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+arXiv:2301.04198v1  [math.CO]  10 Jan 2023
+Sharp thresholds for spanning regular graphs
+Maksim Zhukovskii∗
+Abstract
+Let d ≥ 3 be a constant and let F be a d-regular graph on [n] with not too
+many symmetries. The expectation threshold for the existence of a spanning
+subgraph in G(n, p) isomorphic to F is p∗(n) = (1 + o(1))(e/n)2/d. We give
+a tight bound on the edge expansion of F guaranteeing that the probability
+threshold for the appearance of a copy of F has the same order of magnitude
+as p∗. We also prove that, within a slight strengthening of this bound, the
+probability threshold is asymptotically equal to p∗. In particular, it proves
+the conjecture of Kahn, Narayanan and Park on a sharp threshold for the
+containment of a square of a Hamilton cycle. It also implies that, for d ≥ 4
+and (asymptotically) almost all d-regular graphs F on [n], p(n) = (e/n)2/d is
+a sharp threshold for F-containment.
+1
+Introduction
+Let d ≥ 3 be a fixed constant. Given a d-regular graph Fn on the vertex set [n] :=
+{1, . . . , n}, what is the threshold probability to contain its isomorphic copy by the
+binomial random graph G(n, p) (i.e. the unique p = p(n) such that the probability
+that G(n, p) contains an isomorphic copy of Fn equals 1/2)? Note that the threshold
+probability exists since the considered property is monotone [7, Chapter 1.5].
+If Fn has a small enough automorphism group, then, by the union bound, the
+threshold probability is at least (1 + o(1))(e/n)2/d. Indeed, let Fn be the set of all
+isomorphic copies of Fn on [n], and let the number of automorphisms of Fn be eo(n).
+Clearly |Fn| =
+n!
+eo(n). Let X be the number of graphs from Fn that are subgraphs of
+G(n, p). We get
+EX = |Fn|pdn/2 =
+n!
+eo(n)pdn/2 → 0
+as n → ∞
+if p < (1 − ε)
+� e
+n
+�2/d, implying that with high probability (whp for brevity) G(n, p)
+does not contain any graph from Fn. Let us denote by p∗(n) = (1+o(1))(e/n)2/d the
+∗The University of Sheffield; [email protected]
+1
+
+expectation threshold for the existence of a spanning subgraph in G(n, p) isomorphic
+to Fn (i.e. p∗(n) is the unique solution of the equation EX = 1).
+On the other hand, from the recently resolved “expectation–threshold” conjec-
+ture of Kahn and Kalai [16] it follows that the threshold does not exceed Cp∗(n) log n
+for some constant C > 0. For some specific Fn it is known that the logarithmic fac-
+tor can be removed, and the threshold probability equals Θ(n−2/d): it is true for
+example for powers of a Hamilton cycle [15] and for the square tori T√n×√n [15]. On
+the other hand, if Fn has many small subgraphs with a small edge boundary, this
+is no longer true. More precisely, assume that, for some constant v, every vertex
+of Fn belongs to a subgraph on v vertices with the edge boundary at most d (the
+edge boundary of a subgraph ˜F is the number of edges between ˜F and its vertex
+complement) or, equivalently, with at least dv
+2 − d
+2 edges. Then, a polylogarithmic
+factor arises since in order to contain a copy of Fn, the random graph should have
+every vertex inside a graph with v vertices and at least dv
+2 − d
+2 edges — see [17].
+We prove that, when the number of automorphisms of Fn is small enough, this
+condition on the edge boundary is the only obstacle.
+Theorem 1. Let d ≥ 3 and let Fn be a sequence of d-regular graphs on [n], n ∈ N,
+such that
+• for every ε > 0 and all large enough n the number of automorphisms of Fn is
+less then eεn2/d;
+• for every ˜F ⊂ Fn with 3 ≤ |V ( ˜F)| ≤ n − 3, the edge boundary of ˜F is at least
+d + 1.
+Let ε > 0. If p > (1 + ε)dp∗, then whp (assuming that dn is even) G(n, p) contains
+a copy of Fn.
+It immediately implies that the threshold probability for containing a copy of Fn
+equals p(n) = Θ(n−2/d). As we mentioned above, the restriction on edge boundaries
+is tight — if we allow subgraphs with edge boundary d instead of d + 1, then the
+assertion becomes false.
+Note that a bound on the number of symmetries can not be omitted — as soon
+as the number of automorphisms of Fn becomes larger, the expectation threshold
+p∗ becomes larger as well. In particular, p(n) = (d! log n)
+2
+d(d+1) n−2/(d+1) is a sharp
+threshold for the existence of a Kd+1-factor [14].
+In [15] Riordan proved a general result that for d-regular graphs can be stated
+as follows: p(n) = Θ(n−2/d) is the threshold probability for containing a copy of Fn
+if the d-regular graph Fn (the automorphism group should be at most exponential
+2
+
+in n) satisfies a stronger condition on the edge boundary: for every ˜F ⊂ Fn with
+3 ≤ |V ( ˜F)| ≤ n−3, the edge boundary of ˜F is at least 2d. For powers of a Hamilton
+cycle, this result implies the following: for every k ≥ 3, the threshold probability
+for containing the kth power of a Hamilton cycle equals Θ(n−1/k). However, the
+proof of Riordan does not work for k = 2. In [10], K¨uhn and Osthus proved that
+n−1/2+o(1) is the threshold probability for containing the second power of a Hamilton
+cycle and conjectured that the threshold is actually Θ(n−1/2). In [13], Nenadov and
+ˇSkori´c proved the upper bound n−1/2(log n)4, which was improved to n−1/2(log n)3 by
+Fischer, ˇSkori´c, Steger and Truji´c in [4], and to n−1/2(log n)2 in an unpublished work
+of Montgomery (see [6]). Eventually, the conjecture was solved by Kahn, Narayanan
+and Park in [8]. However, they did non settle a right constant in front of n−1/2 and
+conjectured that the right constant is √e and that the threshold p(n) =
+�
+e/n(1 +
+o(1)) is sharp (i.e., if p > (1 + ε)
+�
+e/n, then whp G(n, p) contains the second power
+of a Hamilton cycle). In this paper, we prove this conjecture and even more: for
+d ≤ 4 the requirement from Theorem 1 guarantees that p(n) = (1 + o(1))
+�
+e/n
+is even sharp; however, for d ≥ 5 we need to strengthen the bound on the edge
+boundary to 2d − 2 (note that this is still better than the condition of Riordan).
+Theorem 2. Let d ≥ 3 and let Fn be a sequence of d-regular graphs on [n], n ∈ N,
+such that
+• for every ε > 0 and all large enough n the number of automorphisms of Fn is
+less then eεn2/d;
+• either d ∈ {3, 4} and, for every ˜F ⊂ Fn with 3 ≤ |V ( ˜F)| ≤ n − 3, the edge
+boundary of ˜F is at least d + 1,
+or d ≥ 5 and, for every ˜F ⊂ Fn with 3 ≤ |V ( ˜F)| ≤ n − 3, the edge boundary
+of ˜F is at least 2d − 2.
+Let ε > 0. If p > (1+ε)
+� e
+n
+�2/d, then whp (assuming that dn is even) G(n, p) contains
+a copy of Fn.
+Kahn, Narayanan and Park in [8] noted that the crucial fact that can be used
+to prove that the threshold for appearance of the second power of a Hamilton cycle
+equals Θ(n−1/2) is that the hypergraph of all copies of the second power of a cycle
+on [n] is (1 + o(1))
+�
+e/n-spread. Actually, they refined the notion of spreadness by
+incorporating the count of the number of components in a subhyperedge. This re-
+fined notion was distilled by D´ıaz and Person in [3], named superspreadness and used
+to generalise the result of Kahn, Narayanan and Park to a wider class of spanning
+subgraphs in G(n, p). In particular, they answered a question of Frieze asked in [5]
+— they showed that the threshold for appearance of spanning 2-overlapping 4-cycles
+(i.e. the copies of C4 are ordered cyclically, two consecutive C4 overlap in exactly
+3
+
+one edge, whereby each C4 overlaps with two copies of C4 in opposite edges) equals
+Θ(n−2/3). Clearly, Theorem 2 implies that p(n) = (e/n)2/3 is a sharp threshold for
+appearance of spanning 2-overlapping 4-cycles.
+Let us call a sequence of d-regular graphs on [n] satisfying the conditions of
+Theorem 2 good. Note that, for every d ≥ 4, almost all d-regular graphs are good
+(see [2, 9, 11]). In particular, if d ≥ 5, then whp in a random d-regular graph on
+[n] there are no subgraphs with 3 ≤ v ≤ n − 3 vertices and the edge boundary at
+most 2d − 2. If d = 4, then whp there are no subgraphs with 3 ≤ v ≤ n − 3 vertices
+and the edge boundary at most d + 1. If d = 3, then whp there are no subgraphs
+with 3 ≤ v ≤ n − 3 vertices and the edge boundary at most d + 1 other than C3, C4
+and their vertex-complements. Since the edge boundary of C4 is exactly d + 1 = 4,
+a random 3-regular graph is good whp under the condition that it does not contain
+triangles.
+Corollary 1. For every good sequence Fn, p(n) =
+� e
+n
+�2/d is a sharp threshold for
+containing a copy of Fn. In particular,
+• for every ℓ ≥ 2, p(n) = (e/n)ℓ is a sharp threshold for containing the ℓth power
+of a Hamilton cycle;
+• p(n) = (e/n)2/3 is a sharp threshold for containing a spanning 2-overlapping
+4-cycle;
+• for every 3 ≤ m ≤ √n, p(n) =
+�
+e/n is a sharp threshold for containing
+rectangular tori Tm×n/m (assuming that n is divisible by m);
+• for every d ≥ 4 and (asymptotically) almost all d-regular graphs Fn on [n], as-
+suming that dn is even, p(n) = (e/n)2/d is a sharp threshold for Fn-containment;
+• for (asymptotically) almost all triangle-free 3-regular graphs Fn on [n], assum-
+ing that n is odd, p(n) = (e/n)2/3 is a sharp threshold for Fn-containment.
+Actually we are able to establish the same sharp threshold for almost all 3-regular
+graphs — the condition of the absence of triangles is redundant, since the number
+of triangles converges in probability to a Poisson random variable [19], and so it is
+bounded in probability. In other words, we may allow Fn to have a bounded number
+of subgraphs with a smaller edge boundary. However, we do not want to overload
+the proof with technical details, and so we formulate Theorem 1 and Theorem 2 as
+well as Corollary 1 in their current laconic forms.
+We prove Theorem 1 using the “planted trick” that in different forms appears
+in many applications — one of them is the well-known and very useful “spread
+4
+
+lemma” [1] which in particular gives good sunflower bounds [18]; in probabilistic
+terms the application of the trick for the “spread lemma” is described in [12]. Kahn,
+Narayanan and Park [8] and further D´ıaz and Person [3] used the “planted trick” to
+prove their results on threshold probabilities as well. In essence, the key idea is to
+“plant” a graph F from the family Fn and to combine it with the noise produced
+by G(n, p). Then, it turns out that whp there exists a graph F ′ ∈ Fn which is
+entirely inside the perturbed planted hyperedge F ∪ G(n, p) such that the size of
+F ′ \ G(n, p) is quite small. This allows to replace Fn with the set of fragments of
+F ∈ Fn equal to F ′ \ G(n, p), to draw independently edges of another G(n, p) and
+to apply the same argument once again. If the number of steps in this procedure
+is bounded by a constant, then we get that the threshold probability has the same
+order of magnitude as p∗.
+In the proof of Theorem 2 we show that it is sufficient to apply this trick only
+once. Actually the usual second moment method (but for the uniform model instead
+of the binomial) works as well. However, we give the proof of Theorem 2 in terms
+of the planted hyperedge for the sake of convenience and coherence. In particular,
+we want to explicitly show the borders between the following three phenomena:
+1) it is sufficient to apply the “planted trick” once, 2) it is sufficient to apply the
+“planted trick” constantly many times, 3) the number of applications of the trick is
+unbounded. We claim that our analysis is optimal, and the method in its current
+form cannot be used to weaken the bound on edge boundaries in Theorem 2 for
+d ≥ 5. Our main achievement is that we make a step beyond the usage of the notions
+of spreadness and superspreadness. We obtain optimal bounds on the number of
+hyperedges containing a given set of edges I (commonly denoted by |Fn ∩ ⟨I⟩|)
+and on the number of subgraphs of Fn with a fixed number of vertices, edges and
+components (see Section 5 and Claim 6). The main ingredient of the proof of Claim 6
+is a very nice property of d-regular graphs satisfying the requirements of Theorem 1:
+for every v, there are not too many subgraphs on v vertices with the maximum
+possible number of edges
+dv
+2 −
+� d+1
+2
+�
+(see Section 2).
+We describe the “planted
+trick” in Section 3. Then we prove both theorems in Section 4. Sections 6 and 7 are
+devoted to the proof of Claim 6 and the key lemma (Lemma 3 from Section 3) that
+validates the application of the planted trick respectively.
+2
+Linearly many closed subgraphs
+Let us call a graph Fn with the second property from the requirement (on the edge
+boundary) in Theorem 1 locally sparse. Note that this (local sparsity) property is
+that the edge boundary of every subgraph ˜F with 3 ≤ |V ( ˜F)| ≤ n − 3 is at least
+d + 1. Clearly d + 1 can be replaced with d + 2 for even d since in this case the edge
+boundary δ( ˜F) cannot be odd. Let ∆ = d+1 for odd d and ∆ = d+2 for even d. It
+5
+
+is easy to see that the condition |δ( ˜F)| ≥ ∆ holds for all ˜F with 2 ≤ |V ( ˜F)| ≤ d − 1
+just due to the d-regularity of Fn. Let us call a subgraph ˜F with the edge boundary
+exactly ∆ closed (note that a closed subgraph is always connected). For j < d, let
+us call a vertex w of a connected subgraph ˜F ⊂ Fn j-free, if its degree in ˜F equals
+j; w is simply free, if it is j-free for some j < d.
+Let F be a locally sparse d-regular graph on [n].
+Claim 1. Every closed subgraph of F with at least 3 vertices has minimum degree
+at least d/2.
+Proof. Assume that ˜F is a closed subgraph of F with a vertex w having degree
+d′ < d/2. If we remove the vertex w from ˜F, then we get the graph ˜F \ w with edge
+boundary δ(F) + 2d′ −d < δ(F) = ∆. This contradicts the local sparsity of F when
+|V ( ˜F)| ≥ 4. Otherwise it contradicts the fact that a subgraph on 2 vertices has the
+edge boundary at least 2d − 2 ≥ ∆.
+Claim 2. For any pair of adjacent vertices x, y in F and for every 3 ≤ v ≤ n − 3,
+there are at most two closed subgraphs in F on v vertices containing x and not
+containing y.
+Proof. Fix adjacent vertices x, y and 3 ≤ v ≤ n − 3.
+A closed graph ˜F ⊂ F sends exactly ∆ edges to F \ ˜F implying that F \ ˜F is
+also closed. Assume that v ≥ n/2, and that there are at least 3 closed graphs on
+v vertices that share x and do not contain y. Then their complements are closed
+graphs on n − v ≤ n/2 vertices that share y and do not share x. Therefore, it is
+sufficient to prove the claim for v ≤ n/2.
+Let H1, H2 be different closed subgraphs of F on v vertices that contain x and do
+not contain y. Note that H1, H2 should have at least one other common vertex since
+otherwise the degree of x is bigger than d due to Claim 1. Then |V (H1) ∪ V (H2)| ≤
+n − 2.
+Let H0 = H1 ∩ H2. Note that |E(H0)| ≤ d
+2|V (H0)| − ∆
+2 implying that |E(Hj) \
+E(H0)| ≥
+d
+2|V (Hj \ H0)| for both j = 1 and j = 2 since H1, H2 are closed. On
+the other hand, if, say |E(H2) \ E(H0)| >
+d
+2|V (H2 \ H0)|, then |E(H1 ∪ H2)| >
+d
+2|V (H1∪H2)|− ∆
+2 which contradicts the local sparsity of F since |V (H1∪H2)| ≤ n−2.
+Therefore, |E(Hj) \ E(H0)| = d
+2|V (Hj \ H0)| for both j = 1 and j = 2, but then H0
+is closed.
+Then, there are exactly ∆ edges between H0 and F \ H0, and one of them is the
+edge between x and y. It means that Hj \ H0, j ∈ {1, 2}, send at most ∆ − 1 edges
+(in total) to H0. This may happen only if |V (Hj \ H0)| = 1 for both j = 1 and
+j = 2. Indeed, |V (H1 \ H0)| = |V (H2 \ H0)|. Moreover, the number of edges that
+6
+
+Hj \ H0 sends to H0 equals
+|E(Hj) \ E(H0)| − |E(Hj \ H0)| ≥ d
+2|V (Hj \ H0)| −
+�d
+2|V (Hj \ H0)| − ∆
+2
+�
+= ∆
+2
+whenever |V (Hj \ H0)| ≥ 2.
+Assume that there exists a closed graph H3 ̸⊂ H1∪H2 on v vertices that contains
+x and does not contain y. From the above it follows that H3 ∩ H1 = H3 ∩ H2 = H0.
+Each vertex of Hj \ H0 sends at least d
+2 edges to H0 due to Claim 1. But then the
+vertices from Hj \ H0 send at least 3d
+2 ≥ ∆ edges to H0 — a contradiction (since
+there is one additional edge {x, y} in the edge boundary of H0). Therefore, any
+other closed graph that contains x and does not contain y should be entirely inside
+H1 ∪ H2. Assume that such a graph H3 exists. Let w1 ∈ H1 \ H0, w2 ∈ H2 \ H0.
+Clearly, H3 contains w1, w2 and all but 1 vertex of H0. In the same way as above
+we get that H1 ∩ H2 = H0, H1 ∩ H3 and H2 ∩ H3 are three closed graphs on v − 1
+vertices that contain x and do not contain y. These three closed graphs on v − 1
+vertices have the property that none of them is inside the union of the other two —
+this is only possible when v − 1 = 2, i.e. v = 3. The only possible closed graph on
+3 vertices is a triangle. Moreover, a triangle is closed only when d = 4. So, H1, H2
+are triangles sharing an edge, but then H3 adds another edge to the union H1 ∪ H2
+implying that H1 ∪ H2 ∪ H3 is a 4-clique. We get a contradiction with the local
+sparsity since the edge boundary of a 4-clique is 4 < ∆ = 6.
+From this, it immediately follows, that for every v, there are at most Cn closed
+subgraphs on v vertices in F for a certain universal constant C. More precisely, the
+following is true.
+Claim 3. Let k ∈ N, and let F ′ be the induced subgraph of F on [k]. For every
+3 ≤ v ≤ n − 3, the number of closed subgraphs of F ′ with v vertices is at most 2dk
+3 .
+Proof. Fix a vertex w in F ′ and let us bound the number (denoted by µ(w)) of
+closed subgraphs of F ′ on v vertices containing w such that the vertex w is free in
+these graphs. Due to Claim 2, µ(w) ≤ 2d. On the other hand, Claim 1 implies
+that every closed subgraph contains at least 3 free vertices. Letting f to be the
+number of closed subgraphs in F ′ on v vertices, by double counting, we get that
+3f ≤ �
+w∈V (F ′) µ(w) ≤ 2dk as needed.
+For d = 3, 4, we need sharper bounds. Let us start from d = 3.
+Claim 4. Let d = 3, k ∈ N. Let F ′ be the induced subgraph of F on [k]. Then for
+every 3 ≤ v ≤ n − 3, there are at most 3
+4k closed subgraphs in F ′ on v vertices.
+7
+
+Proof. Fix a vertex w in F ′ and let us compute the number µ(w) of closed subgraphs
+of F ′ on v vertices containing w such that the vertex w is free in these graphs.
+Reviewing the proof of Claim 2, we may see that in the case d = 3, every vertex
+x may be inside only a single closed subgraph on v vertices that does not contain
+another vertex y — otherwise H1 \ H0 sends at least 2 edges to H0, and the same
+for H2 \ H0 implying that H0 cannot be closed. Then, for every w, µ(w) ≤ 3. On
+the other hand, Claim 1 implies that every closed subgraph contains at least 4 free
+vertices. Letting f to be the number of closed subgraphs in F ′ on v vertices, by
+double counting, we get that 4f ≤ �
+w∈V (F ′) µ(w) ≤ 3k as needed.
+For d = 4, we get the following.
+Claim 5. Let d = 4, k ∈ N. Let F ′ be the induced subgraph of F on [k]. Then for
+every 3 ≤ v ≤ n − 3, there are at most 4
+3k closed subgraphs in F ′ on v vertices.
+Proof. Fix a vertex w in F ′ and let us compute the number of closed subgraphs of
+F ′ on v vertices containing w such that the vertex w is free in these graphs. Let
+µj(w) be the number of closed subgraphs on v vertices such that w is j-free in these
+graphs. Due to Claim 1, µ1(w) = 0. Due to Claim 2, µ3(w) + 2µ2(w) ≤ 8. On the
+other hand, letting f to be the number of closed subgraphs in F ′ on v vertices, since
+every closed graph has the edge boundary equal to 6, we get that 6f is exactly the
+number of pairs (a closed graph ˜F, an edge from the boundary of ˜F). Therefore,
+6f = �
+w(µ3(w) + 2µ2(w)) ≤ 8k implying that the number of closed graphs on v
+vertices is at most 4
+3k as needed.
+3
+Planted hyperedge
+Let dn be even, Fn be a d-regular graph on [n] satisfying the requirements of
+Theorem 1, and let F be a uniformly chosen random element of Fn. Let ε > 0,
+w = (1 + ε + o(1))(e/n)2/d�n
+2
+�
+be a sequence of integers and W be a random graph
+on [n] with w edges chosen uniformly at random. In this section, we review the
+constructions and follow the terminology from [8]. For the sake of convenience, we
+give the argument in full. Our achievement is Lemma 3 that we state in the end of
+the section.
+Fix a non-negative integer ℓ0. Let us call a pair (F ∈ Fn, W ⊂
+�[n]
+2
+�
+) ℓ0-bad, if
+for every ℓ0-subset L ⊂ F (we hereinafter assume that F is the set of edges), we
+have that L ⊔ [W \ F] does not contain a graph from Fn.
+Fix t ∈
+�
+0, 1, . . . , dn
+2
+�
+and let w′ = w − t. Note that, for F ∈ Fn, W ⊂
+�[n]
+2
+�
+,
+such that |F ∩ W| = t, we have
+|F ∪ W| = dn
+2 + w − t = w′ + dn
+2 .
+8
+
+Call Z ∈
+� ([n]
+2 )
+w′+dn/2
+�
+ℓ0-pathological if
+|{F ⊂ Z : F ∈ Fn, (F, Z) is ℓ0-bad}| > 1
+n|Fn|
+��n
+2
+�
+− dn/2
+w′
+�
+/
+�
+�n
+2
+�
+w′ + dn/2
+�
+=: M.
+Note that
+P(|F ∩ W| = t) =
+�dn/2
+t
+���n
+2
+�
+− dn/2
+w′
+�
+/
+��n
+2
+�
+w
+�
+.
+We have therefore
+P((F, W) is ℓ0-bad, F ∪ W is not ℓ0-pathological | |F ∩ W| = t)
+= P((F, W) is ℓ0-bad, F ∪ W is not ℓ0-pathological, |F ∩ W| = t)
+P(|F ∩ W| = t)
+≤
+�
+(n
+2)
+w′+dn/2
+�
+|{F ⊂ Z : F ∈ Fn, (F, Z) is ℓ0-bad}|
+�dn/2
+t
+�
+|Fn|
+�(n
+2)
+w
+��dn/2
+t
+��(n
+2)−dn/2
+w′
+�
+/
+�(n
+2)
+w
+�
+≤ 1
+n,
+where Z is a not ℓ0-pathological hyperedge from
+� ([n]
+2 )
+w′+dn/2
+�
+with maximum possible
+value of |{F ⊂ Z : F ∈ Fn, (F, Z) is ℓ0-bad}|.
+Fix F ∈ Fn and a set B ⊂ F of size t. Let W′ be chosen uniformly at random
+from
+�([n]
+2 )\F
+w′
+�
+. Note that if F ∪ W′ is ℓ0-pathological, then there are at least M
+subgraphs from Fn in F ∪ W′. On the other hand, if (F, W′) is ℓ0-bad, then, for
+every F ′ ∈ Fn such that F ′ ⊂ F ∪ W′,
+|F ′ ∩ F| = |F ′ \ W′| ≥ ℓ0 + 1.
+Let X count the number of F ′ ∈ Fn such that F ′ ⊂ F ∪ W′ and |F ′ ∩ F| ≥ ℓ0 + 1.
+We get that the event “(F, W′) is ℓ0-bad and F ∪ W′ is ℓ0-pathological” implies
+X ≥ M. Then
+P((F, W) is ℓ0-bad, F ∪ W is ℓ0-pathological | |F ∩ W| = t) =
+P((F, W) is ℓ0-bad, F ∪ W is ℓ0-pathological | F = F, F ∩ W = B) =
+P((F, W′) is ℓ0-bad, F ∪ W′ is ℓ0-pathological) ≤ P(X ≥ M) ≤ EX
+M .
+For ℓ ≥ ℓ0 + 1, let πℓ := P(|F ∩ F| = ℓ). Then
+EX =
+�
+ℓ≥ℓ0+1
+|Fn|πℓ
+�
+w′
+dn/2 − ℓ
+�
+/
+��n
+2
+�
+− dn/2
+dn/2 − ℓ
+�
+.
+(1)
+9
+
+We have
+EX
+M =
+�
+ℓ≥ℓ0+1
+|Fn| πℓ
+M
+�
+w′
+dn/2 − ℓ
+�
+/
+��n
+2
+�
+− dn/2
+dn/2 − ℓ
+�
+= n
+�
+ℓ≥ℓ0+1
+πℓ
+�
+w′
+dn/2−ℓ
+�
+/
+�(n
+2)−dn/2
+dn/2−ℓ
+�
+�(n
+2)−dn/2
+w′
+�
+/
+�
+(n
+2)
+w′+dn/2
+�.
+(2)
+Note that
+�
+w′
+dn/2−ℓ
+�
+/
+�(n
+2)−dn/2
+dn/2−ℓ
+�
+�(n
+2)−dn/2
+w′
+�
+/
+�
+(n
+2)
+w′+dn/2
+� ∼
+w′2w′(
+�n
+2
+�
+− dn + ℓ)(n
+2)−dn+ℓ�n
+2
+�(n
+2)
+(w′ − dn/2 + ℓ)w′−dn/2+ℓ(w′ + dn/2)w′+dn/2(
+�n
+2
+�
+− dn/2)2(n
+2)−dn
+< e− (dn/2−ℓ)2
+2w
+− (dn/2)2
+2w
++O(1)
+��
+n
+(1 + ε)e
+�2/d�ℓ
+.
+(3)
+In Section 7, we prove the following.
+Lemma 3. If one of the following two conditions hold
+• ℓ0 =
+�
+d2
+2 ln(1+ε/2)n1−(∆−d)/d�
+, or
+• Fn is good and ℓ0 = 0,
+then
+EX
+M ≤ n
+�
+ℓ≥ℓ0+1
+πℓe− (dn/2−ℓ)2
+2w
+− (dn/2)2
+2w
+��
+n
+(1 + ε)e
+�2/d�ℓ
+= o
+�1
+n
+�
+.
+(4)
+4
+Proofs of Theorems 1, 2
+Lemma 3 implies Theorem 2 immediately.
+It remains to prove Theorem 1 for d ≥ 5. Let p > (1 + ε)d
+� e
+n
+�2/d. We use
+the first assertion of Lemma 3 for that. Consider d independent copies G1, . . . , Gd
+of G(n, p′), p′ = (1 + ε)
+� e
+n
+�2/d. For every F ∈ Fn, consider a minimum possible
+R = R(F) ⊂ F such that R ∪ G1 contains a graph from Fn. By Lemma 3 whp
+|R| ≤
+d2
+2 ln(1+ε/2)n1−(∆−d)/d.
+Let Σ be the set of all F ∈ Fn such that |R(F)| ≤
+d2
+2 ln(1+ε/2)n1−(∆−d)/d. We have that E|Σ| = (1 − o(1))|Fn|. By Markov’s inequality,
+P
+�
+|Σ| ≤ |Fn|
+2
+�
+= P
+�
+|Fn| − |Σ| ≥ |Fn|
+2
+�
+≤ 2(|Fn| − E|Σ|)
+|Fn|
+→ 0.
+10
+
+Let R = {R(F) : F ∈ Σ} be a multiset, i.e. |R| = |Σ|. We may assume that all sets
+R ∈ R have equal cardinality exactly ℓ1 :=
+�
+d2
+2 ln(1+ε/2)n1−(∆−d)/d�
+. We then apply
+the same proof (as in Section 3) but for R instead of Fn.
+Let ℓ0 = 1
+2
+�
+d2
+ln(1+ε/2)
+�2
+n1−2(∆−d)/d. Let us call a pair (R ∈ R, W ∈
+�([n]
+2 )
+w
+�
+) bad, if
+for every ℓ0-subset L ⊂ R, we have that L ⊔ [W \ R] does not contain a graph from
+R. For a fixed size of intersection t, a set Z ∈
+� ([n]
+2 )
+w−t+ℓ1
+�
+is pathological if
+|{R ⊂ Z : R ∈ R, (R, Z) is bad}| > 1
+n|R|
+��n
+2
+�
+− ℓ1
+w − t
+�
+/
+�
+�n
+2
+�
+w − t + ℓ1
+�
+=: M.
+In order to show that for a fixed R ∈ R, whp in R∪G2 there exists a subset R′ ∈ R
+such that |R′ \ G2| ≤ ℓ0, it is sufficient to prove an analogue of the first assertion of
+Lemma 3: let
+• W′ be chosen uniformly at random from
+�([n]
+2 )\R
+w−t
+�
+;
+• X be the number of R′ ∈ R such that R′ ⊂ R ∪ W′ and |R′ ∩ R| ≥ ℓ0 + 1,
+then EX/M = o(1/n). Note that an analogue of the first inequality in (4) holds true
+with dn/2 replaced by ℓ1. Note that R has at most 2ℓ1 vertices. Defining p(ℓ, x, c)
+in the same way as in Section 5 and applying Claim 6, we get
+p(ℓ, x, c) ≤ max
+a
+α2ℓ1(a, ℓ, x, c)β(a, ℓ, x, c)
+|R|
+≤
+�2ℓ1
+c
+�
+[(n − x + c)! + O(1)]
+|R|
+×
+× max
+a
+�x − 2c
+c − a
+��c
+a
+��˜σ + c
+c − a
+��d
+2
+�a �4d
+3
+�c−a
+22d(d+5)˜σ
+max
+o≤(d+4)˜σ
+�x
+o
+�2
+.
+Note that this bound differs from (8) only in the first binomial coefficient with n
+replaced by 2ℓ1. Therefore, applying the same arguments as in Section 7.1, we get
+that, for every ℓ > ℓ0,
+πℓ =
+�
+x,c
+p(x, ℓ, c) ≤ n2 � e
+n
+� 2ℓ
+d (1 + δ)ℓe
+d2
+2
+d2
+ln(1+ε/2) n1−2(∆−d)/d.
+Therefore the analogues of (1), (2) and (3) imply that
+EX
+M ≤
+�
+ℓ>ℓ0
+n3
+�1 + δ
+1 + ε
+�ℓ
+e
+d2
+2
+d2
+2 ln(1+ε/2) n1−2(∆−d)/d = o
+�1
+n
+�
+.
+Applying repeatedly the whole argument
+d
+∆−d − 1 ≤ d − 1 times, we arrive to
+fragments of graphs from Fn of sizes at most ℓd−1 =
+�
+1
+2
+�
+d2
+ln(1+ε/2)
+�d−1
+n(∆−d)/d
+�
+.
+11
+
+Defining R (whp |R| ≥ |Fn|/2d−1) in a usual way as the multiset of fragments of
+size exactly ℓd−1, letting
+M := 1
+n|R|
+��n
+2
+�
+− ℓd−1
+w − t
+�
+/
+�
+�n
+2
+�
+w − t + ℓd−1
+�
+,
+considering a fixed fragment R, a uniformly chosen W′ ∈
+�([n]
+2 )\R
+w−t
+�
+, and defining X
+as the number of R′ ∈ R such that R′ ⊂ R ∪ W′ and |R′ ∩ R| ≥ 1, it remains to
+show that EX/M = o(1/n). We then consider p(ℓ, x, c) and apply Claim 6:
+p(ℓ, x, c) ≤ max
+a
+α2ℓd−1(a, ℓ, x, c)β(a, ℓ, x, c)
+|R|
+≤
+�2ℓd−1
+c
+�
+[(n − x + c)! + O(1)]
+|R|
+×
+× max
+a
+�x − 2c
+c − a
+��c
+a
+��˜σ + c
+c − a
+��d
+2
+�a �4d
+3
+�c−a
+22d(d+5)˜σ
+max
+o≤(d+4)˜σ
+�x
+o
+�2
+.
+In the same way as in Section 7.1, the only non-trivial case is σ < ε′x, x < ε′n,
+where 0 < ε′ ≪ δ is small enough (otherwise, p(ℓ, x, c) is even smaller). In this case,
+for large enough constant C = C(d),
+p(ℓ, x, c) ≤ 2d−1
+�2ℓd−1
+c
+�
+e2ℓ/d+(x−c)2/n+o(n2/d)
+n2ℓ/d+(∆−d)c/d
+(1 + δ/2)ℓ
+�d2
+2 e2/d−5/6
+�c
+≤ 2d−1
+��
+d2
+ln(1 + ε/2)
+�d−1 e
+c
+�c
+e2ℓ/d+(x−c)2/n+o(n2/d)
+n2ℓ/d
+(1 + δ/2)ℓ
+�d2
+2 e2/d−5/6
+�c
+≤ C e2ℓ/d+o(n2/d)
+n2ℓ/d
+(1 + δ)ℓ.
+Finally, for every ℓ ≥ 1,
+πℓ =
+�
+x,c
+p(x, ℓ, c) ≤ n2C e2ℓ/d+o(n2/d)
+n2ℓ/d
+(1 + δ)ℓ.
+Therefore the analogues of (1), (2) and (3) imply that
+EX
+M ≤
+�
+ℓ>ℓ0
+n3
+�1 + δ
+1 + ε
+�ℓ
+e−Θ(n2/d) = o
+�1
+n
+�
+.
+5
+Spread
+We here follow the notations of Section 3: F ∈ Fn is fixed, F ∈ Fn is chosen
+uniformly at random, and πℓ = P(|F ∩ F| = ℓ).
+12
+
+Fix c ∈ [ℓ], x ∈
+� 2ℓ
+d + ∆
+d c, ℓ + c
+�
+. Denote
+σ := d
+2x −
+�
+ℓ + ∆
+2 c
+�
+.
+Let p(ℓ, x, c) be the probability that the intersection of F with F is a graph on x
+vertices with ℓ edges and c connected components (we think about graphs as about
+sets of their edges, so there are no isolated vertices in |F∩F|). Let integers ℓ1, . . . , ℓc
+and x1, . . . , xc be chosen in a way such that
+•
+2ℓi
+d + ∆
+d ≤ xi ≤ ℓi + 1 for all i ∈ [c];
+• �c
+i=1 ℓi = ℓ, �c
+i=1 xi = x.
+Let p(ℓ1, . . . , ℓc, x1, . . . , xc) be the probability that the intersection of F with F con-
+sists of c connected components R1, . . . , Rc such that |V (Ri)| = xi, |E(Ri)| = ℓi.
+Clearly,
+p(ℓ, x, c) =
+�
+ℓi,xi
+p(ℓi, xi, i ∈ [c]),
+(5)
+where the summation is over all unordered choices of ℓ1, . . . , ℓc, x1, . . . , xc. Note
+that, in the case of the ordered choice, the number of ways to choose the values of
+xi ≥ 2 is at most
+�x−c
+c
+�
+. The number of ways to choose the respective ℓi is at most
+�σ+c
+c
+�
+.
+We will use the following claim. Let ˜F be a subgraph of F on k vertices. We
+assume that either k = n and ˜F = F, or k ≪ n.
+Claim 6. The number of ways to choose a subgraph R1 ⊔ . . . ⊔ Rc from ˜F without
+isolated vertices with x vertices, ℓ edges and c components such that a is the number
+of Ri that are either an edge or a full vertex-complement to an edge (that comprises
+n − 2 vertices and dn/2 − (2d − 1) edges) is
+αk(a, ℓ, x, c) ≤
+�k
+c
+��x − 2c
+c − a
+��c
+a
+��˜σ + c
+c − a
+� �d
+2
+�a
+γc−a
+max
+o≤(d+4)˜σ
+�x
+o
+�
+2d(d+5)˜σ,
+(6)
+where
+γ = 2d
+3 I(d ≥ 5) + 4
+3I(d = 4) + 3
+4I(d = 3),
+˜σ = σ − a(d − 1 − ∆/2).
