to the position angle determined
+for the outflow. Moreover, we check the consistency between
+the apparent motion as measured from Gaia and ground-based
+facilities, and the one measured by assuming the motion of the
+modeled binary system. We find that the offset between the two
+epochs (1998 and 2016) corresponds to 72±26 mas, which is
+consistent with the value of 130±66 mas measured via Gaia and
+UCAC4/PPMXL observations, hence justifying the large proper
+motion of T CrA with respect to the Coronet motion as due to the
+motion of the binary system. We will further discuss the results
+from the model in the next Section.
+Fig. 6: Light curve of T CrA (red points) compared to the light
+curves computed with the MC model (black lines) assuming a
+period of 29.6 years.
+4.2. Disk and extended emission
+Thanks to the new images acquired with SPHERE/IRDIS, and
+to the wealth of literature data on this target, we have now a bet-
+ter knowledge of the disk and extended structure around T CrA,
+and it appears very composite. The disk itself is composed by in-
+ner (circumstellar) disk(s) surrounding the primary (secondary)
+star of the binary system, an intermediate (circumbinary) disk,
+slightly visible in scattered light, and an outer (circumbinary)
+disk that is the most prominent in scattered light. Together with
+the extended emission features, we will discuss all these features
+in the following subsections.
+Disks. The outer disk around T CrA is not continuous. The
+scattered light images and the radial profile analysis of the QΦ
+image show that the bright top-side of the outer disk extends up
+to ∼100 au in the N-S direction, and show a gap in the same
+direction that extends down to ∼25 au.
+Evidence of an inner (circumstellar) disk(s) surrounding the
+primary (secondary) star of the binary system comes from the
+several tracers of gas and dust well beyond the dust gap. Pas-
+cucci et al. (2020) analyze the [OI] λ6300 and [NeII] 12.81 µm
+emission lines observed in high-resolution optical and infrared
+spectra, and conclude that they are associated to fast and col-
+limated microjets. In addition, the presence of gas can also be
+inferred from the non-negligible level of mass accretion rate
+( ˙Macc ∼8.1×10−9 M⊙/yr, Dong et al. 2018; Takami et al. 2003).
+This gas is most likely distributed into an inner circumstellar
+disk, that allows accretion onto the system. The presence of the
+inner disk is also highlighted by mid-infrared interferometric
+data of the thermal emission of disk (Varga et al. 2018), and by
+the SED (Sicilia-Aguilar et al. 2013; Sandell et al. 2021).
+The images acquired with SPHERE show the presence of
+scattered light down to the edge of the coronagraph in the E-
+W direction. The origin of such emission, highly inclined with
+respect to the outer disk, is not clear. However, as we will see
+in section 4.4, it might be due to an intermediate circumbinary
+disk, that is a natural transient consequence of the breaking of
+the innermost circumstellar disks due to the different inclination
+of inner and outer disks. Evidence of emission very close to the
+coronagraph edges are also found by Cugno et al. 2022 using
+the NaCo imager with the L′ filter (λ=3.6 µm) within the NaCo-
+ISPY large program.
+Feature 1. The PA of the extended emission identified as
+feature 1 is consistent with the large scale MHOs and coinci-
+Article number, page 7 of 17
+
+12
+(mag)
+1.3
+14
+15
+1900
+1920
+1940
+1960
+1980
+2000
+2020
+Time
+(yr)A&A proofs: manuscript no. TCrA_Rigliaco
+dent with the small scale microjets detected through forbidden
+lines (Pascucci et al. 2020). Hence, we reasonably assume that
+it is representing outflows detected in scattered light, and that
+this feature is orthogonal to the inner and intermediate disk. The
+innermost disks (inner and intermediate) are misaligned with re-
+spect to the outer disk, with a PA for the inner disk of ∼125◦,
+measured as PAoutflow+90◦. Considering the outer disk is seen
+with PAdisk=7◦, the resulting misalignment between innermost
+and outer disk is of the order of 62◦ with an uncertainty of ±10◦.
+This feature is illuminated by the central system. The shape of
+the outflow is due to higher density regions of dust, generated by
+instabilities created by two or more layers of material with dif-
+ferent densities and velocities resulting in a wind-blown cavity
+(Liang et al. 2020). The regions with different physical prop-
+erties are the highly collimated microjet (as seen from the de-
+tection of forbidden lines, e.g., Pascucci et al. 2020), and the
+surrounding wider-angle disk wind, or parent cloud. The impact
+between these two regions, besides carving out a large and slow
+massive outflow cavity into the parent cloud (Frank et al. 2014),
+creates regions of high density where dust grains accumulate, be-
+coming brighter in scattered light. We also notice that there is a
+good agreement between the small scale outflow seen in the po-
+larimetric images, and the large scale outflows determined by the
+MHOs, supporting the scenario of highly collimated jets carving
+a cavity and creating high density regions. We have also tested
+the emission seen in scattered light versus the continuum thermal
+emission at 1.3 mm, and the 12CO emission seen with ALMA.
+In Figure 7, we show the continuum emission and the red- and
+blue-shifted line emission overlayed on the SPHERE scattered
+light image. The continuum emission, shown as white contours,
+is slightly resolved, compact, and it is distinctly different from
+the orientation of the beam. The comparison with the SPHERE
+image is not quite conclusive in the direction of the emission, if
+along the disk or the extended emission identified as feature 1.
+12CO line emission was clearly detected in the channels, consis-
+tent with a structure of ∼ 2.5 arcsec in diameter. The emission is
+most likely due to the combination from emission aligned with
+the disk orientation inferred from SPHERE, and emission from
+the outflowing material in the same direction as the MHOs. The
+gas emission close to the N-S direction might trace the gas in the
+outer disk, and the velocity structure of the line emission is con-
+sistent with Keplerian rotation. The emission from the outflow-
+ing material is in the same direction as the MHOs. The velocities
+of the extended emission span from -3 km s−1 to 11 km s−1. The
+low velocities for the outflowing material confirm that the emis-
+sion must happen close to the plane of the sky, as also found
+by Pascucci et al. (2020). In both cases, either when tracing the
+outer disk or the outflowing material, the N-E side is receding
+and the S-W side is approaching the observer.
+Misalignment between the inner and the outer disks are not
+rare. As an example, Bohn et al. (2021) have recently inves-
+tigated misalignment between inner and outer disks in transi-
+tional disks, finding that out of a sample of 20 objects ana-
+lyzed, six clearly show evidence of misalignment, five do not
+show evidence of misalignment and the others can not be eval-
+uated with the current data. Misaligned disks, and disks whose
+orientations vary with time can be due to their formation in a
+turbulent, chaotic environment (Bate 2018). Moreover, the evo-
+lution of the stellar and disk spin axes during the formation of
+a star which is accreting in a variable fashion from an inher-
+ently chaotic environment might affect the disk orientation as
+well (Bate et al. 2010). Also late infalling events, which carry
+along a specific angular momentum with respect to the star, may
+tilt the pre-existing disk to another rotation axis depending on the
+Fig. 7: Overlay of SPHERE (color scale) and ALMA (contours)
+data. On the left all the extended structure as seen with SPHERE,
+on the right a zoom-in of the innermost 2′′. White contours are
+ALMA 1.3 mm continuum, plotted at contours starting at, and
+increasing with, 3σ=0.37 mJy beam−1. Red and blue contours
+are integrated 12CO 2–1 emission over 10 km s−1 blue- and red-
+shifted relative to the source velocity, taken as VLSR=4.5 km s−1.
+Red and blue contours are also drawn starting at, and increasing
+with, 3σ = 0.12 Jy beam−1 km s−1. The ALMA data are aligned
+with the SPHERE data to have the stellar position at the center
+of the image; the continuum emission peaks ∼ 0.06′′ North of
+that position.
+mass ratio of the mass accreted and the disk (Dullemond et al.
+2019; Kuffmeier et al. 2021). This was indeed recently observed
+within the DESTINYS program for the SU Aur system (Ginski
+et al. 2021), which shows large scale streamers in scattered light,
+similar to those observed in our new observations of T CrA and
+which were shown to trace infalling material. Stellar properties,
+such as strong stellar magnetic dipole, can cause a warp or mis-
+alignment in the innermost region of the disk (e.g., Matsumoto
+& Tomisaka 2004; Machida et al. 2006; Matsumoto et al. 2006;
+Hennebelle & Ciardi 2009; Joos et al. 2012; Krumholz et al.
+2013; Li et al. 2013; Lewis et al. 2015; Lewis & Bate 2017;
+Wurster & Li 2018). Additionally, the presence of a compan-
+ion, either stellar or substellar, can also cause inner and outer
+disks misalignment (e.g., Facchini et al. 2013, 2018; Zhu 2019;
+Nealon et al. 2020), as in the case of HD142527 (Owen & Lai
+2017; Price et al. 2018a).
+Indeed, T CrA and HD142527 show several similarities even
+if the inclinations at which the outer disks are seen are very dif-
+ferent (almost edge-on in the case of T CrA and almost face-
+on for HD142527). HD142527 is a binary system characterized
+by a primary 2.0 M⊙ star surrounded by an inner disk signifi-
+cantly misaligned (59◦) with respect to the outer disk (Balmer
+et al. 2022). For T CrA the outer disk is seen almost edge-on
+and the misalignment between outer and inner disk is coinci-
+dent with the inclination of the inner disk orbit, namely ∼55◦.
+The primary star in both cases is an F-type Herbig. In the case
+of HD142527 all the main observational features (spirals, shad-
+ows seen in scattered light, horseshoe dust structure, radial flows
+and streamers) can be explained by the interaction between the
+disk and the observed binary companion (Price et al. 2018a). The
+analysis done on HD142527 led the authors (Price et al. 2018a)
+to conclude that the disk around this Herbig star is a circumbi-
+nary rather than transitional disk, with an inclined inner disk, and
+Article number, page 8 of 17
+
+103
+102
+1.0"Rigliaco et al.: DESTINYS–TCrA
+with streamers of material connecting the inner and outer disk.
+In the case of T CrA, if we assume that the inner disk is aligned
+perpendicular to the outflowing material, and hence misaligned
+with respect to the outer disk, the configuration is similar. Hints
+of dusty material inside and misaligned with respect to the outer
+disk come from the radial profile of the scattered light signal
+seen from SPHERE/IRDIS and shown in Fig. 4, where in the
+East-side of the disk in the direction orthogonal to the disk there
+is material down to the coronagraph edge. However, we cannot
+say from these images if this material is organized into a disk-
+structure itself, or if it represents a streamer of material accreting
+from the outer disk onto the inner regions of the system. How-
+ever, as opposite to HD142527, we must mention the absence
+of obvious shadowing features in scattered light in T CrA, that
+can nevertheless be due to the different viewing geometry. In the
+following section we will present a 3D hydrodynamical model
+as the one developed for HD142527 to explain the observed fea-
+tures as disk–binary interaction.
+Feature 2. The extended emission identified as feature 2 ap-
+pears very extended and resemble material falling onto the disk
+as in the case of SU Aur (Ginski et al. 2021). Unfortunately,
+the strong foreground contamination due to the overall cloud
+does not allow to clearly detect the 12CO (2–1), 13CO (2–1),
+and C18O (2–1) transitions at distances larger than ∼2.5′′, thus
+we cannot perform a detailed analysis of the kinematics of the
+material, as it was done, for example, in the case of SU Aur
+(Ginski et al. 2021). Indeed, some parts of the CO disk may
+be missing from from Fig. 7 because the cloud contaminates
+the signal. Moreover, the large scale streamers do not show any
+emission due to the removing of any sensitivity to large scale
+emission in the data reduction process. They may exist, but they
+are very hard to image. The disk may also be more extended
+than seen here. Hence we cannot be conclusive on the nature
+of the extended emission in feature 2. It is highly unlikely that
+this emission is itself indicating outflowing material, as feature
+1, but it can be most likely due to streamers of material that is
+falling onto the disk connecting the disk itself to the surrounding
+cloud material, as for SU Aur. To some extent we might con-
+sider the scattered light morphology of T CrA as an edge-on
+view of SU Aur, where we can see the streamers of infalling
+material and at least one tail of accretion. The same stream-
+ers of accretion were already seen, but not interpreted as such,
+by Ward-Thompson et al. (1985); Clark et al. (2000). Ward-
+Thompson et al. (1985) used linear polarization mapping of the
+region in R-band and identified a jet-like structure with a pro-
+jected lengths of 20′′ emerging from T CrA, in the direction of,
+but pointing away from R CrA. Clark et al. (2000) performed
+near-infrared linear imaging polarimetry in J, H and Kn bands,
+and circular imaging polarimetry in the H band and interpreted
+the images as bipolar cavities, where the SE emission is visi-
+ble as far as ∼15′′ from T CrA. They stress the presence of a
+pronounced asymmetry in the polarized intensity images, sug-
+gestive of fairly sudden depolarization of the dust grains caused
+by foreground material in the reflection nebula. The identifica-
+tion of the MHOs, and the analysis of the images acquired with
+NACO and SPHERE is now showing that the features observed
+in the past were not associated with jets but more likely the same
+streamer of accretion seen in scattered light. A possible test to
+ascertain the origin of feature 2 can be done using the SO2 tran-
+sition from ALMA. Garufi et al. (2022) have indeed shown that
+for the source IRAS 04302+2247, the SO2 emission does not
+probe the disk region, but rather originates at the intersection be-
+tween extended streamers and disks. We notice that the presence
+of streamers of material feeding the disk of T CrA would also
+go in the direction of mitigating the issue of the low disk masses
+found in CrA. Indeed, it was found that the average disk mass
+in CrA is significantly lower than that of disks in other young
+(1-3 Myr) star forming regions (Lupus, Taurus, Chamaeleon I,
+and Ophiuchus, Cazzoletti et al. 2019). If there is accretion of
+fresh material onto the disk, one could have lower measured disk
+masses at the beginning, and mitigate the issue (Manara et al.