+Given disjoint non-trivial connected R1, . . . , Rc ⊂ ˜F such that their union has x
+vertices and ℓ edges, and there are exactly a graphs Ri that are either an edge or a
+full vertex-complement to an edge, the number of ways to extend R1 ⊔ . . . ⊔ Rc to a
+graph from Fn is
+β(a, ℓ, x, c) ≤ (n − x + c)!(d − 1)a2c−a
+max
+o≤(d+4)˜σ
+�x
+o
+�
+2d(d+4)˜σ + O(1).
+(7)
+13
+
+6
+Proof of Claim 6
+Fix ℓ1, . . . , ℓc and x1, . . . , xc. Let us compute the number of ways to choose connected
+vertex-disjoint subgraphs R1, . . . , Rc from ˜F with the respective numbers of edges
+and vertices. Let us call Ri dense, if one of the following holds: 1) xi = 2 and ℓi = 1,
+2) Ri is closed, 3) xi = n − 2 and ℓi = d
+2n − (2d − 1), 4) xi = n − 1 and ℓi = d
+2n − d,
+5) xi = n and ℓi = d
+2n.
+For i ∈ [c], set σi =
+d
+2xi − ℓi − ∆
+2 . Note that a is the number of i such that
+xi = 2 and ℓi = 1, or xi = n − 2 and ℓi = d
+2n − (2d − 1). Let us call the respective
+Ri edge-components.
+The number of ways to choose i ∈ [c] such that Ri is an
+edge component equals
+�c
+a
+�
+, while the number of ways to choose the values of the
+remaining xi is at most
+�x−2c
+c−a
+�
+. The number of ways to choose the respective ℓi is at
+most
+�˜σ+c
+c−a
+�
+.
+We first choose dense graphs. If c = 1, x1 = n − 1 and ℓ1 = d
+2n − d, then there
+are exactly n ways to choose R1. If c = 1, x1 = n and ℓ1 = 2n, there is only one way
+to choose R1. Otherwise, we first choose edge-components Ri: for j = 1, 2, . . . , a,
+the jth edge is chosen out of the set of kj remaining vertices in dkj
+2 ways, and then
+kj−1 ≤ kj −2. After that, we choose all the remaining dense graphs from R1, . . . , Rc.
+Note that the remaining dense graphs from R1, . . . , Rc are closed. Assume that we
+want to choose a closed Rj, and kj is the number of remaining vertices. Then by
+Claims 3, 4, and 5, the number of ways to choose Rj is at most γkj. After that,
+kj − |V (Rj)| ≤ kj − 3 vertices remain. Assuming that c − ˜c is the number of dense
+Rj, we get that there are at most
+� d
+2
+�a γc−˜c−a
+n!
+(n−(c−˜c))! number of ways to choose
+dense subgraphs.
+Without loss of generality, we assume that it remains to choose R1, . . . , R˜c.
+Note that the component Ri might have at most ∆ + 2σi free vertices.
+For ev-
+ery i = 1, 2, . . . , ˜c, we expose Ri in ˜F in the following way.
+Assume that F ′ is
+the current graph (obtained by removing all the chosen subgraphs R1, . . . , Ri−1 and
+R˜c+1, . . . , Rc) on k′ vertices,
+1. choose the number of free vertices oi ≤ d + 2 + 2σi;
+2. choose the iterations of the below algorithm (out of the total number of iter-
+ations xi) that produce a free vertex of Ri in
+�xi
+oi
+�
+ways;
+3. choose a vertex w in F ′ which is minimum in Ri (here we mean the ordering
+of the vertices of Ri induced by the ordering of the vertices of F ′) in at most
+k′ ways, and activate it;
+4. at every step, choose the minimum vertex (in the ordering of the vertices from
+F ′) in the set of active vertices:
+14
+
+• if it should be free (in accordance to the above choice), then add to Ri
+some of its neighbours (in at most 2d ways), deactivate it and activate all
+its chosen neighbours,
+• if it should not be free, then add to Ri all its neighbours, deactivate the
+vertex and activate all its neighbours.
+We get that the number of ways to choose Ri is at most k′(d+2+2σi)
+max
+oi≤d+2+2σi
+�xi
+oi
+�
+2doi.
+Eventually we get that the number of ordered choices of components with pa-
+rameters ℓi, xi, i ∈ [c], in ˜F is at most
+c!
+�k
+c
+� �d
+2
+�a
+γc−˜c−a
+˜c�
+i=1
+(d + 2 + 2σi)
+max
+oi≤d+2+2σi
+�xi
+oi
+�
+2doi
+≤ c!
+�k
+c
+� �d
+2
+�a
+γc−˜c−a
+max
+o≤(d+4)˜σ
+�x
+o
+�
+2d(d+5)˜σ.
+Note that this bound does not depend on the order of the choice of all these com-
+ponents R1, . . . , Rc, thus it implies (6).
+Let us now fix connected subgraphs R1, . . . , Rc of F as above. Let us bound
+the number of ways to extend R1 ⊔ . . . ⊔ Rc to an F ′ ∈ Fn. We construct such an
+extension in the following way.
+First, let us consider the two special cases: 1) c = 1, x1 ≥ n − 2 and R1 is dense;
+2) c = 2, x1 = 2, x2 = n − 2 and R1, R2 are dense. In the first case, if x1 = n − 2,
+then there are at most
+�2(d−1)
+d−1
+�
+ways to construct F ′ (the remaining 2 vertices should
+be adjacent — so we should only draw missing edges in constantly many ways); if
+x1 ≥ n − 1, then there is a unique way to construct F ′. In the second case, there
+are also at most
+�2(d−1)
+d−1
+�
+ways to draw F ′.
+We then forget the labels of the vertices from R1, . . . , Rc and assume (without
+loss of generality) that the desired F ′ ∈ Fn is defined on [n] in a way such that
+every i ∈ {2, . . . , n} has a neighbour in [i − 1], denote one such neighbour (chosen
+randomly) by ν(i) (note that the graph F ′ is connected due to the restriction on
+edge boundaries). Let H be obtained from F by deleting all the edges that do not
+belong to R1 ⊔ . . . ⊔ Rc. Then let Z be the set of all components in H (together
+with the isolated vertices), i.e. |Z| = n − x + c. We should compute the number of
+ways to embed the elements of Z in F ′ disjointly.
+Let z1, . . . , zn−x+c be an ordering of Z. At every step i = 1, . . . , n−x+c, consider
+the minimum vertex κi of F ′ such that none of the embedded elements of Z in F ′
+contain this vertex. If zi /∈ {R1, . . . , Rc}, then we assign κi with zi and proceed
+with the next step. Otherwise, we distinguish between the following cases. We let
+zi = R1 without loss of generality.
+15
+
+First, we assume that R1 is dense.
+If |V (R1)| = 2, then there are at most
+d − 1 ways to choose the image of the edge R1, since the edge {κi, ν(κi)} is already
+‘occupied’. If 2 < |V (R1)| < n − 2, then due to Claim 2, there are at most 2 ways
+to choose a copy of R1 in F ′ containing κi and not containing ν(κi), as desired.
+Second, let R1 be not dense. Choose the iterations of the below algorithm (out
+of the total number of iterations xi) that produce a free vertex of Ri in
+�xi
+oi
+�
+ways.
+Activate κi.
+At every step, choose the minimum vertex (in the ordering of the
+vertices from F ′) in the set of active vertices:
+• if it should be free (in accordance to the above choice), then add to the image of
+R1 under construction some of its neighbours (in at most 2d ways), deactivate
+it and activate all its neighbours,
+• if it should not free, then add all its neighbours, deactivate the vertex and
+activate all its neighbours.
+We get that the number of ways to construct the image of R1 is at most
+�xi
+oi
+�
+2doi.
+Eventually we get that there are at most
+(n − x + c)!(d − 1)a2c−˜c−a
+˜c�
+i=1
+max
+oi≤d+2+2σi
+�xi
+oi
+�
+2doi + O(1)
+≤ (n − x + c)!(d − 1)a2c−˜c−a
+max
+o≤(d+4)˜σ
+�x
+o
+�
+2d(d+4)˜σ + O(1).
+ways to expose F ′ as needed.
+7
+Proof of Lemma 3
+Summing up, from (5), (6) and (7), we get that
+p(ℓ, x, c) ≤ max
+a
+αn(a, ℓ, x, c)β(a, ℓ, x, c)
+|Fn|
+≤
+�n
+c
+�
+[(n − x + c)! + O(1)]
+|Fn|
+×
+× max
+a
+�x − 2c
+c − a
+��c
+a
+��˜σ + c
+c − a
+��d
+2
+�a
+(2γ)c−a22d(d+5)˜σ
+max
+o≤(d+4)˜σ
+�x
+o
+�2
+.
+(8)
+We let
+Υ := max
+a
+�˜σ + c
+c − a
+�
+22d(d+5)˜σ
+max
+o≤(d+4)˜σ
+�x
+o
+�2
+.
+Let us find the maximum value of ϕ(x) =
+�x−2c
+c−a
+�
+e∆c/d−x as a function of x. Since
+ϕ(x + 1)
+ϕ(x)
+= 1
+e
+�
+1 +
+c − a
+x + 1 − 3c + a
+�
+,
+16
+
+we get that the maximum value is achieved at x = 2c+(c−a)
+e
+e−1 +O(1). Therefore,
+p(ℓ, x, c) ≤
+�n
+c
+�
+e2σ/d+2ℓ/d+(x−c)2/n+o(n2/d)
+n2ℓ/d+(∆−d)c/d+2σ/d
+Υ×
+× (2γ)c max
+a
+�c
+a
+� �d(d − 1)
+4γ
+�a ��
+(c − a)
+e
+e−1
+�
+c − a
+�
+e−(2−∆/d)c−(c−a)
+e
+e−1.
+Since
+�c
+a
+� �d(d − 1)
+4γ
+�a ��
+(c − a)
+e
+e−1
+�
+c − a
+�
+e−(c−a)
+e
+e−1 ≤
+�c
+a
+� �d(d − 1)
+4γ
+�a �2
+e
+�
+e
+e−1 (c−a)
+≤
+�
+d(d − 1)
+4γ
++
+�2
+e
+�e/(e−1)�c
+,
+we eventually get
+p(ℓ, x, c) ≤
+�n
+c
+�
+e2σ/d+2ℓ/d+(x−c)2/n+o(n2/d)
+n2ℓ/d+(∆−d)c/d+2σ/d
+Υ
+�
+4d
+3 e−(2−∆/d)
+�
+3(d − 1)
+8
++
+�2
+e
+�e/(e−1)��c
+≤
+�n
+c
+�
+e2σ/d+2ℓ/d+(x−c)2/n+o(n2/d)
+n2ℓ/d+(∆−d)c/d+2σ/d
+Υ
+�d2
+2 e2/d−5/6
+�c
+for all d ≥ 5;
+p(ℓ, x, c) ≤
+�n
+c
+�
+eσ/2+ℓ/2+(x−c)2/n+o(√n)
+nℓ/2+c/2+σ/2
+Υ
+�
+8
+3√e
+�
+9
+4 +
+�2
+e
+�e/(e−1)��c
+≤
+�n
+c
+�
+eσ/2+ℓ/2+(x−c)2/n+o(√n)
+nℓ/2+c/2+σ/2
+Υ
+�7.8
+√e
+�c
+for d = 4;
+p(ℓ, x, c) ≤
+�n
+c
+�
+e2σ/3+2ℓ/3+(x−c)2/n+o(n2/3)
+n2ℓ/3+c/3+2σ/3
+Υ
+�
+3
+2e2/3
+�
+2 +
+�2
+e
+�e/(e−1)��c
+≤
+�n
+c
+�
+e2σ/3+2ℓ/3+(x−c)2/n+o(n2/3)
+n2ℓ/3+c/3+2σ/3
+Υ
+� 4
+e2/3
+�c
+for d = 3.
+From now on, we separately proof three assertions of Lemma 3.
+7.1
+d ≥ 5: existence of a fragment
+First we assume that d ≥ 5, ℓ0 =
+d2
+2 ln(1+ε/2)n1−(∆−d)/d, ℓ > ℓ0. Let δ > 0 be small
+enough. Choose ε′ = ε′(δ) > 0 small enough in a way such that σ < ε′x implies
+17
+
+Υ ≤ (1 + δ/3)ℓ and c < ε′x implies
+�x−2c
+c−a
+��c
+a
+��d
+2
+�a(2γ)c−a ≤ (1 + δ/3)ℓ for all a as
+well.
+Assume that σ < ε′x. In this case,
+p(ℓ, x, c) ≤
+�n
+c
+�
+e2ℓ/d+(x−c)2/n+o(n2/d)
+n2ℓ/d+(∆−d)c/d
+(1 + δ)ℓ
+�d2
+2 e2/d−5/6
+�c
+.
+If x > ε′n and c > ε′x, then p(ℓ, x, c) =
+� e
+n
+� 2ℓ
+d exp(−Θ(n ln n)).
+If x > ε′n and c ≤ ε′x, then
+p(ℓ, x, c) ≤
+�n
+c
+�
+n(∆−d)c/d
+� e
+n
+� 2ℓ
+d (1 + δ)ℓ ≤ en1−(∆−d)c/d � e
+n
+� 2ℓ
+d (1 + δ)ℓ.
+Finally, let x ≤ ε′n.
+The maximum value of
+�n
+c
+�
+(d2e2/d−5/6/2)cn−(∆−d)c/d is
+achieved when c = d2
+2 e2/d−5/6n1−(∆−d)/d + O(1) and is at most
+�en
+c
+�c
+(d2e2/d−5/6/2)cn−(∆−d)c/d ≤ exp
+�d2
+2 e2/d−5/6n1−(∆−d)/d + O(1)
+�
+implying that
+p(ℓ, x, c) ≤
+� e
+n
+� 2ℓ
+d (1 + δ)ℓe
+d2
+2 n1−(∆−d)/d.
+Let σ ≥ ε′x. Then, for some large enough C > 0,
+p(ℓ, x, c) ≤
+�n
+c
+�
+[(n − x + c)! + O(1)]
+|Fn|
+23x+σ+2d(d+5)σ
+�d
+2
+�a
+(2γ)c−a
+≤ n−2ℓ/dCx22d(d−5)σ
+n2σ/d
+= o(n−2ℓ/d).
+Summing up, for every ℓ > ℓ0,
+πℓ =
+�
+x,c
+p(x, ℓ, c) ≤ n2 � e
+n
+� 2ℓ
+d (1 + δ)ℓe
+d2
+2 n1−(∆−d)/d.
+Therefore (1), (2) and (3) imply that
+EX
+M ≤
+�
+ℓ>ℓ0
+n3
+�1 + δ
+1 + ε
+�ℓ
+e
+d2
+2 n1−(∆−d)/d = o
+�1
+n
+�
+.
+18
+
+7.2
+d ≥ 5: sharp threshold for a good sequence
+Now, we assume that d ≥ 5, Fn is good, ℓ0 = 0 and ℓ ≥ 1. In this case ˜σ ≥
+(c − a)(d − ∆/2). Therefore,
+(∆ − d)c
+d
++ 2σ
+d ≥ c
+�
+1 − 2
+d
+�
++
+˜σ
+d − ∆/2.
+In the same way as above, if x > ε′n and c > ε′x, then p(ℓ, x, c) =
+� e
+n
+� 2ℓ
+d exp(−Θ(n ln n)).
+If x > ε′n and c ≤ ε′x, then p(ℓ, x, c) ≤ exp
+�
+n(∆−d)c/d� � e
+n
+� 2ℓ
+d (1 + δ)ℓ.
+Finally, let x ≤ ε′n. Assume first that c − a ≤ ε′x. Then
+p(ℓ, x, c) ≤
+�n
+c
+�
+[(n − x + c)! + O(1)]
+|Fn|
+�˜σ + c
+c − a
+��d
+2
+�c
+(1 + δ/2)ℓ22d(d+5)˜σ
+max
+o≤(d+4)˜σ
+�x
+o
+�2
+≤
+�n
+c
+�
+e(x−c)2/n+o(n2/d)
+n2ℓ/d+(∆−d)c/d+2σ/d
+�˜σ + c
+c − a
+��d
+2
+�c
+(1 + δ/2)ℓ22d(d+5)˜σ
+max
+o≤(d+4)˜σ
+�x
+o
+�2
+≤
+� e
+n
+�2ℓ/d
+e(d
+2)(n/e)2/d e(x−c)2/n+o(n2/d)
+n˜σ/(d−∆/2)
+�˜σ + c
+c − a
+�
+(1 + δ/2)ℓ22d(d+5)˜σ
+max
+o≤(d+4)˜σ
+�x
+o
+�2
+≤
+� e
+n
+�2ℓ/d
+e(d
+2)(n/e)2/d+o(n2/d)(1 + δ)ℓ.
+Let c − a > ε′x. Then ˜σ > ε′x(d − ∆/2). Therefore,
+p(ℓ, x, c) ≤
+�n
+c
+�
+[(n − x + c)! + O(1)]
+|Fn|
+2x+˜σ
+�d
+2
+�c
+(2γ)c−a22d(d+5)˜σ
+max
+o≤(d+4)˜σ
+�x
+o
+�2
+≤
+�n
+c
+�
+e(x−c)2/n+o(n2/d)
+n2ℓ/d+c(1−2/d)+˜σ/(d−∆/2) 23x+˜σ
+�d
+2
+�c
+(2γ)c−a22d(d+5)˜σ
+≤
+�1
+n
+�2ℓ/d
+�n
+c
+�
+nc(1−2/d) (1 + δ)ℓ ≤
+�1
+n
+�2ℓ/d
+en2/d(1 + δ)ℓ.
+Summing up, for every ℓ > 0,
+πℓ =
+�
+x,c
+p(x, ℓ, c) ≤ n2 � e
+n
+�2ℓ/d
+e(d
+2)(n/e)2/d+o(n2/d)(1 + δ)ℓ.
+Therefore (1), (2) and (3) imply that (assuming that ε < 1/d)
+EX
+M ≤
+�
+ℓ>0
+n3
+�1 + δ
+1 + ε
+�ℓ
+e− d(1−dε)
+2
+(n/e)2/d = o
+�1
+n
+�
+.
+(9)
+19
+
+7.3
+d = 4
+We now consider d = 4. The maximum value of
+�n
+c
+�
+(7.8/√e)cn−c/2 is achieved when
+c = 7.8
+√e
+√n + O(1) and is at most
+�en
+c
+�c
+(7.8/√e)cn−c/2 ≤
+�7.8√e√n
+c
+�c
+≤ e
+7.8
+√e
+√n+O(1)
+implying that
+p(ℓ, x, c) ≤ e(x−c)2/n+o(√n)
+nℓ/2+σ/2
+Υeℓ/2+σ/2e
+7.8
+√e
+√n.
+Let us assume that σ < ε′ℓ. If 1 ≤ ℓ ≤ ε′n, then
+p(ℓ, c, x) ≤ (1 + δ)ℓe
+�
+7.8
+√e +o(1)
+�√n(e/n)ℓ/2.
+If ℓ > ε′n and c > ε′x, then p(ℓ, c, x) ≤ n−ℓ/2 exp(−Ω(n ln n)). If c ≤ ε′x, then
+�x−2c
+c−a2
+�
+≤ (1 + δ/3)ℓ. Therefore,
+p(ℓ, x, c) ≤
+�n
+c
+�
+[(n − x + c)! + O(1)]
+|Fn|
+(1+2δ/3)ℓ32c ≤
+�n
+c
+�
+e(x−c)2/n+o(√n)
+nℓ/2+c/2+σ/2
+(1+2δ/3)ℓ32c.
+Since
+�n
+c
+�
+n−c/232c ≤ e32√n+O(1), we get
+p(ℓ, x, c) ≤ e32√n+o(√n)e(x−c)2/n(1 + 2δ/3)ℓn−ℓ/2 ≤ (1 + δ)ℓ(e/n)ℓ/2.
+Finally, let σ ≥ ε′ℓ. Note that this is possible only when ℓ is large enough. Since
+�c
+a
+��x − 2c
+c − a
+��σ + c − a
+c − a
+�
+6a
+�8
+3
+�c−a �x
+o
+�2
+≤ 23x+σ3a
+�8
+3
+�c−a
+≤ 23x+σ3c,
+we get that
+p(ℓ, x, c) ≤
+�n
+c
+�
+[(n − x + c)! + O(1)]
+|Fn|
+23x+73σ3c ≤
+�n
+c
+�
+eo(√n)
+nℓ/2+c/2+σ/2 e(x−c)2/n23x+73σ3c
+≤ e
+√n+o(√n)n−ℓ/2−σ/2ex−c23x+73σ3c ≤ e
+√n+o(√n)n−ℓ/2.
+Therefore,
+πℓ =
+�
+x,c
+p(x, ℓ, c) ≤ n2(1 + δ)ℓe
+�
+7.8
+√e +o(1)
+�√n(e/n)ℓ/2.
+Then (1), (2) and (3) imply (9) as needed.
+20
+
+7.4
+d = 3
+It remains to consider d = 3. We only need to consider the case σ < ε′ℓ, 1 ≤ ℓ ≤ ε′n.
+For all the other values of the parameters, the proof is absolutely the same as for
+d = 4. The maximum value of
+�n
+c
+�
+(4/e2/3)cn−c/3 is achieved when c =
+4
+e2/3n2/3+O(1)
+and is at most
+�en
+c
+�c
+(4/e2/3)cn−c/3 ≤
+�4e1/3n2/3
+c
+�c
+≤ e4(n/e)2/3+O(1)
+implying that
+p(ℓ, x, c) ≤ e(x−c)2/n+o(√n)
+n2ℓ/3+2σ/3
+Υe2ℓ/3+2σ/3e4(n/e)2/3 ≤ (1 + δ)ℓe(4/e2/3+o(1))n2/3(e/n)2ℓ/3.
+Therefore,
+πℓ =
+�
+x,c
+p(x, ℓ, c) ≤ n2(1 + δ)ℓe(4/e2/3+o(1))n2/3(e/n)2ℓ/3.
+Then (1), (2) and (3) imply (9) as needed.
+Acknowledgements
+This work was originated when the author was a visitor at Tel Aviv University.
+The author is grateful to Wojciech Samotij for his kind hospitality during the visit
+and for helpful discussions. The author would like to thank Michael Krivelevich for
+helpful remarks and valuable comments on the paper.
+References
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+22
+

4tAzT4oBgHgl3EQf9v5K/content/tmp_files/2301.01923v1.pdf.txt

@@ -0,0 +1,696 @@
+Two-dimensional Heisenberg models with materials-dependent superexchange
+interactions
+Jia-Wen Li,1 Zhen Zhang,2 Jing-Yang You,3 Bo Gu,1, 4, 5, ∗ and Gang Su1, 4, 5, 6, †
+1Kavli Institute for Theoretical Sciences, University of Chinese Academy of Sciences, Beijng 100049, China
+2Key Laboratory of Multifunctional Nanomaterials and Smart Systems,
+Division of Advanced Materials, Suzhou Institute of Nano-Tech and Nano-Bionics,
+Chinese Academy of Sciences, Suzhou, 215123 China
+3Department of Physics, National University of Singapore, Science Drive, Singapore 117551
+4CAS Center for Excellence in Topological Quantum Computation,
+University of Chinese Academy of Sciences, Beijng 100190, China
+5Physical Science Laboratory, Huairou National Comprehensive Science Center, Beijing 101400, China
+6School of Physical Sciences, University of Chinese Academy of Sciences, Beijng 100049, China
+The two-dimensional (2D) van der Waals ferromagnetic semiconductors, such as CrI3 and
+Cr2Ge2Te6, and the 2D ferromagnetic metals, such as Fe3GeTe2 and MnSe2, have been obtained in
+recent experiments and attracted a lot of attentions. The superexchange interaction has been sug-
+gested to dominate the magnetic interactions in these 2D magnetic systems. In the usual theoretical
+studies, the expression of the 2D Heisenberg models were fixed by hand due to experiences. Here, we
+propose a method to determine the expression of the 2D Heisenberg models by counting the possible
+superexchange paths with the density functional theory (DFT) and Wannier function calculations.
+With this method, we obtain a 2D Heisenberg model with six different nearest-neighbor exchange
+coupling constants for the 2D ferromagnetic metal Cr3Te6, which is very different for the crystal
+structure of Cr atoms in Cr3Te6. The calculated Curie temperature Tc = 328 K is close to the
+Tc = 344 K of 2D Cr3Te6 reported in recent experiment. In addition, we predict two stable 2D
+ferromagnetic semiconductors Cr3O6 and Mn3O6 sharing the same crystal structure of Cr3Te6. The
+similar Heisenberg models are obtained for 2D Cr3O6 and Mn3O6, where the calculated Tc is 218
+K and 208 K, respectively. Our method offers a general approach to determine the expression of
+Heisenberg models for these 2D magnetic semiconductors and metals, and builds up a solid basis
+for further studies.
+I.
+INTRODUCTION
+Recently, the successful synthesis of two-dimensional
+(2D) van der Waals ferromagnetic semiconductors in ex-
+periments, such as CrI3 [1] and Cr2Ge2Te6 [2] has at-
+tracted extensive attentions to 2D ferromagnetic materi-
+als. According to Mermin-Wagner theorem [3], the mag-
+netic anisotropy is essential to produce the long-range
+magnetic order in 2D systems. For the 2D magnetic semi-
+conductors obtained in experiments, the Curie tempera-
+ture Tc is still much lower than room temperature. For
+example, Tc = 45 K in CrI3 [1], 30 K in Cr2Ge2Te6 [2],
+34 K in CrBr3 [4], 17 K in CrCl3 [5], 75 K in Cr2S3 [6, 7],
+etc. For applications, the ferromagnetic semiconductors
+with Tc higher than room temperature are highly re-
+quired [8–11]. On the other hand, the 2D van der Waals
+ferromagnetic metals with high Tc have been obtained in
+recent experiments. For example, Tc = 140 K in CrTe
+[12], 300 K in CrTe2 [13, 14], 344 K in Cr3Te6 [15], 160
+K in Cr3Te4 [16], 280 K in CrSe [17], 300 K in Fe3GeTe2
+[18, 19], 270 K in Fe4GeTe2 [20], 229 K in Fe5GeTe2
+[21, 22], 300 K in MnSe2 [23], etc.
+In these 2D van der Waals ferromagnetic materials, the
+superexchange interaction has been suggested to domi-
+∗ [email protected]
+† [email protected]
+nate the magnetic interactions. The superexchange in-
+teraction describes the indirect magnetic interaction be-
+tween two magnetic cations mediated by the neighboring
+non-magnetic anions [24–26]. The superexchange inter-
+action has been discussed in the 2D magnetic semicon-
+ductors.
+Based on the superexchange interaction, the
+strain-enhanced Tc in 2D ferromagnetic semiconductor
+Cr2Ge2Se6 can be understood by the decreased energy
+difference between the d electrons of cation Cr atoms
+and the p electrons of anion Se atoms [27]. The similar
+superexchange picture was obtained in several 2D ferro-
+magnetic semiconductors, including the great enhance-
+ment of Tc in bilayer heterostructures Cr2Ge2Te6/PtSe2
+[28], the high Tc in technetium-based semiconductors
+TcSiTe3, TcGeSe3 and TcGeTe3 [29], and the electric
+field enhanced Tc in the monolayer MnBi2Te4 [30]. The
+superexchange interaction has also been discussed in
+the semiconductor heterostructure CrI3/MoTe2 [31], and
+2D semiconductor Cr2Ge2Te6 with molecular adsorption
+[32].
+In addition, the superexchange interaction has also
+been obtained in the 2D van der Waals ferromagnetic
+metals. By adding vacancies, the angles of the superex-
+change interaction paths of 2D metals VSe2 and MnSe2
+will change, thereby tuning the superexchange coupling
+strength [33]. It is found that biaxial strain changes the
+angle of superexchange paths in 2D metal Fe3GeTe2, and
+affects Tc [34]. Under tensile strain, the ferromagnetism
+of the 2D magnetic metal CoB6 is enhanced, due to the
+arXiv:2301.01923v1  [cond-mat.mtrl-sci]  5 Jan 2023
+
+2
+competition between superexchange and direct exchange
+interactions [35].
+It is important to determine the spin Hamiltonian for
+the magnetic materials, in order to theoretically study
+the magnetic properties, such as Tc. In the usual theoret-
+ical studies, the expression of the spin Hamiltonian needs
+to be fixed by hand according to the experiences. By the
+four-state method and density functional theory (DFT)
+calculations [36–38], the exchange coupling parameters of
+the spin Hamiltonian, such as the nearest neighbor, the
+next nearest neighbor, inter-layer, etc, can be obtained.
+Then the Tc can be estimated through Monte Carlo sim-
+ulations [38]. With different spin Hamiltonians chosen by
+hand, sometimes different results are obtained in calcu-
+lations. Is it possible to determine the spin Hamiltonian
+by the help of calculations rather than by the experiences
+?
+In this paper, we propose a method to establish the 2D
+Heisenberg models for the 2D van der Waals magnetic
+materials, when the superexchange interactions domi-
+nate.
+Through the DFT and Wannier function calcu-
+lations, we can calculate the exchange coupling between
+any two magnetic cations, by counting the possible su-
+perexchange paths.
+By this method, we obtain a 2D
+Heisenberg model with six different nearest-neighbor ex-
+change coupling constants for the 2D van der Waals fer-
+romagnetic metal Cr3Te6 [15], where the calculated Tc
+= 328 K is close to the Tc = 344 K reported in the ex-
+periment. In addition, based on the crystal structure of
+2D Cr3Te6, we predict two 2D magnetic semiconductors
+Cr3O6 and Mn3O6 with Tc of 218 K and 208 K, and
+energy gap of 0.99 eV and 0.75 eV, respectively.
+II.
+COMPUTATIONAL METHODS
+Our calculations were based on the DFT as im-
+plemented in the Vienna ab initio simulation package
+(VASP) [39]. The exchange-correlation potential is de-
+scribed with the Perdew-Burke-Ernzerhof (PBE) form
+of the generalized gradient approximation (GGA) [40].
+The electron-ion potential is described by the projector-
+augmented wave (PAW) method [41].
+We carried out
+the calculation of GGA + U with U = 3.2 eV, a rea-
+sonable U value for the 3d electrons of Cr in Cr3Te6
+[15]. The band structures for 2D Cr3O6 and Mn3O6 were
+calculated in HSE06 hybrid functional [42]. The plane-
+wave cutoff energy is set to be 500 eV. Spin polariza-
+tion is taken into account in structure optimization. To
+prevent interlayer interaction in the supercell of 2D sys-
+tems, the vacuum layer of 16 ˚A is included. The 5×9×1,
+5×9×1 and 7×11×1 Monkhorst Pack k-point meshed
+were used for the Brillouin zone (BZ) sampling for 2D
+Cr3O6, Cr3Te6 and Mn3O6, respectively [43]. The struc-
+tures of 2D Cr3O6 and Mn3O6 were fully relaxed, where
+the convergence precision of energy and force were 10−6
+and 10−3 eV/˚A, respectively. The phonon spectra were
+obtained in a 3×3×1 supercell with the PHONOPY pack-
+age [44]. The Wannier90 code was used to construct a
+tight-binding Hamiltonian [45, 46] to calculate the mag-
+netic coupling constant.
+In the calculation of molecu-
+lar dynamics, a 3×4×1 supercell (108 atoms) was built,
+and we took the NVT ensemble (constant-temperature,
+constant-volume ensemble) and maintained a tempera-
+ture of 250 K with a step size of 3 fs and a total duration
+of 6 ps.
+III.
+Method to determine the 2D Heisenberg
+model: an example of 2D Cr3Te6
+A.
+Calculate exchange coupling J from
+superexchange paths
+The crystal structure of 2D Cr3Te6 is shown in Fig. 1,
+where the space goup is Pm (No.6). In experiment, it
+is a ferromagnetic metal with high Tc = 344 K [15]. To
+theoretically study its magnetic properties, we considered
+seven different magnetic configurations, including a ferro-
+magnetic (FM) , a ferrimagnetic (FIM), and five antifer-
+romagnetic (AFM) configurations, as discussed in Sup-
+plemental Materials [47]. The calculation results show
+that the magnetic ground state is ferromagnetic, consis-
+tent with the experimental results. Since the superex-
+change interaction has been suggested to dominate the
+magnetic interactions in these 2D van der Waals ferro-
+magnetic semiconductors and metals, we study the su-
+perexchange interactions in 2D Cr3Te6.
+The superexchange interaction can be reasonably de-
+scried by a simple Cr-Te-Cr model [48], as shown in Fig.
+2.
+There are two Cr atoms at sites i and j, and one
+Te atom at site k between the two Cr atoms. By the
+perturbation calculation, the superexchange coupling Jij
+between the two Cr atoms can be obtained as [48],
+Jij =( 1
+E2
+↑↓
+−
+1
+E2
+↑↑
+)
+�
+k,p,d
+|Vik|2Jpd
+kj
+= 1
+A
+�
+k,p,d
+|Vik|2Jpd
+kj .
+(1)
+The indirect exchange coupling Jij is consisting of two
+processes. One is the direct exchange process between the
+d electron of Cr at site j and the p electrons of Te at site
+k, presented by Jpd
+kj. The other is the electron hopping
+process between p electrons of Te atom at site k and d
+electrons of Cr atom at site i, presented by —Vik|2/A.
+Vik is the hopping parameter between d electrons of Cr
+atom at site i and p electrons of Te atom at site k. Here, A
+= 1/(1/E2
+↑↓-1/E2
+↑↑), and is taken as a pending parameter.
+E↑↑ and E↑↓ are energies of two d electrons at Cr atom
+at site i with parallel and antiparallel spins, respectively.
+The direct exchange coupling Jpd
+kj can be expressed as
+[27–30]:
+
+3
+FIG. 1. Crystal structure of Cr3Te6 . (a) Top view (b) Side view.
+Jpd
+kj =
+2|Vkj|2
+|Ep
+k − Ed
+j |.
+(2)
+Vkj is the hopping parameter between p electrons of
+Te atom at site k and d electrons of Cr atom at site j.
+Ep
+k is the energy of p electrons of Te atom at site k, and
+Ed
+j is the energy of d electrons of Cr atom at site j.
+FIG. 2. Schematic picture of superexchange interaction by
+a Cr-Te-Cr model. There are two process, one is direct ex-
+change process between Crj and Tek, noted as Jpd
+kj, and the
+other is electron hopping between Tek and Cri, noted as
+|Vik|2/A. See text for details.
+By the DFT and Wannier function calculations, the
+parameters Vik, Vkj, Ep
+k, and Ed
+j in Eqs. (1) and (2) can
+be calculated. The JijA can be obtained by counting all
+the possible k sites of Te atoms, p orbitals of Te atoms,
+and d orbitals of Cr atoms.
+From the calculated results in Table I, it is suggested
+that there are six different nearest-neighbor couplings,
+denoted as J11, J22, J33, J12, J13, and J23, as shown in
+Fig. 3(b). Accordingly, there are three kinds of Cr atoms,
+noted as Cr1, Cr2, and Cr3. Based on the results in Table
+I, the effective spin Hamiltonian can be written as
+H =J11
+�
+n
+⃗S1n · ⃗S1n + J22
+�
+n
+⃗S2n · ⃗S2n + J33
+�
+n
+⃗S3n · ⃗S3n
++J12
+�
+n
+⃗S1n · ⃗S2n + J13
+�
+n
+⃗S1n · ⃗S3n + J23
+�
+n
+⃗S2n · ⃗S3n
++D
+�
+n
+(S2
+1nz + S2
+2nz + S2
+3nz),
+(3)
+where Jij means magnetic coupling between Cri and Crj,
+as indicated in Fig.
+3(b).
+D represents the magnetic
+anisotropy energy (MAE) of Cr3Te6.
+B.
+Determine the parameters D and A
+The single-ion magnetic anisotropy parameter DS2 can
+be obtained by: DS2=(E⊥-E∥)/6, where E⊥ and E∥ are
+energies of Cr3Te6 with out-of-plane and in-plane polar-
+izations in FM state, respectively. It has DS2 = -0.14
+meV/Cr for 2D Cr3Te6, which is in agreement with the
+value of -0.13 meV/Cr reported in the previous study of
+Cr3Te6 [15].
+The parameter A can be calculated in the following
+way. Considering a FM and an AFM configurations, the
+total energy of Eq. (3) without MAE term can be re-
+spectively expressed as [47]:
+EF M = 2J11S2
+1 + 2J22S2
+2 + 2J33S2
+3 + 8J12S1S2
++2J23S2S3 + 8J13S1S3 + E0
+= 11838/A + E0,
+EAF M1 = 2J11S2
+1 + 2J22S2
+2 − 2J33S2
+3 − 8J12S1S2 + E0
+= −2502/A + E0.
+(4)
+The results in Table I are used to obtain the final ex-
+pressions in Eq. (4). Since two parameters A and E0 are
+kept, two spin configurations FM and AFM1 are consid-
+ered here. Discussion on the choice of spin configurations
+
+(b)
+(a)
+y
+Te
+X
+X1
+2
+Tek
+Tpd
+A
+kj
+d1
+P1
+P2
+d24
+FIG. 3. (a) The crystal structure of Cr atoms in 2D Cr3Te6. (b) The magnetic structure of Cr atoms in 2D Cr3Te6, calculated
+by Eqs. (1) and (2).
+TABLE I. For 2D Cr3Te6, the calculated exchange coupling parameters JijA in Eqs.(1) and (2), by the density functional
+theory and Wannier functional calculations. A is a pending parameter. The unit of JijA is meV3.