+2018). The observed increase in disk masses with time (e.g.,
+Testi et al. 2022; Cazzoletti et al. 2019) should otherwise be ex-
+plained with other mechanisms such as planetesimal collisions
+(Bernabò et al. 2022).
+Moreover, the presence of streamers of accretion is also in
+agreement with the orientation of T CrA with respect to R CrA,
+both belonging to the Coronet Cluster. These two stars formed
+within the same filament, which is oriented at PA=124◦ pro-
+jected on sky (this is also the PA of T CrA relative to R CrA).
+This orientation is indeed similar to that of the orbit proposed
+for the central binary of T CrA and very close to perpendicular
+to the PA of the MHO objects (PA=33◦); these values are well
+consistent with the direction of the same structures seen in the
+neighbor star R CrA (Rigliaco et al. 2019; Mesa et al. 2019)).
+This suggests that the bulk of the inflow of material that formed
+the T CrA system was coplanar with this filament and that the
+original disk of T CrA was likely oriented at the PA of the fil-
+ament; this is actually the case also for the disk around R CrA.
+However, the current outer disk of T CrA has a very different
+orientation (PA=7 degree), though it seems to be still fed by the
+same filament. This is because T CrA appears to be presently
+offset by a few hundreds au (a few arcsec on sky) with respect
+to the filament. Considering the age of T CrA (likely 1-3 Myr),
+this offset is indeed very small, corresponding to a minuscule ve-
+locity of only ∼ 1m/s. This suggests that the generation of mis-
+aligned structure is very likely whenever accretion on the disk is
+prolonged over such long intervals of time.
+4.3. Spectral Energy Distribution
+We model the SED of T CrA using the dust radiative transfer
+model developed by Whitney et al. (2003b,a). The code uses a
+Monte Carlo radiative transfer scheme that follows photon pack-
+ets emitted by the central star as they are scattered, absorbed,
+and re-emitted throughout the disk. For the modeling we have
+assumed that the geometry of the star+disk system is comprised
+by a central 2.0 M⊙ source emitting photons and a gapped and
+misaligned circumstellar disk as described above. The total mass
+of the disk Mdisk=10−3M⊙, which is in agreement with Mdust re-
+trieved by Cazzoletti et al. (2019) using the 1.3 mm continuum
+flux, assuming an ISM gas-to-dust ratio of 100. The outcome of
+the model, shown in orange in Fig. 8, well reproduces the ob-
+served photometric points collected in Table 2, suggesting that
+the interpretation of inner and outer disks misaligned with re-
+spect to each other is in very good agreement with the collected
+photometry 4. For comparison, we also show the SED obtained
+with the same parameters, in the case where no misalignment
+between inner and outer disk is assumed (red profile). In this
+case the curve does not well reproduce the observed photome-
+try at wavelengths longer than ∼10–15 µm. We must notice that
+the radiative transfer model does not account for the binary star,
+hence it may cause deviation in the illumination of the disk. In
+particular, in their orbit the two stars spend time above the disk
+midplane, hence illuminating the circumbinary disk from above.
+4 The apparent oscillations of the model at wavelengths longer than
+300 µm is due to low number statistics and has no physical meaning.
+Article number, page 9 of 17
+
+A&A proofs: manuscript no. TCrA_Rigliaco
+λc
+Flux
+Facility
+Reference
+(µm)
+(Jy)
+0.349
+0.00531
+SkyMapper
+Wolf et al. (2018)
+0.444
+0.00792
+CTIO
+Henden et al. (2016)
+0.444
+0.00988
+UCAC4-RPM
+Nascimbeni et al. (2016)
+0.482
+0.0138
+CTIO
+Henden et al. (2016)
+0.497
+0.0126
+SkyMapper
+Wolf et al. (2018)
+0.504
+0.0142
+GAIA
+Gaia Collaboration (2020)
+0.554
+0.0163
+Hamilton
+Herbig & Bell (1988)
+0.554
+0.0171
+CTIO
+Henden et al. (2016)
+0.554
+0.0204
+UCAC4-RPM
+Nascimbeni et al. (2016)
+0.604
+0.0181
+SkyMapper
+Wolf et al. (2018)
+0.762
+0.045
+GAIA
+Gaia Collaboration (2020)
+0.763
+0.0539
+CTIO
+Henden et al. (2016)
+1.24
+0.425
+2MASS-J
+Cutri et al. (2003)
+1.65
+0.871
+2MASS-H
+Cutri et al. (2003)
+2.16
+1.55
+2MASS-K
+Cutri et al. (2003)
+3.55
+1.93
+Spitzer/IRAC
+Gutermuth et al. (2009)
+4.49
+2.07
+Spitzer/IRAC
+Gutermuth et al. (2009)
+5.73
+2.38
+Spitzer/IRAC
+Gutermuth et al. (2009)
+11.6
+3.48
+WISE/W3
+Cutri & et al. (2012)
+19.7
+23.4
+SOFIA
+Sandell et al. (2021)
+22.1
+23.8
+WISE/W4
+Cutri & et al. (2012)
+25.3
+30.7
+SOFIA
+Sandell et al. (2021)
+31.5
+29.0
+SOFIA
+Sandell et al. (2021)
+37.1
+29.3
+SOFIA
+Sandell et al. (2021)
+70.0
+19.3
+Herschel
+Herschel Group et al. (2020)
+100.0
+14.2
+Herschel
+Herschel Group et al. (2020)
+160.0
+5.0
+Herschel
+Herschel Group et al. (2020)
+1300
+0.00499
+ALMA
+Cazzoletti et al. (2019)
+Table 2: List of the fluxes at different wavelengths collected
+from the literature used for the SED.
+In the two SEDs shown in Fig. 8 we do not account for this ef-
+fect.
+4.4. Hydrodynamical Simulation
+We perform a 3D hydrodynamical simulation of the T CrA con-
+figuration considered in this work using the Smoothed Particle
+Hydrodynamics (SPH) code Phantom (Price et al. 2018b; Mon-
+aghan 2005; Price 2012). The initial conditions of the system
+are set following the observational constraints acquired so far.
+T CrA is modeled as a binary system with masses 1.7 M⊙, and
+1.0 M⊙ for the primary and secondary component, respectively.
+Each star is simulated as a sink particle (Price et al. 2018b; Bate
+et al. 1995) with an accretion radius of 0.5 au. The orbit is eccen-
+tric, and the period of the binary star is 29.6 years, correspond-
+ing to a semi-major axis of 13.3 au. The orbit is seen edge-on
+with an inclination of 90◦, and PAorbit is perpendicular to the out-
+flowing material (PAorbit=145◦). The outer disk, extending from
+Rin = 25 au to Rout = 100 au is simulated with 8 × 105 SPH par-
+ticles, resulting in a smoothing length ≈ 0.2 times the disk scale
+height. The inner disk, extending from rin = 1 au to rout = 5 au,
+and co-planar to the orbit of the binary star, is simulated with
+2 × 105 SPH particles, resulting in a smoothing length of about
+the disk scale height. Outflows and inflows are not considered in
+this model. Viscosity is implemented with the artificial viscosity
+method (Lucy 1977; Gingold & Monaghan 1977) that results in
+an Shakura & Sunyaev (1973) α-viscosity as shown by Lodato
+& Price (2010). We use α ≈ 5 × 10−3. We run the full hydrody-
+namical model (with both the outer and the inner disk) for 100
+binary orbits in order to relax the initial condition and to produce
+a synthetic image of the system to compare with the observation.
+To perform a direct comparison with observations of T CrA we
+Fig. 8: SED of TCrA. The black asterisks show the published
+photometry as reported in Table 2. The orange curve shows the
+total emission. The magenta line shows the SED component due
+to stellar origin, in blue the component due to the disk, and in
+green the component due to the envelope. The red curve shows
+the emission if no misalignment between the intermediate and
+outer disk is assumed. The oscillations in the model curves at
+the longest wavelengths are artifacts related to the finite number
+of photon packets considered in the Monte Carlo scheme.
+post-processed our simulation using the Monte Carlo radiative
+transfer code MCFOST (Pinte et al. 2016) in order to produce
+synthetic images of the hydrodynamical model. MCFOST maps
+the physical quantities in the SPH simulation (e.g. dust and gas
+density, temperature) onto a Voronoi mesh directly built around
+the SPH particles, without interpolation. We adopt a gas-to-dust
+mass ratio equals to 100 and we assume micrometer grains to be
+well coupled with the gas. These grains scatter the stellar light
+collected by SPHERE and are assumed to be spherical and ho-
+mogeneous (as in the Mie theory). Their chemical composition
+is 60% astronomical silicates and 15% amorphous carbons (as
+DIANA standard dust composition, Woitke et al. 2016) and they
+have a porosity of 10%. The gas mass is directly taken from the
+SPH simulation. We use the same distance from the source used
+in this paper (149.4 pc) and ≈ 106 photon packets to compute
+the temperature profile of the model and ≈ 1010 photon packets
+to compute the source function of the model in order to produce
+the scattered light image at 2 µm wavelength.
+The total intensity polarized light image obtained with the
+hydrodynamical simulation is show in the left panel of Fig. 9.
+The middle panel is the synthetic image convolved to the
+SPHERE/IRDIS resolution and in the right panel we show the
+observed image. There are a few features that are clearly repro-
+duced in the simulation: the dark lane, the offset of the dark lane
+with respect to the center of the image, the top-surface of the disk
+brighter than the bottom-side of the disk. There are two bright
+spots in the East-West direction on the convolved synthetic im-
+age, that are also observed in the real image. These points are
+due to the intermediate circumbinary disk that breaks from the
+outer regions, precessing as a rigid body, and leading to its evo-
+lution. The breaking of the inner disk generates an intermediate
+disk, that is visible as bright spots at the East and West side of
+the coronagraph. We must notice that the simulation does not
+take into consideration the outflowing material, and does not ac-
+Article number, page 10 of 17
+
+1000
+collectedphotometry
+Itotal
+stellarorigin
+100
+- disk origin
+LL
+:envelopeorigin
+Itotal-nodisksmisalignment
+10
+TTT
+(Jy)
+Flux
+1
+LLL
+0.1
+TT
+0.01
+L
+*
+0.001
+0.1
+1
+10
+100
+1000
+104
+Wavelength (μm)Rigliaco et al.: DESTINYS–TCrA
+count for the replenishment of the outer disk due to the accretion
+streamers (hence slowing down its expansion). A more detailed
+simulation is needed for T CrA, but it is beyond the scope of this
+observational paper and will be discussed in a separate publica-
+tion.
+In order to measure how the circumbinary disk mass dis-
+tributes among the binary stars, we run a second hydrodynamical
+model as the one described above but without the circumprimary
+disk. Indeed, accretion into a binary system happens via the for-
+mation of up to three disks (two circumstellar disks, one around
+each component, and a circumbinary disk, Monin et al. (2007)).
+The two circumstellar disks are periodically replenished by ac-
+cretion streamers pulled from the inner edge of the circumbi-
+nary disks by the stars (Artymowicz & Lubow 1994; Tofflemire
+et al. 2017). In a quasi-steady state regime, the mass flux en-
+tering the Roche lobe of a star via the gas streamers equals the
+star accretion rate. Thus, we can reliably measure the fraction of
+mass accreted onto a star by simulating only the circumbinary
+disk, provided that the stellar Roche lobes are resolved by the
+simulation and the central part of the disk has relaxed (as done
+with SPH simulations e.g. in Young & Clarke 2015 and recently
+tested in Ceppi et al. 2022). In general, simulations of accretion
+into binary systems find that the primordial mass ratio is pushed
+towards unity (that is, closer to equal masses in the binary com-
+ponents) by accretion from a circumbinary disk (Clarke 2012).
+This is due to the ease with which the secondary component ac-
+cretes the infalling gas, as it lies farther from the binary barycen-
+ter and closer to the disk edge. Its differential velocity with re-
+spect to the gas is also low, allowing it to accrete efficiently. In
+the case of T CrA the primary star is still accreting more than the
+secondary (see Fig. 10). This is due to the misalignment between
+inner and outer disk that makes the secondary to be at consider-
+able height over/below the disk for a large fraction of its orbit.
+5. Summary and Conclusions
+We investigate new and archival data of the Herbig Ae/Be star
+T CrA collected with different instruments. The analysis of the
+data shows that T CrA is a very interesting and complex system,
+belonging to one of the nearest and most active region of star
+formation. Combining archival NACO imaging data with pho-
+tometric data, and new and archival SPHERE adaptive optics
+images we study the complex stellar environment around T CrA
+and the stellar properties:
+– the outer disk is seen edge-on as a dark lane elongated ap-
+proximately in the N-S. The dark lane is shifted by 122 mas
+with respect to the center of the image, and it is seen with
+a PA of 7◦. This value is in very good agreement with the
+value recently found by Cugno et al. 2022 using a different
+instrument and set of data;
+– the bright illuminated top-side of the disk surface is clearly
+visible in scattered light;
+– extended emission in the NE–SW direction, identified as fea-
+ture 1, is consistent in direction with the line connecting the
+two-lobed MHOs seen on larger scale. It is most likely out-
+flowing material, with PA=33◦, consistent with the PA of the
+two MHOs.
+– extended emission in the N-S direction, identified as feature
+2, is interpreted as large scale streamers of material likely in-
+falling onto the disk. In the North the streamer extends up to
+∼4.5′′ from the central system, while in the South it extends
+up to the edge of the field of view, and probably beyond, as
+suggested by previous stellar polarization images in the op-
+tical and near-IR;
+– the periodic behavior of the light curve suggests a cen-
+tral binary with a period of 29.6 years. Even if the non-
+coronagraphic images acquired with NACO and SPHERE do
+not show direct evidence of the presence of a stellar compan-
+ion, a detailed comparison of the position of the secondary
+along the proposed orbit at the epochs of the observations
+acquired so far with NACO and SPHERE shows that in all
+of them it was too close to the primary star for detection as
+a separate object. According to our modeling results the two
+components will be at their maximum separation in 2027: ap-
+propriate high-contrast images at that epoch should provide
+direct evidence of the binary system.
+Overall, we find that the binary system and intermediate
+circumbinary disk lay on different geometrical planes, placing
+T CrA among the objects with a misaligned inner disk. Inner
+and outer disk misalignment is not rare, and in very recent years,
+thanks to high-contrast imaging, it is becoming clear that the
+misalignment can also be due to the accretion history of the star-
+forming cloud onto the disk. Indeed in the case of T CrA (as well
+as SU Aur) we found evidences of the presence of streamers of
+accreting material that connect the filament along which the star
+has formed with the outer part of the disk. These streamers have
+an angular momentum with respect to the star whose direction is
+very different from that of the system (in the case of T CrA, this
+is dominated by the binary) causing a misalignment between an
+inner and outer disk.
+Besides characterizing the disk/outflow structures around
+T CrA, we have also modeled its spectral energy distribution,
+showing that the disk geometry obtained is well consistent with
+the observed SED, and such consistency is not reached if we
+do not consider the misalignment between inner and outer disk.
+Moreover, we have performed hydrodynamical simulation of the
+configuration for 100 orbits of the binary star. The model is
+consistent with the observations and the analysis of the accre-
+tion rates of the individual stars shows that the accretion hap-
+pens mainly onto the primary star, rather than on the secondary,
+as a consequence of the inclination between inner/intermediate
+and outer disk. Also the light curve is easily explained assum-
+ing the configuration of two misaligned disks. Comparison of
+the ALMA continuum and 12CO emission have also been per-
+formed. While for the continuum emission we cannot clearly
+point out the region where the dust is located, if along the disk
+or the outflowing material, the gas emission is most likely due
+to the combination from emission aligned with the disk orienta-
+tion inferred from SPHERE, and emission from the outflowing
+material in the same direction as the MHOs.
+The analysis conducted on T CrA has confirmed its ex-
+tremely interesting and complex nature. As in the case of
+HD142527, the misalignment between inner and outer disk can
+be due to the interaction between the disk and the central binary
+system. On the other hand, the large scale streamers observed in
+the N–S direction are very similar to the disk-cloud interaction
+observed for SU Aur, that represents material infalling onto the
+disk, and inner and outer disk misalignment might be caused by
+this interaction. It comes clear the need for high resolution obser-
+vations to disentangle the different effects that shape early plan-
+etary system formation. T CrA is an excellent target/laboratory
+to better understand the impact of binarity and the environment
+in the evolution of protoplanetary disks.
+Acknowledgements. We would like to thank the referee Roubing Dong, whose
+careful and constructive comments improved the quality of this manuscript. E.R.
+was supported by the European Union’s Horizon 2020 research and innovation
+programme under the Marie Skłodowska-Curie grant agreement No 664931.
+This work has been supported by the project PRIN INAF 2016 The Cradle of Life
+Article number, page 11 of 17
+
+A&A proofs: manuscript no. TCrA_Rigliaco
+Fig. 9: Snapshot of the SPH simulation compared to the observed image. The left panel shows the result in total intensity of the
+SPH simulation, with a resolution of 4.0 mas/pixel. In the middle panel the same image convolved to the SPHERE/IRDIS resolution
+(12.25 mas/pixel). On the right the observed total intensity image. All images have a 2′′ field of view.
+Fig. 10: Mass accretion rate ratio of secondary and primary star
+as a function of the number of orbits.
+- GENESIS-SKA (General Conditions in Early Planetary Systems for the rise of
+life with SKA) and by the "Progetti Premiali" funding scheme of the Italian Min-
+istry of Education, University, and Research. C.F.M acknowledges funding from
+the European Union under the European Union’s Horizon Europe Research &
+Innovation Programme 101039452 (WANDA). Views and opinions expressed
+are however those of the author(s) only and do not necessarily reflect those of
+the European Union or the European Research Council. Neither the European
+Union nor the granting authority can be held responsible for them. T.B. acknowl-
+edges funding from the European Research Council (ERC) under the European
+Union’s Horizon 2020 research and innovation programme under grant agree-
+ment No 714769 and funding by the Deutsche Forschungsgemeinschaft (DFG,
+German Research Foundation) under grants 361140270, 325594231, and Ger-
+many’s Excellence Strategy - EXC-2094 - 390783311. A.R. has been supported
+by the UK Science and Technology research Council (STFC) via the consoli-
+dated grant ST/S000623/1 and by the European Union’s Horizon 2020 research
+and innovation programme under the Marie Sklodowska-Curie grant agreement
+No. 823823 (RISE DUSTBUSTERS project). This paper makes use of the fol-
+lowing ALMA data: ADS/JAO.ALMA#2016.0.01058.S. ALMA is a partnership
+of ESO (representing its member states), NSF (USA) and NINS (Japan), together
+with NRC (Canada), MOST and ASIAA (Taiwan), and KASI (Republic of Ko-
+rea), in cooperation with the Republic of Chile. The Joint ALMA Observatory is
+operated by ESO, AUI/NRAO and NAOJ. MRH acknowledges the assistance of
+Allegro, the ARC node in the Netherlands, who assisted with the calibration of
+this data set. This work is partly based on data products produced at the SPHERE
+Data Centre hosted at OSUG/IPAG, Grenoble. We thank P. Delorme and E. La-
+gadec (SPHERE Data Centre) for their efficient help during the data reduction
+process. SPHERE is an instrument designed and built by a consortium consist-
+ing of IPAG (Grenoble, France), MPIA (Heidelberg, Germany), LAM (Marseille,
+France), LESIA (Paris, France), Laboratoire Lagrange (Nice, France), INAF Os-
+servatorio Astronomico di Padova (Italy), Observatoire de Genève (Switzerland),
+ETH Zurich (Switzerland), NOVA (Netherlands), ONERA (France) and AS-
+TRON (Netherlands) in collaboration with ESO. SPHERE was funded by ESO,
+with additional contributions from CNRS (France), MPIA (Germany), INAF
+(Italy), FINES (Switzerland) and NOVA (Netherlands). SPHERE also received
+funding from the European Commission Sixth and Seventh Framework Pro-
+grammes as part of the Optical Infrared Coordination Network for Astronomy
+(OPTICON) under grant number RII3-Ct-2004-001566 for FP6 (2004-2008),
+grant number 226604 for FP7 (2009-2012), and grant number 312430 for FP7
+(2013-2016). This work has made use of data from the European Space Agency
+(ESA) mission Gaia (https://www.cosmos.esa.int/gaia), processed by
+the Gaia Data Processing and Analysis Consortium (DPAC, https://www.
+cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has
+been provided by national institutions, in particular the institutions participating
+in the Gaia Multilateral Agreement. We acknowledge with thanks the variable
+star observations from the AAVSO International Database contributed by ob-
+servers worldwide and used in this research.
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+
+A&A proofs: manuscript no. TCrA_Rigliaco
+Appendix A: SPHERE polarimetric images
+In Figure A.1 we present the Stokes Q and U, as well as the
+derived QΦ and UΦ images of T CrA. The flux calibration was
+carried out by measuring the flux of the central star in the non-
+coronagraphic flux calibration images, taken at the beginning
+and end of the observation sequence. To convert pixel counts
+to physical units we used the 2MASS H-band magnitude of the
+star.
+4
+3
+2
+1
+0
+1
+2
+3
+4
+4
+3
+2
+1
+0
+1
+2
+3
+4
+Q
+4
+3
+2
+1
+0
+1
+2
+3
+4
+4
+3
+2
+1
+0
+1
+2
+3
+4
+U
+4
+3
+2
+1
+0
+1
+2
+3
+4
+4
+3
+2
+1
+0
+1
+2
+3
+4
+Qφ
+4
+3
+2
+1
+0
+1
+2
+3
+4
+4
+3
+2
+1
+0
+1
+2
+3
+4
+Uφ
+∆RA (arcsec)
+∆Dec (arcsec)
+60
+45
+30
+15
+0
+15
+30
+45
+60
+(mJy/arcsec2)
+Fig. A.1: Flux calibrated image of the Q, U, QΦ and UΦ frames.
+Appendix B: Proper motion analysis
+The average proper motion for the on-cloud Coronet cluster
+members obtained using Gaia DR2 data from Galli et al. 2020,
+is µα cos δ = 4.3 mas yr−1 and µδ=-27.3 mas yr−1 with a small
+dispersion of less than 1 mas/yr for the individual objects. As we
+mentioned in the introduction, Gaia does not provide astrometric
+solutions and proper motion for the star T CrA. However, we can
+check for peculiar/transient motion of the star using the follow-
+ing procedure. We collect from UCAC4 (Zacharias et al. 2012),
+PPMXL (Roeser et al. 2010) and Gaia DR3 (Gaia Collaboration
+et al. 2016, 2021) database a list of 10 bright stars in the T CrA
+surroundings for which proper motion are available in all cata-
+logs. These stars, listed in Table B.1, are selected such that they
+have magG ≤14 and lay within 10′ from T CrA. For these stars
+we measure a long term proper motion given by the difference in
+position between the UCAC4, PPMXL, and Gaia DR3 epochs.