+J11A
+J22A
+J33A
+J12A
+J13A
+J23A
+40
+26
+53
+29
+44
+83
+is given in Supplemental Materials [47]. For the FM spin
+configuration, the ground state of Cr3Te6, the total en-
+ergy is taken as EF M = 0 for the energy reference. The
+total energy of AFM1, EAF M1 = 535 meV is obtained by
+the DFT calculation. The parameters A and E0 are ob-
+tained by solving Eq. (4), and the six exchange coupling
+parameters Jij can be obtained by Table I. The results
+are given in Table II.
+C.
+Estimate Tc by Monte Carlo simulation
+To calculate the Curie temperature, we used the Monte
+Carlo program for the Heisenberg-type Hamiltonian in
+Eq. (3) with parameters in Table II. The Monte Carlo
+simulation was performed on a 30
+√
+3 ×30
+√
+3 lattice with
+more than 1×106 steps for each temperature. The first
+two-third steps were discarded, and the last one-thirds
+steps were used to calculate the temperature-dependent
+physical quantities. As shown in Table II and Fig. 4 (d),
+the calculated Tc = 328 K for 2D Cr3Te6, close to the Tc
+= 344 K of 2D Cr3Te6 in the experiment [15]. Discussion
+on the choice of spin configurations and the estimation of
+exchange couplings Jij and Tc is given in Supplemental
+Materials [47].
+IV.
+Prediction of Two High Curie Temperature
+Magnetic Semiconductors Cr3O6 and Mn3O6
+Inspired by the high Tc in the 2D magnetic metal
+Cr3Te6, we explore the possible high Tc magnetic semi-
+conductors with the same crystal structure of Cr3Te6 by
+FIG. 4. (a) Band structures of Cr3O6 with a bandgap of 0.99
+eV. (b) Band structures of Mn3O6 with a bandgap of 0.75 eV.
+(c) Energy gap of Cr3O6 and Mn3O6 under external electric
+field out-plane. (d) The magnetic moment of Cr3Te6, Cr3O6,
+and Mn3O6 varies with temperature.
+the DFT calculations. We obtain two stable ferromag-
+netic semiconductors Cr3O6 and Mn3O6.
+In order to
+study the stability of the 2D Cr3O6 and Mn3O6, we cal-
+culate the phonon spectrum. As shown in Supplemental
+Materials [47], there is no imaginary frequency, indicat-
+ing the dynamical stability. In addition, we performed
+molecular dynamics simulations of Cr3O6 and Mn3O6
+at 250 K, taking the NVT ensemble (constant temper-
+ature and volume) and run for 6 ps. The results show
+that 2D Cr3O6 and Mn3O6 are thermodynamically sta-
+
+(a)
+(b)
+22
+J11
+J23
+J33
+J13
+2(a)
+(b)
+-1.0
+-1.0
+(eV)
+(eV)
+-0.5
+-0.5
+- Spin up
+2DCr306
+2D/Mn.06
+- Spin up
+E
+0.0
+0.0
+ Spin down
+ Spin down
+E
+-0.5
+-0.5
+-1.0
+-1.0
+-1.5
+-1.5
+X
+S
+X
+S
+(c)
+(d)
+1.0
+1.4
+- 2D C
+1.2
+- 2D Mn3O6
+0.8
+Exp
+ (eV)
+1.0
+ref. 15)
+0.6
+0.8
+Gap
+0.6
+Mas
+Cr,Te6"
+0.4
+Cr,O6
+0.2
+0.2
+Mn,O6
+0.0
+-0.3 -0.2 -0.1
+0.3
+400
+0
+0.1 0.2
+100
+200
+300
+500
+Electric field (V/A)
+Temepture (K)5
+TABLE II. For 2D magnetic metal Cr3Te6 and semiconductors Cr3O6 and Mn3O6, the parameter A (in unit of meV−2) in Eq.
+(1), the exchange couping parameters JijS2 and the magnetic anisotropy parameter DS2 (in unit of meV) in the Hamiltonian
+in Eq. (3), and the estimated Curie temperature Tc. See text for details.
+Materials
+A
+J11S2
+J22S2
+J33S2
+J12S2
+J13S2
+J23S2
+DS2
+Tc (K)
+Cr3Te6
+-27
+-17.1
+-11.5
+-24.4
+-12.6
+-19.6
+-37.4
+-0.14
+328
+Cr3O6
+-36
+-18.9
+-14.6
+-10.1
+-18.7
+-1.8
+-3.1
+0.04
+218
+Mn3O6
+-465
+-11.9
+-7.6
+-50.4
+-15.9
+-5.2
+-10.7
+-0.09
+208
+ble [47]. These calculation results suggest that 2D Cr3O6
+and Mn3O6 may be feasible in experiment.
+The band structure of 2D Cr3O6 and Mn3O6 is shown
+in Figs. 4(a) and 4(b), respectively, where the band gap
+is 0.99 eV for Cr3O6 and 0.75 eV for Mn3O6. As shown
+in Figs. 4(a) and (b), the band gap for 2D Cr3O6 and
+Mn3O6 is 0.99 eV and 0.75 eV, respectively. When ap-
+plying an out-of-plane electric field with a range of ± 0.3
+V/˚A, which is possible in experiment [49], the band gap
+of Cr3O6 (Mn3O6) increases (decreases) with increasing
+electric field, as shown in Fig. 4(c). By the same calcula-
+tion method above, the parameter A, the similar Heisen-
+berg models in Eq. 3 with six nearest-neighbor exchange
+coupling Jij are obtained for the 2D Cr3O6 and Mn3O6.
+The parameters A, Jij and D are calculated and shown
+in Table II. The spin polarization of Cr3O6 and Mn3O6
+is in-plane (DS2 = 0.04 meV) and out-of-plane (DS2 =
+-0.09 meV), respectively. Fig. 4(d) shows the magnetiza-
+tion as a function of temperature for 2D Cr3Te6, Cr3O6
+and Mn3O6. The calculated Curie temperature is Tc =
+218 K for 2D Cr3O6 and Tc = 208 K for 2D Mn3O6,
+respectively.
+V.
+CONCLUSION
+Based on the DFT and Wannier function calculations,
+we propose a method for constructing the 2D Heisen-
+berg model with the superexchange interactions. By this
+method, we obtain a 2D Heisenberg model with six differ-
+ent nearest-neighbor exchange couplings for the 2D fer-
+romagnetic metal Cr3Te6. The calculated Curie temper-
+ature Tc = 328 K is close to the Tc = 344 K of Cr3Te6 in
+the experiment. In addition, we predicted two 2D mag-
+netic semiconductors: Cr3O6 with band gap of 0.99 eV
+and Tc = 218 K, and Mn3O6 with band gap of 0.75 eV
+and Tc = 208 K, where the similar 2D Heisenberg models
+are obtained. The complex Heisenberg model developed
+from the simple crystal structure shows the power of our
+method to study the magnetic properties in these 2D
+magnetic metals and semiconductors.
+ACKNOWLEDGEMENTS
+This work is supported in part by the National Natu-
+ral Science Foundation of China (Grants No. 12074378
+and No. 11834014), the Beijing Natural Science Foun-
+dation (Grant No.
+Z190011), the National Key R&D
+Program of China (Grant No.
+2018YFA0305800), the
+Beijing Municipal Science and Technology Commission
+(Grant No. Z191100007219013), the Chinese Academy of
+Sciences (Grants No. YSBR-030 and No. Y929013EA2),
+and the Strategic Priority Research Program of Chinese
+Academy of Sciences (Grants No. XDB28000000 and No.
+XDB33000000).
+[1] B. Huang, G. Clark, E. Navarro-Moratalla, D. R. Klein,
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+arXiv:2301.04881v1  [math.CO]  12 Jan 2023
+Strengthening the Directed Brooks’ Theorem for oriented graphs
+and consequences on digraph redicolouring *
+Lucas Picasarri-Arrieta
+Universit´e Cˆote d’Azur, CNRS, I3S, INRIA, Sophia Antipolis, France
+[email protected]
+Abstract
+Let D = (V, A) be a digraph. We define ∆max(D) as the maximum of {max(d+(v), d−(v)) | v ∈ V } and
+∆min(D) as the maximum of {min(d+(v), d−(v)) | v ∈ V }. It is known that the dichromatic number of D
+is at most ∆min(D) + 1. In this work, we prove that every digraph D which has dichromatic number exactly
+∆min(D) + 1 must contain the directed join of ←→
+Kr and ←→
+Ks for some r, s such that r + s = ∆min(D) + 1. In
+particular, every oriented graph ⃗G with ∆min( ⃗G) ≥ 2 has dichromatic number at most ∆min( ⃗G).
+Let ⃗G be an oriented graph of order n such that ∆min( ⃗G) ≤ 1. Given two 2-dicolourings of ⃗G, we show that
+we can transform one into the other in at most n steps, by recolouring one vertex at each step while maintaining
+a dicolouring at any step. Furthermore, we prove that, for every oriented graph ⃗G on n vertices, the distance
+between two k-dicolourings is at most 2∆min( ⃗G)n when k ≥ ∆min( ⃗G) + 1.
+We then extend a theorem of Feghali to digraphs. We prove that, for every digraph D with ∆max(D) =
+∆ ≥ 3 and every k ≥ ∆ + 1, the k-dicolouring graph of D consists of isolated vertices and at most one further
+component that has diameter at most c∆n2, where c∆ = O(∆2) is a constant depending only on ∆.
+1
+Introduction
+1.1
+Graph (re)colouring
+Given a graph G = (V, E), a k-colouring of G is a function c : V −→ {1, . . ., k} such that, for every edge xy ∈ E,
+we have c(x) ̸= c(y). So for every i ∈ {1, . . ., k}, c−1(i) induces an independent set on G. The chromatic number
+of G, denoted by χ(G), is the smallest k such that G admits a k-colouring. The maximum degree of G, denoted
+by ∆(G), is the degree of the vertex with the greatest number of edges incident to it. A simple greedy procedure
+shows that, for any graph G, χ(G) ≤ ∆(G) + 1. The celebrated theorem of Brooks [6] characterizes the graphs
+for which equality holds.
+Theorem 1 (Brooks, [6]). A connected graph G satisfies χ(G) = ∆(G) + 1 if and only if G is an odd cycle or a
+complete graph.
+For any k ≥ χ(G), the k-colouring graph of G, denoted by Ck(G), is the graph whose vertices are the k-
+colourings of G and in which two k-colourings are adjacent if they differ by the colour of exactly one vertex. A
+path between two given colourings in Ck(G) corresponds to a recolouring sequence, that is a sequence of pairs
+composed of a vertex of G, which is going to receive a new colour, and a new colour for this vertex. If Ck(G)
+is connected, we say that G is k-mixing. A k-colouring of G is k-frozen if it is an isolated vertex in Ck(G). The
+graph G is k-freezable if it admits a k-frozen colouring. In the last fifteen years, since the papers of Cereceda,
+van den Heuvel and Johnson [8, 7], graph recolouring has been studied by many researchers in graph theory. We
+refer the reader to the PhD thesis of Bartier [2] for a complete overview on graph recolouring and to the surveys
+of van Heuvel [12] and Nishimura [15] for reconfiguration problems in general. Feghali [9] proved the following
+analogue of Brooks’ Theorem for graphs recolouring.
+*Research supported by research grant DIGRAPHS ANR-19-CE48-0013 and by the French government, through the EUR DS4H Invest-
+ments in the Future project managed by the National Research Agency (ANR) with the reference number ANR-17-EURE-0004.
+1
+
+Theorem 2 (Feghali, [9]). Let G = (V, E) be a connected graph with ∆(G) = ∆ ≥ 3, k ≥ ∆ + 1, and α, β two
+k-colourings of G. Then at least one of the following holds:
+• α is k-frozen, or
+• β is k-frozen, or
+• there is a recolouring sequence of length at most c∆|V |2 between α and β, where c∆ = O(∆) is a constant
+depending on ∆.
+1.2
+Digraph (re)dicolouring.
+In this paper, we are looking for extensions of the previous results on graphs colouring and recolouring to digraphs.
+Let D be a digraph. A digon is a pair of arcs in opposite directions between the same vertices. A simple arc
+is an arc which is not in a digon. For any two vertices x, y ∈ V (D), the digon {xy, yx} is denoted by [x, y]. The
+digon graph of D is the undirected graph with vertex set V (D) in which uv is an edge if and only if [u, v] is a
+digon of D. An oriented graph is a digraph with no digon. The bidirected graph associated to a graph G, denoted
+by ←→
+G , is the digraph obtained from G, by replacing every edge by a digon. The underlying graph of D, denoted
+by UG(D), is the undirected graph G with vertex set V (D) in which uv is an edge if and only if uv or vu is an
+arc of D.
+Let v be a vertex of a digraph D. The out-degree (resp. in-degree) of a vertex v, denoted by d+(v) (resp.
+d−(v)), is the number of arcs leaving (resp. entering) v. We de��ne the maximum degree of v as dmax(v) =
+max{d+(v), d−(v)}, and the minimum degree of v as dmin(v) = min{d+(v), d−(v)}. We can then define the cor-
+responding maximum degrees of D: ∆max(D) = maxv∈V (D)(dmax(v)) and ∆min(D) = maxv∈V (D)(dmin(v)).
+A digraph D is ∆-diregular if, for every vertex v ∈ V (D), d−(v) = d+(v) = ∆.
+In 1982, Neumann-Lara [14] introduced the notions of dicolouring and dichromatic number, which generalize
+the ones of colouring and chromatic number. A k-dicolouring of D is a function c : V (D) → {1, . . ., k} such
+that c−1(i) induces an acyclic subdigraph in D for each i ∈ {1, . . . , k}. The dichromatic number of D, denoted
+by ⃗χ(D), is the smallest k such that D admits a k-dicolouring. There is a one-to-one correspondence between
+the k-colourings of a graph G and the k-dicolourings of the associated bidirected graph ←→
+G , and in particular
+χ(G) = ⃗χ(←→
+G ). Hence every result on graph colourings can be seen as a result on dicolourings of bidirected
+graphs, and it is natural to study whether the result can be extended to all digraphs.
+The directed version of Brooks’ Theorem has been first proved by Mohar in [13], but people discovered a flaw
+in the proof. Harutyunyan and Mohar then gave a stronger result in [10]. Finally, Aboulker and Aubian gave four
+new proofs of the following theorem in [1].
+Theorem 3 (DIRECTED BROOKS’ THEOREM). Let D be a connected digraph. Then ⃗χ(D) ≤ ∆max(D) + 1 and
+equality holds if and only if one of the following occurs:
+• D is a directed cycle, or
+• D is a bidirected odd cycle, or
+• D is a bidirected complete graph (of order at least 4).
+It is easy to prove, by induction on |V (D)|, that every digraph D can be dicoloured with ∆min(D) + 1
+colours. Hence, one can wonder if Brooks’ Theorem can be extended to digraphs using ∆min(D) instead of
+∆max(D). Unfortunately, Aboulker and Aubian [1] proved that, given a digraph D, deciding whether D is
+∆min(D)-dicolourable is NP-complete. Thus, unless P=NP, we cannot expect an easy characterization of digraphs
+satisfying ⃗χ(D) = ∆min(D) + 1.
+Let the maximum geometric mean of a digraph D be ˜∆(D) = max{
+�
+d+(v)d−(v)|v ∈ V (D)}. By definition
+we have ∆min(D) ≤ ˜∆(D) ≤ ∆max(D). Restricted to oriented graphs, Harutyunyan and Mohar [11] have
+strengthened Theorem 3 by proving the following.
+2
+
+Theorem 4 (Harutyunyan and Mohar [11]). There is an absolute constant ∆1 such that every oriented graph ⃗G
+with ˜∆(⃗G) ≥ ∆1 has ⃗χ(⃗G) ≤ (1 − e−13) ˜∆(⃗G).
+In Section 2, we give another strengthening of Theorem 3 on a large class of digraphs which contains oriented
+graphs. The directed join of H1 and H2, denoted by H1 ⇒ H2, is the digraph obtained from disjoint copies of H1
+and H2 by adding all arcs from the copy of H1 to the copy of H2 (H1 or H2 may be empty).
+Theorem 5. Let D be a digraph. If D is not ∆min(D)-dicolourable, then one of the following holds:
+• ∆min(D) ≤ 1, or
+• ∆min(D) = 2 and D contains ←→
+K2, or
+• ∆min(D) ≥ 3 and D contains ←→
+Kr ⇒ ←→
+Ks, for some r, s such that r + s = ∆min(D) + 1.
+In particular, the following is a direct consequence of Theorem 5.
+Corollary 6. Let D be a digraph. If ⃗χ(D) = ∆min(D) + 1, then D contains the complete bidirected graph on
+� ∆min+1
+2
+�
+vertices as a subdigraph.
+Moreover, since an oriented graph does not contain any digon, Corollary 6 implies the following:
+Corollary 7. Let ⃗G be an oriented graph. If ∆min(⃗G) ≥ 2, then ⃗χ(⃗G) ≤ ∆min(⃗G).
+Corollary 6 is best possible: if we restrict D to not contain the complete bidirected graph on
+� ∆min+1
+2
+�
++ 1,
+then we show that deciding whether ⃗χ(D) ≤ ∆min(D) remains NP-complete (Theorem 11).
+For any k ≥ ⃗χ(D), the k-dicolouring graph of D, denoted by Dk(D), is the graph whose vertices are the
+k-dicolourings of D and in which two k-dicolourings are adjacent if they differ by the colour of exactly one vertex.
+Observe that Ck(G) = Dk(←→
+G ) for any bidirected graph ←→
+G . A redicolouring sequence between two dicolourings is
+a path between these dicolourings in Dk(D). The digraph D is k-mixing if Dk(D) is connected. A k-dicolouring of
+D is k-frozen if it is an isolated vertex in Dk(D). The digraph D is k-freezable if it admits a k-frozen dicolouring.
+A vertex v is blocked to its colour in a dicolouring α if, for every colour c ̸= α(v), recolouring v to c in α creates
+a monochromatic directed cycle.
+Digraph redicolouring was first introduced in [5], where the authors generalized different results on graph re-
+colouring to digraphs, and proved some specific results on oriented graphs redicolouring. In particular, they studied
+the k-dicolouring graph of digraphs with bounded degeneracy or bounded maximum average degree, and they show
+that finding a redicolouring sequence between two given k-dicolouring of a digraph is PSPACE-complete. Dealing
+with the maximum degree of a digraph, they proved that, given an orientation of a subcubic graph ⃗G on n ver-
+tices, its 2-dicolouring graph D2(⃗G) is connected and has diameter at most 2n and they asked if this bound can be
+improved. We answer this question in Section 3 by proving the following theorem.
+Theorem 8. Let ⃗G be an oriented graph of order n such that ∆min(⃗G) ≤ 1. Then D2(⃗G) is connected and has
+diameter exactly n.
+In particular, if ⃗G is an orientation of a subcubic graph, then ∆min(⃗G) ≤ 1 (because d+(v) + d−(v) ≤ 3 for
+every vertex v), and so D2(⃗G) has diameter exactly n. Furthermore, we prove the following as a consequence of
+Corollary 7 and Theorem 8.
+Corollary 9. Let ⃗G be oriented graph of order n with ∆min(⃗G) = ∆ ≥ 1, and let k ≥ ∆ + 1. Then Dk(⃗G) is
+connected and has diameter at most 2∆n.
+Corollary 9 does not hold for digraphs in general: indeed, ←→
+Pn, the bidirected path on n vertices, satisfies
+∆min(←→
+Pn) = 2 and D3(←→
+Pn) = C3(Pn) has diameter Ω(n2), as proved in [4].
+In Section 4, we extend Theorem 2 to digraphs.
+Theorem 10. Let D = (V, A) be a connected digraph with ∆max(D) = ∆ ≥ 3, k ≥ ∆ + 1, and α, β two
+k-dicolourings of D. Then at least one of the following holds:
+3
+
+• α is k-frozen, or
+• β is k-frozen, or
+• there is a redicolouring sequence of length at most c∆|V |2 between α and β, where c∆ = O(∆2) is a
+constant depending only on ∆.
+Furthermore, we prove that a digraph D is k-freezable only if D is bidirected and its underlying graph is
+k-freezable. Thus, an obstruction in Theorem 10 is exactly the bidirected graph of an obstruction in Theorem 2.
+2
+Strengthening of Directed Brooks’ Theorem for oriented graphs
+A digraph D is k-dicritical if ⃗χ(D) = k and for every vertex v ∈ V (D), ⃗χ(D − v) < k. Observe that every
+digraph with dichromatic number at least k contains a k-dicritical subdigraph.
+Let F2 be {←→
+K2}, and for each ∆ ≥ 3, we define F∆ = {←→
+Kr ⇒ ←→
+Ks | r, s ≥ 0 and r + s = ∆ + 1}. A digraph
+D is F∆-free if it does not contain F as a subdigraph, for any F ∈ F∆. Theorem 5 can then be reformulated as
+follows:
+Theorem 5. Let D be a digraph with ∆min(D) = ∆ ≥ 2. If D is F∆-free, then ⃗χ(D) ≤ ∆.
+Proof. Let D be a digraph such that ∆min(D) = ∆ ≥ 2 and ⃗χ(D) = ∆ + 1. We will show that D contains some
+F ∈ F∆ as a subdigraph.
+Let (X, Y ) be a partition of V (D) such that for each x ∈ X, d+(x) ≤ ∆, and for each y ∈ Y , d−(y) ≤ ∆.
+We define the digraph ˜D as follows:
+• V ( ˜D) = V (D),
+• A( ˜D) = A(D⟨X⟩) ∪ A(D⟨Y ⟩) ∪ {xy, yx | xy ∈ A(D), x ∈ X, y ∈ Y }.
+Claim 5.1: ⃗χ( ˜D) ≥ ∆ + 1.
+Proof of claim. Assume for a contradiction that there exists a ∆-dicolouring c of ˜D. Then D, coloured with c,
+must contain a monochromatic directed cycle C. Now C is not contained in X nor Y , for otherwise C would be a
+monochromatic directed cycle of D⟨X⟩ or D⟨Y ⟩ and so a monochromatic directed cycle of ˜D. Thus C contains
+an arc xy from X to Y . But then, [x, y] is a monochromatic digon in ˜D, a contradiction.
+♦
+Since ⃗χ( ˜D) ≥ ∆ + 1, there is a (∆ + 1)-dicritical subdigraph H of ˜D. By dicriticality of H, for every vertex
+v ∈ V (H), d+
+H(v) ≥ ∆ and d−
+H(v) ≥ ∆, for otherwise a ∆-dicolouring of H − v could be extended to H by
+choosing for v a colour which is not appearing in its out-neighbourhood or in its in-neighbourhood. We define XH
+as X ∩ V (H) and YH as Y ∩ V (H). Note that both H⟨XH⟩ and H⟨YH⟩ are subdigraphs of D.
+Claim 5.2: H is ∆-diregular.
+Proof of claim. Let ℓ be the number of digons between XH and YH in H. Observe that, by definition of X and
+H, for each vertex x ∈ XH, d+
+H(x) = ∆. Note also that, in H, ℓ is exactly the number of arcs leaving XH and
+exactly the number of arcs entering XH. We get:
+∆|XH| =
+�
+x∈XH
+d+
+H(x)
+= ℓ + |A(H⟨XH⟩)|
+=
+�
+x∈XH
+d−
+H(x)
+which implies, since H is dicritical, d+
+H(x) = d−
+H(x) = ∆ for every vertex x ∈ XH. Using a symmetric argument,
+we prove that ∆|YH| = �
+y∈YH d+
+H(y), implying d+
+H(y) = d−
+H(y) = ∆ for every vertex y ∈ YH.
+♦
+4
+
+Since H is ∆-diregular, then in particular ∆max(H) = ∆. Hence, because ⃗χ(H) = ∆ + 1, by Theorem 3,
+either ∆ = 2 and H is a bidirected odd cycle, or ∆ ≥ 3 and H is the bidirected complete graph on ∆ + 1 vertices.
+• If ∆ = 2 and H is a bidirected odd cycle, then at least one digon of H belongs to H⟨XH⟩ or H⟨YH⟩, for oth-
+erwise H would be bipartite (with bipartition (XH, YH)). Since both H⟨XH⟩ and H⟨YH⟩ are subdigraphs
+of D, this shows, as desired, that D contains a copy of ←→
+K2.
+• If k ≥ 3 and H is the bidirected complete graph on ∆ + 1 vertices, let AH be all the arcs from YH to XH.
+Then D⟨V (H)⟩ \ AH is a subdigraph of D which belongs to F∆.
+Now we will justify that Corollary 6 is best possible. To do so, we prove that given a digraph D which does
+not contain the bidirected complete graph on
+�
+∆min(D)+1
+2
+�
++ 1 vertices, deciding if it is ∆min(D)-dicolourable is
+NP-complete. We shall use a reduction from k-DICOLOURABILITY which is defined as follows:
+k-DICOLOURABILITY
+Input: A digraph D
+Question: Is D k-dicolourable ?
+k-DICOLOURABILITY is NP-complete for every fixed k ≥ 2 [3]. It remains NP-complete when we restrict to
+digraphs D with ∆min(D) = k [1].
+Theorem 11. For all k ≥ 2, k-DICOLOURABILITY remains NP-complete when restricted to digraphs D satisfying
+∆min(D) = k and not containing the bidirected complete graph on
+� k+1
+2
+�
++ 1 vertices.
+Proof. Let D = (V, A) be an instance of k-DICOLOURABILITY for some fixed k ≥ 2. Then we build D′ =
+(V ′, A′) as follows:
+• For each vertex x ∈ V , we associate a copy of S−
+x ⇒ S+
+x where S−
+x is the bidirected complete graph on
+� k+1
+2
+�
+vertices, and S+
+x is the bidirected complete graph on
+� k+1
+2
+�
+vertices.
+• For each arc xy ∈ A, we associate all possible arcs x+y− in A′, such that x+ ∈ S+
+x and y− ∈ S−
+y .
+First observe that ∆min(D′) = k. Let v be a vertex of D′, if v belongs to some S+
+x , then d−(v) = k, otherwise
+it belongs to some S−
+x and then d+(v) = k. Then observe that D′ does not contain the bidirected complete graph
+on
+� k+1
+2
+�
++ 1 vertices since every digon in D′ is contained in some S+
+x or S−
+x . Thus we only have to prove that
+⃗χ(D) ≤ k if and only if ⃗χ(D′) ≤ k to get the result.
+• Let us first prove that ⃗χ(D) ≤ k implies ⃗χ(D′) ≤ k.
+Assume that ⃗χ(D) ≤ k. Let φ : V −→ {1, . . . , k} be a k-dicolouring of D. Let φ′ be the k-dicolouring of
+D′ defined as follows: for each vertex x ∈ V , choose arbitrarily x− ∈ S−
+x , x+ ∈ S+
+x , and set φ′(x−) =
+φ′(x+) = φ(x). Then choose a distinct colour for every other vertex v in S−
+x ∪ S+
+x , and set φ′(v) to
+this colour. We get that φ′ must be a k-dicolouring of D′: for each x ∈ V , every vertex but x− in S−
+x
+must be a sink in its colour class, and every vertex but x+ in S+
+x must be a source in its colour class.
+Thus if D′, coloured with φ′, contains a monochromatic directed cycle C′, then C′ must be of the form
+x−
+1 x+
+1 x−
+2 x+
+2 · · · x−
+ℓ x+
+ℓ x−
+1 . But then C = x1x2 · · · xℓx1 is a monochromatic directed cycle in D coloured
+with φ: a contradiction.
+• Reciprocally, let us prove that ⃗χ(D′) ≤ k implies ⃗χ(D) ≤ k.
+Assume that ⃗χ(D′) ≤ k. Let φ′ : V ′ −→ {1, . . ., k} be a k-dicolouring of D′. Let φ be the k-dicolouring
+of D defined as follows. For each vertex x ∈ V , we know that |S+
+x ∪ S−
+x | = k + 1, thus there must be
+two vertices x+ and x− in S+
+x ∪ S−
+x such that φ′(x+) = φ′(x−). Moreover, since both S+
+x and S−
+x are
+bidirected, one of these two vertices belongs to S+
+x and the other one belongs to S−
+x . We assume without
+loss of generality x+ ∈ S+
+x and x− ∈ S−
+x . Then we set φ(x) = φ′(x+). We get that φ must be a k-
+dicolouring of D. If D, coloured with φ, contains a monochromatic directed cycle C = x1x2 · · · xℓx1, then
+C′ = x−
+1 x+
+1 x−
+2 x+
+2 · · · x−
+ℓ x+
+ℓ x−
+1 is a monochromatic directed cycle in D′ coloured with φ′, a contradiction.
+5
+
+3
+Redicolouring oriented graphs
+In this section, we restrict to oriented graphs. We first prove Theorem 8, let us restate it.
+Theorem 8. Let ⃗G be an oriented graph of order n such that ∆min(⃗G) ≤ 1. Then D2(⃗G) is connected and has
+diameter exactly n.
+Observe that, if D2(⃗G) is connected, then its diameter must be at least n: for any 2-dicolouring α, we can
+define its mirror ¯α where, for every vertex v ∈ V (⃗G), α(v) ̸= ¯α(v); then every redicolouring sequence between α
+and ¯α has length at least n.
+Lemma 12. Let C be a directed cycle of length at least 3. Then D2(C) is connected and has diameter exactly n.
+Proof. Let α and β be any two 2-dicolourings of C. Let x = diff(α, β) = |{v ∈ V (C) | α(v) ̸= β(v)}|. By
+induction on x ≥ 0, let us show that there exists a path of length at most x from α to β in D2(C). This clearly holds
+for x = 0 (i.e., α = β). Assume x > 0 and the result holds for x − 1. Let v ∈ V (C) be such that α(v) ̸= β(v).
+If v can be recoloured in β(v), then we recolour it and reach a new 2-dicolouring α′ such that diff(α′, β) = x−1
+and the result holds by induction. Else if v cannot be recoloured, then recolouring v must create a monochromatic
+directed cycle, which must be C. Then there must be a vertex v′, different from v, such that β(v) = α(v′) ̸= β(v′),
+and v′ can be recoloured. We recolour it and reach a new 2-dicolouring α′ such that diff(α′, β) = x − 1 and the
+result holds by induction.
+We are now ready to prove Theorem 8.
+Proof of Theorem 8. Let α and β be any two 2-dicolourings of ���G. We will show that there exists a redicolouring
+sequence of length at most n between α and β. We may assume that ⃗G is strongly connected, otherwise we
+consider each strongly connected component independently. This implies in particular that ⃗G does not contain any
+sink nor source. Let (X, Y ) be a partition of V (⃗G) such that, for every x ∈ X, d+(x) = 1, and for every y ∈ Y ,
+d−(y) = 1.
+Assume first that ⃗G⟨X⟩ contains a directed cycle C. Since every vertex in X has exactly one out-neighbour,
+there is no arc leaving C. Thus, since ⃗G is strongly connected, ⃗G must be exactly C, and the result holds by
+Lemma 12. Using a symmetric argument, we get the result when ⃗G⟨Y ⟩ contains a directed cycle.
+Assume now that both ⃗G⟨X⟩ and ⃗G⟨Y ⟩ are acyclic. Thus, since every vertex in X has exactly one out-
+neighbour, ⃗G⟨X⟩ is the union of disjoint and independent in-trees, that are oriented trees in which all arcs are
+directed towards the root. We denote by Xr the set of roots of these in-trees. Symmetrically, ⃗G⟨Y ⟩ is the union of
+disjoint and independent out-trees (oriented trees in which all arcs are directed away from the root), and we denote
+by Yr the set of roots of these out-trees. Set Xℓ = X \ Xr and Yℓ = Y \ Yr. Observe that the arcs from X to Y
+form a perfect matching directed from Xr to Yr. We denote by Mr this perfect matching. Observe also that there
+can be any arc from Y to X. Now we define X1
+r and Y 1
+r two subsets of Xr and Yr respectively, depending on the
+two 2-dicolourings α and β, as follows:
+X1
+r = {x | xy ∈ Mr, α(x) = β(y) ̸= α(y) = β(x)}
+Y 1
+r = {y | xy ∈ Mr, α(x) = β(y) ̸= α(y) = β(x)}
+Set X2
+r = Xr \ X1
+r and Y 2
+r = Yr \ Y 1
+r . We denote by M 1
+r (respectively M 2
+r ) the perfect matching from X1
+r to Y 1
+r
+(respectively from X2
+r to Y 2
+r ). Figure 1 shows a partitioning of V (⃗G) into X1
+r , X2
+r, Xℓ, Y 1
+r , Y 2
+r , Yℓ.
+Claim 8.1: There exists a redicolouring sequence of length sα from α to some 2-dicolouring α′ and a redicolouring
+sequence of length sβ from β to some 2-dicolouring β′ such that each of the following holds:
+(i) For any arc xy ∈ Mr, α′(x) ̸= α′(y) and β′(x) ̸= β′(y),
+(ii) For any arc xy ∈ M 2
+r , α′(x) = β′(x) (and so α′(y) = β′(y) by (i)), and
+(iii) sα + sβ ≤ |X2
+r| + |Y 2
+r |.
+6
+
+X1
+r
+X2
+r
+Xℓ
+Y 1
+r
+Y 2
+r
+Yℓ
+⃗G dicoloured with α
+X1
+r
+X2
+r
+Xℓ
+Y 1
+r
+Y 2
+r
+Yℓ
+⃗G dicoloured with β
+Figure 1: The partitioning of V (⃗G) into X1
+r, X2
+r , Xℓ, Y 1
+r , Y 2
+r , Yℓ.
+Proof of claim. We consider the arcs xy of M 2
+r one after another and do the following recolourings depending on
+the colours of x and y in both α and β to get α′ and β′.
+• If α(x) = α(y) = β(x) = β(y), then we recolour x in both α and β;
+• Else if α(x) = α(y) ̸= β(x) = β(y), then we recolour x in α and we recolour y in β;
+• Else if α(x) = β(x) ̸= α(y) = β(y), then we do nothing;
+• Else if α(x) ̸= α(y) = β(x) = β(y), then we recolour x in β;
+• Finally if α(y) ̸= α(x) = β(x) = β(y), then we recolour y in β.
+Each of these recolourings is valid because, when a vertex in X2
+r (respectively Y 2
+r ) is recoloured, it gets a colour
+different from its only out-neighbour (respectively in-neighbour). Let α′ and β′ be the the two resulting 2-
+dicolourings. By construction, α′ and β′ agree on X2
+r ∪ Y 2
+r . For each arc xy ∈ M 2
+r , either α(x) = α′(x) or
+α(y) = α′(y), and the same holds for β and β′. This implies that sα + sβ ≤ 2|M 2
+r | = |X2
+r | + |Y 2
+r |.
+♦
+Claim 8.2: There exists a redicolouring sequence from α′ to some 2-dicolouring ˜α of length s′
+α and a redicolouring
+sequence from β′ to some 2-dicolouring ˜β of length s′
+β such that each of the following holds:
+(i) ˜α and ˜β agree on V (⃗G) \ (X1
+r ∪ Y 1
+r ),
+(ii) α′ and ˜α agree on Xr ∪ Yr,
+(iii) β′ and ˜β agree on Xr ∪ Yr,
+(iv) Xℓ ∪ Yℓ is monochromatic in ˜α (and in ˜β by (i)), and
+(v) s′
+α + s′
+β ≤ |Xℓ| + |Yℓ|.
+Proof of claim. Observe that in both 2-dicolourings α′ and β′, we are free to recolour any vertex of Xℓ ∪ Yℓ since
+there is no monochromatic arc from X to Y and both ⃗G⟨X⟩ and ⃗G⟨Y ⟩ are acyclic. Let n1 (respectively n2) be the
+number of vertices in Xℓ ∪ Yℓ that are coloured 1 (respectively 2) in both α′ and β′. Without loss of generality,
+assume that n1 ≤ n2. Then we set each vertex of Xℓ ∪ Yℓ to colour 2 in both α′ and β′. Let ˜α and ˜β the resulting
+2-dicolouring. Then s′
+α + s′
+β is exactly |Xℓ| + |Yℓ| + n1 − n2 ≤ |Xℓ| + |Yℓ|.
+♦
+7
+
+Claim 8.3: There is a redicolouring sequence between ˜α and ˜β of length |X1
+r| + |Y 1
+r |.
+Proof of claim. By construction of ˜α and ˜β, we only have to exchange the colours of x and y for each arc xy ∈ M 1
+r .
+Without loss of generality, we may assume that the colour of all vertices in Xℓ ∪ Yℓ by ˜α and ˜β is 1.
+We first prove that, by construction, we can recolour any vertex of X1
+r ∪Y 1
+r from 1 to 2. Assume not, then there
+is such a vertex x ∈ X1
+r ∪Y 1
+r such that recolouring x from 1 to 2 creates a monochromatic directed cycle C. Since
+both ⃗G⟨X⟩ and ⃗G⟨Y ⟩ are acyclic, C must contain an arc of Mr. Since Mr does not contain any monochromatic
+arc in ˜α, then this arc must be incident to x. Now observe that colour 2, in ˜α, induces an independent set on both
+⃗G⟨X⟩ and ⃗G⟨Y ⟩. This implies that C must contain at least 2 arcs in Mr. This is a contradiction since recolouring
+x creates exactly one monochromatic arc in Mr.