+The long term proper motion, defined as the motion of the star
+between different epochs of observations is measured as:
+µα cos δ = (RAEpoch1 − RAEpoch2) ∗ cos(DECEpoch1)
+(Epoch1 − Epoch2)
+(B.1)
+µδ = (DECEpoch1 − DECEpoch2)
+(Epoch1 − Epoch2)
+(B.2)
+The analysis of the proper motion between the various epochs of
+the selected stars allows us to find a systematic offset between the
+average of the coordinate systems of UCAC4 and PPMXL with
+respect to Gaia DR3, that averages to 0.41±2.46 mas yr−1 in RA
+and 9.67 mas yr−1±2.93 mas yr−1 in DEC for this specific region
+of the sky. For T CrA we obtain an estimate of the proper motion
+of the star by correcting the long term proper motion with the
+systematic offset, finding as values µα cos δ = 8.5 ± 2.5 mas yr−1
+and µδ=-33.5±2.9 mas yr−1. Considering the average proper
+motion of the on-cloud members we find for T CrA an ap-
+parent motion µα cos δ = 4.2 ± 2.5 mas yr−1 in RA and µδ=-
+6.2±2.9 mas yr−1 in DEC.
+Fig. B.1: Proper motion of the ten stars reported in Table B.1.
+The cyan square represents the average proper motion for the on-
+cloud Coronet cluster obtained using Gaia DR2 data (Galli et al.
+2020). The blue star is the calculated apparent proper motion
+of T CrA after correcting the long term proper motion for the
+systematic offset. In orange the direction of the systematic offset
+of the proper motion due to the different coordinate system.
+Article number, page 14 of 17
+
+-10
+PMfromGaiafortheselectedstars
+average PM of the on-cloud members
+apparent PM of TCrA
+-20
+μ(mas/yr)
+-30
+40
+average systematic offset
+betweenGaiaandUCAC4/PPMXL
+50
+40
+-20
+0
+20
+μαcosd(mas/yr)Rigliaco et al.: DESTINYS–TCrA
+T CrA
+V709 CrA
+HD176269
+HD176270
+TY CrA
+HD176423
+V702 CrA
+HD176386
+HD176497
+HD176018
+CD-36 13202
+(UCAC4)
+RA
+285.494904
+285.395229
+285.263532
+285.267908
+285.420107
+285.460076
+285.508229
+285.412217
+285.528297
+284.930902
+284.920385
+Dec
+-36.963871
+-37.015723
+-37.060898
+-37.061555
+-36.876063
+-36.664651
+-37.128761
+-36.890712
+-36.361622
+-36.788004
+-36.588797
+Ep. RA
+1997.40
+1985.45
+1991.25
+1991.25
+1991.09
+1990.50
+1985.88
+1991.25
+1990.57
+1988.91
+1995.62
+Ep. Dec
+1997.77
+1985.43
+1991.25
+1991.25
+1990.52
+1989.91
+1984.44
+1991.25
+1990.28
+1988.04
+1995.74
+(PPMXL)
+RA
+285.494908
+285.395224
+285.263532
+285.267908
+285.420102
+285.460076
+285.508229
+285.412219
+285.528303
+284.930902
+284.920394
+Dec
+-36.963869
+-37.015722
+-37.060898
+-37.061555
+-36.876064
+-36.664651
+-37.128761
+-36.890703
+-36.361625
+-36.788007
+-36.588800
+Ep. RA
+1999.95
+1988.00
+1991.73
+1991.18
+1991.53
+1991.41
+1997.44
+1991.14
+1991.32
+1991.23
+1997.69
+Ep. Dec
+1999.95
+1986.70
+1991.64
+1991.19
+1991.76
+1991.62
+1998.14
+1991.09
+1991.38
+1991.64
+1998.44
+(Gaia DR3)
+RA
+285.494959
+285.395278
+285.263578
+285.267966
+285.420142
+285.460103
+285.508268
+285.412238
+285.528324
+284.930856
+284.920226
+Dec
+-36.963983
+-37.015845
+-37.061040
+-37.061682
+-36.876201
+-36.664770
+-37.128870
+-36.890838
+-36.361752
+-36.788188
+-36.589008
+Ep. RA
+2016.0
+2016.0
+2016.0
+2016.0
+2016.0
+2016.0
+2016.0
+2016.0
+2016.0
+2016.0
+2016.0
+Ep. Dec
+2016.0
+2016.0
+2016.0
+2016.0
+2016.0
+2016.0
+2016.0
+2016.0
+2016.0
+2016.0
+2016.0
+Table B.1: List of the stars used to measure the proper motion offset.
+Article number, page 15 of 17
+
+A&A proofs: manuscript no. TCrA_Rigliaco
+Epoch
+Offset B-A (mas)
+dH (mag)
+2007.54 (NACO)
+-26.0±7.0
+1.0±0.6
+2016.25 (NACO)
+-72.0±5.0
+0.0±0.7
+2016.60 (SPHERE)
+-69.0±5.0
+0.2±0.7
+2018.36 (SPHERE )
+-44.0±7.0
+0.3±0.6
+2021.50 (SPHERE)
+11.0±7.0
+1.0±0.5
+Table C.1: Relative position of the secondary star (B) with re-
+spect to the primary star (A), and relative contrast (dH) in H-
+band of the secondary star with respect to the primary star, for a
+period of 29.6 years. The offset is defined in the direction of the
+semi-major axis of the stellar orbit.
+Appendix C: Binarity and light curve
+The light curve of T CrA appears to be periodic. The period is
+found to be 29.6 years and it can be due to the presence of a
+binary star at the center of the T CrA system with a mass ratio
+q∼0.5±0.2, that is partially obscured by a disk seen edge-on, that
+has an offset with respect to the photocenter of the binary star of
+∼90 mas. The model of this binary system, described in Sect 3.1,
+is also able to account for the large apparent proper motion mea-
+sured in the period between 1998 and 2016. None of the images
+acquired in recent years with NACO (in 2007, 2016 and 2017)
+and SPHERE (in 2016, 2018 and 2021) shows clear evidence of
+a binary system for T CrA. Hence, we have checked what was
+the relative position of the secondary star with respect to the pri-
+mary for every single epoch for which we have an image, and
+the H-band contrast that should be observed. These quantities
+are shown in Table C.1. These value are all consistent with the
+fact that the binary system is not clearly resolved. Indeed, in the
+2007, 2018 and 2021 epochs the separation between the two stars
+is too small to see the two sources separately. On the contrary,
+the two 2016 epochs have a larger separation, though still within
+2×λ/D, that is so close that the secondary cannot be clearly sep-
+arated from the primary. We notice however, that in both images
+acquired around this epoch with NACO and SPHERE, the PSF
+appears elongated in the NW-SE direction, that corresponds to
+the direction of the major axis of the orbital motion of the bi-
+nary system. The average position angle of the elongated PSFs
+acquired in 2016 is 130±15◦, in very good agreement with the
+direction of the peculiar proper motion (PAPM) measured, and
+with the hypothesis that the orbit of the binary system is seen
+edge-on, and perpendicular to the outflow. This elongation in
+different epochs supports then the scenario of a binary star. In
+a few years, namely in 2027, when the system is at its highest
+separation, the secondary component should be detectable with
+high-contrast images.
+We have also considered that the period of the system
+might be double than the period measured in Sect. 2.3, namely
+59.2 years. While the light curve can be, also in this case, eas-
+ily reproduced, there are several observational shortcomings in
+this interpretation. First of all we must notice that in this case
+the model predicts a mass ratio q as high as 0.9, and an off-
+set of the disk of ∼10 mas. This last quantity is in disagree-
+ment with the observations, that instead show that the disk
+dark lane has an offset ten times larger. Moreover, the position
+of the center of the binary system as retrieved by assuming a
+59.2 years period is not consistent with the motion of the system
+obtained from UCAC4/PPMXL and Gaia DR3 data. Addition-
+ally, in the SPHERE image acquired in 2021, the predicted sep-
+aration between the primary and secondary component should
+be 108±6 mas, with a contrast dH=0.5±0.1 mag, making it visi-
+ble as a separate point source in the image. The relative position
+Epoch
+Offset B-A (mas)
+dH (mag)
+2007.54 (NACO)
+91.0±7.0
+1.0±0.2
+2016.25 (NACO)
+-39.0±8.0
+0.0±0.3
+2016.60 (SPHERE)
+-44.0±8.0
+0.0±0.2
+2018.36 (SPHERE)
+-70.0±7.0
+0.0±0.2
+2021.50 (SPHERE)
+-108.0±6..0
+0.0 ± 0.1
+Table C.2: Relative position of the secondary star (B) with re-
+spect to the primary star (A), and relative contrast (dH) in H-
+band of the secondary star with respect to the primary star, for a
+period of 59.2 years. The offset is defined in the direction of the
+semi-major axis of the stellar orbit.
+of the secondary star with respect to the primary for every sin-
+gle epoch for which we have an image, and the H-band contrast
+that should be observed are reported in Table C.2. The image
+does not reveal the presence of the secondary star. Given these
+shortcomings between observations and the output of the model,
+we exclude that the period of the binary star is 59.2 years. Fig-
+ure C.1 and C.2 show the corner plot of the derived quantities
+of the model used in Sect. 4.1 to model the light curve assuming
+a period of 29.6 years or the double (59.2 years). The light curve
+for the period of 59.2 years is shown in Fig. C.3.
+Fig. C.1: Corner plot showing the results of the MC parameters
+estimation for the model described in the paper when a period of
+29.2 years is considered. The plots show the 2D joints posterior
+densities of all couple of parameters.
+Article number, page 16 of 17
+
+2009
+2008
+2007
+ 2006
+2005
+2004
+2003
+-1.0-0.8-0.6-0.4-0.2 0.0
+log q
+(mag)
+12
+Max absorption 
+10
+8
+6
+4
+1.0-0.8-0.6-0.4-0.2 0.0
+log q
+2003
+2004
+2005
+2006
+2007
+2008
+2009
+TO
+Disk Thickness (mas)
+Disk Thickness (mas)
+3688885
+40
+140
+120
+8688867
+100E
+Thickness
+80 E
+60 E
+40
+20 E
+1.0-0.8-0.6-0.4-0.2 0.0
+6
+8
+10
+12
+log q
+20032004
+2005
+2006
+2007
+20082009
+Max absorption (mag)
+TO
+148
+160E
+(mas)
+140E
+offset (mas)
+160
+(mas)
+160
+148
+(sDw)
+120 E
+2888898
+offset
+Disk offset
+100日
+offset
+80 E
+Disk
+60 E
+Disk
+40 E
+1.0-0.8-0.6-0.4-0.2 0.0
+20日
+4
+6
+8
+10
+12
+20 40 60 80100120140
+b 6ol
+2003
+20042005
+2006
+200720082009
+Max absorption (mag)
+Disk thickness (mas)
+TORigliaco et al.: DESTINYS–TCrA
+Fig. C.2: Corner plot showing the results of the MC parameters
+estimation for the model described in the paper when a period of
+59.6 years is considered. The plots show the 2D joints posterior
+densities of all couple of parameters.
+Fig. C.3: Light curve of T CrA (red points) compared to the light
+curves computed with the MC model (black lines) assuming a
+period of 59.2 years.
+Article number, page 17 of 17
+
+2017
+2016
+2015
+ 2014
+2013
+2012
+2011
+1.0-0.8-0.6-0.4-0.2 0.0
+log q
+absorption
+XDW
+.5
+1.0-0.8-0.6-0.4-0.2 0.0
+log q
+2011
+2012
+2013
+2014
+2015
+20162017
+TO
+(mas)
+(mas)
+140
+40
+140
+2888898
+120
+Disk Thickness 
+Thickness
+00
+100
+80
+80
+60
+60
+40
+MSI
+40
+Disk
+20
+1.0-0.8-0.6-0.4-0.2 0.0
+4.0 4.5 5.0 5.5 6.0 6.5 7.0
+log q
+2011
+2012
+2013
+2014
+2015
+2016
+2017
+Max absorption (mag)
+TO
+< offset (mas)
+(mas)
+(mas)
+40
+40
+40
+40
+SDU
+20
+20
+20
+offset
+< offset
+0
+0
+0
+Disk
+Disk
+20
+Disk
+Disk
+20
+-1.0-0.8-0.6-0.4-0.2 0.0
+4.0 4.5 5.0 5.5 6.0 6.5 7.0
+20 40 60 80100120140
+log q
+201120122013
+2014
+201520162017
+Max absorption (mag)
+Disk thickness (mas)
+TO12
+mag
+13
+15
+1900
+1920
+1940
+1960
+1980
+2000
+2020
+Time
+(yr

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+arXiv:2301.11793v1  [hep-th]  27 Jan 2023
+1
+Schwinger-Dyson equation in complex plane
+− The (1 + 1)-dimensional Gross-Neveu model −
+Hidekazu Tanaka ∗) and Shuji Sasagawa
+Rikkyo University, Tokyo 171-8501, Japan
+ABSTRACT
+Effective mass and energy of fermions are investigated using the Schwinger-
+Dyson equation (SDE) in the complex plane. As a simple example, we solve the
+SDE for the (1+1)-dimensional Gross-Neveu model and study some properties of
+the effective mass and energy of fermions in the complex plane.
+∗) E-mail:[email protected]
+
+2
+§1.
+Introduction
+Behavior of effective mass and energy in non-perturbative region is one of in-
+teresting problems to be studied, because they are related to the properties of the
+propagator in non-perturbative region.
+Particularly, interesting phenomena are expected in Minkowski space. In some
+studies, it has been pointed out that the positivity of the gluon spectral function in
+quantum chromodynamics (QCD) appears to be violated in strong coupling region.
+[1,2] This indicates that gluons do not have asymptotic states, suggesting that gluons
+are confined to hadrons.
+Unfortunately, lattice simulations for studying non-perturbative region do not
+allow direct evaluation of the imaginary part of the effective mass in Minkowski space.
+One useful tool for studying non-perturbative phenomena is the Schwinger-Dyson
+equation (SDE) [3,4]. The structure of the gluon propagator has been evaluated by
+the SDE, in which the squared momentum for the gluon is extended to the complex
+value. [5,6]. They found that the gluon propagator has poles not on the real axis
+in the squared momentum plane at zero temperature. In their framework, they also
+showed that the spectral function of the gluon violates positive value condition.
+In evaluations using the SDE, one of difficulties in Minkowski space is the exis-
+tence of poles in propagator. This requires knowledge of the precise pole positions of
+the propagator in the self-energy calculation. To avoid this, it is computed by Wick-
+rotating the axis of integration from the real axis to the imaginary axis. However,
+the Wick rotation requires the location of the poles to be known in advance, but the
+value of the mass in the non-perturbative region is non-trivial.
+In this paper, as a starting point for thinking about these problems, we examine
+the (1 + 1)-dimensional (one dimension of time and one dimension of space) Gross-
+Neveu (GN) model at zero temperature. [7] We extend the SDE to the complex
+plane, and integrate the loop momentum around poles of the propagator in the self-
+energy with two different integration paths in the complex energy plane. Then we
+examine the properties of the solutions obtained by the SDE in the complex plane.
+In Section 2, we formulate the SDE for the (1 + 1)-dimensional GN model in
+terms of complex mass and energy.
+In Section 3, we discuss analytical solutions
+for effective mass and energy in the complex plane with finite cutoff values of the
+momentum. In Section 4, we numerically calculate the effective mass and energy
+using the SDE. Section 5 is devoted to the summary and some comments. Explicit
+expressions of the complex mass and energy implemented in calculations are given
+in Appendix.
+
+3
+§2.
+The SDE for effective mass of fermion in complex plane
+The Lagrangian density of the GN model is given by
+L = i ¯ψ∂/ψ + g2
+2 ( ¯ψψ)2,
+(2.1)
+where ψ and g2 are the 2-component fermion field in (1 + 1) dimensions and the
+coupling constant of 4-fermion interaction, respectively.
+In this paper, we evaluate the fermion effective mass M using the SDE. In order
+to obtain the effective mass, we calculate the one-loop self-energy Σ of the fermion
+in (1 + 1) dimensions, which is given by
+Σ = i
+g2
+(2π)2
+�
+d2QTr[S(Q)].
+(2.2)
+In Eq.(2 · 2), S(Q) is an effective propagator of the fermion with momentum Q =
+(q0, q), which is given by
+iS(Q) =
+i
+Q/ − Σ + iε
+(2.3)
+Here, we define Σ ≡ M, because the wave-function renormalization constant of the
+fermion is √Z2 = 1 in one-loop order of perturbation.
+Therefore, the SDE for the effective mass M is given by
+M = i 2g2
+(2π)2
+�
+d2Q
+M
+Q2 − M2 + iε = iλ
+�
+dq0dq
+M
+q2
+0 − q2 − M2 + iε,
+(2.4)
+where we define λ ≡ 2g2/(2π)2 for simplicity. The propagator S(Q) has poles, which
+satisfies q2
+0 − q2 − M2 + iε = 0.
+In this paper, we extend q0 as a complex value z and the effective mass M is also
+extended as a complex value. Explicitly, they are written as q0 = (q0)R + i(q0)I ≡
+zR + izI = z and M ≡ MR + iMI, respectively. Here, we write the denominator of
+the fermion propagator S(Q) as
+z2 − q2 − M2 + iε ≡ z2 − E2(q) = (z − E(q))(z + E(q))
+(2.5)
+with
+E(q) ≡
+�
+E2(q) =
+�
+q2 + M2 − iε ≡ ER(q) + iEI(q).
+(2.6)
+Therefore, the poles are located at z = ±E(q) in the complex z plane. Here, we define
+ER(q) > 0. Explicit relations among the complex values are given in Appendix.
+The SDE for the effective fermion mass in terms of the complex values is written
+as
+M = iλ
+�
+dq
+�
+C
+dz
+M
+z2 − q2 − M2 + iε = iλ
+�
+dq
+�
+C
+dz
+M
+(z − E(q))(z + E(q)).(2.7)
+
+4
+Here, we write above equation as
+M = 1
+2M(+) + 1
+2M(−),
+(2.8)
+where
+M(±) = iλ
+�
+dq
+�
+C
+dz
+1
+z − z±
+�
+M
+z + z±
+�
+≡ iλ
+�
+dq
+�
+C
+dz
+1
+z − z±
+f (±)(z, q) (2.9)
+with z± = ±E(q) and
+f (±)(z, q) =
+M
+z + z±
+.
+(2.10)
+In our calculation, we integrate Eq.(2·9) around z = z± with following two
+integral paths.
+(1) Integral path including the imaginary axis
+In this case, we separate the integral path around the poles z± = ±E(q) to C1
+and C2 as follows:
+For the integral path around z+ = E(q), we take −iΛ0 − η < z < iΛ0 − η as
+the path C1, and the path C2 is defined as clockwise rotation in right-half on the
+complex energy plane with z = Λ0eiθ, where we take the integration from θ = π/2
+to θ = −π/2.
+On the other hand, for the integral path around z− = −E(q), we take −iΛ0+η <
+z < iΛ0 + η as the path C1, and the path C2 is defined as anticlockwise rotation in
+left-half on the complex energy plane with z = Λ0eiθ, where we take the integration
+from θ = π/2 to θ = 3π/2. ∗)
+Integrating over the integral path C around the pole z± = ±E(q) in the right-
+hand side of Eq. (2·9), we have
+M(±) = iλ
+�
+dq(∓2πi)f (±)(z±, q) = πλ
+�
+dq M
+E(q)
+(2.11)
+for Λ0 → ∞. Therefore, the SDE for the effective mass is given by
+M = 1
+2M(+) + 1
+2M(−) = πλ
+�
+dq M
+E(q).
+(2.12)
+For η → 0, this case corresponds to the SDE for Euclidian momentum integration
+with the complex mass M, which is given as
+M = λ
+�
+dq
+� ∞
+−∞
+dq4
+M
+q2
+4 + q2 + M2 − iε
+(2.13)
+with z = iq4 in Eq. (2·7).
+∗) In order to evaluate the contributions from the singular poles on the imaginary axis, we sift
+the integral path by ∓η from the imaginary axis.
+
+5
+(2) Integral path including the real axis
+In this case, we separate the integral path around the poles z± = ±E(q) to C1
+and C2 as follows:
+For the integral path around z+ = E(q), we take −Λ0 − iη < z < Λ0 − iη as the
+path C1 if EI > 0, and the path C2 is defined as anticlockwise rotation in upper-half
+on the complex energy plane with z = Λ0eiθ, where we take the integration from
+θ = 0 to θ = π. If EI < 0, we take −Λ0 + iη < z < Λ0 + iη as the path C1, and the
+path C2 is defined as clockwise rotation in lower-half on the complex energy plane
+with z = Λ0eiθ, where we take the integration from θ = 0 to θ = −π.
+For the integral path around z− = −E(q), we take −Λ0 −iη < z < Λ0 −iη as the
+path C1 if EI < 0, and the path C2 is defined as anticlockwise rotation in upper-half
+on the complex energy plane with z = Λ0eiθ, where we take the integration from
+θ = 0 to θ = π. If EI > 0, we take −Λ0 + iη < z < Λ0 + iη as the path C1, and the
+path C2 is defined as clockwise rotation in lower-half on the complex energy plane
+with z = Λ0eiθ, where we take the integration from θ = 0 to θ = −π. ∗)
+Integrating over the integral path C around the pole z± = ±E(q) in the right-
+hand side of Eq. (2·9), we have
+M(±) = iλ
+�
+dq(±2πi)
+� EI(q)
+|EI(q)|
+�
+f (±)(z±, q) = −πλ
+�
+dq
+� EI(q)
+|EI(q)|
+� M
+E(q)(2.14)
+for Λ0 → ∞.
+Therefore, the effective mass is given as
+M = 1
+2M(+) + 1
+2M(−) = −πλ
+�
+dq
+� EI(q)
+|EI(q)|
+� M
+E(q).
+(2.15)
+For η → 0, this case corresponds to the SDE for Minkowski momentum integra-
+tion with the complex mass M, which is given as
+M = iλ
+�
+dq
+� ∞
+−∞
+dq0
+M
+q2
+0 − q2 − M2 + iε
+(2.16)
+with z = q0 in Eq. (2·7).
+§3.
+Analytical solutions
+We can find analytical solutions for the SDE obtained in the previous section.
+Here, we write the SDE for two different integral paths as
+M = πsλ
+�
+dq M
+E(q)
+(3.1)
+with s = 1 for the case (1) and s = −EI(q)/|EI(q)| = −[(M2)I − ε]/|(M2)I − ε| for
+the case (2), respectively.
+∗) In order to evaluate the contributions from the singular poles on the real axis, we sift the
+integral path by ∓iη from the real axis.
+
+6
+From Eq.(3·1), nontrivial solutions with M ̸= 0 are given by solving the equation
+1 − πsλ
+�
+dq
+1
+E(q) = 0.
+(3.2)
+Thus, the real and imaginary parts of Eq.(3·2) satisfy
+1 − πsλ
+�
+dq ER(q)
+|E(q)|2 = 0
+(3.3)
+and
+πsλ
+�
+dq EI(q)
+|E(q)|2 = 0,
+(3.4)
+respectively.
+Here, Eq. (3·4) is written as
+πsλ
+�
+dq EI(q)
+|E(q)|2 = πsλ((M2)I − ε)
+�
+dq
+1
+2ER(q)|E(q)|2 = 0.
+(3.5)
+For ER(q) > 0, we obtain (M2)I − ε = 0 for sλ ̸= 0, which gives EI(q) = 0.∗)
+Moreover, from
+(M2)I − ε = 2MRMI − ε = 0,
+(3.6)
+the imaginary part of the effective mass MI is given by
+MI =
+ε
+2MR
+.
+(3.7)
+In the calculation below, we neglect the imaginary part of the effective mass for small
+ε. Thus, we approximate as (M2)R ≃ M2
+R for simplicity.
+Using EI(q) = 0, and introducing a ultraviolet cutoff Λ and an infrared cutoff δ
+for the momentum q, we write Eq. (3·3) as
+1 = πsλ
+� Λ
+−Λ
+dq 1
+ER
+θ(|q| − δ) = 2πsλ
+� Λ
+δ
+dq
+1
+�
+q2 + M2
+R
+.
+(3.8)
+Here, θ(|q| − δ) denotes the step function for restriction of the momentum q.
+Eq. (3·8) gives
+1 = 2πsλ log
+������
+Λ +
+�
+Λ2 + M2
+R
+δ +
+�
+δ2 + M2
+R
+������
+,
+(3.9)
+which is satisfied if sλ > 0. Therefore, for λ > 0, s = −EI(q)/|EI(q)| = 1 should be
+satisfied for the case (2).
+∗) As shown in the next section, EI(q) is determined by an asymptotic value, which is numerically
+calculated by the SDE with a given initial value of the mass M.