+Then, for each arc xy ∈ M 1
+r , we can first recolour the vertex coloured 1 and then the vertex coloured 2.
+Note that we maintain the invariant that colour 2 induces an independent set on both ⃗G⟨X⟩ and ⃗G⟨Y ⟩. We get a
+redicolouring sequence from ˜α to ˜β in exactly 2|M 1
+r | = |X1
+r| + |Y 1
+r | steps.
+♦
+Combining the three claims, we finally proved that there exists a redicolouring sequence between α and β of length
+at most n.
+In the following, when α is a dicolouring of a digraph D, and H is a subdigraph of D, we denote by α|H the
+restriction of α to H. We will prove Corollary 9, let us restate it.
+Corollary 9. Let ⃗G be an oriented graph of order n with ∆min(⃗G) = ∆ ≥ 1, and let k ≥ ∆ + 1. Then Dk(⃗G) is
+connected and has diameter at most 2∆n.
+Proof. We will show the result by induction on ∆.
+Assume first that ∆ = 1, let k ≥ 2. Let α be any k-dicolouring of ⃗G and γ be any 2-dicolouring of ⃗G. To
+ensure that Dk(⃗G) is connected and has diameter at most 2n, it is sufficient to prove that there is a redicolouring
+sequence between α and γ of length at most n. Let H be the digraph induced by the set of vertices coloured 1 or
+2 in α, and let J be V (⃗G) \ V (H). By Theorem 8, since ∆min(H) ≤ ∆min(⃗G) ≤ 1, we know that there exists
+a redicolouring sequence, in H, from α|H to γ|H of length at most |V (H)|. This redicolouring sequence extends
+in ⃗G because it only uses colours 1 and 2. Let α′ be the obtained dicolouring of ⃗G. Since α′(v) = γ(v) for every
+v ∈ H, we can recolour each vertex in J to its colour in γ. This shows that there is a redicolouring sequence
+between α and γ of length at most |V (H)| + |J| = |V (⃗G)|. This ends the case ∆ = 1.
+Assume now that ∆ ≥ 2 and let k ≥ ∆ + 1. Let α and β be two k-dicolourings of ⃗G. By Corollary 7, we
+know that ⃗χ(⃗G) ≤ ∆ ≤ k − 1. We first show that there is a redicolouring sequence of length at most 2n from
+α to some (k − 1)-dicolouring γ of ⃗G. From α, whenever it is possible we recolour each vertex coloured 1, 2 or
+k with a colour of {3, . . . , k − 1} (when k = 3 we do nothing). Let ˜α be the obtained dicolouring, and let M be
+the set of vertices coloured in {3, . . . , k − 1} by ˜α (when k = 3, M is empty). We get that H = ⃗G − M satisfies
+∆min(H) ≤ 2, since every vertex in H has at least one in-neighbour and one out-neighbour coloured c for every
+c ∈ {3, . . ., k − 1}. By Corollary 7, there exists a 2-dicolouring γ|H of H. From ˜α|H, whenever it is possible,
+we recolour a vertex coloured 1 or 2 to colour k. Let ˆα be the resulting dicolouring, and ˆH be the subdigraph of
+H induced by the vertices coloured 1 or 2 in ˆα. We get that ∆min( ˆH) ≤ 1 since every vertex in ˆH has, in ⃗G, at
+least one in-neighbour and one out-neighbour coloured c for every c ∈ {3, . . ., k}. In at most |V ( ˆH)| steps, using
+Theorem 8, we can recolour the vertices of V ( ˆH) to their colour in γ|H (using only colours 1 and 2). Then we can
+recolour each vertex coloured k to its colour in γ|H. This results in a redicolouring sequence of length at most 2n
+from α to some (k − 1)-dicolouring γ of ⃗G , since colour k is not used in the resulting dicolouring (recall that M
+is coloured with {3, . . ., k − 1}).
+Now, from β, whenever it is possible we recolour each vertex to colour k. Let ˜β be the obtained k-dicolouring,
+and let N be the set of vertices coloured k in ˜β. We get that J = ⃗G − N satisfies ∆min(J) ≤ ∆ − 1. Thus,
+by induction, there exists a redicolouring sequence from ˜β|J to γ|J, in at most 2(∆ − 1)|V (J)| steps (using only
+colours {1, . . . , k − 1}). Since N is coloured k in ˜β, this extends to a redicolouring sequence in ⃗G. Now, since γ
+does not use colour k, we can recolour each vertex in N to its colour in γ. We finally get a redicolouring sequence
+from β to γ of length at most 2(∆ − 1)n. Concatenating the redicolouring sequence from α to γ and the one from
+γ to β, we get a redicolouring sequence from α to β in at most 2∆n steps.
+8
+
+4
+An analogue of Brook’s theorem for digraph redicolouring
+Let us restate Theorem 10.
+Theorem 10. Let D be a connected digraph with ∆max(D) = ∆ ≥ 3, k ≥ ∆ + 1, and α, β two k-dicolourings
+of D. Then at least one of the following holds:
+• α is k-frozen, or
+• β is k-frozen, or
+• there is a redicolouring sequence of length at most c∆|V |2 between α and β, where c∆ = O(∆2) is a
+constant depending only on ∆.
+An L-assignment of a digraph D is a function which associates to every vertex a list of colours. An L-
+dicolouring of D is a dicolouring α where, for every vertex v of D, α(v) ∈ L(v). An L-redicolouring sequence is
+a redicolouring sequence γ1, . . . , γr, such that for every i ∈ {1, . . . , r}, γi is an L-dicolouring of D.
+Lemma 13. Let D = (V, A) be a digraph and L be a list-assignment of D such that, for every vertex v ∈ V ,
+|L(v)| ≥ dmax(v) + 1. Let α be an L-dicolouring of D. If u ∈ V is blocked in α, then for each colour c ∈ L(u)
+different from α(u), u has exactly one out-neighbour u+
+c and one in-neighbour u−
+c coloured c. Moreover, if
+u+
+c ̸= u−
+c , there must be a monochromatic directed path from u+
+c to u−
+c . In particular, u is not incident to a
+monochromatic arc.
+Proof. Since u is blocked to its colour in α, for each colour c ∈ L(u) different from α(u), recolouring u to c must
+create a monochromatic directed cycle C. Let v be the out-neighbour of u in C and w be the in-neighbour of u in
+C. Then α(v) = α(w) = c, and there is a monochromatic directed path (in C) from v to w.
+This implies that, for each colour c ∈ L(u) different from α(u), u has at least one out-neighbour and at least
+one in-neighbour coloured c. Since |L(u)| ≥ dmax(u) + 1, then |L(u)| = dmax(u) + 1, and u must have exactly
+one out-neighbour and exactly one in-neighbour coloured c. In particular, u cannot be incident to a monochromatic
+arc.
+Lemma 14. Let D = (V, A) be a digraph such that for every vertex v ∈ V , N +(v) \ N −(v) ̸= ∅ and N −(v) \
+N +(v) ̸= ∅. Let L be a list assignment of D, such that for every vertex v ∈ V , |L(v)| ≥ dmax(v) + 1.
+Then for any pair of L-dicolourings α, β of D, there is an L-redicolouring sequence of length at most (|V | +
+3)|V |.
+Proof. Let x = diff(α, β) = |{v ∈ V | α(v) ̸= β(v)}|. We will show by induction on x that there is an L-
+redicolouring sequence from α to β of length at most (|V | + 3)x. The result clearly holds for x = 0 (i.e. α = β).
+Let v ∈ V be such that α(v) ̸= β(v). We denote α(v) by c and β(v) by c′. If v can be recoloured to c′, then we
+recolour it and we get the result by induction.
+Assume now that v cannot be recoloured to c′. Whenever v is contained in a directed cycle C of length at least
+3, such that every vertex of C but v is coloured c′, we do the following: we choose w a vertex of C different from
+v, such that β(w) ̸= c′. We know that such a w exists, for otherwise C would be a monochromatic directed cycle
+in β. Now, since w is incident to a monochromatic arc in C, and because |L(w)| ≥ dmax(w) + 1, by Lemma 13,
+we know that w can be recoloured to some colour different from c′. Thus we recolour w to this colour. Observe
+that it does not increase x.
+After repeating this process, maybe v cannot be recoloured to c′ because it is adjacent by a digon to some
+vertices coloured c′. We know that these vertices are not coloured c′ in β. Thus, whenever such a vertex can be
+recoloured, we recolour it. After this, let η be the obtained dicolouring. If v can be recoloured to c′ in η, we are
+done. Otherwise, there must be some vertices, blocked to colour c′ in η, adjacent to v by a digon. Let S be the set
+of such vertices. Observe that, by Lemma 13, for every vertex s ∈ S, c belongs to L(s), for otherwise s would not
+be blocked in η. We distinguish two cases, depending on the size of S.
+9
+
+• If |S| ≥ 2, then by Lemma 13, v can be recoloured to a colour c′′, different from both c and c′, because v is
+adjacent by a digon with two neighbours coloured c′. Hence we can successively recolour v to c′′, and every
+vertex of S to c . This does not create any monochromatic directed cycle because for each s ∈ S, since s is
+blocked in η, by Lemma 13 v must be the only neighbour of s coloured c in η.
+We can finally recolour v to c′.
+• If |S| = 1, let w be the only vertex in S. If v can be recoloured to any colour (different from c′ since w is
+coloured c′), then we first recolour v, allowing us to recolour w to c, because v is the single neighbour of w
+coloured c in η by Lemma 13. We finally can recolour v to c′.
+Assume then that v is blocked to colour c in η. Let us fix w+ ∈ N +(w) \ N −(w). Since w is blocked to c′
+in η, by Lemma 13, there exists exactly one vertex w− ∈ N −(w) \ N +(w) such that η(w+) = η(w−) = c′′
+and there must be a monochromatic directed path from w+ to w−.
+Since v is blocked to colour c in η, either vw− /∈ A or w+v /∈ A, otherwise, by Lemma 13, there must be
+a monochromatic directed path from w− to w+, which is blocking v to its colour. But since there is also a
+monochromatic directed path from w+ to w− (blocking w) there would be a monochromatic directed cycle,
+a contradiction (see Figure 2).
+w
+v
+w+
+w−
+Figure 2: The vertices v, w, w+ and w−.
+We distinguish the two possible cases:
+– if vw− /∈ A, then we start by recolouring w− with a colour that does not appear in its in-neighbourhood.
+This is possible because w− has a monochromatic entering arc, and because |L(w−)| ≥ dmax(w−)+1.
+We first recolour w with c′′, since c′′ does not appear in its in-neighbourhood anymore (w− was the
+only one by Lemma 13). Next we recolour v with c′: this is possible because v does not have any
+out-neighbour coloured c′ since w was the only one by Lemma 13 and w− is not an out-neighbour of
+v. We can finally recolour w to colour c and w− to c′′. After all these operations, we exchanged the
+colours of v and w.
+– if w+v /∈ A, then we use a symmetric argument.
+Observe that we found an L-redicolouring sequence from α to a α′, in at most |V |+3 steps, such that diff(α′, β) <
+diff(α, β). Thus by induction, we get an L-redicolouring sequence of length at most (|V | + 3)x between α and
+β.
+We are now able to prove Theorem 10. The idea of the proof is to divide the digraph D into two parts. One of
+them is bidirected and we will use Theorem 2 as a black box on it. In the other part, we know that each vertex is
+incident to at least two simple arcs, one leaving and one entering, and we will use Lemma 14 on it.
+Proof of Theorem 10. Let D = (V, A) be a connected digraph with ∆max(D) = ∆, k ≥ ∆ + 1. Let α and β be
+two k-dicolourings of D. Assume that neither α nor β is k-frozen.
+We first make a simple observation. For any simple arc xy ∈ A, we may assume that N +(y) \ N −(y) ̸= ∅
+and N −(x) \ N +(x) ̸= ∅. If this is not the case, then every directed cycle containing xy must contain a digon,
+implying that the k-dicolouring graph of D is also the k-dicolouring graph of D \ {xy}. Then we may look for a
+redicolouring sequence in D \ {xy}.
+10
+
+Let X = {v ∈ V | N +(v) = N −(v)} and Y = V \ X. Observe that D⟨X⟩ is bidirected, and thus the
+dicolourings of D⟨X⟩ are exactly the colourings of UG(D⟨X⟩). We first show that α|D⟨X⟩ and β|D⟨X⟩ are not
+frozen k-colourings of D⟨X⟩. If Y is empty, then D⟨X⟩ = D and α|D⟨X⟩ and β|D⟨X⟩ are not k-frozen by
+assumption. Otherwise, since D is connected, there exists x ∈ X such that, in D⟨X⟩, d+(x) = d−(x) ≤ ∆ − 1,
+implying that x is not blocked in any dicolouring of D⟨X⟩. Thus, by Theorem 2, there is a redicolouring sequence
+γ′
+1, . . . , γ′
+r in D⟨X⟩ from α|D⟨X⟩ to β|D⟨X⟩, where r ≤ c∆|X|2, and c∆ = O(∆) is a constant depending on ∆.
+We will show that, for each i ∈ {1, . . . , r − 1}, if γi is a k-dicolouring of D which agrees with γ′
+i on X, then
+there exist a k-dicolouring γi+1 of D that agrees with γ′
+i+1 on X and a redicolouring sequence from γi to γi+1 of
+length at most ∆ + 2.
+Observe that α agrees with γ′
+1 on X. Now assume that there is such a γi, which agrees with γ′
+i on X, and
+let vi ∈ X be the vertex for which γ′
+i(vi) ̸= γ′
+i+1(vi). We denote by c (respectively c′) the colour of vi in γ′
+i
+(respectively γ′
+i+1). If recolouring vi to c′ in γi is valid then we have the desired γi+1. Otherwise, we know that
+vi is adjacent with a digon (since vi is only adjacent to digons) to some vertices (at most ∆) coloured c′ in Y .
+Whenever such a vertex can be recoloured to a colour different from c′, we recolour it. Let ηi be the reached
+k-dicolouring after these operations. If vi can be recoloured to c′ in ηi we are done. If not, then the neighbours of
+vi coloured c′ in Y are blocked to colour c′ in ηi. We denote by S the set of these neighbours. We distinguish two
+cases:
+• If |S| ≥ 2, then by Lemma 13, vi can be recoloured to a colour c′′, different from both c and c′, because vi
+has two neighbours with the same colour. Then we successively recolour vi to c′′, and every vertex of S to
+c. This does not create any monochromatic directed cycle because, by Lemma 13, for each s ∈ S, vi is the
+only neighbour of s coloured c in ηi. We can finally recolour vi to c′ to reach the desired γi+1.
+• If |S| = 1, let y be the only vertex in S. Since y belongs to Y and is blocked to its colour in ηi, by Lemma 13,
+we know that y has an out-neighbour y+ ∈ N +(y)\N −(y) and an in-neighbour y− ∈ N −(y)\N +(y) such
+that there is a monochromatic directed path from y+ to y−. Observe that both y+ and y− are recolourable
+in ηi by Lemma 13, because there are incident to a monochromatic arc.
+– If vi is not adjacent to y+, then we recolour y+ to any possible colour, and we recolour y to ηi(y+).
+We can finally recolour vi to c′ to reach the desired γi+1.
+– If vi is not adjacent to y−, then we recolour y− to any possible colour, and we recolour y to ηi(y−).
+We can finally recolour vi to c′ to reach the desired γi+1.
+– Finally if vi is adjacent to both y+ and y−, since ηi(y+) = ηi(y−), then vi can be recoloured to a
+colour c′′ different from c and c′. This allows us to recolour y to c, and we finally can recolour vi to c′
+to reach the desired γi+1.
+We have shown that there is a redicolouring sequence of length at most (∆ + 2)c∆n2 from α to some α′ that
+agrees with β on X. Now we define the list-assignment: for each y ∈ Y ,
+L(y) = {1, . . . , k} \ {β(x) | x ∈ N(y) ∩ X}.
+Observe that, for every y ∈ Y ,
+|L(y)| ≥ k − |N +(y) ∩ X| ≥ ∆ + 1 − (∆ − d+
+Y (y)) ≥ d+
+Y (y) + 1.
+Symmetrically, we get |L(y)| ≥ d−
+Y (y) + 1. This implies, in D⟨Y ⟩, |L(y)| ≥ dmax(y) + 1. Note also that
+both α′
+|D⟨Y ⟩ and β|D⟨Y ⟩ are L-dicolourings of D⟨Y ⟩. Note finally that, for each y ∈ Y , N +(y) \ N −(y) ̸= ∅
+and N +(y) \ N −(y) ̸= ∅ by choice of X and Y and by the initial observation. By Lemma 14, there is an L-
+redicolouring sequence in D⟨Y ⟩ between α′
+|D⟨Y ⟩ and β|D⟨Y ⟩, with length at most (|Y | + 3)|Y |. By choice of L,
+this extends directly to a redicolouring sequence from α′ to β on D of the same length.
+The concatenation of the redicolouring sequence from α to α′ and the one from α′ to β leads to a redicolouring
+sequence from α to β of length at most c′
+∆|V |2, where c′
+∆ = O(∆2) is a constant depending on ∆.
+11
+
+Remark 15. If α is a k-frozen dicolouring of a digraph D, with k ≥ ∆max(D) + 1, then D must be bidirected.
+If D is not bidirected, then we choose v a vertex incident to a simple arc. If v cannot be recoloured in α, by
+Lemma 13, since v is incident to a simple arc, there exists a colour c for which v has an out-neighbour w and an
+in-neighbour u both coloured c, such that u ̸= w and there is a monochromatic directed path from w to u. But
+then, every vertex on this path is incident to a monochromatic arc, and it can be recoloured by Lemma 13. Thus, α
+is not k-frozen. This shows that an obstruction of Theorem 10 is exactly the bidirected graph of an obstruction of
+Theorem 2.
+5
+Further research
+In this paper, we established some analogues of Brooks’ Theorem for the dichromatic number of oriented graphs
+and for digraph redicolouring. Many open questions arise, we detail a few of them.
+Restricted to oriented graphs, Mcdiarmid and Mohar (see [11]) conjectured that the Directed Brooks’ Theorem
+can be improved to the following.
+Conjecture 16 (Mcdiarmid and Mohar). Every oriented graph ⃗G has ⃗χ(⃗G) = O
+�
+∆max
+log(∆max)
+�
+.
+Concerning digraph redicolouring, we believe that Corollary 9 and Theorem 10 can be improved. We pose the
+following two conjectures.
+Conjecture 17. There is an absolute constant c such that for every integer k and every oriented graph ⃗G such that
+k ≥ ∆min(⃗G) + 1, the diameter of Dk(⃗G) is bounded by cn.
+Conjecture 18. There is an absolute constant d such that for every integer k and every digraph D with k ≥
+∆max(D) + 1, the diameter of Dk(D) is bounded by dn2.
+Given an orientation ⃗G of a planar graph, a celebrated conjecture from Neumann-Lara [14] states that the
+dichromatic number of ⃗G is at most 2.
+It is known that it must be 4-mixing because planar graphs are 5-
+degenerate [5]. It is also known that there exists 2-freezable orientations of planar graphs [5]. Thus the following
+problem, stated in [5], remains open:
+Question 19. Is every oriented planar graph 3-mixing ?
+Acknowledgement
+I am grateful to Fr´ed´eric Havet and Nicolas Nisse for stimulating discussions.
+References
+[1] Pierre Aboulker and Guillaume Aubian.
+Four proofs of the directed Brooks’ Theorem.
+arXiv preprint
+arXiv:2109.01600, 2021.
+[2] Valentin Bartier. Combinatorial and Algorithmic aspects of Reconfiguration. PhD thesis, Universit´e Grenoble
+Alpes, 2021.
+[3] D. Bokal, G. Fijavz, M. Juvan, P.M. Kayll, and B. Mohar. The circular chromatic number of a digraph. J.
+Graph Theory, 46(3):227–240, 2004.
+[4] Marthe Bonamy, Matthew Johnson, Ioannis Lignos, Viresh Patel, and Daniel Paulusma. Reconfiguration
+graphs for vertex colourings of chordal and chordal bipartite graphs. Journal of Combinatorial Optimization,
+27(1):132–143, 2014.
+12
+
+[5] Nicolas Bousquet, Fr´ed´eric Havet, Nicolas Nisse, Lucas Picasarri-Arrieta, and Amadeus Reinald. Digraph
+redicolouring. arXiv preprint arXiv:2301.03417, 2023.
+[6] R. L. Brooks. On colouring the nodes of a network. Mathematical Proceedings of the Cambridge Philosoph-
+ical Society, 37(2):194–197, 1941.
+[7] Luis Cereceda, Jan Van den Heuvel, and Matthew Johnson. Mixing 3-colourings in bipartite graphs. Euro-
+pean Journal of Combinatorics, 30(7):1593–1606, 2009.
+[8] Luis Cereceda, Jan van den Heuvel, and Matthew Johnson. Finding paths between 3-colorings. Journal of
+Graph Theory, 67(1):69–82, 2011.
+[9] Carl Feghali, Matthew Johnson, and Dani¨el Paulusma. A reconfigurations analogue of Brooks’ Theorem and
+its consequences. Journal of Graph Theory, 83(4):340–358, 2016.
+[10] Ararat Harutyunyan and Bojan Mohar. Gallai’s theorem for list coloring of digraphs. SIAM Journal on
+Discrete Mathematics, 25(1):170–180, 2011.
+[11] Ararat Harutyunyan and Bojan Mohar. Strengthened Brooks' theorem for digraphs of girth at least three. The
+Electronic Journal of Combinatorics, 18(1), October 2011.
+[12] Jan van den Heuvel. The complexity of change, page 127–160. London Mathematical Society Lecture Note
+Series. Cambridge University Press, 2013.
+[13] Bojan Mohar. Eigenvalues and colorings of digraphs. Linear Algebra and its Applications, 432(9):2273–
+2277, 2010.
+[14] Victor Neumann-Lara. The dichromatic number of a digraph. J. Combin. Theory Ser. B., 33:265–270, 1982.
+[15] Naomi Nishimura. Introduction to reconfiguration. Algorithms, 11(4), 2018.
+13
+

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+A Multi-View Joint Learning Framework for Embedding Clinical Codes and Text
+Using Graph Neural Networks
+Lecheng Kong, Christopher King, Bradley Fritz, Yixin Chen
+Washington University in St. Louis
+One Brookings Drive
+St. Louis, Missouri 63130, USA
+{jerry.kong, christopherking, bafritz, ychen25}@wustl.edu
+Abstract
+Learning to represent free text is a core task in many clini-
+cal machine learning (ML) applications, as clinical text con-
+tains observations and plans not otherwise available for in-
+ference. State-of-the-art methods use large language models
+developed with immense computational resources and train-
+ing data; however, applying these models is challenging be-
+cause of the highly varying syntax and vocabulary in clinical
+free text. Structured information such as International Clas-
+sification of Disease (ICD) codes often succinctly abstracts
+the most important facts of a clinical encounter and yields
+good performance, but is often not as available as clinical text
+in real-world scenarios. We propose a multi-view learning
+framework that jointly learns from codes and text to com-
+bine the availability and forward-looking nature of text and
+better performance of ICD codes. The learned text embed-
+dings can be used as inputs to predictive algorithms indepen-
+dent of the ICD codes during inference. Our approach uses a
+Graph Neural Network (GNN) to process ICD codes, and Bi-
+LSTM to process text. We apply Deep Canonical Correlation
+Analysis (DCCA) to enforce the two views to learn a similar
+representation of each patient. In experiments using planned
+surgical procedure text, our model outperforms BERT models
+fine-tuned to clinical data, and in experiments using diverse
+text in MIMIC-III, our model is competitive to a fine-tuned
+BERT at a tiny fraction of its computational effort.
+We also find that the multi-view approach is beneficial for
+stabilizing inferences on codes that were unseen during train-
+ing, which is a real problem within highly detailed coding
+systems. We propose a labeling training scheme in which
+we block part of the training code during DCCA to improve
+the generalizability of the GNN to unseen codes. In experi-
+ments with unseen codes, the proposed scheme consistently
+achieves superior performance on code inference tasks.
+1
+Introduction
+An electronic health record (EHR) stores a patient’s com-
+prehensive information within a healthcare system. It pro-
+vides rich contexts for evaluating the patient’s status and fu-
+ture clinical plans. The information in an EHR can be clas-
+sified as structured or unstructured. Over the past decade,
+ML techniques have been widely applied to uncover pat-
+terns behind structured information such as lab results (Yu,
+Copyright © 2023, Association for the Advancement of Artificial
+Intelligence (www.aaai.org). All rights reserved.
+Beam, and Kohane 2018; Shickel et al. 2017; Goldstein et al.
+2017). Recently, the surge of deep learning and large-scale
+pre-trained networks has allowed unstructured data, mainly
+clinical notes, to be effectively used for learning (Huang, Al-
+tosaar, and Ranganath 2019; Lee et al. 2020; Si et al. 2019).
+However, most methods focus on either structured or un-
+structured data only.
+A particularly informative type of structured data is the
+International Classification of Diseases (ICD) codes. ICD is
+an expert-identified hierarchical medical concept ontology
+used to systematically organize medical concepts into cate-
+gories and encode valuable domain knowledge about a pa-
+tient’s diseases and procedures.
+Because ICD codes are highly specific and unambigu-
+ous, ML models that use ICD codes to predict procedure
+outcomes often yield more accurate results than those do
+not (Deschepper et al. 2019; Liu et al. 2020a). However,
+the availability of ICD codes is not always guaranteed. For
+example, billing ICD codes are generated after the clinical
+encounter, meaning that we cannot use the ICD codes to
+predict post-operative outcomes before the surgery. A more
+subtle but crucial drawback of using ICD codes is that there
+might be unseen codes during inference. When a future pro-
+cedure is associated with a code outside the trained subset,
+most existing models using procedure codes cannot accu-
+rately represent the case. Shifts in coding practices can also
+cause data during inference to not overlap the trained set.
+On the other hand, unstructured text data are readily and
+consistently available. Clinical notes are generated as free
+text and potentially carry a doctor’s complete insight about
+a patient’s condition, including possible but not known di-
+agnoses and planned procedures. Unfortunately, the clinical
+text is a challenging natural language source, containing am-
+biguous abbreviations, input errors, and words and phrases
+rarely seen in pre-training sources. It is consequently diffi-
+cult to train a robust model that predicts surgery outcomes
+from the large volume of free texts. Most current models
+rely on large-scale pre-trained models (Huang, Altosaar, and
+Ranganath 2019; Lee et al. 2020). Such methods require
+a considerable corpus of relevant texts to fine-tune, which
+might not be available at a particular facility. Hence, mod-
+els that only consider clinical texts suffer from poor perfor-
+mance and incur huge computation costs.
+To overcome the problems of models using only text or
+arXiv:2301.11608v1  [cs.CL]  27 Jan 2023
+
+codes, we propose to learn from the ICD codes and clini-
+cal text in a multi-view joint learning framework. We ob-
+serve that despite having different formats, the text and code
+data are complementary and broadly describe the same un-
+derlying facts about the patient. This enables each learner
+(view) to use the other view’s representation as a regulariza-
+tion function where less information is present. Under our
+framework, even when one view is missing, the other view
+can perform inference independently and maintain the effec-
+tive data representation learned from the different perspec-
+tives, which allows us to train reliable text models without a
+vast corpus and computation cost required by other text-only
+models.
+Specifically, we make the following contributions in this
+paper. (1) We propose a multi-view learning framework us-
+ing Deep Canonical Correlation Analysis (DCCA) for ICD
+codes and clinical notes. (2) We propose a novel tree-like
+structure to encode ICD codes by relational graph and ap-
+ply Relational Graph Convolution Network (RGCN) to em-
+bed ICD codes. (3) We use a two-stage Bi-LSTM to en-
+code lengthy clinical texts. (4) To solve the unseen code pre-
+diction problem, we propose a labeling training scheme in
+which we simulate unseen node prediction during training.
+Combined with the DCCA optimization process, the training
+scheme teaches the RGCN to discriminate between unseen
+and seen codes during inference and achieves better perfor-
+mance than plain RGCN.
+2
+Related Works
+Deep learning on clinical notes. Many works focus on
+applying deep learning to learn representations of clini-
+cal texts for downstream tasks. Early work (Boag et al.
+2018) compared the performance of classic NLP meth-
+ods including bag-of-words (Zhang, Jin, and Zhou 2010),
+Word2Vec (Mikolov et al. 2013), and Long-Short-Term-
+Memory (LSTM) (Hochreiter and Schmidhuber 1997) on
+clinical prediction tasks. These methods solely learn from
+the training text, but as the clinical texts are very noisy, they
+either tend to overfit the data or fail to uncover valuable pat-
+terns behind the text. Inspired by large-scale pre-trained lan-
+guage models such as BERT (Devlin et al. 2018), a series of
+works developed transformer models pre-trained on medical
+notes, including ClinicalBERT (Huang, Altosaar, and Ran-
+ganath 2019), BioBERT (Lee et al. 2020), and PubBERT
+(Alsentzer et al. 2019). These models fine-tune general lan-
+guage models on a large corpus of clinical texts and achieve
+superior performance. Despite the general nature of these
+models, the fine-tuning portion may not translate well to new
+settings. For example, PubBERT is trained on the clinical
+texts of a single tertiary hospital, and the colloquial terms
+used and procedures typically performed may not map to
+different hospitals. BioBERT is trained on Pubmed abstracts
+and articles, which also is likely poorly representative of the
+topics and terms used to, for example, describe a planned
+surgery.
+Some other models propose to use joint learning models
+to learn from the clinical text, and structured data (e.g., mea-
+sured blood pressure and procedure codes) (Wei et al. 2016;
+Zhang et al. 2020a). Since the structured data are less noisy,
+these models can produce better and more stable results.
+However, most assume the co-existence of text and struc-
+tured data at the inference time, while procedure codes for a
+patient are frequently incomplete until much later.
+Machine learning and procedure codes. Procedure
+codes are a handy resource for EHR data mining. Most
+works focus on automatic coding, using machine learning
+models to predict a patient’s diagnostic codes from clini-
+cal notes (Pascual, Luck, and Wattenhofer 2021; Li and Yu
+2020). Some other works directly use the billing code to pre-
+dict clinical outcomes (Liu et al. 2020a; Deschepper et al.
+2019), whereas our work focuses on using the high correla-
+tion of codes and text data to augment the performance of
+each. Most of these works exploit the code hierarchies by
+human-defined logic based on domain knowledge. In con-
+trast, our proposed framework uses GNN and can encode
+arbitrary relations between codes.
+Graph neural networks. A series of works (Xu et al.
+2018; Gilmer et al. 2017) summarize GNN structures in
+which each node iteratively aggregates neighbor nodes’ em-
+bedding and summarizes information in a neighborhood.
+The resulting node embeddings can be used to predict down-
+stream tasks. RGCN (Schlichtkrull et al. 2018) generalizes
+GNN to heterogeneous graphs where nodes and edges can
+have different types. Our model utilizes such heterogeneous
+properties on our proposed hierarchy graph encoding. Some
+works (Liu et al. 2020b; Choi et al. 2020) applied GNN to
+model interaction between EHRs, whereas our model uses
+GNN on the code hierarchy.
+Privileged information. Our approach is related to the
+Learning Under Privileged Information (LUPI) (Vapnik and
+Vashist 2009) paradigm, where the privileged information
+is only accessible during training (in this case, billing code
+data). Many works have applied LUPI to other fields like
+computer vision (Lambert, Sener, and Savarese 2018) and
+metric learning (Fouad et al. 2013).
+3
+Methods
+Admissions with ICD codes and clinical text can be repre-
+sented as D = {(C1, A1, y1), ..., (Cn, An, yn)}, where Ci
+is a set of ICD codes for admission i, Ai is a set of clin-
+ical texts, and yi is the desired task label (e.g. mortality,
+re-admission, etc.). The ultimate goal is to minimize task-
+appropriate losses L defined as:
+min
+fC,gC
+�
+i
+L(fC(gC(Ci)), yi)
+(1)
+and
+min
+fA,gA
+�
+i
+L(fA(gA(Ai)), yi),
+(2)
+where gC and gA embed codes and texts to vector repre-
+sentations respectively, and fC and fA map representations
+to the task labels. Note that (gC, fC) and (gA, fA) should
+operate independently during inference, meaning that even
+when one type of data is missing, we can still make accurate
+predictions.
+In this section, we first propose a novel ICD ontology
+graph encoding method and describe how we use Graph
+
+Figure 1: Overall multi-view joint learning framework.
+Blue boxes/arrows represent the text prediction pipeline, and
+green represents the code prediction pipeline. Dashed boxes
+and arrows denote processes only happening during training.
+By removing the dashed parts, text and code pipelines can
+predict tasks independently.
+Neural Network (GNN) to parameterize gC. We then de-
+scribe the two-stage Bi-LSTM (gA) to embed lengthy clini-
+cal texts. We then describe how to use DCCA on the repre-
+sentation from gC and gA to generate representations that are
+less noisy and more informative, so the downstream models
+fC and fA are able to make accurate predictions. Figure 1
+shows the overall architecture of our multi-view joint learn-
+ing framework.
+3.1
+ICD Ontology as Graphs
+The ICD ontology has a hierarchical scheme. We can rep-
+resent it as a tree graph as shown in Figure 2, where each
+node is a medical concept and a node’s children are finer di-
+visions of the concept. All top-level nodes are connected to a
+root node. In this tree graph, only the leaf nodes correspond
+to observable codes in the coding system, all other nodes are
+the hierarchy of the ontology. This representation is widely
+adopted by many machine learning systems (Zhang et al.
+2020b; Li, Ma, and Gao 2021) as a refinement of the earlier
+approach of grouping together all codes at the top level of the
+hierarchy. A tree graph is ideal for algorithms based on mes-
+sage passing. It allows pooling of information within disjoint
+groups, and encodes a compact set of neighbors. However,
+it (1) ignores the granularity of different levels of classifica-
+tion, and (2) cannot encode similarities of nodes that are dis-
+tant from each other. This latter point comes about because
+a tree system may split on factors that are not the most rele-
+Figure 2: Top: Conventional encoding of ICD ontology. Bot-
+tom Left: ICD ontology encoded with relations. Relation
+types for different levels are denoted by different colors.
+Bottom Right: Jump connection creates additional edges to
+leaf nodes’ predecessors, denoted by dashed color lines.
+vant for a given task, such as the same procedure in an arm
+versus a leg, or because cross-system concepts are empiri-
+cally very correlated in medical syndromes, such as kidney
+failure and certain endocrine disorders.
+To overcome the aforementioned problems, we propose
+to augment the tree graph with edge types and jump connec-
+tions. Unlike conventional tree graphs, where all edges have
+the same edge type, we use different edge types for connec-
+tions between different levels in the tree graph as shown in
+the bottom left of Figure 2. For example, ICD-10 codes have
+seven characters and hence eight levels in the graph (includ-
+ing the root level). The edges between the root node and its
+children have edge Type 1, and the edges between the sev-
+enth level and the last level (actual code level) have edge
+Type 7. Different edge types not only encode whether two
+procedures are related but also encode the level of similarity
+between codes.
+With multiple edge types introduced to the graph, we are
+able to further extend the graph structure by jump connec-
+tions. For each leaf node, we add one additional edge be-
+tween the node and each of its predecessors up to the root
+node, as shown in the bottom right of Figure 2. The edge
+type depends on the level that the predecessor resides. For
+example, in the ICD-10 tree graph, a leaf node will have
+seven additional connections to its predecessors. Its edge to
+the root node will have Type 8 (the first seven types are used
+to represent connections between levels), and its edge to the
+third level node will have Type 10. Jump connections signifi-
+cantly increase the connectivity of the graph. Meanwhile, we
+still maintain the good hierarchical information of the origi-
+
+Adm 1: Posterior Cervical Decompression.
+Adm 1: 0QSH06Z
+Adm 2: Thoracic Laminectomy for.
+Adm 2: 00CU0ZZ,009U3ZX,02HV33Z
+Adm 3: ...
+Adm 3: ...
+Word2Vec
+Code to Node Index
+Two-Stage LSTM
+RGCN
+1
+Codes Embedding
+Text Embedding
+Generated from
+Sum/Max Pooling
+Text Projection Matrix
+DCCA
+Code Projection Matrix
+Projected Text
+Projected Codes
+Embedding
+Embedding
+MLP
+MLP
+Downstream Task Predictior
+Downstream Task PredictionRoot node r connects
+to all level-1 ontology
+Bottom level nodes
+represent actual codes
+0
+8
+0RGAOT0
+ORGA0T1
+5A09357
+5A09358
+Relation-Augmented Graph
+Jump Connection Graphnal tree graph because the jump connections are represented
+by a different set of edge types. Using jump connection helps
+uncover relationships between codes that are not presented
+in the ontology. For example, the relationship between ane-
+mia and renal failure can be learned using jump connec-
+tion even though these diverge at the root node in ICD-9
+and ICD-10. Moreover, GNNs suffer from over-smoothing,
+where all node representations converge to the same value
+when the GNN has too many layers (Li, Han, and Wu 2018).