+
+7
+Defining mR = MR/Λ, ¯δ = δ/Λ and
+1 +
+�
+1 + m2
+R
+¯δ +
+�
+¯δ2 + m2
+R
+= e1/(2πsλ) ≡ ζ,
+(3.10)
+Eq. (3·10) is written as
+m2
+R(Am2
+R − B) = 0
+(3.11)
+with A = (1 − ζ2)2 and B = 4ζ(1 − ¯δζ)(ζ − ¯δ).
+The solution for m2
+R ̸= 0 is given as
+m2
+R = B
+A = 4ζ(1 − ¯δζ)(ζ − ¯δ)
+(1 − ζ2)2
+.
+(3.12)
+For sλ > 0, ζ − ¯δ > 0 is satisfied. Moreover, m2
+R > 0 demands 1 − ¯δζ > 0, which
+gives
+ζ = e1/(2πsλ) < 1
+¯δ = Λ
+δ .
+(3.13)
+Eq. (3·13) restricts the coupling constant λ as
+λ >
+1
+2π log Λ
+δ
+≡ λc
+(3.14)
+with s = 1.
+For above restriction of λ, the real part of the effective mass is given as
+mR = MR
+Λ
+= ±
+�
+4ζ(1 − ¯δζ)(ζ − ¯δ)
+(1 − ζ2)2
+.
+(3.15)
+§4.
+Numerical solutions
+In this section, we calculate the SDE for two different integral paths. The SDE
+is given in Eq. (3·1). In numerical calculation, we write the SDE for the real and
+imaginary parts of the mass as
+MR = 2πsλ
+� Λ
+δ
+dq[M(E(q))∗]R
+|E(q)|2
+= 2πsλ
+� Λ
+δ
+dqMRER(q) + MIEI(q)
+|E(q)|2
+(4.1)
+and
+MI = 2πsλ
+� Λ
+δ
+dq[M(E(q))∗]I
+|E(q)|2
+= 2πsλ
+� Λ
+δ
+dqMIER(q) − MREI(q)
+|E(q)|2
+,
+(4.2)
+,respectively with |E(q)|2 = E2
+R(q) + E2
+I (q).
+
+8
+We solve the SDE by iteration method from some initial input values for the
+real and imaginary parts of the effective mass denoted by MR(0) and MI(0).
+For the case (1), we can start from any values of the mass to solve the SDE, since
+s is independent on the mass. However, for the case (2), the SDE has non-trivial
+solutions only for s = −EI(q)/|EI(q)| = −[(M2)I−ε]/|(M2)I−ε| = 1. Since (M2)I =
+2MRMI, we set initial input values of the real and imaginary parts of the mass, which
+satisfy (M2)I(0) = 2MR(0)MI(0) < 0.
+1e-010
+1e-008
+1e-006
+0.0001
+0.01
+1
+100
+0
+20
+40
+60
+80
+100
+ |M|/Λ 
+ I
+λ=0.020 
+λ=0.025 
+λ=0.030 
+Fig. 1.
+The convergence behaviors of |M|/Λ for λ = 0.020, 0.025, 0.030 with MR(0) = −MI(0) =
+0.01Λ. The horizontal axis denotes the number of iterations.
+In Fig.1, we present the convergence behaviors of |M|/Λ =
+�
+M2
+R + M2
+I /Λ near
+the critical coupling constant λc denoted in Eq. (3·14) with δ/Λ = 10−3, which gives
+λ > 0.023. Here, we set the input values of the mass as MR(0) = −MI(0) = 0.01Λ.∗)
+From Fig.1, we can conclude that λc locates between λ = 0.020 and λ = 0.025.
+0.001
+0.01
+0.1
+1
+10
+100
+0
+0.5
+1
+1.5
+2
+2.5
+3
+  |M|/Λ 
+ λ
+Solution by SDE
+Analytical solution
+Fig. 2.
+The λ dependence of |M|/Λ for 0.03 ≤ λ ≤ 3 with MR(0) = −MI(0) = 0.01Λ. The dotted
+curve denotes the calculated result by the analytical solution divided by Λ.
+∗) We set ε = 10−5Λ2.
+
+9
+In Fig.2, we present the λ dependence of the absolute value of the effective mass
+|M|/Λ for 0.03 ≤ λ ≤ 3 with MR(0) = −MI(0) = 0.01Λ. The dotted curve denotes the
+calculated result using Eqs. (3·7) and (3·15).
+In the following calculations, we set four initial values for the mass as MR(0) =
+±0.01Λ and MI(0) = ±0.01Λ, respectively.
+0
+0.5
+1
+1.5
+2
+2.5
+3
+0
+5
+10
+15
+20
+ ER/Λ 
+ I
+MR(0)>0 MI(0)>0
+MR(0)>0 MI(0)<0
+Analytical soltion for MR>0
+MR(0)<0 MI(0)>0
+MR(0)<0 MI(0)<0
+Analytical soltion for MR<0
+Fig. 3.
+The convergence behavior of ER(q)/Λ with q = 0. The straight lines denote the energy
+divided by Λ calculated using the analytical solutions of the mass. The horizontal axis denotes
+the number of iterations.
+-2e-005
+-1.5e-005
+-1e-005
+-5e-006
+0
+5e-006
+1e-005
+1.5e-005
+2e-005
+0
+5
+10
+15
+20
+ EI/Λ 
+ I
+MR(0)>0 MI(0)>0
+MR(0)>0 MI(0)<0
+Analytical soltion for MR>0
+MR(0)<0 MI(0)>0
+MR(0)<0 MI(0)<0
+Analytical soltion for MR<0
+Fig. 4.
+The convergence behavior of EI(q)/Λ with q = 0. The straight lines denote the analytical
+solutions of energy, which is EI(q)/Λ = 0. The horizontal axis denotes the number of iterations.
+In Figs.3 and 4, we show the convergence behaviors of the real and imaginary
+parts of the energy with the momentum q = 0, respectively. The straight lines denote
+the energy calculated using the analytical solutions of the mass given in Eqs.(3·7)
+and (3·15).(See Appendix.)
+Since the real part of the energy is defined to be positive, the numerical results
+do not depend on the sign of initial values of the mass. The imaginary part of the
+energy converges EI → 0, in which the convergence behavior depends on the sign of
+
+10
+(M2)I(0). The calculated results shown in Figs. 1-4 are common for the two integral
+paths (1) and (2).
+On the other hand, the convergence behaviors for the effective mass are different
+for the two integral paths.
+-3
+-2
+-1
+0
+1
+2
+3
+4
+5
+0
+5
+10
+15
+20
+ MR/Λ 
+ I
+MR(0)>0 MI(0)>0
+MR(0)>0 MI(0)<0
+Analytical soltion for MR>0
+MR(0)<0 MI(0)>0
+MR(0)<0 MI(0)<0
+Analytical soltion for MR<0
+Fig. 5.
+The convergence behavior of MR/Λ for the case (1). The straight lines denote the analytical
+solutions of the real part of the effective mass divided by Λ. The horizontal axis denotes the
+number of iterations.
+-2e-005
+-1.5e-005
+-1e-005
+-5e-006
+0
+5e-006
+1e-005
+1.5e-005
+2e-005
+0
+5
+10
+15
+20
+ MI/Λ 
+ I
+MR(0)>0 MI(0)>0
+MR(0)>0 MI(0)<0
+Analytical soltion for MI>0 
+MR(0)<0 MI(0)>0
+MR(0)<0 MI(0)<0
+Analytical soltion for MI<0
+Fig. 6.
+The convergence behavior of MI/Λ for the case (1). The straight lines denote the analytical
+solutions of the imaginary part of the effective mass divided by Λ. The horizontal axis denotes
+the number of iterations.
+For the case (1), the real and imaginary parts of the effective mass calculated
+by the SDE are shown in Figs.5 and 6, respectively.
+As shown in Fig.
+5, the
+convergent solution splits into two values depending on the sign of MR(0)/Λ. As
+shown in Fig.6, the imaginary part of the effective mass is small and it depends
+on ε. Moreover, MI/Λ initially behaves according to the sign of the initial value
+of MI(0)/Λ, but the convergent solution depends on the sign of the initial value
+MR(0)/Λ.
+
+11
+-3
+-2
+-1
+0
+1
+2
+3
+4
+5
+0
+5
+10
+15
+20
+ MR/Λ 
+ I
+MR(0)>0 MI(0)>0
+MR(0)>0 MI(0)<0
+Analytical soltion for MR>0
+MR(0)<0 MI(0)>0
+MR(0)<0 MI(0)<0
+Analytical soltion for MR<0
+Fig. 7.
+The convergence behavior of MR/Λ for the case (2). The straight lines denote the analytical
+solutions of the real part of the effective mass divided by Λ. The horizontal axis denotes the
+number of iterations.
+-2e-005
+-1.5e-005
+-1e-005
+-5e-006
+0
+5e-006
+1e-005
+1.5e-005
+2e-005
+0
+5
+10
+15
+20
+ MI/Λ 
+ I
+MR(0)>0 MI(0)>0
+MR(0)>0 MI(0)<0
+Analytical soltion for MI>0 
+MR(0)<0 MI(0)>0
+MR(0)<0 MI(0)<0
+Analytical soltion for MI<0
+Fig. 8.
+The convergence behavior of MI/Λ for the case (2). The straight lines denote the analytical
+solutions of the imaginary part of the effective mass divided by Λ. The horizontal axis denotes
+the number of iterations.
+For the case (2), the convergence behaviors of the effective mass calculated by
+the SDE are shown in Figs.7 and 8, respectively. As shown in Figs. 7 and 8, the
+convergent solution splits into two values depending on the sign of the initial value
+of MR(0)/Λ for (M2)I(0) < 0. However, for (M2)I(0) > 0, the iterated values are
+oscillated. In this case, since s < 0, the SDE has no non-trivial solution.
+§5.
+Summary and Comments
+In this paper, we examined the (1 + 1)-dimensional Gross-Neveu (GN) model at
+zero temperature and solved the Schwinger-Dyson equation (SDE) in the complex
+plane. We compered the effective mass and energy calculated in two different integral
+paths in the complex energy plane. Then we examined the properties of the solutions
+
+12
+obtained by the SDE.
+First, we investigated the effect of the momentum cutoff on chiral symmetry
+breaking. Though the cutoff on the momentum is an artificial parameter for numer-
+ical calculations, this example suggests a possibility of changing the critical point in
+a physical system with restricted momentum.
+In the model treated in this paper, the imaginary part of the energy is zero and
+the poles of the effective propagator are on the real axis, which is different situation
+in QCD pointed out in Ref.[5,6], in which the poles are not on the real axis.
+We also investigated the dependence of the solutions obtained by the SDE on
+the initial input parameters. The effective mass obtained by the SDE depends on
+the sign of the input initial input values. Our calculations suggest that the SDE
+may lead to multiple solutions depending on the initial input values. Moreover, it
+can be seen that, for the integral path including the real axis, which corresponds to
+the SDE in Minkowski space, the input values leading to chiral symmetry broken
+phases are limited than the case with the integral path including the imaginary axis,
+which corresponds to the SDE in Euclidean space. This result suggests that the
+calculation of SDE requires careful selection of input values. On the other hand, in
+our example, when an oscillating solution exists, there exists a solution with broken
+chiral symmetry for input values of appropriate sign.
+The SDE extended to complex plane may be useful for investigating a wider
+class of non-perturbative solutions. Although further computational techniques will
+be required, it is expected that the method presented in this paper can be applied
+to other models such as non-perturbative QCD in Minkowski space.
+Appendix. Complex mass and energy
+In order to solve the SDE in complex energy plane, we need the explicit forms
+ofcomplex mass and energy.
+We define the complex mass as M = MR + iMI and the squared of the mass as
+M2 = (M2)R + i(M2)I. Here, (M2)R and (M2)I are given by
+(M2)R = M2
+R − M2
+I ,
+(M2)I = 2MRMI.
+The squared energy E2 is defined by
+E2 = q2 + M2 − iε ≡ (E2)R + i(E2)I.
+with
+(E2)R = q2 + (M2)R,
+(E2)I = (M2)I − ε.
+On the other hand, using the complex energy E = ER + iEI, (E2)R and (E2)I are
+also written as
+(E2)R = E2
+R − E2
+I ,
+(E2)I = 2EREI.
+Therefore the imaginary part of the energy is written as
+EI = (E2)I
+2ER
+.
+
+13
+Substituting above equation to (E2)R = E2
+R − E2
+I , we have a quadratic equation for
+E2
+R as
+(E2
+R)2 − E2
+R(E2)R − (E2)2
+I /4 = 0.
+The solution of the equation for E2
+R > 0 is given by
+E2
+R = (E2)R + |E2|
+2
+with |E2| =
+�
+[(E2)R]2 + [(E2)I]2.
+Therefore, we have the solution
+ER =
+�
+(E2)R + |E2|
+2
+,
+EI = (E2)I
+2ER
+for ER > 0.
+-
+References
+1) D.Dudal,O.Oliveira and P.J.Silva, Phys.Rev.D89,014010(2014) [arXiv:1310:4069 [hep-
+lat]].
+2) F.Siringo, Phys.Rev.D94,0114036(2016) [arXiv:1605:07357 [hep-ph]].
+3) F.J.Dyson, Phys.Rev.75 (1949) ,1736.
+4) J.S.Schwinger,Proc.Nat.Acad.Sci.37 (1951),452.
+5) S.Strauss,
+C.S.Fischer
+and
+C.Kellermann,
+Phys.
+Rev.
+Lett.
+109
+(2012),252001
+[arXiv:1208:6239 [hep-ph]].
+6) C.S.Fischer and M.Q.Huber, Phys. Rev. D102 (2020),094005 [arXiv:2007.11505].
+7) D.J.Gross and A.Neveu, Phys. Rev. D10 (1974),3235.
+

BdFKT4oBgHgl3EQfXC6p/content/tmp_files/load_file.txt

@@ -0,0 +1,314 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf,len=313
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='11793v1  [hep-th]  27 Jan 2023  1  Schwinger-Dyson equation in complex plane  − The (1 + 1)-dimensional Gross-Neveu model −  Hidekazu Tanaka ∗) and Shuji Sasagawa  Rikkyo University, Tokyo 171-8501, Japan  ABSTRACT  Effective mass and energy of fermions are investigated using the Schwinger-  Dyson equation (SDE) in the complex plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content=' As a simple example, we solve the  SDE for the (1+1)-dimensional Gross-Neveu model and study some properties of  the effective mass and energy of fermions in the complex plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='  ∗) E-mail:tanakah@rikkyo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='jp  2  §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='  Introduction  Behavior of effective mass and energy in non-perturbative region is one of in-  teresting problems to be studied, because they are related to the properties of the  propagator in non-perturbative region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='  Particularly, interesting phenomena are expected in Minkowski space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content=' In some  studies, it has been pointed out that the positivity of the gluon spectral function in  quantum chromodynamics (QCD) appears to be violated in strong coupling region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='  [1,2] This indicates that gluons do not have asymptotic states, suggesting that gluons  are confined to hadrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='  Unfortunately, lattice simulations for studying non-perturbative region do not  allow direct evaluation of the imaginary part of the effective mass in Minkowski space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='  One useful tool for studying non-perturbative phenomena is the Schwinger-Dyson  equation (SDE) [3,4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content=' The structure of the gluon propagator has been evaluated by  the SDE, in which the squared momentum for the gluon is extended to the complex  value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content=' [5,6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content=' They found that the gluon propagator has poles not on the real axis  in the squared momentum plane at zero temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content=' In their framework, they also  showed that the spectral function of the gluon violates positive value condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='  In evaluations using the SDE, one of difficulties in Minkowski space is the exis-  tence of poles in propagator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content=' This requires knowledge of the precise pole positions of  the propagator in the self-energy calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content=' To avoid this, it is computed by Wick-  rotating the axis of integration from the real axis to the imaginary axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content=' However,  the Wick rotation requires the location of the poles to be known in advance, but the  value of the mass in the non-perturbative region is non-trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='  In this paper, as a starting point for thinking about these problems, we examine  the (1 + 1)-dimensional (one dimension of time and one dimension of space) Gross-  Neveu (GN) model at zero temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content=' [7] We extend the SDE to the complex  plane, and integrate the loop momentum around poles of the propagator in the self-  energy with two different integration paths in the complex energy plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content=' Then we  examine the properties of the solutions obtained by the SDE in the complex plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='  In Section 2, we formulate the SDE for the (1 + 1)-dimensional GN model in  terms of complex mass and energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='  In Section 3, we discuss analytical solutions  for effective mass and energy in the complex plane with finite cutoff values of the  momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content=' In Section 4, we numerically calculate the effective mass and energy  using the SDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content=' Section 5 is devoted to the summary and some comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content=' Explicit  expressions of the complex mass and energy implemented in calculations are given  in Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='  3  §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='  The SDE for effective mass of fermion in complex plane  The Lagrangian density of the GN model is given by  L = i ¯ψ∂/ψ + g2  2 ( ¯ψψ)2,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='1)  where ψ and g2 are the 2-component fermion field in (1 + 1) dimensions and the  coupling constant of 4-fermion interaction, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='  In this paper, we evaluate the fermion effective mass M using the SDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content=' In order  to obtain the effective mass, we calculate the one-loop self-energy Σ of the fermion  in (1 + 1) dimensions, which is given by  Σ = i  g2  (2π)2  �  d2QTr[S(Q)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='2)  In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content=' (2 · 2), S(Q) is an effective propagator of the fermion with momentum Q =  (q0, q), which is given by  iS(Q) =  i  Q/ − Σ + iε  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='3)  Here, we define Σ ≡ M, because the wave-function renormalization constant of the  fermion is √Z2 = 1 in one-loop order of perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='  Therefore, the SDE for the effective mass M is given by  M = i 2g2  (2π)2  �  d2Q  M  Q2 − M2 + iε = iλ  �  dq0dq  M  q2  0 − q2 − M2 + iε,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='4)  where we define λ ≡ 2g2/(2π)2 for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content=' The propagator S(Q) has poles, which  satisfies q2  0 − q2 − M2 + iε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='  In this paper, we extend q0 as a complex value z and the effective mass M is also  extended as a complex value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content=' Explicitly, they are written as q0 = (q0)R + i(q0)I ≡  zR + izI = z and M ≡ MR + iMI, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content=' Here, we write the denominator of  the fermion propagator S(Q) as  z2 − q2 − M2 + iε ≡ z2 − E2(q) = (z − E(q))(z + E(q))  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='5)  with  E(q) ≡  �  E2(q) =  �  q2 + M2 − iε ≡ ER(q) + iEI(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='6)  Therefore, the poles are located at z = ±E(q) in the complex z plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content=' Here, we define  ER(q) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content=' Explicit relations among the complex values are given in Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='  The SDE for the effective fermion mass in terms of the complex values is written  as  M = iλ  �  dq  �  C  dz  M  z2 − q2 − M2 + iε = iλ  �  dq  �  C  dz  M  (z − E(q))(z + E(q)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='7)  4  Here, we write above equation as  M = 1  2M(+) + 1  2M(−),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='8)  where  M(±) = iλ  �  dq  �  C  dz  1  z − z±  �  M  z + z±  �  ≡ iλ  �  dq  �  C  dz  1  z − z±  f (±)(z, q) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='9)  with z± = ±E(q) and  f (±)(z, q) =  M  z + z±  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='10)  In our calculation, we integrate Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content=' (2·9) around z = z± with following two  integral paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='  (1) Integral path including the imaginary axis  In this case, we separate the integral path around the poles z± = ±E(q) to C1  and C2 as follows:  For the integral path around z+ = E(q), we take −iΛ0 − η < z < iΛ0 − η as  the path C1, and the path C2 is defined as clockwise rotation in right-half on the  complex energy plane with z = Λ0eiθ, where we take the integration from θ = π/2  to θ = −π/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='  On the other hand, for the integral path around z− = −E(q), we take −iΛ0+η <  z < iΛ0 + η as the path C1, and the path C2 is defined as anticlockwise rotation in  left-half on the complex energy plane with z = Λ0eiθ, where we take the integration  from θ = π/2 to θ = 3π/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content=' ∗)  Integrating over the integral path C around the pole z± = ±E(q) in the right-  hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content=' (2·9), we have  M(±) = iλ  �  dq(∓2πi)f (±)(z±, q) = πλ  �  dq M  E(q)  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='11)  for Λ0 → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content=' Therefore, the SDE for the effective mass is given by  M = 1  2M(+) + 1  2M(−) = πλ  �  dq M  E(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='12)  For η → 0, this case corresponds to the SDE for Euclidian momentum integration  with the complex mass M, which is given as  M = λ  �  dq  � ∞  −∞  dq4  M  q2  4 + q2 + M2 − iε  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='13)  with z = iq4 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content=' (2·7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='  ∗) In order to evaluate the contributions from the singular poles on the imaginary axis, we sift  the integral path by ∓η from the imaginary axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='  5  (2) Integral path including the real axis  In this case, we separate the integral path around the poles z± = ±E(q) to C1  and C2 as follows:  For the integral path around z+ = E(q), we take −Λ0 − iη < z < Λ0 − iη as the  path C1 if EI > 0, and the path C2 is defined as anticlockwise rotation in upper-half  on the complex energy plane with z = Λ0eiθ, where we take the integration from  θ = 0 to θ = π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content=' If EI < 0, we take −Λ0 + iη < z < Λ0 + iη as the path C1, and the  path C2 is defined as clockwise rotation in lower-half on the complex energy plane  with z = Λ0eiθ, where we take the integration from θ = 0 to θ = −π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='  For the integral path around z− = −E(q), we take −Λ0 −iη < z < Λ0 −iη as the  path C1 if EI < 0, and the path C2 is defined as anticlockwise rotation in upper-half  on the complex energy plane with z = Λ0eiθ, where we take the integration from  θ = 0 to θ = π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content=' If EI > 0, we take −Λ0 + iη < z < Λ0 + iη as the path C1, and the  path C2 is defined as clockwise rotation in lower-half on the complex energy plane  with z = Λ0eiθ, where we take the integration from θ = 0 to θ = −π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content=' ∗)  Integrating over the integral path C around the pole z± = ±E(q) in the right-  hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content=' (2·9), we have  M(±) = iλ  �  dq(±2πi)  � EI(q)  |EI(q)|  �  f (±)(z±, q) = −πλ  �  dq  � EI(q)  |EI(q)|  � M  E(q)(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='14)  for Λ0 → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='  Therefore, the effective mass is given as  M = 1  2M(+) + 1  2M(−) = −πλ  �  dq  � EI(q)  |EI(q)|  � M  E(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='15)  For η → 0, this case corresponds to the SDE for Minkowski momentum integra-  tion with the complex mass M, which is given as  M = iλ  �  dq  � ∞  −∞  dq0  M  q2  0 − q2 − M2 + iε  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='16)  with z = q0 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content=' (2·7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='  §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='  Analytical solutions  We can find analytical solutions for the SDE obtained in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='  Here, we write the SDE for two different integral paths as  M = πsλ  �  dq M  E(q)  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='1)  with s = 1 for the case (1) and s = −EI(q)/|EI(q)| = −[(M2)I − ε]/|(M2)I − ε| for  the case (2), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='  ∗) In order to evaluate the contributions from the singular poles on the real axis, we sift the  integral path by ∓iη from the real axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='  6  From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content=' (3·1), nontrivial solutions with M ̸= 0 are given by solving the equation  1 − πsλ  �  dq  1  E(q) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='2)  Thus, the real and imaginary parts of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content=' (3·2) satisfy  1 − πsλ  �  dq ER(q)  |E(q)|2 = 0  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='3)  and  πsλ  �  dq EI(q)  |E(q)|2 = 0,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='4)  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='  Here, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content=' (3·4) is written as  πsλ  �  dq EI(q)  |E(q)|2 = πsλ((M2)I − ε)  �  dq  1  2ER(q)|E(q)|2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='5)  For ER(q) > 0, we obtain (M2)I − ε = 0 for sλ ̸= 0, which gives EI(q) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='∗)  Moreover, from  (M2)I − ε = 2MRMI − ε = 0,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='6)  the imaginary part of the effective mass MI is given by  MI =  ε  2MR  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='7)  In the calculation below, we neglect the imaginary part of the effective mass for small  ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content=' Thus, we approximate as (M2)R ≃ M2  R for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='  Using EI(q) = 0, and introducing a ultraviolet cutoff Λ and an infrared cutoff δ  for the momentum q, we write Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content=' (3·3) as  1 = πsλ  � Λ  −Λ  dq 1  ER  θ(|q| − δ) = 2πsλ  � Λ  δ  dq  1  �  q2 + M2  R  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='8)  Here, θ(|q| − δ) denotes the step function for restriction of the momentum q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content=' (3·8) gives  1 = 2πsλ log  ������  Λ +  �  Λ2 + M2  R  δ +  �  δ2 + M2  R  ������  ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='9)  which is satisfied if sλ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content=' Therefore, for λ > 0, s = −EI(q)/|EI(q)| = 1 should be  satisfied for the case (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='  ∗) As shown in the next section, EI(q) is determined by an asymptotic value, which is numerically  calculated by the SDE with a given initial value of the mass M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='  7  Defining mR = MR/Λ, ¯δ = δ/Λ and  1 +  �  1 + m2  R  ¯δ +  �  ¯δ2 + m2  R  = e1/(2πsλ) ≡ ζ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='10)  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content=' (3·10) is written as  m2  R(Am2  R − B) = 0  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='11)  with A = (1 − ζ2)2 and B = 4ζ(1 − ¯δζ)(ζ − ¯δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='  The solution for m2  R ̸= 0 is given as  m2  R = B  A = 4ζ(1 − ¯δζ)(ζ − ¯δ)  (1 − ζ2)2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='12)  For sλ > 0, ζ − ¯δ > 0 is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content=' Moreover, m2  R > 0 demands 1 − ¯δζ > 0, which  gives  ζ = e1/(2πsλ) < 1  ¯δ = Λ  δ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='13)  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content=' (3·13) restricts the coupling constant λ as  λ >  1  2π log Λ  δ  ≡ λc  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='14)  with s = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='  For above restriction