+If we do not employ jump connections, the maximal distance
+between one leaf node to another is twice the number of
+levels in the graph. To capture the connection between the
+nodes, we will need a GNN with that many layers, which
+is computationally expensive and prone to over-smoothing.
+Jump connections make the distance between two leaf nodes
+two, and this ensures that the GNN is able to embed any
+correlation between two nodes. We will discuss this in more
+detail in Section 3.2.
+3.2
+Embedding ICD Codes using GNN
+We use GNN to embed medical concepts in the ICD ontol-
+ogy. Let G = {V, E, R} be a graph, where V is its set of the
+vertex (medical concepts in the ICD graph), E ⊆ {V ×V } is
+its set of edges (connects medical concept to its sub-classes),
+and R is the set of edge type in the graph (edges in different
+levels and jump connection). As each ICD code corresponds
+to one node in the graph, we use code and node interchange-
+ably.
+We adopt RGCN (Schlichtkrull et al. 2018), which itera-
+tively updates a node’s embedding from its neighbor nodes.
+Specifically, the kth layer of RGCN on node u ∈ V is:
+h(k+1)
+u
+= σ
+�
+��
+r∈R
+�
+v∈N r
+u
+1
+cu,r
+W (k)
+r
+h(k)
+v
++ W (k)h(k)
+u
+�
+� (3)
+where N r
+i is the set of neighbors of i that connects to i by re-
+lation r, h(k)
+i
+is the embedding of node i after k GNN layers,
+h0
+i is a randomly initialized trainable embedding, W (k)
+r
+is a
+linear transformation on embeddings of nodes in N r
+i , W (k)
+updates the embedding of u, and σ is a nonlinear activation
+function. We have c = |N r
+i | as a normalization factor.
+After T iterations, h(T )
+u
+can be used to learn down-
+stream tasks. Since a patient can have a set of codes, Ci =
+{vi1, vi2, vi3, ...} ⊆ V , we use sum and max pooling to sum-
+marize Ci in an embedding function gC:
+gC(Ci) =
+�
+v∈Ci
+h(T )
+v
+⊕ max({h(T )
+v
+|v ∈ Ci}),
+(4)
+where max is the element-wise maximization, and ⊕ rep-
+resents vector concatenation. Summation more accurately
+summarizes the codes’ information, while maximization
+provides regularization and stability in DCCA, which we
+will discuss in Section 3.4.
+Training RGCN helps embed the ICD codes into vectors
+based on the defined ontology. Nodes that are close together
+in the graph will be assigned similar embeddings because
+of their similar neighborhood. Moreover, distant nodes that
+appear together frequently in the health record can also be
+assigned correlated embeddings because the jump connec-
+tion keeps the maximal distance between two nodes at two.
+Consider a set of codes C = {u, v}, because of the sum-
+mation in the code pooling, using a 2-layer RGCN, we will
+have non-zero gradients of hT
+u and hT
+v with respect to h0
+v
+and h0
+u, respectively, which connects the embeddings of u
+and v. In contrast, applying RGCN on a graph without jump
+connections will result in zero gradients when the distance
+between u and v is greater than two.
+3.3
+Embedding Clinical Notes using Bi-LSTM
+Patients can have different numbers of clinical texts in each
+encounter. Where applicable, we sort the texts in an en-
+counter in ascending order by time, and have a set of texts
+Ai = (ai1, ai2, ..., ain). In our examples, we concatenate
+the texts together to a single document Hi, and we have
+Hi = CAT(Ai) = �
+j={1...n} aij. We leave to future work
+the possibility of further modeling the collection.
+The concatenated text might be very lengthy with over
+ten thousands word tokens, and RNN suffers from dimin-
+ishing gradients with LSTM-type modifications. While at-
+tention mechanisms are effective for arbitrary long-range
+dependence, they require large sample size and expensive
+computational resources. Hence, following previously suc-
+cessful approach (Huang et al. 2019), we adopt a two-stage
+model which stacks a low-frequency RNN on a local RNN.
+Given Hi, we first split it into blocks of equal size b, Hi =
+{Hi1, Hi2, ..., HiK}. The last block HiK is padded to length
+b. The two-stage model first generates block-wise text em-
+beddings by
+lHik = LSTM({w(Hik1), w(Hik2), ..., w(Hikb)}),
+(5)
+where w(·) is a Word2Vec (Mikolov et al. 2013) trainable
+embedding function. The representation of Ai is given by
+gA(Ai) = LSTM({lHi1, ..., lHiK}).
+(6)
+The two-stage learning scheme minimizes the effect of di-
+minishing gradients while maintaining the temporal order of
+the text.
+3.4
+DCCA between Graph and Text Data
+As previously mentioned, ICD codes may not be available at
+the time when models would be most useful, but are struc-
+tured and easier to analyze, while the clinical text is read-
+ily available but very noisy. Despite different data formats,
+they usually describe the same information: the main diag-
+noses and treatments for an encounter. Borrowing ideas from
+multi-view learning, we can use them to supplement each
+other. Many existing multi-view learning methods require
+the presence of both views during inference and are not able
+to adapt to the applications we envision. Specifically, we use
+DCCA (Andrew et al. 2013; Wang et al. 2015) on gA(Ai)
+and gC(Ci) to learn a joint representation. DCCA solves the
+
+following optimization problem,
+max
+gC,gA,U,V
+1
+N tr(U T M T
+C MAV )
+s.t.
+U T ( 1
+N M T
+C MC + rCI)U = I,
+V T ( 1
+N M T
+AMA + rAI)V = I,
+uT
+i M T
+C MAvj = 0,
+∀i ̸= j,
+1 ≤ i, j ≤ L
+MC = stack{gC(Ci)|∀i},
+MA = stack{gA(Ai)|∀i},
+(7)
+where MC and MA are the matrices stacked by vector rep-
+resentations of codes and texts, (rC, rA) > 0 are regulariza-
+tion parameters. U = [u1, ..., uL] and V = [v1, ..., vL] maps
+GNN and Bi-LSTM output to maximally correlated embed-
+ding, and L is a hyper-parameter controlling the number of
+correlated dimensions. We use gC(Ci)U, gA(Ai)V as the fi-
+nal embedding of codes and texts. By maximizing their cor-
+relation, we force the weak learner (usually the LSTM) to
+learn a similar representation as the strong learner (usually
+the GNN) and to filter out inputs unrelated to the structured
+data. Hence, when a record’s codes can yield correct results,
+its text embedding is highly correlated with that of the codes,
+and the text should also be likely to produce correct predic-
+tions.
+During development, we found that a batch of ICD data
+often contains many repeated codes with the same embed-
+ding and that a SUM pooling tended to obtain a less than
+full rank embedding matrix MC and MA, which causes in-
+stability in solving the optimization problem. A nonlinear
+max pooling function helps prevent this.
+The above optimization problem suggests full-batch train-
+ing. However, the computation graph will be too large for
+the text and code data. Following (Wang et al. 2015), we use
+large mini-batches to train the model, and from the experi-
+mental results, they sufficiently represent the overall distri-
+bution. After training, we stack MC, MA again from all data
+output and obtain U and V as fixed projection matrix from
+equation (7).
+After obtaining the projection matrices and embedding
+models, we attach two MLPs (fA and fC) to the embedding
+models as the classifier, and train/fine-tune fA (fC) and gA
+(gC) together in an end-to-end fashion with respect to the
+learning task using the loss functions in (1) and (2).
+4
+Predicting Unseen Codes
+In the previous section, we discuss the formulation of ICD
+ontology and how we can use DCCA to generate embed-
+dings that share representations across views. In this section,
+we will demonstrate another use case for DCCA-regularized
+embeddings. In real-world settings, the set of codes that re-
+searchers observe in training is usually a small subset of the
+entire ICD ontology. In part, this is due to the extreme speci-
+ficity of some ontologies, with ICD-10-PCS having 87,000
+distinct procedures and ICD-10-CM 68,000 diagnostic pos-
+sibilities before considering that some codes represent a fur-
+ther modification of another entity. In even large training
+samples, some codes will likely be seen zero or a small num-
+ber of times in training. Traditional models using indepen-
+dent code embedding are expected to function poorly on rare
+codes and have arbitrary output on previously unseen nodes,
+even if similar entities are contained in the training data.
+Our proposed model and the graph-embedded hierarchy
+can naturally address the above challenge. Its two features
+enable predictions of novel codes at inference:
+• Relational embedding. By embedding the novel code
+in the ontology graph, we are able to exploit the repre-
+sentation of its neighbors. For example, a rare diagnostic
+procedure’s embedding is highly influenced by other pro-
+cedures that are nearby in the ontology.
+• Jump connection. While other methods also exploit
+the proximity defined by the hierarchy, as we suggested
+above, codes can be highly correlated but remain distant
+in the graph. Jump connections increase the graph con-
+nectivity; hence, our model can seek the whole hierarchy
+for potential connection to the missing code. Because the
+connections across different levels are assigned different
+relation types, our GNN can also differentiate the likeli-
+hood of connections across different levels and distances.
+Meanwhile, during inference, the potential problem is that
+the model does not automatically differentiate between the
+novel and the previously seen codes. Because the model
+never uses novel codes to generate any gC(Ci), the embed-
+dings of the seen and novel nodes experience different gra-
+dient update processes and hence are from different distri-
+butions. Nevertheless, during inference, the model will treat
+them as if they are from the same distribution. However,
+such transferability and credibility of novel node embed-
+dings are not guaranteed, and applying them homogeneously
+may result in untrustworthy predictions.
+Hence, we propose a labeling training scheme to teach
+the model how to handle novel nodes during inference. Let
+G = {V, E, R} be the ICD graph and U be the set of unique
+nodes in the training set, U ⊆ V . We select a random subset
+Us from U to form the seen nodes during training, and Uu =
+V \ Us be treated as unseen nodes. We augment the initial
+node embeddings with 1-0 labels, formally,
+h0+
+u
+= h0
+u ⊕ 1
+∀u ∈ Us
+h0+
+v
+= h0
+v ⊕ 0
+∀v ∈ V \ Us
+(8)
+Note that we still use h0
+u as the trainable node embedding,
+while the input to the RGCN is augmented to h0+
+u . We fur-
+ther extract data that only contain the seen nodes to form the
+seen data: Ds = {(Ci, Ai, yi)|c ∈ Us∀c ∈ Ci}.
+We, again, use DCCA on Ds to maximize the correlation
+between the text representation and the code representation.
+After obtaining the projection matrix, we train on the en-
+tire dataset D to minimize the prediction loss. Note that D
+contains nodes that do not appear in the DCCA process and
+are labeled differently from the seen nodes. The different la-
+bels allow the RGCN to tell whether a node is unseen during
+the DCCA process. If unseen nodes hurt the prediction, it
+will be reflected in the prediction loss. Intuitively, if unseen
+nodes are less credible, data with more 0-labeled nodes will
+
+Method
+Local Data
+MIMIC-III
+DEL
+DIA
+TH
+D30
+MORT
+R30
+Corr.
+17.3 ± 1.3
+16.8 ± 2.6
+16.8 ± 2.6
+16.8 ± 2.6
+10.4 ± 1.7
+12.7 ± 2.3
+BERT
+65.2 ± 0.6
+76.3 ± 1.2
+62.1 ± 1.1
+74.6 ± 1.8
+88.4 ± 1.8
+69.2 ± 1.9
+ClinicalBERT
+66.3 ± 0.5
+77.0 ± 0.9
+62.7 ± 0.8
+74.9 ± 1.5
+90.5 ± 1.3
+71.4 ± 1.8
+Bi-LSTM
+64.6 ± 0.2
+76.8 ± 1.8
+61.3 ± 1.2
+73.9 ± 1.9
+87.3 ± 1.7
+68.4 ± 2.6
+DCCA+Bi-LSTM
+66.9 ± 0.8
+78.9 ± 1.1
+61.6 ± 1.1
+76.5 ± 1.3
+87.2 ± 1.6
+71.1 ± 1.4
+RGCN
+76.4 ± 1.2
+97.2 ± 1.1
+75.9 ± 3.0
+91.5 ± 1.0
+90.4 ± 1.0
+68.6 ± 1.4
+DCCA+RGCN
+78.9 ± 1.3
+98.4 ± 0.9
+77.6 ± 1.2
+91.5 ± 1.3
+90.5 ± 1.5
+67.2 ± 2.5
+RGCN+Bi-LSTM
+79.5 ± 1.7
+97.1 ± 1.4
+75.6 ± 0.8
+90.8 ± 0.8
+91.3 ± 1.2
+69.5 ± 1.2
+DCCA+RGCN+Bi-LSTM
+78.7 ± 2.3
+98.2 ± 1.3
+77.1 ± 2.9
+91.0 ± 0.9
+90.1 ± 1.3
+71.2 ± 1.0
+Table 1: DCCA Joint Learning and baseline AUROC (%). Top 4 lines use clinical notes only during inference, middle 2 ICD
+codes only, and bottom 2 both. Corr = Sum of correlation of latent representations over 20 dimensions.
+Method
+Local Data
+MIMIC-III
+DEL
+DIA
+TH
+D30
+MORT
+R30
+RGCN
+74.6 ± 1.2
+87.3 ± 13.1
+67.4 ± 6.9
+82.8 ± 3.7
+84.5 ± 3.6
+60.4 ± 2.8
+RGCN+Labling
+73.2 ± 0.6
+87.4 ± 14.9
+68.5 ± 3.4
+83.8 ± 2.1
+85.7 ± 3.6
+61.3 ± 2.3
+DCCA+RGCN
+74.9 ± 1.0
+89.1 ± 12.5
+70.8 ± 0.9
+83.5 ± 1.9
+85.1 ± 4.1
+61.7 ± 2.6
+DCCA+RGCN+Labeling
+75.3 ± 1.1
+95.4 ± 0.7
+70.6 ± 3.2
+84.4 ± 1.4
+86.4 ± 4.2
+63.4 ± 2.8
+Table 2: Ablation Study of the Labeling Training Scheme under Unseen Code Setting in AUROC (%).
+have poor prediction results; GNN can capture this charac-
+teristic and reflect it in the prediction by assigning less pos-
+itive/negative scores to queries with more 0-labeled nodes.
+The labeling training scheme essentially blocks a part of the
+training code during DCCA and thus obtains embeddings
+for Us and Uu from different distributions. And we train on
+the entire training dataset so that the model learns to handle
+seen and unseen codes heterogeneously. This setup mimics
+the actual inference scenario. Note that despite being differ-
+ent, the distributions of seen and unseen node embeddings
+can be similar and overlapped. Thus, the additional 1-0 la-
+beling is necessary to differentiate them.
+5
+Experimental Results
+Datasets. We use two datasets to evaluate the performance
+of our framework: Proprietary Dataset. This dataset con-
+tains medical records of 38,551 admissions at the local Hos-
+pital from 2018 to 2021. Each entry is also associated with
+a free text procedural description and a set of ICD-10 pro-
+cedure codes. We aim to use our framework to predict a set
+of post-operative outcomes, including delirium (DEL), dial-
+ysis (DIA), troponin high (TH), and death in 30 days (D30).
+MIMIC-III dataset (Johnson et al. 2016). This dataset con-
+tains medical records of 58,976 unique ICU hospital ad-
+mission from 38,597 patients at the Beth Israel Deaconess
+Medical Center between 2001 and 2012. Each admission
+record is associated with a set of ICD-9 diagnoses codes
+and multiple clinical notes from different sources, includ-
+ing case management, consult, ECG, discharge summary,
+general nursing, etc. We aim to predict two outcomes from
+the codes and texts: (1) In-hospital mortality (MORT). We
+use admissions with hospital expire flag=1 in the MIMIC-
+III dataset as the positive data and sample the same number
+of negative data to form the final dataset. All clinical notes
+generated on the last day of admission are filtered out to
+avoid directly mentioning the outcome. We use all clinical
+notes ordered by time and take the first 2,500-word tokens
+as the input text. (2) 30-day readmission (R30). We follow
+(Huang, Altosaar, and Ranganath 2019), label admissions
+where a patient is readmitted within 30 days as positive, and
+sample an equal number of negative admissions. Newborn
+and death admissions are filtered out. We only use clinical
+notes of type Discharge Summary and take the first 2,500-
+word tokens as the input text. Sample sizes can be found in
+Table 3.
+Effectiveness of DCCA training. We split the dataset
+with a train/validation/test ratio of 8:1:1 and use 5-fold
+cross-validation to evaluate our model. GNN and Bi-LSTM
+are optimized in the DCCA process using the training set.
+The checkpoint model with the best validation correlation
+is picked to compute the projection matrix only from the
+training dataset. Then we attach an MLP head to the tar-
+get prediction model (either the GNN or the Bi-LSTM) and
+fine-tune the model in an end-to-end fashion to minimize the
+prediction loss.
+For this task, we compare our framework to popular pre-
+trained models ClinicalBERT and BERT. We also compare
+it to the base GNN and Bi-LSTM to show the effective-
+ness of our proposed framework. We additionally provide
+experimental results where both text and code embedding
+
+are used to make predictions. We compare our model with a
+vanilla multi-view model without DCCA. For all baselines,
+we report their Area Under Receiver Operating Characteris-
+tic (AUROC) as evaluation metrics, and Average Precision
+(AP) can be found in Appendix A. For all datasets, we set
+L, the number of correlated dimensions to 20, and report the
+total amount of correlation obtained (Corr).
+Table 1 shows the main results. For clinical notes predic-
+tion, we can see that the codes augmented model can con-
+sistently outperform the base Bi-LSTM, with an average rel-
+ative performance increase of 2.4% on the proprietary data
+and 1.6% on the MIMIC-III data. Our proposed method out-
+performs BERT on most tasks and achieves very competi-
+tive performance compared to that of ClinicalBERT. Note
+that our model only trains on the labeled EHR data without
+unsupervised training on extra data like BERT and Clini-
+calBERT do. ClinicalBERT has been previously trained and
+fine-tuned on the entire MIMIC dataset, including the dis-
+charge summaries, and therefore these results may overesti-
+mate its performance.
+For ICD code prediction, we see that DCCA brings a 1.5%
+performance increase on the proprietary data. Since the
+codes model significantly outperforms the language model
+on all tasks, the RGCN is a much stronger learner and has
+less information to learn from the text model. Comparing the
+results of the proprietary and the MIMIC datasets, we can
+see that DCCA brings a more significant performance boost
+to the proprietary dataset, presumably because of the larger
+amount of correlation obtained in the proprietary dataset
+(85% versus 58%). Moreover, an important difference in
+these datasets is the ontology used: MIMIC-III uses ICD-9
+and the proprietary dataset uses ICD-10. The ICD-9 ontol-
+ogy tree has a height of four, which is much smaller than that
+of ICD-10 and is more coarsely classified. This may also ex-
+plain the smaller performance gains in MIMIC-III.
+The combined model with DCCA only brings a slight per-
+formance boost compared to the one without because the
+amount of information for the models to learn is equiva-
+lent. Nevertheless, the DCCA model encourages the two
+views’ embeddings to agree and allows independent predic-
+tion. In contrast, a vanilla multi-view model does not help
+the weaker learner learn from the stronger learner.
+Unseen Codes Experiments. We identify the set of
+unique codes in the dataset. We split the codes into k-fold
+and ran k experiments on each split. For each experiment,
+we pick one fold as the unseen code set. Data that contain
+at least one unseen code are used as the evaluation set. The
+evaluation set is split into two halves as the valid and test
+sets. The rest of the data forms the training set. We pick an-
+other fold from the code split as the DCCA unseen code
+set. Training set data that do not contain any DCCA unseen
+code form the DCCA training set. Then, the entire training
+set is used for task fine-tuning. Because the distribution of
+codes is not uniform, the number of data for each split is not
+equal across different folds. We use k=10 for the proprietary
+dataset and k=20 for the MIMIC-III dataset to generate a
+reasonable data division. We provide average split sizes in
+Appendix C.
+For this task, we compare our method with the base GNN,
+# Admission
+# Pos. Samples
+# Unique codes
+DEL
+11,064
+5,367
+5,637
+DIA
+38,551
+1,387
+9,320
+TH
+38,551
+1,235
+9,320
+D30
+38,551
+1,444
+9,320
+MORT
+5,926
+2,963
+4,448
+R30
+10,998
+5,499
+3,645
+Table 3: Statistics of different datasets and tasks.
+base GNN augmented with the same labeling training strat-
+egy, and DCCA-optimized GNN to demonstrate the out-
+standing performance of our framework. Similarly, we re-
+port AUROC and include AP in Appendix A.
+Table 2 summarizes the results of the unseen codes exper-
+iments. Note that all test data contain at least one code that
+never appears in the training process. In such a more diffi-
+cult inference scenario, comparing the plain RGCN with the
+DCCA-augmented RGCN, we see a 2.2% average relative
+performance increase on the proprietary dataset. With the
+labeling learning method, we can further improve the per-
+formance gain to 4.2%. On the MIMIC-III dataset, the per-
+formance boost of our model over the plain RGCN is 3.6%,
+demonstrating our method’s ability to differentiate seen and
+unseen codes. We also notice that DCCA alone only slightly
+improves the performance on the MIMIC-III dataset (1.4%).
+We suspect that while the labeling training scheme helps dis-
+tinguish seen and unseen codes, the number of data used
+in the DCCA process is also reduced. As MORT and R30
+datasets are smaller and a small DCCA training set may not
+faithfully represent the actual data distribution, the regular-
+ization effect of DCCA diminishes.
+6
+Conclusions
+Predicting patient outcomes from EHR data is an essen-
+tial task in clinical ML. Conventional methods that solely
+learn from clinical texts suffer from poor performance, and
+those that learn from codes have limited application in real-
+world clinical settings. In this paper, we propose a multi-
+view framework that jointly learns from the clinical notes
+and ICD codes of EHR data using Bi-LSTM and GNN. We
+use DCCA to create shared information but maintain each
+view’s independence during inference. This allows accurate
+prediction using clinical notes when the ICD codes are miss-
+ing, which is commonly the case in pre-operative analysis.
+We also propose a label augmentation method for our frame-
+work, which allows the GNN model to make effective infer-
+ences on codes that are not seen during training, enhancing
+generalizability. Experiments are conducted on two different
+datasets. Our methods show consistent effectiveness across
+tasks. In the future, we plan to incorporate more data types in
+the EHR and combine them with other multi-view learning
+methods to make more accurate predictions.
+
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+
+A
+Average Precision Score Results
+AP results demonstrate a similar pattern to AUROC results,
+where DCCA augmented model can consistently outper-
+form the base model while achieving very competitive re-
+sults compared to ClinicalBERT for the text data as shown
+in Table 5. The proposed labeling training scheme can also
+consistently improve our model’s performance on the un-
+seen codes experiments, as shown in Table 6.
+B
+Hyperparameters
+We use grid search for hyperparameter tuning. For missing
+view experiments on text, we fix the number of RGCN layers
+to be 3. We use 32 for all hidden dimensions as we found that
+varying hidden size has minimal impact on the performance
+of the data. Text and Code represent the hyperparameters
+used for text and code inference tasks. Table 7 summarizes
+the set of hyperparameters used for tuning.
+C
+Unseen Code Sample Size
+We use 10-fold code split for the local data and 20-fold code
+split for the MIMIC-III data so that the split sizes are reason-
+able for training. We report the average number of samples
+for all tasks in Table 4.
+DCCA Train
+Full Train
+Test
+DEL
+3,458.9
+4,624.5
+3,219.4
+DIA
+19,305.4
+23,717.8
+7,416.3
+TH
+19,305.4
+23,717.8
+7,416.3
+D30
+19,305.4
+23,717.8
+7,416.3
+MORT
+4,603.1
+6,148.2
+2,424.7
+R30
+2,528.1
+3,264.0
+1,330.8
+Table 4: Average Split Size in Unseen Codes Experiment
+.
+D
+Data And Implementation
+We adopted the local dataset because it is the only dataset we
+have access to that uses both clinical free texts and ICD-10
+codes. The implementation details of the MIMIC-III dataset
+experiments can be found in the supplementary material
+(code).
+
+Local Data
+MIMIC-III
+DEL
+DIA
+TH
+D30
+MORT
+R30
+Corr.
+17.3 ± 1.3
+16.8 ± 2.6
+16.8 ± 2.6
+16.8 ± 2.6
+10.4 ± 1.7
+12.7 ± 2.3
+BERT
+65.4 ± 0.7
+23.6 ± 1.2
+6.5 ± 1.2
+15.1 ± 1.6
+85.9 ± 1.9
+67.7 ± 1.6
+ClinicalBERT
+66.0 ± 0.6
+23.5 ± 0.7
+7.0 ± 0.8
+15.8 ± 2.1
+88.6 ± 1.2
+70.1 ± 2.2
+LSTM
+64.4 ± 0.5
+22.1 ± 1.6
+6.3 ± 0.9
+14.3 ± 0.7
+85.4 ± 1.4
+66.2 ± 2.8
+DCCA+LSTM
+65.4 ± 0.4
+24.6 ± 0.8
+6.3 ± 1.4
+15.9 ± 1.0
+84.9 ± 1.6
+70.4 ± 2.1
+RGCN
+74.2 ± 1.7
+91.9 ± 2.1
+11.7 ± 0.3
+60.4 ± 4.8
+90.1 ± 1.7
+67.4 ± 2.2
+DCCA+RGCN
+76.6 ± 1.7
+90.6 ± 1.6
+14.8 ± 0.5
+60.8 ± 4.9
+90.2 ± 1.2
+68.4 ± 1.9
+RGCN+Bi-LSTM
+78.6 ± 2.6
+90.3 ± 1.6
+12.6 ± 0.7
+62.1 ± 1.5
+90.2 ± 1.4
+66.8 ± 1.5
+DCCA+RGCN+Bi-LSTM
+77.2 ± 2.1
+91.6 ± 2.0
+15.8 ± 0.6
+61.7 ± 1.2
+89.6 ± 1.7
+67.6 ± 1.6
+Table 5: Effect of DCCA Joint Learning Compared to Different Baselines in AP (%).
+Method
+Local Data
+MIMIC-III
+DEL
+DIA
+TH
+D30
+MORT
+R30
+RGCN
+72.6 ± 1.5
+82.1 ± 9.4
+9.2 ± 4.1
+53.4 ± 7.1
+87.6 ± 3.6
+63.7 ± 3.1
+RGCN+Labling
+73.2 ± 0.9
+83.6 ± 9.6
+9.1 ± 4.7
+54.2 ± 9.1
+88.5 ± 3.9
+65.4 ± 3.6
+DCCA+RGCN
+73.8 ± 1.2
+85.3 ± 8.1
+12.6 ± 1.3
+53.6 ± 8.2
+88.7 ± 3.0
+65.0 ± 2.9
+DCCA+RGCN+Labeling
+74.5 ± 1.1
+89.4 ± 1.3
+12.7 ± 3.0
+53.5 ± 6.9
+89.9 ± 3.1
+65.1 ± 3.4
+Table 6: Ablation Study of the Labeling Training Scheme under Unseen Code Setting in AP (%).
+Hyperparameter
+Local-Text
+Local-Code
+MIMIC-III-Text
+MIMIC-III-Code
+GNN
+#layers
+3
+{2,3,4}
+3
+{2,3,4}
+LSTM
+block size(b)
+-
+-
+30
+30
+MLP
+#layers
+2
+2
+1
+1
+dropout
+{0,0.2,0.4}
+{0,0.2,0.4}
+{0.2,0.4,0.6,0.8}
+{0.2,0.4,0.6,0.8}
+DCCA
+learning rate
+0.001
+0.001
+0.001
+0.001
+batch size
+1024
+1024
+400
+400
+Task
+learning rate
+{1e-3,1e-4,1e-5}
+{1e-3,1e-4,1e-5}
+{1e-3,1e-4,1e-5}
+{1e-3,1e-4,1e-5}
+batch size
+256
+256
+32
+32
+Table 7: Hyperparameters used for tuning.
+

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+arXiv:2301.02193v1  [physics.app-ph]  5 Jan 2023
+Universal scaling between wave speed and size
+enables nanoscale high-performance reservoir
+computing based on propagating spin-waves
+Satoshi Iihama,1,2 Yuya Koike,2,3,5 Shigemi Mizukami,2,4 Natsuhiko Yoshinaga2,5∗
+1Frontier Research Institute for Interdisciplinary Sciences (FRIS), Tohoku University,
+Sendai, 980-8578, Japan
+2WPI Advanced Institute for Materials Research (AIMR), Tohoku University,
+Katahira 2-1-1, Sendai, 980-8577, Japan
+3Department of Applied Physics, Tohoku University,
+Sendai, 980-8579, Japan
+4Center for Science and Innovation in Spintronics (CSIS), Tohoku University,
+Sendai, 980-8577, Japan
+5MathAM-OIL, AIST, Sendai, 980-8577, Japan
+∗To whom correspondence should be addressed; E-mail: [email protected].
+Neuromorphic computing using spin waves is promising for high-speed nanoscale
+devices, but the realization of high performance has not yet been achieved.
+Here we show, using micromagnetic simulations and simplified theory with
+response functions, that spin-wave physical reservoir computing can achieve
+miniaturization down to nanoscales keeping high computational power com-
+parable with other state-of-art systems. We also show the scaling of system
+sizes with the propagation speed of spin waves plays a key role to achieve high
+performance at nanoscales.
+1
+
+Introduction
+Non-local magnetization dynamics in a nanomagnet, spin-waves, can be used for processing in-
+formation in an energy-efficient manner since spin-waves carry information in a magnetic ma-
+terial without Ohmic losses (1). The wavelength of the spin-wave can be down to the nanometer
+scale, and the spin-wave frequency becomes several GHz to THz frequency, which are promis-
+ing properties for nanoscale and high-speed operation devices. Recently, neuromorphic com-
+puting using spintronics technology has attracted great attention for the development of future
+low-power consumption artificial intelligence (2). Spin-waves can be created by various means
+such as magnetic field, spin-transfer torque, spin-orbit torque, voltage induced change in mag-
+netic anisotropy and can be detected by the magnetoresistance effect (3). Therefore, neuromor-
+phic computing using spin waves may have a potential of realisable devices.
+Reservoir computing (RC) is a promising neuromorphic computation framework. RC is a
+variant of recurrent neural networks (RNNs) and has a single layer, referred to as a reservoir, to
+transform an input signal into an output (4). In contrast with the conventional RNNs, RC does
+not update the weights in the reservoir. Therefore, by replacing the reservoir of an artificial
+neural network with a physical system, for example, magnetization dynamics, we may realize a
+neural network device to perform various tasks, such as time-series prediction (4,5), short-term
+memory (6, 7), pattern recognition, and pattern generation. Several physical RC has been pro-
+posed: spintronic oscillators (8,9), optics (10), photonics (11,12), fluids, soft robots, and others
+(see reviews (13–15)). Among these systems, spintronic RC has the advantage in its potential
+realization of nanoscale devices at high speed of GHz frequency with low power consumption,
+which may outperform conventional electric computers in future. So far, spintronic RC has
+been considered using spin-torque oscillators (8, 9), magnetic skyrmion (16), and spin waves
+in garnet thin films (17–19). However, the current performance of spintronic RC still remains
+2
+
+poor compared with the Echo State Network (ESN) (6, 7), idealized RC systems. The biggest
+issue is a lack of our understanding of how to achieve high performance in the RC systems.
+To achieve high performance, the reservoir has to have a large degree of freedom, N. How-
+ever, in practice, it is difficult to increase the number of physical nodes, Np, because it requires
+more wiring of multiple inputs. In this respect, wave-based computation in continuum media
+has attracting features. The dynamics in the continuum media have large, possibly infinite, de-
+grees of freedom. In fact, several wave-based computations have been proposed (20, 21). The
+challenge is to use the advantages of both wave-based computation and RC to achieve high-
+performance computing of time-series data. For spin wave-based RC, so far, the large degrees
+of freedom are extracted only by using a large number of input and/or output nodes (19, 22).
+Here, to propose a realisable spin wave RC, we use an alternative route; we extract the informa-
+tion from the continuum media using a small number of physical nodes.
+Along this direction, using Nv virtual nodes for the dynamics with delay was proposed to
+increase N in (23). This idea was applied in optical fibres with a long delay line (11) and a net-
+work of oscillators with delay (24). Nevertheless, the mechanism of high performance remains
+elusive, and no unified understanding has been made. The increase of N = NpNv with Nv does
+not necessarily improve performance. In fact, RC based-on STO struggles with insufficient per-
+formance both in experiments (9) and simulations (25). The photonic RC requires a large size
+of devices due to the long delay line (11,12).
+In this work, we show nanoscale and high-speed RC based on spin wave propagation with a
+small number of inputs can achieve performance comparable with the ESN and other state-of-art
+RC systems. More importantly, by using a simple theoretical model, we clarify the mechanism
+of the high performance of spin wave RC. We show the scaling between wave speed and system
+size to make virtual nodes effective.
+3
+
+Results
+Reservoir computing using wave propagation
+The basic task of RC is to transform an input signal Un to an output Yn for the discrete step n =
+1, 2, . . . , T at the time tn. For example, for speech recognition, the input is an acoustic wave,
+and the output is a word corresponding to the sound. Each word is determined not only by the
+instantaneous input but also by the past history. Therefore, the output is, in general, a function
+of all the past input, Yn = g ({Um}n
+m=1) as in Fig. 1(a). The RC can also be used for time-series
+prediction by setting the output as Yn = Un+1 (4). In this case, the state at the next time step
+is predicted from all the past data; namely, the effect of delay is included. The performance of
+the input-output transformation g can be characterized by how much past information does g
+have, and how much nonlinear transformation does g perform. We will discuss that the former
+is expressed by memory capacity (MC) (26), whereas the latter is measured by information
+processing capacity(IPC) (27).
+We propose physical computing based on a propagating wave (see Fig. 1(b,c)). Time series
+of an input signal Un can be transformed into an output signal Yn (Fig. 1(a)). As we will discuss
+below, this transformation requires large linear and nonlinear memories; for example, to predict
+Yn, we need to memorize the information of Un−2 and Un−1. The input signal is injected in
+the first input node and propagates in the device to the output node spending a time τ1 as in
+Fig. 1(b). Then, the output may have past information at tn − τ1 corresponding to the step
+n − m1. The output may receive the information from another input at different time tn − τ2.
+The sum of the two peices of information is mixed and transformed as Un−m1Un−m2 either by
+nonlinear readout or by nonlinear dynamics of the reservoir (see also Sec. B in Supplementary
+Information). We will demonstrate the wave propagation can indeed enhances memory capacity
+and learning performance of the input-output relationship.
+4
+
+Figure 1:
+Illustration of physical reservoir computing and reservoir based on propa-
+gating spin-wave network. (a) Schematic illustration of output function prediction by using
+time-series data. Output signal Y is transformed by past information of input signal U. (b)
+Schematic illustration of reservoir computing with multiple physical nodes. The output signal
+at physical node A contains past input signals in other physical nodes, which are memorized by
+the reservoir. (c) Schematic illustration of reservoir computing based on propagating spin-wave.
+Propagating spin-wave in ferromagnetic thin film (m ∥ ez) is excited by spin-transfer torque at
+multiple physical nodes with reference magnetic layer (m ∥ ex). x-component of magnetization
+is detected by the magnetoresistance effect at each physical node.
+5
+
+a
+(b)
+Y(t) α U(t - T1) · U(t - t2)
+output
+input
+g((U(t)))
+U(t - T1)
+X(t) α U(t - T1) + U(t - T2)
+input
+reservoir
+U(t - T2)
+Spin injector and detector
+(c)
+Ferromagnetic thin filmBefore explaining our learning strategy, we discuss how to achieve accurate learning of the
+input-output relationship Yn = g ({Um}n
+m=1) from the data. Here, the output may be dependent
+on a whole sequence of the input {Um}n
+m=1 = (U1, . . . , Un). Even when both Un and Yn are
+one-variable time-series data, the input-output relationship g(·) may be T-variable polynomials,
+where T is the length of the time series. Formally, g(·) can be expanded in a polynomial
+series (Volterra series) such that g ({Um}n
+m=1) = �
+k1,k2,··· ,kt βk1,k2,··· ,ktUk1
+1 Uk2
+2 · · · Ukn
+n with the
+coefficients βk1,k2,··· ,kn. Therefore, even for the linear input-output relationship, we need T
+coefficients in g(·), and as the degree of powers in the polynomials increases, the number of
+the coefficients increases exponentially. This observation implies that a large number of data is
+required to estimate the input-output relationship. Nevertheless, we may expect a dimensional
+reduction of g(·) due to its possible dependence on the time close to t and on the lower powers.
+Still, our physical computers should have degrees of freedom N ≫ 1, if not exponentially large.
+The reservoir computing framework is used to handle time-series data of the input U and the
+output Y (6). In this framework, the input-output relationship is learned through the reservoir
+dynamics X(t), which in our case, is magnetization at the detectors. The reservoir state at a
+time tn is driven by the input at the nth step corresponding to tn as
+X(tn+1) = f (X(tn), Un)
+(1)
+with nonlinear (or possibly linear) function f(·). The output is approximated by the readout
+operator ψ(·) as
+ˆYn = ψ (X(tn)) .
+(2)
+Our study uses the nonlinear readout ψ (X(t)) = W1X(t) + W2X2(t) (5, 28). The weight
+matrices W1 and W2 are estimated from the data of the reservoir dynamics X(t) and the true
+output Yn, where X(t) is obtained by (1). With the nonlinear readout, the RC with linear
+6
+
+dynamics can achieve nonlinear transformation, as Fig.1(b). We stress that the system also
+works with linear readout when the RC has nonlinear dynamics. We discuss this case in Sec.B.
+Spin wave reservoir computing
+We consider a magnetic device of a thin rectangular system with cylindrical injectors (see
+Fig.1(c)). The size of the device is L × L × D. Under the uniform external magnetic field,
+the magnetization is along the z direction. Electric current is injected at the Np injectors with
+the radius a and the same height with the device. The spin-torque by the current drives mag-
+netization m(x, t) and propagating spin-waves as schematically shown in Fig.1(c). The actual
+demonstration of the spin-wave reservoir computing is shown in Fig. 2. We demonstrate the
+spin-wave RC using two methods: the micromagnetic simulations and the theoretical model
+using a response function.
+In the micromagnetic simulations, we analyze the Landau-Lifshitz-Gilbert (LLG) equation
+with the effective magnetic field Heff = Hext+Hdemag+Hexch consists of the external field, de-
+magnetization, and the exchange interaction (see Theoretical analysis using response function
+in Methods). The spin waves are driven by Slonczewski spin-transfer torque (29). The driving
+term is proportional to the DC current j(t) at the nanocontact. We inject the DC current propor-
+tional to the input time series U with a pre-processing filter. From the resulting spatially inho-
+mogeneous magnetization m(x, t), we measure the averaged magnetization at ith nanocontact
+mi(t). We use the method of time multiplexing with Nv virtual nodes (23). We choose the x-
+component of magnetization mx,i as a reservoir state, namely, Xn = {mx,i(tn,k)}i∈[1,Np],k∈[1,Nv]
+(see (14) in Methods for its concrete form). For the output transformation, we use ψ(mi,x) =
+W1,imi,x + W2,im2
+i,x. Therefore, the dimension of our reservoir is 2NpNv. The nonlinear output
+transformation can enhance the nonlinear transformation in reservoir (5), and it was shown that
+even under the linear reservoir dynamics, RC can learn any nonlinearity (28, 30). In Sec. B in
+7
+
+0
+0.5
+0
+0.5
+0
+1.6
+3.2
+4.8
+6.4
+8
+0
+0.5
+・
+・
+・
+�i
+n
+n+1
+n+2
+n+3
+n+4
+0
+0.5
+Time step
+higher damping region
+cylindrical region to apply 
+spin-transfer torque
+(a)
+(b)
+Training
+�
+Input, �
+Time
+Binary mask, �i 
+�n,�
+�n
+��
+�n
+��
+�n��
+�
+��
+Time (ns)
+・
+・
+・
+Masked input, 
+Time step
+Input, �
+Time step
+Output, �
+Time step
+� �
+Time step
+Time
+�
+0
+�
+�n
+�n+1
+�n,3
+�n,5
+�n,7
+(t)
+(t)
+Figure 2:
+Dimension of spin-wave reservoir and prediction of NARMA10 task.
+(a)
+Input signals U are multiplied by binary mask Bi(t) and transformed into injected current
+j(t) = 2jc ˜Ui(t) for the ith physical node. Current is injected into each physical node with
+the cylindrical region to apply spin-transfer torque and to excite spin-wave. Higher damping
+regions in the edges of the rectangle are set to avoid reflection of spin-waves. (b) Prediction of
+NARMA10 task. x-component of magnetization at each physical and virtual node are collected
+and output weights are trained by linear regression.
+8
+
+0.02
+0.01
+0
+-0.01
+Node (1, 1)
+-0.02
+0.02
+0.01
+0
+-0.01
+(2,1)
+-0.02
+0.02
+0.01
+0
+-0.01
+(Ny,Np
+-0.02
+1000
+2000
+3000
+4000
+5000
+6000- OQutput,
+Predicted
+0.8
+0.6
+0.4
+0.2
+Error
+0
+6000
+7000
+8000
+9000
+10000
+110000.8
+0.6
+0.4
+0.2
+0
+1000
+2000
+3000
+4000
+5000
+60000.5
+1000
+2000
+3000
+4000
+5000
+60001000nm
+1000nm
+Q
+500nm
+4nmSupplementary Information, we also discuss the linear readout, but including the z-component
+of magnetization X = (mx, mz). In this case, mz plays a similar role to m2
+x. The performance
+of the RC is measured by three tasks: MC, IPC, and NARMA10. The weights in the readout
+are trained by reservoir variable X and the output Y (Fig.2(b), see also Methods).
+To understand the mechanism of high performance of learning by spin wave propagation,
+we also consider a simplified model using the response function of the spin wave dynamics. By
+linearizing the magnetization around m = (0, 0, 1) without inputs, we may express the linear
+response of the magnetization at the ith readout mi = mx,i + imy,i to the input as(see Methods)
+mi(t) =
+Np
+�
+j=1
+�
+dt′Gij(t, t′)U(j)(t′).
+(3)
+Here, U(j)(t) is the input time series at jth nanocontact. The response function has a self part
+Gii, that is, input and readout nanocontacts are the same, and the propagation part Gij, where
+the distance between the input and readout nanocontacts is |Ri − Rj|. We use the quadratic
+nonlinear readout, which has a structure
+m2
+i (t) =
+Np
+�
+j1=1
+Np
+�
+j2=1
+�
+dt1
+�
+dt2G(2)
+ij1j2(t, t1, t2)U(j1)(t1)U(j2)(t2).
+(4)
+The response function of the nonlinear readout is G(2)
+ij1j2(t, t1, t2) ∝ Gij1(t, t1)Gij2(t, t2). The
+same structure as (4) appears when we use a second-order perturbation for the input (see Meth-
+ods). In general, we may include the cubic and higher-order terms of the input. This expansion
+leads to the Volterra series of the output in terms of the input time series, and suggests how the
+spin wave RC works (see Sec. A.1 in Supplementary Information for more details). Once the
+magnetization at each nanocontact is computed, we may estimate MC and IPC.
+Figure 3 shows the results of the three tasks. When the time scale of the virtual node θ is
+small and the damping is small, the performance of spin wave RC is high. As Fig. 3(a) shows,
+we achieve MC ≈ 60 and IPC ≈ 60. Accordingly, we achieve a small error in the NARMA10
+9
+
+task, NRMSE ≈ 0.2 (Fig. 3(c)). Theses performances are comparable with state-of-the-art ESN
+with the number of nodes ∼ 100. When the damping is stronger, both MC and IPC become
+smaller. Because the NARMA10 task requires the memory with the delay steps ≈ 10 and the
+second order nonlinearity with the delay steps ≈ 10 (see Sec.A in Supplementary Information),
+the NRMSE becomes larger when MC ≲ 10 and IPC ≲ 102/2.
+The results of the micromagnetic simulations are semi-quantitatively reproduced by the the-
+oretical model using the response function, as shown in Fig. 3(b). This result suggests that
+the linear response function G(t, t′) captures the essential feature of delay t − t′ due to wave
+propagation.
+(a)
+(b)
+Non-linear, IPC
+Linear, MC
+0
+20
+40
+60
+80
+5
+2.5
+α = 5×10-4
+Frequency, 1/θ (GHz)
+0
+20
+40
+60
+80
+α = 5×10-3
+Linear and non-linear memory capacity
+0.2
+0.4
+0
+20
+40
+60
+80
+α = 5×10-2
+Distance of virtual nodes, θ (ns)
+(c)
+Normalized root mean square error, NRMSE
+for NARMA10 task
+0
+0.5
+1
+5
+2.5
+α = 5×10-4
+ Training, 
+ Test
+Frequency, 1/θ (GHz)
+0
+0.5
+1
+α = 5×10-3
+ 
+0.2
+0.4
+0
+0.5
+1
+α = 5×10-2
+Distance of virtual nodes, θ (ns)
+0
+20
+40
+60
+80
+5
+2.5
+α = 5×10-4
+Frequency, 1/θ (GHz)
+0
+20
+40
+60
+80
+α = 5×10-3
+Linear and non-linear memory capacity
+0.2
+0.4
+0
+20
+40
+60
+80
+α = 5×10-2
+Distance of virtual nodes, θ (ns)
+Figure 3: Effect of virtual node distance on performance of spin-wave reservoir computing
+obtained with 8 physical nodes and 8 virtual nodes. Memory capacity MC and information
+processing capacity IPC obtained by (a) micromagnetics simulation and (b) response function
+method plotted as a function of virtual node distance θ with different damping parameters α.
+(c) Normalized root mean square error, NRMSE for NARMA10 task is plotted as a function of
+θ with different α.
+To confirm the high MC and IPC are due to spin-wave propagation, we perform micromag-
+netic simulations with damping layers between nodes (Fig. 4(a)). The damping layers inhibit
+spin wave propagation. The result of Fig. 4(b) shows that the memory capacity is substantially
+lower than that without damping, particularly when θ is small. The NARMA10 task shows a
+10
+
+larger error (Fig. 4(d)). When θ is small, the suppression is less effective. This may be due to
+incomplete suppression of wave propagation.
+We also analyze the theoretical model with the response function by neglecting the inter-
+action between two physical nodes, namely, Gij = 0 for i ̸= j. In this case, information
+transmission between two physical nodes is not allowed. We obtain smaller MC and IPC than
+the system with wave propagation, supporting our claim (see (Fig. 4(c))).
+Our spin wave RC also works for the prediction of time-series data. In the study of (5),
+the functional relationship between the state at t + ∆t and the states before t is learned by the
+ESN. The trained ESN can estimate the state at t + ∆t from the past states, and therefore, it can
+predict the dynamics without the data. In (5), the prediction for the chaotic time-series data was
+demonstrated. Figure 5 shows the prediction using our spin wave RC for the Lorenz model. We
+can demonstrate that the RC shows short-time prediction and, more importantly, reconstruct the
+chaotic attractor.
+Scaling of system size and wave speed
+To clarify the mechanism of the high performance of our spin wave RC, we investigate MC
+and IPC of the system with different characteristic length scales L and different wave propagat-
+ing speed v. The characteristic length scale is controlled by the radius of the circle on which
+inputs are located (see Fig. 2(a)). We use our theoretical model with the response function to
+compute MC and IPC in the parameter space (v, R). This calculation can be done because the
+computational cost of our model is much cheaper than numerical micromagnetic simulations.
+Figure 6(a,b) shows that both MC and IPC have maximum when L ∝ v. To obtain a deeper
+understanding of the result, we perform the same analyzes for the further simplified model, in
+11
+
+(�
+�
+(c)
+(d)
+Linear, MC
+Non-linear, IPC
+0
+20
+40
+60
+80
+5
+2.5
+α = 5 × 10-4
+C �
+ 
 
+ 
+ 
+Frequency, 1/θ (GHz)
+0.2
+0.4
+0
+20
+40
+60
+80
+No connection
+Linear and non-linear memory capacity
+Distance of virtual nodes, θ (ns)
+Normalized root mean square error, NRMSE
+for NARMA10 task
+0
+0.5
+1
+5
+2.5
+   
+!"#$ %&'
+ Training, 
+ Test
+Frequency, 1/θ (GHz)
+0.2
+0.4
+0
+0.5
+1
+ 
+No-connection
+Distance of virtual nodes, θ (ns)
+(a)
+)*+
+-./ 123
+4
+56789 :
+No-connection
+0
+20
+40
+60
+80
+5
+2.5
+α = 5 × 10-4
+; <= >?@ ABD
+EFG
+H
+I JK
+Frequency, 1/θ (GHz)
+0.2
+0.4
+0
+20
+40
+60
+80
+No connection (G
+iL = 0 )
+Linear and non-linear memory capacity
+Distance of virtual nodes, θ (ns)
+Figure 4:
+Effect of the network connection on the performance of reservoir computing.
+(a) Schematic illustration of the network of physical nodes connected through propagating spin-
+wave [left] and physical nodes with no connection [right]. Memory capacity MC and informa-
+tion processing capacity IPC obtained using a connected network with 8 physical nodes [top]
+and physical nodes with no connection [bottom] calculated by (a) micromagnetics simulation
+and (b) response function method plotted as a function of virtual node distance θ. 8 virtual
+nodes are used. (c) Normalized root mean square error, NRMSE for NARMA10 task obtained
+by micromagnetics simulation is plotted as a function of θ with a connected network [top] and
+physical nodes with no connection [bottom].
+12
+
+ground truth
+prediction
+time
+-10
+0
+10
+20
+30
+40
+5
+15
+25
+35
+45
+-20
+-10
+0
+10
+20
+-20
+-10
+10
+20
+0
+training
+prediction
+A1
+A1
+A3
+A3
+A1
+A2
+A3
+Figure 5: Prediction of time-series data for the Lorenz system using the RC with micro-
+magnetic simulations. The parameters are θ = 0.4ns and α = 5.0×10−4. (a) The ground truth
+(A1(t), A2(t), A3(t)) and the estimated time series ( ˆ
+A1(t), ˆ
+A1(t), ˆ
+A3(t)) are shown in blue and
+red, respectively. The training steps are during t < 0, whereas the prediction steps are during
+t > 0. (b) The attractor in the A1A3 plane for the ground truth and during the prediction steps.
+13
+
+45
+40
+35
+30
+25
+20
+15
+10
+5
+-20
+-10
+0
+10
+2045
+40
+35
+30
+25
+20
+15
+10
+5
+-20
+-10
+0
+10
+20which the response function is replaced by the Gaussian function
+Gij(t) = exp
+�
+− 1
+2w2
+�
+t − Rij
+v
+�2�
+(5)
+where Rij is the distance between ith and jth physical nodes, and w is the width of the function.
+Even in this simplified model, we obtain MC≈ 40 and IPC≈ 60, and also the maximum when
+L ∝ v (Fig. 6(c,d)). From this result, the origin of the optimal ratio between the length and
+speed becomes clearer; when L ≪ v, the response functions under different Rij overlap so
+that different physical nodes cannot carry the information of different delay times. On the
+other hand, when L ≫ v, the characteristic delay time L/v exceeds the maximum delay time
+to compute MC and IPC, or exceeds the total length of the time series. Note that we set the
+maximum delay time as 100, which is much longer than the value necessary for the NARMA10
+task.
+The result suggests the universal scaling between the size of the system and the speed of
+the RC based on wave propagation. Our system of the spin wave has a characteristic length
+L ∼ 500 nm and a speed of v ∼ 200 m s−1. In fact, the reported photonic RC has characteristic
+length scale of optical fibres close to the scaling in Fig. 7.
+Discussion
+Figure 7 shows reports of reservoir computing in literature with multiple nodes plotted as a
+function of the length of nodes L and products of wave speed and delay time vτ0 for both
+photonic and spintronic RC. For the spintronic RC, the dipole interaction is considered for wave
+propagation in which speed is proportional to both saturation magnetization and thickness of the
+film (31)(See supplementary information sec. C). For the photonic RC, the characteristic speed
+is the speed of light, v ∼ 108 m s−1. Symbol size corresponds to MC taken from the literature
+[See details of plots in supplementary information sec. D]. Plots are roughly on a broad oblique
+14
+
+(a)
+(b)
+(c)
+(d)
+wave speed (log m/s)
+characteristic size (log nm)
+1.0
+3.0
+2.0
+2.0
+3.0
+4.0
+MC
+10
+20
+30
+40
+50
+60
+damping time
+1.0
+wave speed (log m/s)
+characteristic size (log nm)
+1.0
+3.0
+2.0
+2.0
+3.0
+4.0
+IPC
+10
+20
+30
+40
+50
+60
+damping time
+1.0
+wave speed (log m/s)
+characteristic size (log nm)
+1.0
+3.0
+2.0
+2.0
+3.0
+4.0
+MC
+10
+20
+30
+40
+1.0
+4.0
+5.0
+wave speed (log m/s)
+characteristic size (log nm)
+1.0
+3.0
+2.0
+2.0
+3.0
+4.0
+IPC
+10
+20
+30
+40
+1.0
+4.0
+5.0
+60
+50
+time 
+response function
+11( )
+G
+t
+12( )
+G
+t
+13( )
+G
+t
+14( )
+G
+t
+15( )
+G
+t
+time 
+dense
+sparse
+memorise
+(e)
+Figure 6:
+Scaling between characteristic size and propagating wave speed obtained by
+response function method. MC (a,c) and IPC (b,d) as a function of the characteristic length
+scale between physical nodes R and the speed of wave propagation v. The results with the
+response function for the dipole interaction (a,b) and for the Gaussian function (5) (c,d) are
+shown. (e) Schematic illustration of the response function and its relation to wave propagation
+between physical nodes. When the speed of the wave is too fast, all the response functions are
+overlapped (dense regime), while the response functions cannot cover the time windows when
+the speed of the wave is too slow (sparse regime).
+15
+
+line with a ratio L/(vτ0) ∼ 1. Therefore, the photonic RC requires a larger system size, as
+long as the delay time of the input τ0 = Nvθ is the same order (τ0 = 0.3 − 3 ns in our spin
+wave RC). As can be seen in Fig. 6, if one wants to reduce the length of physical nodes, one
+must reduce wave speed or delay time; otherwise the information is dense, and the reservoir
+cannot memorize many degrees of freedom (See Fig. 6(e)). Reducing delay time is challenging
+since the experimental demonstration of the photonic reservoirs has already used the short delay
+close to the instrumental limit. Also, reducing wave speed in photonics systems is challenging.
+On the other hand, the wave speed of propagating spin-wave is much lower than the speed of
+light and can be tuned by configuration, thickness and material parameters. If one reduces wave
+speed or delay time over the broad line in Fig. 7, information becomes sparse and cannot be
+used efficiently(See Fig. 6(e)). Therefore, there is an optimal condition for high-performance
+RC.
+The performance is comparable with other state of the art techniques, which are summa-
+rized in Fig. 8. For example, for the spintronic RC, MC ≈ 30 (19) and NRMSE ≈ 0.2 (22) in
+the NARMA10 task are obtained using Np ≈ 100 physical nodes. The spintronic RC with one
+physical node but with 101 − 102 virtual nodes do not show high performance; MC is less than
+10 (the bottom left points in Fig. 8). This fact suggests that the spintronic RC so far cannot use
+virtual nodes effectively. On the other hand, for the photonic RC, comparable performances are
+achieved using Nv ≈ 50 virtual nodes, but only one physical node. As we discussed, however,
+the photonic RC requires mm system sizes. Our system achieves comparable performances
+using ≲ 10 physical nodes, and the size is down to nanoscales keeping the 2 − 50 GHz compu-
+tational speed. We also demonstrate that the spin wave RC can perform time-series prediction
+and reconstruction of an attractor for the chaotic data. To our knowledge, this has not been done
+in nanoscale systems.
+Our results of micromagnetic simulations suggest that our system can be physically im-
+16
+
+0
+10
+20
+30
+40
+50
+60
+MC
+This work (
+t
+0
+ = 1.6 ns) 
+This work (
+t
+0
+ = 0.16 ns)
+  Spintronic RC 
+ ( 19 ) 
+ ( 22 )
+  Photonic RC
+ ( 46 ) 
+ ( 47 )
+(48 )
+(12 )
+ ( 50 )
+vt
+0 (m)
+Length, 
+L (m)
+Dense
+Sparse
+Figure 7: Reports of reservoir computing using multiple nodes are plotted as a function
+of the length between nodes and characteristic wave speed (v) times delay time (τ0) for
+photonics system (open symbols) and spintronics system (solid symbols). The size of sym-
+bols corresponds to memory capacity, which is taken from literature (12,19,22,32–35) and this
+work. The gray scale represents memory capacity evaluated by using the response function
+method [Eq. (5)].
+17
+
+VJKhJK-Jh-J1
+10
+100
+0.1
+1
+ This work (Calc., Nv = 8, θ = 0.2 ns)
+ This work (Calc., Nv = 8, θ = 0.02 ns)
+  Spintronic RC (Calc.)
+ (22)
+  Photonic RC
+ (45) (Nv = 50)
+(48) (Nv = 50)
+ (49) (Nv = 50)
+Normalized root mean square error, NRMSE 
+for NARMA10 task
+Number of physical nodes, Np
+1
+10
+100
+10
+100
+This work (Calc., Nv = 8, θ = 0.2 ns)
+This work (Calc., Nv = 8, θ = 0.02 ns)
+  Spintronic RC (Calc.)
+(44)
+(19) 
+(22)
+  Spintronic RC (Exp.)
+(9) (Nv = 250) 
+(51) (Nv = 40)
+  Photonic RC
+(46) (Nv = 50) 
+(47) (Nv = 50)
+(48) (Nv = 50)
+Memory capacity, MC
+Number of physical nodes, Np
+(a)
+(b)
+Figure 8: Reservoir computing performance compared with different systems. (a) Memory
+capacity, MC reported plotted as a function of physical nodes Np. (b) Normalized root mean
+square error, NRMSE for NARMA10 task is plotted as a function of Np. Open blue symbols are
+values reported using photonic RC while solid red symbols are values reported using spintronic
+RC. MC and NRMSE for NARMA10 task are taken from Refs. (9,19,22,36,37) for spintronic
+RC and Refs. (32–34,38,39) for photonic RC.
+plemented.
+All the parameters in this study are feasible using realistic materials (40–43).
+Nanoscale propagating spin waves in a ferromagnetic thin film excited by spin-transfer torque
+using nanometer electrical contacts have been observed (44–46). Patterning of multiple elec-
+trical nanocontacts into magnetic thin films was demonstrated in mutually synchronized spin-
+torque oscillators (46). In addition to the excitation of propagating spin-wave in a magnetic thin
+film, its non-local magnetization dynamics can be detected by tunnel magnetoresistance effect
+at each electrical contact, as schematically shown in Fig. 1(c), which are widely used for the
+development of spintronics memory and spin-torque oscillators. In addition, virtual nodes are
+effectively used in our system by considering the speed of propagating spin-wave and distance
+of physical nodes; thus, high-performance reservoir computing can be achieved with the small
+number of physical nodes, contrary to many physical nodes used in previous reports. This work
+provides a way to realize nanoscale high-performance reservoir computing based on propagat-
+ing spin-wave in a ferromagnetic thin film.
+There is an interesting connection between our study to the recently proposed next-generation
+18
+
+RC (28, 47), in which the linear ESN is identified with the NVAR (nonlinear vectorial autore-
+gression) method to estimate a dynamical equation from data. Our formula of the response func-
+tion (3) results in the linear input-output relationship with a delay Yn+1 = anUn+an−1Un−1+. . .
+(see Sec. A in Supplementary Information). More generally, with the nonlinear readout or with
+higher-order response functions, we have the input-output relationship with delay and non-
+linearity Yn+1 = anUn + an−1Un−1 + . . . + an,nUnYn + an,n−1UnUn−1 + . . . (see Sec. B in
+Supplementary Information). These input-output relations are nothing but Volterra series of the
+output as a function of the input with delay and nonlinearity (48). The coefficients of the ex-
+pansion are associated with the response function. Therefore, the performance of RC falls into
+the independent components of the matrix of the response function, which can be evaluated by
+how much delay the response functions between two nodes cover without overlap. The results
+would be helpful to a potential design of the network of the physical nodes.
+We should note that the polynomial basis of the input-output relation in this study originates
+from spin wave excitation around the stationary state mz = 1. When the input data has a hier-
+archical structure, another basis may be more efficient than the polynomial expansion. Another
+setup of magnetic systems may lead to a different basis. We believe that our study shows simple
+but clear intuition of the mechanism of high-performance RC, that can lead to the exploration
+of another setup for more practical application of the physical RC.
+19
+
+Materials and Methods
+Micromagnetic simulations
+We analyze the LLG equation using the micromagnetic simulator mumax3 (49). The LLG
+equation for the magnetization M(x, t) yields
+∂tM(x, t) = −
+γµ0
+1 + α2M × Heff −
+αγµ0
+Ms(1 + α2)M × (M × Heff)
++
+ℏPγ
+4M2s eDJ(x, t)M × (M × mf) .
+(6)
+We consider the effective magnetic field as
+Heff = Hext + Hdemag + Hexch,
+(7)
+Hext = H0ez
+(8)
+Hms = − 1
+4π
+�
+∇∇
+1
+|r − r′|dr′
+(9)
+Hexch = 2Aex
+µ0Ms
+∆M,
+(10)
+where Hext is the external magnetic field, Hms is the magnetostatic interaction, and Hexch is the
+exchange interaction with the exchange parameter Aex.
+The size of our system is L = 1000 nm and D = 4 nm. The number of mesh points is
+200 in the x and y directions, and 1 in the z direction. We consider Co2MnSi Heusler alloy
+ferromagnet, which has a low Gilbert damping and high spin polarization with the parameter
+Aex = 23.5 pJ/m, Ms = 1000 kA/m, and α = 5 × 10−4 (40,41,41–43). Out-of-plane magnetic
+field µ0H0 = 1.5 T is applied so that magnetization is pointing out-of-plane. The spin-polarized
+current field is included by the Slonczewski model (29) with polarization parameter P = 1 and
+spin torque asymmetry parameter λ = 1 with the reduced Planck constant ℏ and the charge of
+an electron e. The uniform fixed layer magnetization is mf = ex. We use absorbing boundary
+layers for spin waves to ensure the magnetization vanishes at the boundary of the system (50).
+We set the initial magnetization as m = ez.
+20
+
+The reference time scale in this system is τ0 = 1/γµ0Ms ≈ 5 ps, where γ is the gyromag-
+netic ratio, µ0 is permeability, and Ms is saturation magnetization. The reference length scale is
+the exchange length l0 ≈ 5 nm. The relevant parameters are Gilbert damping α, the time scale
+of the input time series θ, and the characteristic length between the input nodes R0.
+The injectors and detectors of spin are placed as cylindrical nanocontacts embedded in the
+region with their radius a and height D. We set a = 20nm unless otherwise stated. The
+input time series is uniform random noise Un ∈ U(0, 0.5). The injected density current is
+set as j(tn) = 2jcUn with jc = 2 × 10−4/(πa2)A/m2. Under a given input time series of
+the length T, we apply the current during the time θ, and then update the current at the next
+step. The same input current with different filters is injected for different virtual nodes (see
+Learning with reservoir computing). The total simulation time is, therefore, TθNv.
+Learning with reservoir computing
+Our RC architecture consists of reservoir state variables
+X(t + ∆t) = f (X(t), U(t))
+(11)
+and the readout
+Yn = W · ˜˜X(tn).
+(12)
+In our spin wave RC, the reservoir state is chosen as x-component of the magnetization
+X =
+�
+mx,1(tn), . . . , mx,i(tn), . . . , mx,Np(tn)
+�T ,
+(13)
+for the indices for the physical nodes i = 1, 2, . . . , Np. Here, Np is the number of physical
+nodes, and each mx,i(tn) is a T-dimensional row vector with n = 1, 2, . . . , T. We use a time-
+multiplex network of virtual nodes in RC (23), and use Nv virtual nodes with time interval θ.
+21
+
+The expanded reservoir state is expressed by NpNv × T matrix ˜X as (see Fig.2(b))
+˜X = (mx,1(tn,1), mx,1(tn,2), . . . , mx,1(tn,k), . . . , mx,1(tn,Nv),
+. . . , mx,i(tn,1), mx,i(tn,2), . . . , mx,i(tn,k), . . . , mx,i(tn,Nv), . . . ,
+mx,Np(tn,1), mx,Np(tn,2), . . . , mx,Np(tn,k), . . . , mx,Np(tn,Nv)
+�T ,
+(14)
+where tn,k = ((n − 1)Nv − (k − 1))θ for the indices of the virtual nodes k = 1, 2, . . . , Nv. The
+total number of rows is N = NpNv. We use the nonlinear readout by augmenting the reservoir
+state as
+˜˜X =
+�
+˜X
+˜X ◦ ˜X
+�
+,
+(15)
+where ˜X(t) ◦ ˜X(t) is the Hadamard product of ˜X(t), that is, component-wise product. The
+readout weight W is trained by the data of the output Y (t)
+W = Y · ˜˜X†
+(16)
+where X† is pseudo-inverse of X.
+In the time-multiplexing approach, the input time-series U = (U1, U2, . . . , UT) ∈ RT is
+translated into piece-wise constant time-series ˜U(t) = Un with t = (n − 1)Nvθ + s under
+k = 1, . . . , T and s = [0, Nvθ) (see Fig. 2(a)). This means that the same input remains during
+the time period τ0 = Nvθ. To use the advantage of physical and virtual nodes, the actual input
+Ji(t) at the ith physical node is ˜U(t) multiplied by τ0-periodic random binary filter Bi(t). Here,
+Bi(t) ∈ {0, 1} is piece-wise constant during the time θ. At each physical node, we use different
+realizations of the binary filter as in Fig. 2(a).
+Unless otherwise stated, We use 1000 steps of the input time-series as burn-in. After these
+steps, we use 5000 steps for training and 5000 steps for test for the MC, IPC, and NARMA10
+tasks.
+22
+
+NARMA task
+The NARMA10 task is based on the discrete differential equation,
+Yn+1 = αYn + βYn
+9
+�
+p=0
+Yn−p + γUnUn−9 + δ.
+(17)
+Here, Un is an input taken from the uniform random distribution U(0, 0.5), and yk is an output.
+We choose the parameter as α = 0.3, β = 0.05, γ = 1.5, and δ = 0.1. In RC, the input is
+U = (U1, U2, . . . , UT) and the output Y = (Y1, Y2, . . . , YT). The goal of the NARMA10 task
+is to estimate the output time-series Y from the given input U. The training of RC is done by
+tuning the weights W so that the estimated output ˆY (tn) is close to the true output Yn in terms
+of squared norm | ˆY (tn) − Yn|2.
+The performance of the NARMA10 task is measured by the deviation of the estimated time
+series ˆY = W · ˜˜X from the true output Y. The normalized root-mean-square error (NRMSE)
+is
+NRMSE ≡
+��
+n( ˆY (tn) − Yn)2
+�
+n Y 2
+n
+.
+(18)
+Performance of the task is high when NRMSE ≈ 0. In the ESN, it was reported that NRMSE ≈
+0.4 for N = 50 and NRMSE ≈ 0.2 for N = 200 (51). The number of node N = 200 was used
+for the speech recognition with ≈ 0.02 word error rate (51), and time-series prediction of sptio-
+temporal chaos (5). Therefore, NRMSE ≈ 0.2 is considered as reasonably high performance in
+practical application. We also stress that we use the same order of nodes (virtual and physical
+nodes) N = 128 to achieve NRMSE ≈ 0.2.
+Memory capacity and information processing capacity
+Memory capacity (MC) is a measure of the short-term memory of RC. This was introduced
+in (6). For the input Un of random time series taken from the uniform distribution, the network
+23
+
+is trained for the output Yn = Un−k. The MC is computed from
+MCk = ⟨Un−k, W · X(tn)⟩2
+⟨U2
+n⟩⟨(W · X(tn))2⟩.
+(19)
+This quantity is decaying as the delay k increases, and MC is defined as
+MC =
+kmax
+�
+k=1
+MCk.
+(20)
+Here, kmax is a maximum delay, and in this study we set it as kmax = 100. The advantage of MC
+is that when the input is independent and identically distributed (i.i.d.), and the output function
+is linear, then MC is bounded by N, the number of internal nodes.
+Information processing capacity (IPC) is a nonlinear version of MC (27). In this task, the
+output is set as
+Yn =
+�
+k
+Pdk(Un−k)
+(21)
+where dk is non-negative integer, and Pdk(x) is the Legendre polynomials of x order dk. We
+may define
+IPCd0,d1,...,dT −1 =
+⟨Yn, W · X(tn)⟩2
+⟨Y 2
+n ⟩⟨(W · X(tn))2⟩.
+(22)
+and then compute jth order IPC as We may define
+IPCj =
+�
+dks.t.j=�
+k dk
+IPCd1,d2,...,dT .
+(23)
+When j = 1, the IPC is, in fact, equivalent to MC, because P0(x) = 1 and P1(x) = x. In
+this case, Yn = Un−k for di = 1 when i = k and di = 0 otherwise. (23) takes the sum over
+all possible delay k, which is nothing but MC. When j > 1, IPC captures all the nonlinear
+transformation and delays up to the jth polynomial order. For example, when j = 2, the output
+can be Yn = Un−k1Un−k2 or Yn = U2
+n−k + const. In this study, we focus on j = 2 because
+24
+
+the second-order nonlinearity is essential for the NARMA10 task (see Sec. A in Supplementary
+Information).
+The relevance of MC and IPC is clear by considering the Volterra series of the input-output
+relation,
+Yn =
+�
+k1,k2,··· ,kt
+βk1,k2,··· ,knUk1
+1 Uk2
+2 · · · Ukn
+n .
+(24)
+Instead of polynomial basis, we may use orthonormal basis such as the Legendre polynomials
+Yn =
+�
+k1,k2,··· ,kn
+βk1,k2,··· ,knPk1(U1)Pk2(U2) · · ·Pkn(Un).
+(25)
+Each term in (25) is characterized by the non-negative indices (k1, k2, . . . , kn). Therefore, the
+terms corresponding to j = �
+i ki = 1 in Yn have information on linear terms with time
+delay. Similarly, the terms corresponding to j = �
+i ki = 2 have information of second-order
+nonlinearity with time delay. In this view, the estimation of the output Y (t) is nothing but the
+estimation of the coefficients βk1,k2,...,kn. In RC, the readout of the reservoir state at ith node
+(either physical or virtual node) can also be expanded as the Volterra series
+˜˜X(i)(tn) =
+�
+k1,k2,··· ,kn
+˜˜β(i)
+k1,k2,··· ,knUk1
+1 Uk2
+2 · · · Ukn
+n .
+(26)
+Therefore, MC and IPC are essentially a reconstruction of βk1,k2,··· ,kn from ˜˜β(i)
+k1,k2,··· ,kn with
+i ∈ [1, N]. This can be done by regarding βk1,k2,··· ,kn as a T + T(T − 1)/2 + · · · -dimensional
+vector, and using the matrix M associated with the readout weights as
+βk1,k2,··· ,kn = M ·
+
+
+
+
+
+
+˜˜β(1)
+k1,k2,··· ,kn
+˜˜β(2)
+k1,k2,··· ,kn
+...
+˜˜β(N)
+k1,k2,··· ,kn
+
+
+
+
+
+
+.
+(27)
+MC corresponds to the reconstruction of βk1,k2,··· ,kn for �
+i ki = 1, whereas the second-order
+IPC is the reconstruction of βk1,k2,··· ,kn for �
+i ki = 2. If all of the reservoir states are indepen-
+25
+
+dent, we may reconstruct N components in βk1,k2,··· ,kn. In realistic cases, the reservoir states are
+not independent, and therefore, we can estimate only < N components in βk1,k2,··· ,kn.
+Prediction of chaotic time-series data
+Following (5), we perform the prediction of time-series data from the Lorenz model. The model
+is a three-variable system of (A1(t), A2(t), A3(t)) yielding the following equation
+dA1
+dt = 10(A2 − A1)
+(28)
+dA2
+dt = A1(28 − A3) − A2
+(29)
+dA3
+dt = A1A2 − 8
+3A3.
+(30)
+The parameters are chosen such that the model exhibits chaotic dynamics. Similar to the other
+tasks, we apply the different masks of binary noise for different physical nodes, B(l)
+i (t) ∈
+{−1, 1}. Because the input time series is three-dimensional, we use three independent masks
+for A1, A2, and A3, therefore, l ∈ {1, 2, 3}. The input for the ith physical node after the mask is
+given as Bi(t) ˜Ui(t) = B(1)
+i (t)A1(t)+B(2)
+i (t)A2(t)+B(3)
+i (t)A3(t). Then, the input is normalized
+so that its range becomes [0, 0.5], and applied as an input current. Once the input is prepared,
+we may compute magnetization dynamics for each physical and virtual node, as in the case of
+the NARMA10 task. We note that here we use the binary mask of {−1, 1} instead of {0, 1}
+used for other tasks. We found that the {0, 1} does not work for the prediction of the Lorenz
+model, possibly because of the symmetry of the model.
+The ground-truth data of the Lorenz time-series is prepared using the Runge-Kutta method
+with the time step ∆t = 0.025. The time series is t ∈ [−60, 75], and t ∈ [−60, −50] is used for
+relaxation, t ∈ (−50, 0] for training, and t ∈ (0, 75] for prediction. During the training steps,
+we compute the output weight by taking the output as Y = (A1(t + ∆t), A2(t + ∆t), A3(t +
+∆t)). After training, the RC learns the mapping (A1(t), A2(t), A3(t)) → (A1(t + ∆t), A2(t +
+26
+
+∆t), A3(t + ∆t)). For the prediction steps, we no longer use the ground-truth input but the
+estimated data ( ˆ
+A1(t), ˆ
+A2(t), ˆ
+A3(t)). Using the fixed output weights computed in the training
+steps, the time evolution of the estimated time-series ( ˆ
+A1(t), ˆ
+A2(t), ˆ
+A3(t)) is computed by the
+RC.