of λ, the real part of the effective mass is given as  mR = MR  Λ  = ±  �  4ζ(1 − ¯δζ)(ζ − ¯δ)  (1 − ζ2)2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='15)  §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='  Numerical solutions  In this section, we calculate the SDE for two different integral paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content=' The SDE  is given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content=' (3·1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content=' In numerical calculation, we write the SDE for the real and  imaginary parts of the mass as  MR = 2πsλ  � Λ  δ  dq[M(E(q))∗]R  |E(q)|2  = 2πsλ  � Λ  δ  dqMRER(q) + MIEI(q)  |E(q)|2  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='1)  and  MI = 2πsλ  � Λ  δ  dq[M(E(q))∗]I  |E(q)|2  = 2πsλ  � Λ  δ  dqMIER(q) − MREI(q)  |E(q)|2  ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='2)  ,respectively with |E(q)|2 = E2  R(q) + E2  I (q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='  8  We solve the SDE by iteration method from some initial input values for the  real and imaginary parts of the effective mass denoted by MR(0) and MI(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='  For the case (1), we can start from any values of the mass to solve the SDE, since  s is independent on the mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content=' However, for the case (2), the SDE has non-trivial  solutions only for s = −EI(q)/|EI(q)| = −[(M2)I−ε]/|(M2)I−ε| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content=' Since (M2)I =  2MRMI, we set initial input values of the real and imaginary parts of the mass, which  satisfy (M2)I(0) = 2MR(0)MI(0) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='  1e-010  1e-008  1e-006  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='0001  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='01  1  100  0  20  40  60  80  100  |M|/Λ  I  λ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='020  λ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='025  λ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='030  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='  The convergence behaviors of |M|/Λ for λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='020, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='025, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='030 with MR(0) = −MI(0) =  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='01Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content=' The horizontal axis denotes the number of iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='1, we present the convergence behaviors of |M|/Λ =  �  M2  R + M2  I /Λ near  the critical coupling constant λc denoted in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content=' (3·14) with δ/Λ = 10−3, which gives  λ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content=' Here, we set the input values of the mass as MR(0) = −MI(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='01Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='∗)  From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='1, we can conclude that λc locates between λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='020 and λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='025.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='001  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='1  1  10  100  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='5  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='5  3  |M|/Λ  λ  Solution by SDE  Analytical solution  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='  The λ dependence of |M|/Λ for 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='03 ≤ λ ≤ 3 with MR(0) = −MI(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='01Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content=' The dotted  curve denotes the calculated result by the analytical solution divided by Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='  ∗) We set ε = 10−5Λ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='  9  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='2, we present the λ dependence of the absolute value of the effective mass  |M|/Λ for 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='03 ≤ λ ≤ 3 with MR(0) = −MI(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='01Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content=' The dotted curve denotes the  calculated result using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content=' (3·7) and (3·15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='  In the following calculations, we set four initial values for the mass as MR(0) =  ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='01Λ and MI(0) = ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='01Λ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='5  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='5  3  0  5  10  15  20  ER/Λ  I  MR(0)>0 MI(0)>0  MR(0)>0 MI(0)<0  Analytical soltion for MR>0  MR(0)<0 MI(0)>0  MR(0)<0 MI(0)<0  Analytical soltion for MR<0  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='  The convergence behavior of ER(q)/Λ with q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content=' The straight lines denote the energy  divided by Λ calculated using the analytical solutions of the mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content=' The horizontal axis denotes  the number of iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='  2e-005  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='5e-005  1e-005  5e-006  0  5e-006  1e-005  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='5e-005  2e-005  0  5  10  15  20  EI/Λ  I  MR(0)>0 MI(0)>0  MR(0)>0 MI(0)<0  Analytical soltion for MR>0  MR(0)<0 MI(0)>0  MR(0)<0 MI(0)<0  Analytical soltion for MR<0  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='  The convergence behavior of EI(q)/Λ with q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content=' The straight lines denote the analytical  solutions of energy, which is EI(q)/Λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content=' The horizontal axis denotes the number of iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='  In Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='3 and 4, we show the convergence behaviors of the real and imaginary  parts of the energy with the momentum q = 0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content=' The straight lines denote  the energy calculated using the analytical solutions of the mass given in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content=' (3·7)  and (3·15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content=' (See Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content=')  Since the real part of the energy is defined to be positive, the numerical results  do not depend on the sign of initial values of the mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content=' The imaginary part of the  energy converges EI → 0, in which the convergence behavior depends on the sign of  10  (M2)I(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content=' The calculated results shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content=' 1-4 are common for the two integral  paths (1) and (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='  On the other hand, the convergence behaviors for the effective mass are different  for the two integral paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='  3  2  1  0  1  2  3  4  5  0  5  10  15  20  MR/Λ  I  MR(0)>0 MI(0)>0  MR(0)>0 MI(0)<0  Analytical soltion for MR>0  MR(0)<0 MI(0)>0  MR(0)<0 MI(0)<0  Analytical soltion for MR<0  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='  The convergence behavior of MR/Λ for the case (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content=' The straight lines denote the analytical  solutions of the real part of the effective mass divided by Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content=' The horizontal axis denotes the  number of iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='  2e-005  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='5e-005  1e-005  5e-006  0  5e-006  1e-005  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='5e-005  2e-005  0  5  10  15  20  MI/Λ  I  MR(0)>0 MI(0)>0  MR(0)>0 MI(0)<0  Analytical soltion for MI>0  MR(0)<0 MI(0)>0  MR(0)<0 MI(0)<0  Analytical soltion for MI<0  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='  The convergence behavior of MI/Λ for the case (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content=' The straight lines denote the analytical  solutions of the imaginary part of the effective mass divided by Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content=' The horizontal axis denotes  the number of iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='  For the case (1), the real and imaginary parts of the effective mass calculated  by the SDE are shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='5 and 6, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='  As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='  5, the  convergent solution splits into two values depending on the sign of MR(0)/Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content=' As  shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='6, the imaginary part of the effective mass is small and it depends  on ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content=' Moreover, MI/Λ initially behaves according to the sign of the initial value  of MI(0)/Λ, but the convergent solution depends on the sign of the initial value  MR(0)/Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='  11  3  2  1  0  1  2  3  4  5  0  5  10  15  20  MR/Λ  I  MR(0)>0 MI(0)>0  MR(0)>0 MI(0)<0  Analytical soltion for MR>0  MR(0)<0 MI(0)>0  MR(0)<0 MI(0)<0  Analytical soltion for MR<0  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='  The convergence behavior of MR/Λ for the case (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content=' The straight lines denote the analytical  solutions of the real part of the effective mass divided by Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content=' The horizontal axis denotes the  number of iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='  2e-005  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='5e-005  1e-005  5e-006  0  5e-006  1e-005  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='5e-005  2e-005  0  5  10  15  20  MI/Λ  I  MR(0)>0 MI(0)>0  MR(0)>0 MI(0)<0  Analytical soltion for MI>0  MR(0)<0 MI(0)>0  MR(0)<0 MI(0)<0  Analytical soltion for MI<0  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='  The convergence behavior of MI/Λ for the case (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content=' The straight lines denote the analytical  solutions of the imaginary part of the effective mass divided by Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content=' The horizontal axis denotes  the number of iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='  For the case (2), the convergence behaviors of the effective mass calculated by  the SDE are shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='7 and 8, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content=' As shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content=' 7 and 8, the  convergent solution splits into two values depending on the sign of the initial value  of MR(0)/Λ for (M2)I(0) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content=' However, for (M2)I(0) > 0, the iterated values are  oscillated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content=' In this case, since s < 0, the SDE has no non-trivial solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='  §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='  Summary and Comments  In this paper, we examined the (1 + 1)-dimensional Gross-Neveu (GN) model at  zero temperature and solved the Schwinger-Dyson equation (SDE) in the complex  plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content=' We compered the effective mass and energy calculated in two different integral  paths in the complex energy plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content=' Then we examined the properties of the solutions  12  obtained by the SDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='  First, we investigated the effect of the momentum cutoff on chiral symmetry  breaking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content=' Though the cutoff on the momentum is an artificial parameter for numer-  ical calculations, this example suggests a possibility of changing the critical point in  a physical system with restricted momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='  In the model treated in this paper, the imaginary part of the energy is zero and  the poles of the effective propagator are on the real axis, which is different situation  in QCD pointed out in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content=' [5,6], in which the poles are not on the real axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='  We also investigated the dependence of the solutions obtained by the SDE on  the initial input parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content=' The effective mass obtained by the SDE depends on  the sign of the input initial input values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content=' Our calculations suggest that the SDE  may lead to multiple solutions depending on the initial input values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content=' Moreover, it  can be seen that, for the integral path including the real axis, which corresponds to  the SDE in Minkowski space, the input values leading to chiral symmetry broken  phases are limited than the case with the integral path including the imaginary axis,  which corresponds to the SDE in Euclidean space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content=' This result suggests that the  calculation of SDE requires careful selection of input values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content=' On the other hand, in  our example, when an oscillating solution exists, there exists a solution with broken  chiral symmetry for input values of appropriate sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='  The SDE extended to complex plane may be useful for investigating a wider  class of non-perturbative solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content=' Although further computational techniques will  be required, it is expected that the method presented in this paper can be applied  to other models such as non-perturbative QCD in Minkowski space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='  Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content=' Complex mass and energy  In order to solve the SDE in complex energy plane, we need the explicit forms  ofcomplex mass and energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='  We define the complex mass as M = MR + iMI and the squared of the mass as  M2 = (M2)R + i(M2)I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content=' Here, (M2)R and (M2)I are given by  (M2)R = M2  R − M2  I ,  (M2)I = 2MRMI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='  The squared energy E2 is defined by  E2 = q2 + M2 − iε ≡ (E2)R + i(E2)I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='  with  (E2)R = q2 + (M2)R,  (E2)I = (M2)I − ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='  On the other hand, using the complex energy E = ER + iEI, (E2)R and (E2)I are  also written as  (E2)R = E2  R − E2  I ,  (E2)I = 2EREI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='  Therefore the imaginary part of the energy is written as  EI = (E2)I  2ER  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='  13  Substituting above equation to (E2)R = E2  R − E2  I , we have a quadratic equation for  E2  R as  (E2  R)2 − E2  R(E2)R − (E2)2  I /4 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='  The solution of the equation for E2  R > 0 is given by  E2  R = (E2)R + |E2|  2  with |E2| =  �  [(E2)R]2 + [(E2)I]2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='  Therefore, we have the solution  ER =  �  (E2)R + |E2|  2  ,  EI = (E2)I  2ER  for ER > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content=' References  1) D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
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+page_content='Schwinger,Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
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+page_content='Strauss,  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
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+page_content='Fischer  and  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
+page_content='Kellermann,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
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+page_content='  6) C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
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+page_content='Fischer and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
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+page_content='Huber, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'}
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