+Theoretical analysis using response function
+We consider the Landau-Lifshitz-Gilbert equation for the magnetization field m(x, t),
+∂tm(x, t) = −m × heff − m × (m × heff) + σ(x, t)m × (m × mf)
+(31)
+We normalize both the magnetic and effective fields by saturation magnetization as m =
+M/Ms and heff = Heff/Ms. This normalization applies to all the fields including external
+and anisotropic fields. We also normalize the current density as σ(x, t) = J(x, t)/j0 for the
+current density J(x) and the unit of current density j0 =
+4M2
+s eπa2Dµ0
+ℏP
+. We apply the current
+density at the nanocontact as
+J(x, t) = 2jc ˜U(t)
+Np
+�
+i=1
+χa(|x − Ri|)
+(32)
+Here χa(x) is a characteristic function χa(x) = 1 when x ≤ a and χa(x) = 0 otherwise.
+We expand the solution of (31) around the uniform magnetization m(x, t) = (0, 0, 1) with-
+out current injection as
+m(x, t) = m0(x, t) + ǫm(1)(x, t) + O(ǫ2).
+(33)
+Here, m0(x, t) = (0, 0, 1) and ǫ ≪ 1 is a small parameter corresponding to the magnitude of the
+input σ(x, t). The first-order term corresponds to a linear response of the magnetization to the
+input σ, whereas the higher-order terms describe nonlinear responses, for example, m(2)(x, t) ∼
+σ(x1, t1)σ(x2, t2). Because our input is driven by the spin torque with fixed layer magnetization
+in the x-direction, mf = ex, only mx and my appear in the first-order term O(ǫ). Deviation of
+27
+
+mz from mz = 1 appears in O(ǫ2). Therefore, for the first-order term m(1), we may define the
+complex magnetization
+m = mx + imy.
+(34)
+Here, we will show the magnetization is expressed by the response function Gij(t). The
+input at the jth physical node affects the magnetization at the ith physical node as
+mi(t)
+=
+�
+dτGii(t − τ)σi(τ) + �
+i̸=j
+�
+dτGij(t − τ)σj(τ).
+(35)
+The input for the jth physical node is expressed by σj(t) = 2jcBj(t) ˜Uj(t). Because different
+physical nodes have different masks discussed in Learning with reservoir computing in Meth-
+ods. When the wave propagation is dominated by the exchange interaction, the response func-
+tion for the same node is
+Gii(t − τ) = 1
+2πe−˜h(α+i)(t−τ)
+�
+1 − e−
+a2
+4(α+i)(t−τ)
+�
+(36)
+and for different nodes, it becomes
+Gij(t − τ) = a2
+2πe−˜h(α+i)(t−τ)e−
+|Ri−Rj|2
+4(α+i)(t−τ)
+1
+2(α + i)(t − τ).
+(37)
+When the wave propagation is dominated by the dipole interaction, the response function for
+the same node is
+Gii(t − τ) = 1
+2πe−˜h(α+i)(t−τ) −1 +
+�
+1 +
+a2
+(d/4)2(α+i)2(t−τ)2
+�
+1 +
+a2
+(d/4)2(α+i)2(t−τ)2
+(38)
+and for different nodes it becomes
+Gij(t − τ) = a2
+2πe−˜h(α+i)(t−τ)
+×
+1
+(d/4)2(α + i)2(t − τ)2
+�
+1 +
+|Ri−Rj|2
+(d/4)2(α+i)2(t−τ)2
+�3/2.
+(39)
+28
+
+Clearly, Gii(0) → 1 and Gij(0) → 0, while Gii(∞) → 0 and Gij(∞) → 0.
+Once the magnetization is expressed in the form of (35), we may compute the reservoir state
+X under the input U. Then, we may use the same method as in Learning with reservoir computing,
+and estimate the output ˆY. Similar to the micromagnetic simulations, we evaluate the perfor-
+mance by MC, IPC, and NARMA10 tasks.
+We may extend the analyzes for the higher-order terms in the expansion of (33). In Sec.B in
+Supplementary Materials, we show the second-order term m(2)(x, t) has only the z-component,
+and moreover, it is dependent only on the first-order terms. As a result, the second-order term
+is expressed as
+m(2)
+z (x, t) = −1
+2
+�
+(m(1)
+x )2 + (m(1)
+y )2�
+.
+(40)
+To compute the response functions, we linearize (31) for the complex magnetization m(x, t)
+as
+∂tm(x, t) = Lm + σ(x, t),
+(41)
+where the linear operator is expressed as
+L =
+�
+−˜h + ∆
+�
+(α + i) .
+(42)
+In the Fourier space, the linearized equation becomes
+∂tmk(t) = Lkmk + σk(t),
+(43)
+with
+Lk = −
+�
+˜h + k2�
+(α + i) .
+(44)
+The solution of ((43)) is obtained as
+mk(t) =
+�
+dτeLk(t−τ)σk(τ).
+(45)
+29
+
+We have Np cylindrical shape inputs with radius a and the ith input is located at Ri. The input
+function is expressed as
+σ(x) =
+Np
+�
+i=1
+χa (|x − Ri|) .
+(46)
+We are interested in the magnetization at the input mi(t) = m(x = Ri, t), which is
+mi =
+1
+(2π)2
+�
+j
+�
+dτe−˜h(α+i)(t−τ)
+�
+dke−k2(α+i)(t−τ)eik·(Ri−Rj)2πaJ1(ka)σj(t)
+= a
+2π
+�
+j
+�
+dτe−˜h(α+i)(t−τ)
+�
+dke−k2(α+i)(t−τ)J0 (k|Ri − Rj|) J1(ka)σj(t)
+(47)
+For the same node, |Ri − Rj| = 0, and we may compute the integral explicitly as (36). When
+ka ≪ 1, we may assume J1(ka) ≈ ka/2, and finally, come up with (37).
+When the thickness d of the material is thin, the dispersion relation becomes
+Lk = −˜h(α + i)
+��
+1 + k2
+˜h
+� �
+1 + k2
+˜h
++ βk
+˜h
+�
+(48)
+where
+β = d
+2.
+(49)
+We assume for k ≪ β
+�
+˜h, then the linearized operator becomes
+Lk = −(α + i)
+�
+˜h + kd
+4
+�
+(50)
+leading to (38) and (39).
+Acknowledgements:
+S. M. thanks to CSRN at Tohoku University. Numerical simulations in this work were carried
+out in part by AI Bridging Cloud Infrastructure (ABCI) at National Institute of Advanced In-
+dustrial Science and Technology (AIST), and by the supercomputer system at the information
+30
+
+initiative center, Hokkaido University, Sapporo, Japan.
+Funding:
+This work is support by JSPS KAKENHI grant numbers 21H04648, 21H05000 to S.M., by
+JST, PRESTO Grant Number JPMJPR22B2 to S.I., X-NICS, MEXT Grant Number JPJ011438
+to S.M., and by JST FOREST Program Grant Number JPMJFR2140 to N.Y.
+Author Contributions
+S.M., N.Y., S.I. conceived the research. S.I., Y.K., N.Y. carried out simulations. N.Y., S.I. an-
+alyzed the results. N.Y., S.I., S.M. wrote the manuscript. All the authors discussed the results
+and analysis.
+Competing Interests
+The authors declare that they have no competing financial interests.
+Data and materials availability:
+All data are available in the main text or the supplementary materials.
+A
+Connection between the NARMA10 task and MC/IPC
+In this section, we discuss the necessary properties of reservoir computing to achieve high per-
+formance of the NARMA10 task. In short, the NARMA10 task is dominated by the memory of
+31
+
+nine step previous data and second-order nonlinearity. We discuss these properties in two meth-
+ods. The first method is based on the extended Dynamic Mode Decomposition (DMD) (52) and
+the higher-order DMD (53). The second method is a regression of the input-output relationship.
+We will discuss the details of the two methods. Our results are consistent with previous studies;
+the requirement of memory was discussed in (54), and the second-order nonlinear terms with a
+time delay in (55).
+The NARMA10 task is based on the discrete differential equation,
+Yn+1 = αYn + βYn
+9
+�
+i=0
+Yn−i + γUnUn−9 + δ.
+(51)
+Here, Un is an input at the time step n taken from the uniform random distribution U(0, 0.5),
+and Yn is an output. We choose the parameter as α = 0.3, β = 0.05, γ = 1.5, and δ = 0.1.
+In the first method, we estimate the transition matrix A from the state variable Yn =
+(Y1, Y2, . . . , Yn) to Yn+1 = (Y2, Y3, . . . , Yn+1) yielding
+Yn+1 = A · Yn.
+(52)
+We may extend the notion of the state variable to contain delayed data and polynomials of the
+output with time delay as
+Yn = (Yn, Yn−1, . . . , Y1, YnYn, YnYn−1, . . . , Y1Y1) .
+(53)
+Including the delay terms following from the higher-order DMD (53), while the polynomial
+nonlinear terms are used as a polynomial dictionary in the extended DMD (52). Here, (53)
+contains all the combination of the second-order terms with time delay, Yn−i1Yn−i2 with the
+integers i1 and i2 in 0 ≤ i1 ≤ l2 ≤ n − 1. We may straightforwardly include higher-order terms
+in powers in (53). In the NARMA10 task, the output Yn+1 is also affected by the input Un.
+Therefore, the extended DMD is generalized to include the control as (56)
+Yn+1 = (A B) ·
+�Yn
+Un
+�
+,
+(54)
+32
+
+where the state variable corresponding to the input includes time delay and nonlinearity, and is
+described as
+Un = (Un, Un−1, . . . , U1, UnUn, UnUn−1, . . . , U1U1) .
+(55)
+We denote the generalized transition matrix as
+Ξ = (A B) .
+(56)
+The idea of DMD is to estimate the transition matrix from the data. This is done by taking
+pseudo inverse of the state variables as
+ˆΞ = Yk+1 ·
+�
+Yk
+Uk
+�†
+.
+(57)
+Here, M† is the pseudoinverse of the matrix M. This is nothing but a least-square estimation
+for the cost function of l.h.s minus r.h.s of (54). We may include the Tikhonov regularization
+term.
+Note that for the extended DMD (52) and the higher-order DMD (53), the transition matrix
+Ξ is further decomposed into characteristic modes associated with its eigenvalues. The decom-
+position gives us a dimensional reduction of the system. The estimation of the transition matrix
+is also called nonlinear system identification, particularly, nonlinear autoregression with exoge-
+nous inputs (NARX). In this work, we focus on the estimation of the input-output relationship,
+and do not discuss the dimensional reduction. For time-series prediction, we estimate the func-
+tion Yn+1 = f(Yn, Yn−1, . . . , Y1), and we do not need the input Un in (54). Even in this case,
+we may consider a similar estimation of Ξ (in fact, A). This estimation is the method used in
+the next-generation RC (47).
+The second method is based on the Volterra series of the state variable Yn by the input Un.
+In this method, we assume that the state variable is independent of its initial condition. Then,
+33
+
+we may express the state variable as
+Yn = G · Un.
+(58)
+Note that Un includes the input and its polynomials with a time delay as in (55). Similar to the
+first method, we estimate G by
+ˆG = Yt · U†
+t.
+(59)
+The estimated ˆG gives us information on which time delay and nonlinearity dominate the state
+variable.
+0
+5
+10
+15
+20
+25
+30
+delay
+NRMSE
+0
+0.2
+0.4
+0.6
+0.8
+1.0
+test
+training
+linear
+(A)
+(B)
+(C)
+(D)
+(E)
+(F)
+0
+5
+10
+15
+20
+25
+30
+delay
+NRMSE
+0
+0.2
+0.6
+0.8
+1.0
+0.4
+second-order nonlinearity
+0
+5
+10
+15
+20
+25
+30
+delay
+NRMSE
+0
+0.2
+0.6
+0.8
+1.0
+0.4
+third-order nonlinearity
+0
+5
+10
+15
+20
+25
+30
+delay
+NRMSE
+0
+0.2
+0.4
+0.6
+0.8
+1.0
+linear
+0
+5
+10
+15
+20
+25
+30
+delay
+NRMSE
+0
+0.2
+0.6
+0.8
+1.0
+0.4
+second-order nonlinearity
+0
+5
+10
+15
+20
+25
+30
+delay
+NRMSE
+0
+0.2
+0.6
+0.8
+1.0
+0.4
+third-order nonlinearity
+Figure 9: (A-C) the estimation based on the extended DMD, (D-F) the estimation based on the
+Volterra series. The dictionary of each case is (A,D) first-order (linear) delay terms, (B,E) up to
+second-order delay terms, and (C,F) up to third-order delay terms.
+The results of the two estimation methods are shown in Fig. 9. Both approaches suggest
+that memory of ≈ 10 steps is enough to get high performance, and further memory does not
+improve the error. The second-order nonlinear term shows a reasonably small NRMSE of ≈
+0.01. Including the third-order nonlinearity improves the error, but there is a sign of overfitting
+34
+
+at a longer delay because the number of the state variables is too large. It should also be noted
+that even with the linear terms, the NRMSE becomes ≈ 0.35. This result implies that although
+NRMSE ≈ 0.35 is often considered good performance, nonlinearity of the data is not learned
+at the error of this order.
+A.1
+The MC and IPC tasks as Volterra series for linear and nonlinear
+readout
+In (3) and (4) in the main text, we show that the magnetization at the input region is expressed
+by the response function. The magnetization at the time tn corresponding to the input Un at the
+n step is expressed as
+m(tn) = anUn + an−1Un−1 + · · · ,
+(60)
+where the coefficients an can be computed from the response function. We first consider the
+linear case, but we will generalize the expression for the nonlinear case. Because we use virtual
+nodes, the input Un at the step n continues during the time period t ∈ [tn, tn+1) discretized by
+Nv steps as (tn,1, tn,2, . . . , tn,Nv), and is multiplied by the filter of the binary noise (see Fig.2 and
+Methods in the main text). Therefore, the magnetization is expressed by the response functions
+G(t − t′) is formally expressed as
+m(tn) =
+Np
+�
+i
+[(G(0) + G(θ) + · · · G(θ(Nv − 1))) σi(tn)
++ (G(θNv) + G(θ(Nv + 1)) + · · · G(θ(2Nv − 1))) σi(tn−1)
++ · · ·] ,
+(61)
+where σi(tn) ∝ Un is the non-dimensionalized current injection at the time tn at the ith physical
+node, which is proportional to Un. Therefore, (61) results in the expression of (60). Our input
+is taken from a uniform random distribution. Therefore, the inner product of the reservoir state,
+35
+
+which is nothing but magnetization, and (delayed) input to learn MC is
+⟨m(tn), Un⟩ =
+T
+�
+n=1
+m(tn)Un = an⟨U2
+n⟩ + O(1/T).
+(62)
+Similarly, the variance of the magnetization is equal to the variance of the input with the coef-
+ficient associated with m(tn).
+We may express the MC and IPC tasks in a matrix form as
+˜S ≈ W · G · (S ◦ Win) .
+(63)
+Here, S is the matrix associated with the original input, and ˜S is the delayed one. The output
+weight is denoted by W, and Win is the matrix associated with the mask of binary noise. The
+goal of MC and IPC tasks is to approximate the delayed input ˜S by the reservoir states G · S.
+Here, the reservoir states are expressed by the response function G and input denoted by S. We
+define delayed input ˜S ∈ RK×T
+˜S =
+
+
+
+
+
+Un
+Un+1
+Un+2
+· · ·
+Un−1
+Un
+Un+1
+· · ·
+Un−2
+Un−1
+Un
+· · ·
+...
+...
+...
+...
+
+
+
+
+ .
+(64)
+Here, T is the number of the time series, and K is the total length of the delay that we consider.
+The ith row shows the i−1 delayed time series. The input S ∈ RTNv×T to compute the reservoir
+states are expressed as
+S =
+
+
+
+
+
+
+
+
+
+
+
+
+σ(tn)
+σ(tn+1)
+σ(tn+2)
+· · ·
+...
+...
+...
+...
+σ(tn)
+σ(tn+1)
+σ(tn+2)
+· · ·
+σ(tn−1)
+σ(tn)
+σ(tn+1)
+· · ·
+...
+...
+...
+...
+σ(tn−2)
+σ(tn−1)
+σ(tn)
+· · ·
+...
+...
+...
+...
+
+
+
+
+
+
+
+
+
+
+
+
+.
+(65)
+Note that σ(tn) ∝ Un upto constant. Due to time multiplexing, each row is repeated Nv times,
+and then the time series is delayed in the next row. After multiplying the input filter Win, the
+36
+
+input is fed into the response function. The input filter Win ∈ RTNv×T is a stack of constant row
+vectors with the length T. The Nv different realizations of row vectors are taken from binary
+noise, and then the resulting Nv × T matrix is repeated T times in the row direction. This input
+is multiplied by the coefficients of the Volterra series G ∈ RN×TNv
+G =
+
+
+
+
+
+G(1)(0)
+· · ·
+G(1)(θ(Nv − 1))
+G(1)(θNv)
+· · ·
+G(1)(θ(2Nv − 1))
+· · ·
+G(2)(0)
+· · ·
+G(2)(θ(Nv − 1))
+G(2)(θNv)
+· · ·
+G(2)(θ(2Nv − 1))
+· · ·
+...
+...
+...
+...
+...
+...
+...
+G(N)(0)
+· · ·
+G(N)(θ(Nv − 1))
+G(N)(θNv)
+· · ·
+G(N)(θ(2Nv − 1))
+· · ·
+
+
+
+
+
+(66)
+(63) implies that by choosing the appropriate W, we can get a canonical form of G. If
+the canonical form has N × N identity matrix in the left part of W · G, then the reservoir
+reproduces the time series up to N − 1 delay. This means that the rank of the matrix G, or the
+number of independent rows, is the maximum number of steps of the delay. This is consistent
+with the known fact that MC is bounded by the number of independent components of reservoir
+variables (6).
+Next we extend the Volterra series of the magnetization, including nonlinear terms. The
+magnetization is expressed as
+m(tn) = anσ(tn) + an−1σ(tn−1) + · · · + an,nσ(tn)σ(tn) + an,n−1σ(tn)σ(tn−1) + · · · .
+(67)
+The delayed input ˜S is rewritten as
+˜S =
+
+
+
+
+
+
+
+
+
+Un
+Un+1
+Un+2
+· · ·
+Un−1
+Un
+Un+1
+· · ·
+...
+...
+...
+...
+UnUn
+Un+1Un+1
+Un+2Un+2
+· · ·
+UnUn−1
+Un+1Un
+Un+2Un+1
+· · ·
+...
+...
+...
+...
+
+
+
+
+
+
+
+
+
+.
+(68)
+The matrix ˜S contains all the nonlinear combinations of the input series (Un, Un+1, · · ·). Ac-
+cordingly, we should modify S and also G to include the nonlinear response functions. Note
+that to guarantee the orthogonality, Legendre polynomials (or other orthogonal polynomials)
+37
+
+should be used instead of polynomials in powers. Nevertheless, up to the second order of
+nonlinearity, which is relevant to consider the performance of NARMA10 (see Sec. A), the dif-
+ference is only in the constant terms (P2(x) = x2 − 1
+2). Because we subtract the mean value of
+the time series of all the input, output, and reservoir states, these constant terms do not change
+our conclusion. With nonlinear terms, (66) is extended as G = (Glin, Gnonl). Still, the rank of
+the matrix remains N at most. This is the reason why the total sum of IPC, including all the lin-
+ear and nonlinear delays, is bounded by the number of independent reservoir variables. When
+Gnonl = 0, the reservoir can memorize only the linear delay terms, but MC can be maximized
+to be N. On the other hand, when Gnonl ̸= 0, it is possible that MC is less than N, but the
+reservoir may have finite IPC.
+When the readout is nonlinear, we use the reservoir state variable as
+X =
+�
+M
+M ◦ M
+�
+,
+(69)
+where ◦ is the Hadamard product. If M is linear in the input, G has a structure of
+G =
+�
+Glin
+0
+0
+Gnonlin
+�
+.
+(70)
+In this case, rank(G) = rank(Glin) + rank(Gnonlin).
+B
+Learning with multiple variables
+In the main text, we use only mx for the readout as in (13)-(15). The readout is nonlinear and
+has both the information of mx and m2
+x. In this section, we consider the linear readout, but
+use both mx and mz for the output in micromagnetic simulations. We begin with the linear
+readout only with mx. The results of the MC and IPC tasks are shown in Fig. 10(a,b). We
+obtain a similar performance for the MC task with the result in the main text (Fig. 3). On the
+other hand, the performance for the IPC task in Fig. 10(a) is significantly poorer than the result
+38
+
+in Fig. 3(a). This result demonstrates that the linear readout only with mx does not learn the
+nonlinearity effectively. Note that in the theoretical model with the response function, the IPC
+is exactly zero when we use the linear readout only with mx. The discrepancy arises from the
+expansion (33) around m0 = (0, 0, 1) in the main text. Strictly speaking, the expansion should
+be made around m0 under the constant input ⟨σ⟩ averaged over time at the input nanocontact.
+This reference state is inhomogeneous in space, and is hard to compute analytically. Due to this
+effect, mx in the micromagnetic simulations contain small nonlinearity.
+Next, we consider the linear readout with mx and mz. As seen in Fig. 10(c,d), mz carries
+nonlinear information, and enhances the IPC and learning performance of NARMA10 com-
+pared with linear readout only with mx (Fig. 10 (a,b)). The performance is IPC ≈ 60 under
+α = 5 × 10−4, which is comparable value with the results in the main text (Fig. 3(a,c)) where
+the readout is (mx, m2
+x). Also, high performance for NARMA10 task, NRMSE ≈ 0.2, can be
+obtained using variables (mx, mz). These results show that adding mz into the readout has a
+similar effect to adding m2
+x.
+Similarity between m2
+x and mz can be understood by using the theoretical formula with the
+response function in the main text. We continue the expansion (33) at the second order, and
+obtain
+∂tm(2)(x, t) = − m(1) × ∆m(1) − αm(1) ×
+��
+˜hm(1) − ∆m(1)�
+× ez
+�
++ σ(x, t)m(1) × ey.
+(71)
+This result suggests that m(2) contains only the z component, and is slaved by m(1), which does
+not have z component. Therefore, m(2)
+z
+can be computed as
+m(2)
+z (x, t) = −1
+2
+�
+(m(1)
+x )2 + (m(1)
+y )2�
+.
+(72)
+Because mx and my carry similar information, mz in the readout has a similar effect with m2
+x
+in the readout.
+39
+
+C
+Speed of propagating spin wave using dipole interaction
+Propagating spin wave when magnetization is pointing along film normal is called magneto-
+static forward volume mode, and its dispersion relation can be described by the following equa-
+tion (31).
+ω(k) = γµ0
+�
+(H0 − Ms)
+�
+H0 − Ms
+1 − e−kd
+kd
+�
+.
+(73)
+Then, one can obtain the group velocity at k ∼ 0 as,
+vg = dω
+dk (k = 0) = γµ0Msd
+4
+.
+(74)
+In the magneto-static spin wave driven by dipole interaction, group velocity is proportional to
+both Ms and d. vg ∼ 200 m/s is obtained when the following parameters are used: µ0H = 1.5
+T, Ms = 1.0 × 106 A/m, d = 4 nm. The same estimation is used for calculating the speed of
+information propagation for spin reservoirs in Refs. (19) and (22), which are used to plot Fig. 7
+in the main text.
+D
+Details of reservoir computing scaling compared with lit-
+erature
+In this section, details of Fig. 7 shown in the main text are described. MC and NRMSE for
+NARMA10 tasks using photonic and spintronic RC are reported in Refs. (12,32–35,38,39) for
+photonic RC and (9,19,22,25,36,37,57,58) for spintronic RC. Table 1 and 2 shows reports of
+MC for photonic and spintronic RC with different length scales, which are plotted in Fig. 7 in
+the main text.
+40
+
+Table 1: Report of photonic RC with different length scales used in Fig. 7 in the main text
+Reports
+Length, L
+Time interval, τ0
+vτ0
+N
+MC
+Duport et al. (32)
+1.6 km
+8 µs
+2.4 km
+50
+21
+Dejonckheere et al. (33)
+1.6 km
+8 µs
+2.4 km
+50
+37
+Vincker et al. (34)
+230 m
+1.1 µs
+340 m
+50
+21
+Takano et al. (12)
+11 mm
+200 ps
+60 mm
+31
+1.5
+Sugano et al. (35)
+10 mm
+240 ps
+72 mm
+240
+10
+Note: speed of light, v = 3 × 108 m/s is used.
+Table 2: Report of spin reservoirs with different length scales used in Fig. 7 in the main text
+Reports
+L
+τ0
+v
+vτ0
+N
+MC
+Nakane et al. (19)
+5 µm
+2 ns
+2.4 km/s
+4.8 µm
+72
+21
+Dale et al. (22)
+50 nm
+10 ps
+200 m/s
+2 nm
+100
+35
+This work
+500 nm
+1.6 ns
+200 m/s
+320 nm
+64
+26
+Note: v is calculated based on magneto-static spin wave using Eq. 74.
+E
+Other data
+E.1
+Nv and Np dependence of performance
+Fig. 11 shows Nv and Np dependencies of MC, IPC and NRMSE for NARMA10 task. As Nv
+and Np are increased, MC and IPC increase. Then, NARMA10 prediction task becomes better
+with increasing Nv and Np. MC and NRMSE for NARMA10 with different Np with fixed Nv
+= 8 are compared with other reservoirs shown in Fig. 8 in the main text.
+E.2
+exchange interaction
+In the main text, we use the dipole interaction to compute the response function as (38) and
+(39). In this section, we show the result using the exchange interaction shown in (36) and (37).
+Figure 12 shows the results.
+41
+
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+48
+
+(a) (mx), MC and IPC
+MNO
+Pm
+x), NARMA10
+MC
+IPC
+0
+20
+40
+60
+80
+5
+2.5
+α = 5×10-4
+Frequency, 1/θ (GHz)
+0
+20
+40
+60
+80
+α = 5×10-3
+Linear and non-linear memory capacity
+0.2
+0.4
+0
+20
+40
+60
+80
+α = 5×10-2
+Distance of virtual nodes, θ (ns)
+0
+0.5
+1
+5
+2.5
+α = 5×10-4
+ Training, 
+ Test
+Frequency, 1/θ (GHz)
+0
+0.5
+1
+α = 5×10-3
+Normalized root mean square error, NRMSE 
+for NARMA10 task
+0.2
+0.4
+0
+0.5
+1
+α = 5×10-2
+Distance of virtual nodes, θ (ns)
+(c) (m
+Q, mz) MC and IPC
+(d) (m
+R, mz), NARMA10
+MC
+IPC
+0
+20
+40
+60
+80
+5
+2.5
+α = 5×10-4
+Frequency, 1/θ (GHz)
+0
+20
+40
+60
+80
+α = 5×10-3
+Linear and non-linear memory capacity
+0.2
+0.4
+0
+20
+40
+60
+80
+α = 5×10-2
+Distance of virtual nodes, θ (ns)
+0
+0.5
+1
+5
+2.5
+α = 5×10-4
+ Training, 
+ Test
+Frequency, 1/θ (GHz)
+0
+0.5
+1
+α = 5×10-3
+Normalized root mean square error, NRMSE 
+for NARMA10 task
+0.2
+0.4
+0
+0.5
+1
+α = 5×10-2
+Distance of virtual nodes, θ (ns)
+Figure 10:
+Reservoir computing with various parameter combinations obtained using micro-
+magnetic Mumax3 simulation. Linear memory capacity, MC and nonlinear memory capacity,
+IPC plotted as a function of θ obtained using linear mx output only (a) and using mx, mz (c).
+Normalized root mean square error, NRMSE for NARMA10 task plotted as a function of θ
+obtained using linear mx output only (b) and using mx, mz (d).
+49
+
+2
+4
+6
+8
+2
+4
+6
+8
+Number of physical nodes, Np
+Number of virtual nodes, Nv
+0
+5
+10
+15
+20
+25
+30
+Memory capacity, MC
+2
+4
+6
+8
+2
+4
+6
+8
+Number of physical nodes, Np
+Number of virtual nodes, Nv
+0
+10
+20
+30
+40
+50
+60
+S T
+Nonlinear memory capacity, IPC
+2
+4
+6
+8
+2
+4
+6
+8
+Number of physical nodes, Np
+Number of virtual nodes, Nv
+0.00
+0.20
+0.40
+0.60
+0.80
+1.00
+Normalized mean square error, NRMSE
+for NARMA10 task
+(a)
+U VW
+(c)
+Figure 11: (a) Memory capacity, MC (b) Nonlinear memory capacity, IPC and (c) Normalized
+root mean square error, NRMSE for NARMA10 task plotted as a function of the number of
+virtual and physical nodes. The parameters used in the simulation are α = 5 × 10−4, θ = 0.2
+ns.
+(a)
+(b)
+(c)
+wave speed (log m/s)
+characteristic size (log nm)
+3.0
+2.0
+2.0
+3.0
+4.0
+MC
+20
+30
+40
+50
+damping time
+1.0
+5.0
+4.0
+wave speed (log m/s)
+characteristic size (log nm)
+2.0
+4.0
+3.0
+2.0
+3.0
+4.0
+IPC
+20
+30
+40
+50
+damping time
+1.0
+5.0
+0
+20
+40
+60
+80
+5
+2.5
+� = 5×10-4
+Frequency, 1/� (GHz)
+0
+20
+40
+60
+80
+� = 5×10-3
+Linear and non-linear memory capacity
+0.2
+0.4
+0
+20
+40
+60
+80
+� = 5×10-2
+Distance of virtual nodes, � (ns)
+Figure 12: (a) Memory capacity, MC (solid symbols) and nonlinear memory capacity, IPC
+(open symbols) obtained using the response function method for exchange interaction plotted
+as a function of θ with different damping parameters α. (b) MC and (c) IPC plotted as a function
+of characteristic size and wave speed.
+50
+

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+Learning to Maximize Mutual Information for Dynamic Feature Selection
+Ian Covert 1 Wei Qiu 1 Mingyu Lu 1 Nayoon Kim 1 Nathan White 2 Su-In Lee 1
+Abstract
+Feature selection helps reduce data acquisition
+costs in ML, but the standard approach is to train
+models with static feature subsets. Here, we con-
+sider the dynamic feature selection (DFS) prob-
+lem where a model sequentially queries features
+based on the presently available information. DFS
+is often addressed with reinforcement learning
+(RL), but we explore a simpler approach of greed-
+ily selecting features based on their conditional
+mutual information. This method is theoretically
+appealing but requires oracle access to the data
+distribution, so we develop a learning approach
+based on amortized optimization. The proposed
+method is shown to recover the greedy policy
+when trained to optimality and outperforms nu-
+merous existing feature selection methods in our
+experiments, thus validating it as a simple but
+powerful approach for this problem.
+1. Introduction
+A machine learning model’s inputs can be costly to obtain,
+and feature selection is often used to reduce data acquisition
+costs. In applications where information is gathered sequen-
+tially, a natural option is to select features adaptively based
+on the currently available information rather than using a
+fixed feature set. This setup is known as dynamic feature
+selection (DFS)1, and the problem has been considered by
+several works in the last decade (Saar-Tsechansky et al.,
+2009; Dulac-Arnold et al., 2011; Chen et al., 2015b; Early
+et al., 2016a; He et al., 2016a; Kachuee et al., 2018).
+Compared to static feature selection with a fixed feature
+set (Li et al., 2017; Cai et al., 2018), DFS can offer better
+performance given a fixed budget. This is easy to see, be-
+cause selecting the same features for all instances (e.g., all
+1Paul G. Allen School of Computer Science & Engineering,
+University of Washington 2Department of Emergency Medicine,
+University of Washington.
+Correspondence to:
+Ian Covert
+<[email protected]>.
+1The problem is also sometimes referred to as sequential fea-
+ture selection or active feature acquisition.
+patients visiting a hospital’s emergency room) is suboptimal
+when the most informative features vary across individuals.
+Although it should in theory offer better performance, DFS
+also presents a more challenging learning problem, because
+it requires learning both a feature selection policy and how
+to make predictions given variable feature sets.
+Prior work has approached DFS in several ways, though of-
+ten using reinforcement learning (RL) (Dulac-Arnold et al.,
+2011; Shim et al., 2018; Kachuee et al., 2018; Janisch et al.,
+2019; Li & Oliva, 2021). RL is a natural approach for se-
+quential decision-making problems, but current methods are
+difficult to train and do not reliably outperform static fea-
+ture selection methods (Henderson et al., 2018; Erion et al.,
+2021). Our work therefore explores a simpler approach:
+greedily selecting features based on their conditional mutual
+information (CMI) with the response variable.
+The greedy approach is known from prior work (Fleuret,
+2004; Chen et al., 2015b; Ma et al., 2019) but is difficult
+to use in practice, because calculating CMI requires oracle
+access to the data distribution (Cover & Thomas, 2012).
+Our focus is therefore developing a practical approximation.
+Whereas previous work makes strong assumptions about the
+data (e.g., binary features in Fleuret 2004) or approximates
+the data distribution with generative modeling (Ma et al.,
+2019), we develop a flexible approach that directly predicts
+the optimal selection at each step. Our method is based on a
+variational perspective on the greedy CMI policy, and it uses
+a technique known as amortized optimization (Amos, 2022)
+to enable training using only a standard labeled dataset.
+Notably, the model is trained with an objective that recovers
+the greedy policy when it is trained to optimality.
+Our contributions in this work are the following:
+1. We derive a variational, or optimization-based perspec-
+tive on the greedy CMI policy, which shows it to be
+equivalent to minimizing the one-step-ahead prediction
+loss given an optimal classifier.
+2. We develop a learning approach based on amortized op-
+timization, where a policy network is trained to directly
+predict the greedy selection at each step. Rather than re-
+quiring a dataset that indicates the correct selections, our
+training approach is based on a standard labeled dataset
+and an objective function whose global optimizer is the
+arXiv:2301.00557v1  [cs.LG]  2 Jan 2023
+
+Learning to Maximize Mutual Information for Dynamic Feature Selection
+greedy CMI policy.
+3. We propose a continuous relaxation for the inherently
+discrete learning objective, which enables efficient and
+architecture-agnostic training with stochastic gradient
+descent.
+Our experiments evaluate the proposed method on numer-
+ous datasets, and the results show that it outperforms many
+recent dynamic and static feature selection methods. Over-
+all, our work shows that when learned properly, the greedy
+CMI policy is a simple and powerful method for DFS.
+2. Problem formulation
+In this section, we describe the DFS problem and introduce
+notation used throughout the paper.
+2.1. Notation
+Let x denote a vector of input features and y a response
+variable for a supervised learning task. The input consists
+of d distinct features, or x = (x1, . . . , xd). We use the nota-
+tion s ⊆ [d] ≡ {1, . . . , d} to denote a subset of indices and
+xs = {xi : i ∈ s} a subset of features. Bold symbols x, y
+represent random variables, the symbols x, y are possible
+values, and p(x, y) denotes the data distribution.
+Our goal is to design a policy that controls which features
+are selected given the currently available information. The
+selection policy can be viewed as a function π(xs) ∈ [d],
+meaning that it receives a subset of features as its input
+and outputs the next feature index to query. The policy is
+accompanied by a predictor f(xs) that can make predic-
+tions given the set of available features; for example, if y
+is discrete then predictions lie in the probability simplex,
+or f(xs) ∈ ∆K−1 for K classes. The notation f(xs ∪ xi)
+represents the prediction given the combined features. We
+initially consider policy and predictor functions that operate
+on feature subsets, and Section 4 proposes an implementa-
+tion using a mask variable m ∈ [0, 1]d where the functions
+operate on x ⊙ m.
+2.2. Dynamic feature selection
+The goal of DFS is to select features with minimal budget
+that achieve maximum predictive accuracy. Having access
+to more features generally makes prediction easier, so the
+challenge is selecting a small number of informative features.
+There are multiple formulations for this problem, including
+non-uniform feature costs and different budgets for each
+sample (Kachuee et al., 2018), but we focus on the setting
+with a fixed budget and uniform costs. Our goal is to handle
+data samples at test-time by beginning with no features,
+sequentially selecting features xs such that |s| = k for a
+fixed budget k < d, and finally making accurate predictions
+for the response variable y.
+Given a loss function that measures the discrepancy between
+predictions and labels ℓ(ˆy, y), a natural scoring criterion is
+the expected loss after selecting k features. The scoring is
+applied to a policy-predictor pair (π, f), and we define the
+score for a fixed budget k as follows,
+vk(π, f) = Ep(x,y)
+�
+ℓ
+�
+f
+�
+{xit}k
+t=1
+�
+, y
+��
+,
+(1)
+where feature indices are chosen sequentially for each (x, y)
+according to in = π({xit}n−1
+t=1 ). The goal is to minimize
+vk(π, f), or equivalently, to maximize our final predictive
+accuracy.
+One approach is to frame this as a Markov decision process
+(MDP) and solve it using standard RL techniques, so that
+π and f are trained to optimize a reward function based on
+eq. (1). Several recent works have designed such formula-
+tions (Shim et al., 2018; Kachuee et al., 2018; Janisch et al.,
+2019; Li & Oliva, 2021). However, these approaches are
+difficult to train effectively, so our work focuses on a greedy
+approach that is easier to learn and simpler to interpret.
+3. Greedy information maximization
+This section first defines the greedy CMI policy, and then
+describes an existing approximation strategy based on gen-
+erative modeling.
+3.1. The greedy selection policy
+As an idealized approach to DFS, we are interested in the
+greedy algorithm that selects the most informative feature
+at each step. This feature can be defined in multiple ways,
+but we focus on the information-theoretic perspective that
+the most useful feature has maximum CMI with the re-
+sponse variable (Cover & Thomas, 2012). CMI, denoted as
+I(xi; y | xs), quantifies how much information an unknown
+feature xi provides about the response y when accounting
+for the current features xs, and it is defined as the KL diver-
+gence between the joint and factorized distributions:
+I(xi; y | xs) = DKL
+�
+p(xi, y | xs) || p(xi | xs)p(y | xs)
+�
+.
+Based on this, we define the greedy CMI policy as π∗(xs) ≡
+arg maxi I(xi; y | xs), so that features are sequentially se-
+lected to maximize our information about the response vari-
+able. We can alternatively understand the policy as perform-
+ing greedy uncertainty minimization, because this is equiva-
+lent to minimizing y’s conditional entropy at each step, or
+π∗(xs) = arg mini H(y | xi, xs) (Cover & Thomas, 2012).
+For a complete characterization of this idealized approach
+to DFS, we also consider that the policy is paired with the
+Bayes classifier as a predictor, or f ∗(xs) = p(y | xs).
+
+Learning to Maximize Mutual Information for Dynamic Feature Selection
+Maximizing the information about y at each step is intuitive
+and should be effective in many problems. Prior work has
+considered the same idea, but from two perspectives that
+differ from ours. First, Chen et al. (2015b) take a theoretical
+perspective and prove that the greedy algorithm has bounded
+suboptimality relative to the optimal policy-predictor pair;
+the proof requires specific distributional assumptions, but
+we find that the greedy algorithm performs well with many
+real datasets (Section 6). Second, from an implementation
+perspective, two works aim to provide practical approxi-
+mations; however, these suffer from several limitations, so
+our work aims to develop a simple and flexible alternative
+(Section 4). In these works, Fleuret (2004) requires binary
+features, and Ma et al. (2019) requires a conditional gen-
+erative model of the data distribution, which we discuss
+next.
+3.2. Estimating conditional mutual information
+The greedy policy is trivial to implement if we can directly
+calculate CMI, but this is rarely the case in practice. Instead,
+one option to to estimate it. We now describe a procedure to
+do so iteratively for each feature, assuming for now that we
+have oracle access to the response distributions p(y | xs)
+for all s ⊆ [d] and the feature distributions p(xi | xs) for
+all s ⊆ [d] and i ∈ [d].
+At any point in the selection procedure, given the current
+features xs, we can estimate the CMI for a feature xi where
+i /∈ s as follows. First, we can sample multiple values for
+xi from its conditional distribution, or xj
+i ∼ p(xi | xs) for
+j ∈ [n]. Next, we can generate Bayes optimal predictions
+for each sampled value, or p(y | xs, xj
+i). Finally, we can
+calculate the mean prediction and the mean KL divergence
+relative to the mean prediction, which yields the following
+CMI estimator:
+In
+i = 1
+n
+n
+�
+j=1
+DKL
+�
+p(y | xs, xj
+i) || 1
+n
+n
+�
+l=1
+p(y | xs, xl
+i)
+�
+.
+(2)
+This score measures the variability among predictions and
+captures whether different xi values significantly affect y’s
+conditional distribution. The estimator can be used to select
+features, or we can set π(xs) = arg maxi In
+i , due to the
+following limiting result (see Appendix A):
+lim
+n→∞ In
+i = I(y; xi | xs).
+(3)
+This procedure thus provides a way to identify the correct
+greedy selections by estimating the CMI. Prior work has
+explored similar ideas for scoring features based on sam-
+pled predictions (Saar-Tsechansky et al., 2009; Chen et al.,
+2015a; Early et al., 2016a;b), but the implementation choices
+in these works prevent them from performing greedy infor-
+mation maximization. In eq. (2), is it important that our
+estimator uses the Bayes classifier, that we sample features
+from the conditional distribution p(xi | xs), and that we
+use the KL divergence as a measure of prediction variability.
+However, this estimator is impractical because we typically
+lack access to both p(y | xs) and p(xi | xs).
+In practice, we would instead require learned substitutes
+for each distribution. For example, we can use a a classi-
+fier that approximates f(xs) ≈ p(y | xs) and a generative
+model that approximates samples from p(xi | xs). Simi-
+larly, Ma et al. (2019) propose jointly modeling (x, y) with
+a conditional generative model, which is implemented via
+a modified VAE (Kingma et al., 2015). This approach is
+limited for several reasons, including (i) the difficulty of
+training an accurate conditional generative model, (ii) the
+challenge of modeling mixed continuous/categorical fea-
+tures (Ma et al., 2020; Nazabal et al., 2020), and (iii) the
+slow CMI estimation process. In our approach, which we
+discuss next, we bypass all three of these challenges by
+directly predicting the best selection at each step.
+4. Proposed method
+We now introduce our approach, a practical approximation
+the greedy policy trained using amortized optimization. Un-
+like prior work that estimates CMI as an intermediate step,
+we develop a variational perspective on the greedy policy,
+which we then leverage to train a policy network that directly
+predicts the optimal selection given the current features.
+4.1. A variational perspective on CMI
+For our purpose, it is helpful to recognize that the greedy
+policy can be viewed as the solution to an optimization
+problem. Section 3 provides a conventional definition of
+CMI as a KL divergence, but this is difficult to integrate into
+an end-to-end learning approach. Instead, we now consider
+the one-step-ahead prediction achieved by a policy π and
+predictor f, and we determine the behavior that minimizes
+their loss. Given the current features xs and a selection
+i = π(xs), the expected one-step-ahead loss is:
+Ey,xi|xs
+�
+ℓ
+�
+f(xs ∪ xi), y
+��
+.
+(4)
+The variational perspective we develop here consists of
+two main results regarding this expected loss. The first
+result concerns the predictor, and we show that the loss-
+minimizing predictor can be defined independently of the
+policy π. We formalize this in the following proposition for
+classification tasks, and our results can also be generalized
+to regression tasks (see proofs in Appendix A).
+Proposition 1. When y is discrete and ℓ is cross-entropy
+loss, eq. (4) is minimized for any policy π by the Bayes
+classifier, or f ∗(xs) = p(y | xs).
+
+Learning to Maximize Mutual Information for Dynamic Feature Selection
+The above property requires that features are selected with-
+out knowledge of the remaining features or response vari-
+able, which is a valid assumption for DFS, but not in scenar-
+ios where selections are based on the full feature set (Chen
+et al., 2018; Yoon et al., 2018; Jethani et al., 2021). Now,
+assuming that we use the Bayes classifier f ∗ as a predictor,
+our second result concerns the selection policy. As we show
+next, the loss-minimizing policy is equivalent to making
+selections based on CMI.
+Proposition 2. When y is discrete, ℓ is cross-entropy
+loss and the predictor is the Bayes classifier f ∗, eq. (4)
+is minimized by the greedy CMI policy, or π∗(xs) =
+arg maxi I(y; xi | xs).
+With this, we can see that the greedy CMI policy defined
+in Section 3 is equivalent to minimizing the one-step-ahead
+prediction loss. Next, we exploit this variational perspec-
+tive to develop a joint learning procedure for a policy and
+predictor network.
+4.2. An amortized optimization approach
+Instead of estimating each feature’s CMI to identify the next
+selection, we now develop an approach that directly pre-
+dicts the best selection at step. The greedy policy implicitly
+requires solving an optimization problem at each step, or
+arg maxi I(y, xi; xs), but since we lack access to this ob-
+jective, we now formulate an approach that directly predicts
+the solution. Following a technique known as amortized
+optimization (Amos, 2022), we do so by casting our varia-
+tional perspective on CMI from Section 4.1 as an objective
+function to be optimized by a learnable network.
+First, because it facilitates gradient-based optimization, we
+now consider that the policy outputs a distribution over fea-
+ture indices. With slight abuse of notation, this section lets
+the policy be a function π(xs) ∈ ∆d−1, which generalizes
+the previous definition π(xs) ∈ [d]. Using this stochas-
+tic policy, we can now formulate our objective function as
+follows.
+Let the selection policy be parameterized by a neural
+network π(xs; φ) and the predictor by a neural network
+f(xs; θ). Let p(s) represent a distribution over subsets with
+p(s) > 0 for all |s| < d. Then, our objective function
+L(θ, φ) is defined as
+L(θ, φ) = Ep(x,y)Ep(s)
+�
+Ei∼π(xs;φ)
+�
+ℓ
+�
+f(xs ∪ xi; θ), y
+���
+.
+(5)
+Intuitively, eq. (5) represents generating a random feature
+set xs, sampling a feature index according to i ∼ π(xs; φ),
+and then measuring the loss of the prediction f(xs ∪ xi; θ).
+Our objective thus optimizes for individual selections and
+predictions rather than the entire trajectory, which lets us
+build on Proposition 1-2. We describe this as an implemen-
+tation of the greedy approach because it recovers the greedy
+CMI selections when it is trained to optimality. In the clas-
+sification case, we show the following result under a mild
+assumption that there is a unique optimal selection.
+Theorem 1. When y is discrete and ℓ is cross-entropy loss,
+the global optimum of eq. (5) is a predictor that satisfies
+f(xs; θ∗) = p(y | xs) and a policy π(xs; φ∗) that puts all
+probability mass on i∗ = arg maxi I(y; xi | xs).
+If we relax the assumption of a unique optimal selection,
+the optimal policy π(xs; φ∗) will simply split probability
+mass among the best indices. A similar result holds in the
+regression case, where we can interpret the greedy policy as
+performing conditional variance minimization.
+Theorem 2. When y is continuous and ℓ is squared error
+loss, the global optimum of eq. (5) is a predictor that satisfies
+f(xs; θ∗) = E[y | xs] and a policy π(xs; φ∗) that puts all
+probability mass on i∗ = arg mini Exi|xs[Var(y | xi, xs)].
+Proofs for these results are in Appendix A. This approach
+has two key advantages over the CMI estimation procedure
+from Section 3.2. First, we avoid modeling the feature
+conditional distributions p(xi | xs) for all (s, i). Modeling
+these distributions is a difficult intermediate step, and our
+approach instead aims to directly output the optimal index.
+Second, our approach is faster because each selection is
+made in a single forward pass: selecting k features using
+the Ma et al. (2019) procedure requires O(dk) scoring steps,
+but our approach requires only k forward passes through the
+policy π(xs; φ).
+Furthermore, compared to a policy trained by RL, the
+greedy approach is easier to learn. Our training proce-
+dure can be viewed as a form of reward shaping (Sutton
+et al., 1998; Randløv & Alstrøm, 1998), where the reward
+accounts for the loss after each step and provides a strong
+signal about whether each selection is helpful. In compar-
+ison, observing the reward only after selecting k features
+provides a comparably weak signal to the policy network
+(see eq. (1)). RL methods generally face a challenging
+exploration-exploitation trade-off, but learning the greedy
+policy is simpler because it only requires finding the locally
+optimal choice at each step.
+4.3. Training with a continuous relaxation
+Our objective in eq. (5) yields the correct greedy policy
+when it is perfectly optimized, but L(θ, φ) is difficult to
+optimize by gradient descent. In particular, gradients are
+difficult to propagate through the policy network given a
+sampled index i ∼ π(xs; φ). The REINFORCE trick
+(Williams, 1992) is one way to get stochastic gradients,
+but high gradient variance can make it ineffective in many
+problems. There is a robust literature on reducing gradient
+
+Learning to Maximize Mutual Information for Dynamic Feature Selection
+𝜋 ⋅ ; 𝜙
+𝑓 ⋅ ; 𝜃
+Policy
+Predictor
+#𝑦
+Repeat for 𝑘 selection steps
+Concrete 𝛼, 𝜏
+Update masked
+input
+0
+𝑥#
+0
+𝑥%
+𝑥"
+𝑥#
+0
+𝑥%
+≈
+Figure 1. Diagram of our training approach. Left: features are selected by making repeated calls to the policy network using masked
+inputs. Right: predictions are made after each selection using the predictor network. Only solid lines are backpropagated through when
+performing gradient descent.
+variance in this setting (Tucker et al., 2017; Grathwohl et al.,
+2018), but we propose a simple alternative: the Concrete
+distribution (Maddison et al., 2016).
+An index sampled according to i ∼ π(xs; φ) can be rep-
+resented by a one-hot vector m ∈ {0, 1}d indicating the
+chosen index, and with the Concrete distribution we instead
+sample an approximately one-hot vector in the probability
+simplex, or m ∈ ∆d−1. This continuous relaxation lets us
+calculate gradients using the reparameterization trick (Mad-
+dison et al., 2016; Jang et al., 2016). Relaxing the subset
+s ⊆ [d] to a continuous vector also requires relaxing the
+policy and predictor functions, so we let these operate on
+a masked input x, or the element-wise product x ⊙ m. To
+avoid ambiguity about whether features are zero or masked,
+we can also pass the mask as an input.
+Training with the Concrete distribution requires specifying
+a temperature parameter τ > 0 to control how discrete the
+samples are. Previous works have typically trained with a
+fixed temperature or annealed it over a pre-determined num-
+ber of epochs (Chang et al., 2017; Chen et al., 2018; Balın
+et al., 2019), but we instead train with a sequence of τ values
+and perform early stopping at each step. This removes the
+temperature and number of epochs as important hyperpa-
+rameters to tune. Our training procedure is summarized in
+Figure 1, and in more detail by Algorithm 1.
+There are also several optional steps that we found can
+improve optimization:
+• Parameters can be shared between the predictor and pol-
+icy networks f(x; θ), π(x, φ). This does not complicate
+their joint optimization, and learning a shared represen-
+tation in the early layers can in some cases help the
+networks optimize faster.
+• Rather than training with a random subset distribution
+p(s), we generate subsets using features selected by
+the policy π(x; φ). This allows the models to focus on
+subsets likely to be encountered at inference time, and
+it does not affect the globally optimal policy/predictor:
+gradients are not propagated between selections, so both
+eq. (5) and this sampling approach treat each feature
+set as an independent optimization problem, only with
+different weights (see Appendix D).
+• We pre-train the predictor f(x; θ) using random subsets
+before jointly training the policy-predictor pair. This
+works better than optimizing L(θ, φ) from a random ini-
+tialization, because a random predictor f(x; θ) provides
+no signal to π(x; φ) about which features are useful.
+5. Related work
+Prior work has frequently addressed DFS using RL. For
+example, Dulac-Arnold et al. (2011); Shim et al. (2018);
+Janisch et al. (2019); Li & Oliva (2021) optimize a reward
+based on the final prediction accuracy, and Kachuee et al.
+(2018) use a reward that accounts for prediction uncertainty.
+RL is a natural approach for sequential decision-making
+problems, but it can be difficult to optimize in practice:
+RL requires complex architectures and training routines, is
+slow to converge, and is highly sensitive to its initialization
+(Henderson et al., 2018). As a result, RL-based DFS does
+not reliably outperform static feature selection, as shown by
+Erion et al. (2021) and confirmed in our experiments.
+Several other approaches include imitation learning (He
+et al., 2012; 2016a) and iterative feature scoring methods
+(Melville et al., 2004; Saar-Tsechansky et al., 2009; Chen
+et al., 2015a; Early et al., 2016b;a). Imitation learning casts
+DFS as supervised classification, whereas our training ap-
+proach bypasses the need for an oracle policy. Most existing
+feature scoring techniques are greedy methods, like ours,
+but they use scoring heuristics unrelated to maximizing
+CMI (see Section 3.2). Two feature scoring methods are
+specifically designed to calculate CMI, but they suffer from
+practical limitations: Fleuret (2004) requires binary features,
+and Ma et al. (2019) relies on difficult-to-train generative
+models. Our approach is simpler, faster and more flexi-
+ble, because the selection logic is contained within a policy
+network that avoids the need for generative modeling.
+
+Learning to Maximize Mutual Information for Dynamic Feature Selection
+0
+5
+10
+15
+20
+25
+# Selected Features
+0.55
+0.60
+0.65
+0.70
+0.75
+AUROC
+Bleeding AUROC Comparison
+0
+5
+10
+15
+20
+25
+# Selected Features
+0.65
+0.70
+0.75
+0.80
+0.85
+AUROC
+Respiratory AUROC Comparison
+2
+4
+6
+8
+10
+# Selected Features
+0.70
+0.75
+0.80
+0.85
+AUROC
+Fluid AUROC Comparison
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+IntGrad
+DeepLift
+SAGE
+Perm Test
+CAE
+Opportunistic (OL)
+CMI (Marginal)
+CMI (PVAE)
+Greedy (Ours)
+0
+5
+10
+15
+20
+25
+# Selected Features
+0.70
+0.75
+0.80
+0.85
+0.90
+0.95
+AUROC
+Spam AUROC Comparison
+0
+5
+10
+15
+20
+25
+# Selected Features
+0.65
+0.70
+0.75
+0.80
+0.85
+0.90
+0.95
+AUROC
+MiniBooNE AUROC Comparison
+2
+4
+6
+8
+10
+# Selected Features
+0.75
+0.80
+0.85
+0.90
+0.95
+AUROC
+Diabetes AUROC Comparison
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+IntGrad
+DeepLift
+SAGE
+Perm Test
+CAE
+Opportunistic (OL)
+CMI (Marginal)
+CMI (PVAE)
+Greedy (Ours)
+Figure 2. Evaluating the greedy approach on six tabular datasets. The results for each method are the average across five runs.
+Static feature selection is a long-standing problem (Guyon
+& Elisseeff, 2003; Cai et al., 2018). There are no default ap-
+proaches for neural networks, but one option is ranking fea-
+tures by local or global importance scores (Breiman, 2001;
+Shrikumar et al., 2017; Sundararajan et al., 2017; Covert
+et al., 2020). In addition, several prior works have leveraged
+continuous relaxations to learn feature selection strategies
+by gradient descent: for example, Chang et al. (2017); Balın
+et al. (2019); Yamada et al. (2020); Lee et al. (2021) perform
+static feature selection, and Chen et al. (2018); Jethani et al.
+(2021) perform instance-wise feature selection given all the
+features. Our work uses a similar continuous relaxation
+for optimization but in the DFS context, where our method
+learns a selection policy rather than a static selection layer.
+Finally, several works have examined greedy feature selec-
+tion algorithms from a theoretical perspective. For example,
+Das & Kempe (2011); Elenberg et al. (2018) show that
+weak submodularity implies near-optimal performance in
+the static feature selection setting. Chen et al. (2015b) find
+that the related notion of adaptive submodularity (Golovin
+& Krause, 2011) does not not apply to DFS when evaluated
+via mutual information, but manage to provide performance
+guarantees under specific distributional assumptions.
+6. Experiments
+We now demonstrate the use of our greedy approach on
+several datasets. We first explore tabular datasets of vari-
+ous sizes, including four medical diagnosis tasks, and we
+then consider two image classification datasets. Several
+of the tasks are natural candidates for DFS, and the re-
+maining ones serve as useful tasks to test the effectiveness
+of our approach. Code for reproducing our experiments
+is available online: https://github.com/iancovert/
+dynamic-selection.
+We evaluate our method by comparing to both dynamic and
+static feature selection methods. We also ensure consistent
+comparisons by only using methods applicable to neural
+networks. As static baselines, we use permutation tests
+(Breiman, 2001) and SAGE (Covert et al., 2020) to rank
+features by their importance to model accuracy, as well as
+per-prediction DeepLift (Shrikumar et al., 2017) and Int-
+Grad (Sundararajan et al., 2017) scores aggregated across
+the dataset. We then use a supervised version of the Con-
+crete Autoencoder (CAE, Balın et al. 2019), a state-of-the-
+art static feature selection method. As dynamic baselines,
+we use two versions of the CMI estimation procedure de-
+scribed in Section 3.2. First, we use the PVAE generative
+model from Ma et al. (2019) to sample unknown features,
+and second, we instead sample unknown features from their
+marginal distribution; in both cases, we use a classifier
+trained with random feature subsets to make predictions.
+Finally, we also use the RL-based Opportunistic Learning
+(OL) approach (Kachuee et al., 2018). Appendix C provides
+more information about each of the baselines.
+6.1. Tabular datasets
+We first applied our method to three medical diagnosis tasks
+derived from an emergency medicine setting. The tasks
+involve predicting a patient’s bleeding risk via a low fibrino-
+gen concentration (bleeding), whether the patient requires
+endotracheal intubation for respiratory support (respiratory),
+and whether the patient will be responsive to fluid resusci-
+
+Learning to Maximize Mutual Information for Dynamic Feature Selection
+Table 1. AUROC averaged across budgets of 1-10 features (with 95% confidence intervals).
+Spam
+MiniBooNE
+Diabetes
+Bleeding
+Respiratory
+Fluid
+Static
+IntGrad
+82.84 ± 0.68
+89.10 ± 0.33
+88.91 ± 0.24
+66.70 ± 0.27
+81.10 ± 0.04
+79.94 ± 0.94
+DeepLift
+90.16 ± 1.24
+88.62 ± 0.30
+95.42 ± 0.13
+67.75 ± 0.49
+76.05 ± 0.35
+76.96 ± 0.56
+SAGE
+89.70 ± 1.10
+92.64 ± 0.03
+95.43 ± 0.01
+71.34 ± 0.19
+82.92 ± 0.26
+83.27 ± 0.53
+Perm Test
+85.64 ± 3.58
+92.19 ± 0.15
+95.46 ± 0.02
+68.89 ± 1.06
+81.56 ± 0.28
+81.35 ± 1.04
+CAE
+92.28 ± 0.27
+92.76 ± 0.41
+95.91 ± 0.07
+70.69 ± 0.57
+83.10 ± 0.45
+79.40 ± 0.86
+Dynamic
+Opportunistic (OL)
+85.94 ± 0.20
+69.23 ± 0.64
+83.07 ± 0.82
+60.63 ± 0.55
+74.44 ± 0.42
+78.13 ± 0.31
+CMI (Marginal)
+86.57 ± 1.54
+92.21 ± 0.40
+95.48 ± 0.05
+70.57 ± 0.46
+79.62 ± 0.62
+81.97 ± 0.93
+CMI (PVAE)
+89.01 ± 1.40
+88.94 ± 1.25
+90.50 ± 5.16
+70.17 ± 0.74
+74.12 ± 3.50
+80.27 ± 1.02
+Greedy (Ours)
+93.91 ± 0.17
+94.46 ± 0.12
+96.03 ± 0.02
+72.64 ± 0.31
+84.48 ± 0.08
+86.59 ± 0.25
+tation (fluid). See Appendix B for more details about the
+datasets. In each scenario, gathering all possible inputs at
+test-time is challenging due to time and resource constraints,
+thus making DFS a natural solution.
+We use fully connected networks for all methods, and we
+use dropout to reduce overfitting (Srivastava et al., 2014).
+Figure 2 (top) shows the results of applying each method
+with various feature budgets. The classification accuracy is
+measured via AUROC, and the greedy method achieves the
+best results for nearly all feature budgets on all three tasks.
+Among the baselines, several static methods are sometimes
+close, but the CMI estimation method is rarely competitive.
+Additionally, OL provides unstable and weak results. The
+greedy method’s advantage is often largest when selecting a
+small number of features, and it usually becomes narrower
+once the accuracy saturates.
+Next, we conducted experiments using three publicly avail-
+able tabular datasets: spam classification (Dua & Graff,
+2017), particle identification (MiniBooNE) (Roe et al.,
+2005) and diabetes diagnosis (Miller, 1973). The diabetes
+task is a natural application for DFS and was used in prior
+work (Kachuee et al., 2018). We again tested various num-
+bers of features, and Figure 2 (bottom) shows plots of the
+AUROC for each feature budget. On these tasks, the greedy
+method is once again most accurate for nearly all numbers
+of features. Table 1 summarizes the results via the mean
+AUROC across k = 1, . . . , 10 features, further emphasizing
+the benefits of the greedy method across all six datasets.
+Appendix E shows larger versions of the AUROC curves
+(Figure 4 and Figure 5), as well as plots demonstrating the
+variability of selections within each dataset.
+The results with these datasets reveal that, perhaps surpris-
+ingly, dynamic methods can be outperformed by static meth-
+ods. Interestingly, this point was not highlighted in prior
+work where strong static baselines were not used (Kachuee
+et al., 2018; Janisch et al., 2019). For example, OL is never
+competitive on these datasets, and the two versions of the
+CMI estimation method are not consistently among the top
+baselines. Dynamic methods are in principle capable of
+performing better, so the sub-par results from these methods
+underscore the difficulty of learning both a selection policy
+and a prediction function that works for multiple feature
+sets. In these experiments, our approach is the only dynamic
+method to do both successfully.
+6.2. Image classification datasets
+Next, we considered two standard image classification
+datasets: MNIST (LeCun et al., 1998) and CIFAR-10
+(Krizhevsky et al., 2009). Our goal is to begin with a blank
+image, sequentially reveal multiple pixels or patches, and
+ultimately make a classification using a small portion of
+the image. Although this is not an obvious use case for
+DFS, it represents a challenging problem for our method,
+and similar tasks were considered in several earlier works
+(Karayev et al., 2012; Mnih et al., 2014; Early et al., 2016a;
+Janisch et al., 2019).
+For MNIST, we use fully connected architectures for both
+the policy and predictor, and we treat pixels as individual
+features, where d = 784. For CIFAR-10, we use a shared
+ResNet backbone (He et al., 2016b) for the policy and pre-
+dictor networks, and each network uses its own output head.
+The 32 × 32 images are coarsened into d = 64 patches
+of size 4 × 4, so the selector head generates logits corre-
+sponding to each patch, and the predictor head generates
+probabilities for each class.
+Figure 3 shows our method’s accuracy for different feature
+budgets. For MNIST, we use the previous baselines but ex-
+clude the CMI estimation method due to its computational
+cost. We observe a large benefit for our method, particu-
+larly when making a small number of selections: our greedy
+method reaches nearly 90% accuracy with just 10 pixels,
+which is roughly 10% higher than the best baseline and con-
+siderably higher than prior work (Balın et al., 2019; Yamada
+et al., 2020; Covert et al., 2020). OL yields the worst results,
+and it also trains slowly due to the large number of states.
+For CIFAR-10, we use two simple baselines: center crops
+and random masks of various sizes. For each method, we
+plot the mean and 95% confidence intervals determined
+from five trials. Our greedy approach is slightly less ac-
+curate with 1-2 patches, but it reaches significantly higher
+
+Learning to Maximize Mutual Information for Dynamic Feature Selection
+10
+20
+30
+40
+50
+# Selected Pixels
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+Top-1 Accuracy
+MNIST Accuracy Comparison
+IntGrad
+DeepLift
+SAGE
+Perm Test
+CAE
+Opportunistic (OL)
+Greedy (Ours)
+0
+5
+10
+15
+20
+25
+30
+# Selected Patches
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+Top-1 Accuracy
+CIFAR-10 Accuracy Comparison
+Center Crop
+Random Mask
+Greedy (Ours)
+Horse
+Truck
+Airplane
+Ship
+Frog
+Horse
+Ship
+Deer
+Dog
+Bird
+Prob = 55.34%
+Prob = 98.69%
+Prob = 99.98%
+Prob = 80.30%
+Prob = 97.80%
+Prob = 20.27%
+Prob = 99.01%
+Prob = 51.61%
+Prob = 52.17%
+Prob = 99.86%
+Figure 3. Greedy feature selection for image classification. Top left: accuracy comparison on MNIST with results averaged across five
+runs. Top right: accuracy comparison on CIFAR-10 with 95% confidence intervals. Bottom: example selections and predictions for the
+greedy method with 10 out of 64 patches for CIFAR-10 images.
+accuracy when using 5-20 patches. Figure 3 (bottom) also
+shows qualitative examples of our method’s predictions af-
+ter selecting 10 out of 64 patches, and Appendix E shows
+similar plots with different numbers of patches.
+7. Conclusion
+In this work, we explored a greedy algorithm for DFS that se-
+lects features based on their CMI with the response variable.
+We proposed an approach to approximate this policy by di-
+rectly predicting the optimal selection at each step, and we
+conducted experiments that show our method outperforms
+a variety of existing feature selection methods, including
+both dynamic and static baselines. Future work on this
+topic may include incorporating non-uniform features costs,
+determining the feature budget on a per-sample basis, and
+further characterizing the greedy suboptimality gap: some
+progress has been made in analyzing the greedy algorithm’s
+suboptimality in the dynamic setting (Chen et al., 2015b),
+but more general characterizations remain an open topic for
+future work.
+Acknowledgements
+We thank Samuel Ainsworth, Kevin Jamieson, Mukund
+Sudarshan and the Lee Lab for helpful discussions. This
+work was funded by NSF DBI-1552309 and DBI-1759487,
+NIH R35-GM-128638 and R01-NIA-AG-061132.
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+
+Learning to Maximize Mutual Information for Dynamic Feature Selection
+A. Proofs
+In this section, we re-state and prove our main theoretical results. We begin with our proposition regarding the optimal
+predictor for an arbitrary policy π.
+Proposition 1. When y is discrete and ℓ is cross-entropy loss, eq. (4) is minimized for any policy π by the Bayes classifier,
+or f ∗(xs) = p(y | xs).
+Proof. Given the predictor inputs xs, our goal is to determine the prediction that minimizes the expected loss. Because
+features are selected sequentially by π with no knowledge of the non-selected values, there is no other information to
+condition on; for the predictor, we do not even need to distinguish the order in which features were selected. We can
+therefore derive the optimal prediction ˆy ∈ ∆K−1 for a discrete response y ∈ [K] as follows:
+f ∗(xs) = arg min
+ˆy
+Ey|xs
+�
+ℓ(ˆy, y)
+�
+= arg min
+ˆy
+�
+i∈Y
+p(y = i | xs) log ˆyi
+= arg min
+ˆy
+DKL
+�
+p(y | xs) || ˆy
+�
++ H(y | xs)
+= p(y | xs).
+In the case of a continuous response y ∈ R with squared error loss, we have a similar result involving the response’s
+conditional expectation:
+f ∗(xs) = arg min
+ˆy
+Ey|xs
+�
+(ˆy − y)2�
+= arg min
+ˆy
+Ey|xs
+�
+(ˆy − E[y | xs])2�
++ Var(y | xs)
+= E[y | xs].
+Proposition 2. When y is discrete, ℓ is cross-entropy loss and the predictor is the Bayes classifier f ∗, eq. (4) is minimized
+by the greedy CMI policy, or π∗(xs) = arg maxi I(y; xi | xs).
+Proof. Following eq. (4), the policy network’s selection i = π(xs) incurs the following expected loss with the distribution
+p(y, xi | xs):
+Ey,xi|xs
+�
+ℓ(f ∗(xs ∪ xi), y)
+�
+= Ey,xi|xs
+�
+ℓ(p(y | xi, xs), y)
+�
+= Exi|xs
+�
+Ey|xi,xs[ℓ(p(y | xi, xs), y)]
+�
+= Exi|xs
+�
+H(y | xi, xs)
+�
+= H(y | xs) − I(y; xi | xs).
+Note that H(y | xs) is a constant that does not depend on i. When identifying the index that minimizes the expected loss,
+we thus have the following result:
+arg min
+i
+Ey,xi|xs
+�
+ℓ(f ∗(xs ∪ xi), y)
+�
+= arg max
+i
+I(y; xi | xs).
+
+Learning to Maximize Mutual Information for Dynamic Feature Selection
+In the case of a continuous response with squared error loss and an optimal predictor given by f ∗(xs) = E[y | xs], we have
+a similar result:
+Ey,xi|xs
+�
+(f ∗(xs ∪ xi) − y)2�
+= Ey,xi|xs
+�
+(E[y | xi, xs] − y)2�
+= Exi|xs
+�
+Ey|xi,xs[(E[y | xi, xs] − y)2]
+�
+= Exi|xs[Var(y | xi, xs)].
+When we aim to minimize the expected loss, our selection is thus the index that yields the lowest expected conditional
+variance:
+arg min
+i
+Exi|xs[Var(y | xi, xs)].
+We also prove the limiting result presented in eq. (3), which states that In
+i → I(y; xi | xs).
+Proof. Conditional mutual information I(y; xi | xs) is defined as follows (Cover & Thomas, 2012):
+I(y; xi | xs) = DKL
+�
+p(xi, y | xs) || p(xi | xs)p(y | xs)
+�
+= Ey,xi|xs
+�
+log
+p(y, xi | xs)
+p(xi | xs)p(y | xs)
+�
+.
+Rearranging terms, we can write this as an expected KL divergence with respect to xi:
+I(y; xi | xs) = Exi|xsEy|xs,xi
+�
+log
+p(y, xi | xs)
+p(xi | xs)p(y | xs)
+�
+= Exi|xsEy|xs,xi
+�
+log p(y | xi, xs)
+p(y | xs)
+�
+= Exi|xs
+�
+DKL
+�
+p(y | xi, xs) || p(y | xs)
+��
+Now, when we sample multiple values x1
+i , . . . , xn
+i ∼ p(xi | xs) and make predictions using the Bayes classifier, we have
+the following mean prediction as n becomes large:
+lim
+n→∞
+1
+n
+n
+�
+j=1
+p(y | xs, xj
+i) = Exi|xs
+�
+p(y | xi, xs)
+�
+= p(y | xs).
+Calculating the mean KL divergence across the predictions, we arrive at the following result:
+lim
+n→∞ In
+i = Exi|xs
+�
+DKL
+�
+p(y | xi, xs) || p(y | xs)
+��
+= I(y; xi | xs).
+Theorem 1. When y is discrete and ℓ is cross-entropy loss, the global optimum of eq. (5) is a predictor that satisfies
+f(xs; θ∗) = p(y | xs) and a policy π(xs; φ∗) that puts all probability mass on i∗ = arg maxi I(y; xi | xs).
+
+Learning to Maximize Mutual Information for Dynamic Feature Selection
+Proof. We first consider the predictor network f(xs; θ). When the predictor is given the feature values xs, it means that
+one index i ∈ s was chosen by the policy according to π(xs\i; φ) and the remaining indices s \ i were sampled from p(s).
+Because s is sampled independently from (x, y), and because π(xs\i; φ) is not given access to (x[d]\s, xi, y), the predictor’s
+expected loss must be considered with respect to the distribution y | xs. The globally optimal predictor f(xs; θ∗) is thus
+defined as follows, regardless of the selection policy π(xs; φ) and which index i was selected last:
+f(xs; θ∗) = arg min
+ˆy
+Ey|xs
+�
+ℓ(ˆy, y)
+�
+= p(y | xs).
+The above result follows from our proof for Proposition 1. Now, given the optimal predictor f(xs; θ∗), we can define the
+globally optimal policy by minimizing the expected loss for a fixed input xs. Denoting the probability mass placed on each
+index i ∈ [d] as πi(xs; φ), where π(xs; φ) ∈ ∆d−1, the expected loss is the following:
+Ei∼π(xs;φ)Ey,xi|xs
+�
+ℓ(f(xs ∪ xi; θ∗), y)
+�
+=
+�
+i∈[d]
+πi(xs; φ)Ey,xi|xs
+�
+ℓ
+�
+f(xs ∪ xi; θ∗), y
+��
+=
+�
+i∈[d]
+πi(xs; φ)Exi|xs[H(y | xi, xs)].
+The above result follows from our proof for Proposition 2. If there exists a single index i∗ ∈ [d] that yields the lowest
+expected conditional entropy, or
+Exi∗|xs[H(y | xi∗, xs)] < Exi|xs[H(y | xi, xs)]
+∀i ̸= i∗,
+then the optimal predictor must put all its probability mass on i∗, or πi∗(xs; φ∗) = 1. Note that the corresponding feature
+xi∗ has maximum conditional mutual information with y, because we have
+I(y; xi∗ | xs) = H(y | xs)
+�
+��
+�
+Constant
+−Exi∗|xs[H(y | xi∗, xs)].
+To summarize, we derived the global optimum to our objective L(θ, φ) by first considering the optimal predictor f(xs; θ∗),
+and then considering the optimal policy π(xs; φ∗) when we assume that we use the optimal predictor.
+Theorem 2. When y is continuous and ℓ is squared error loss, the global optimum of eq. (5) is a predictor that satisfies
+f(xs; θ∗) = E[y | xs] and a policy π(xs; φ∗) that puts all probability mass on i∗ = arg mini Exi|xs[Var(y | xi, xs)].
+Proof. Our proof follows the same logic as our proof for Theorem 1. For the optimal predictor given an arbitrary policy, we
+have:
+f(xs; θ∗) = arg min
+ˆy
+Ey|xs
+�
+(ˆy − y)2�
+= E[y | xs].
+Then, for the policy’s expected loss, we have:
+Ei∼π(xs;φ)Ey,xi|xs
+��
+f(xs ∪ xi; θ∗) − y
+�2�
+=
+�
+i∈[d]
+πi(xs; φ)Exi|xs[Var(y | xi, xs)].
+If there exists an index i∗ ∈ [d] that yields the lowest expected conditional variance, then the optimal policy must put all its
+probability mass on i∗, or πi∗(xs; φ∗) = 1.
+
+Learning to Maximize Mutual Information for Dynamic Feature Selection
+B. Datasets
+The datasets used in our experiments are summarized in Table 2. Three of the tabular datasets and the two image classification
+datasets are publicly available, and the three emergency medicine tasks were privately curated from the Harborview Medical
+Center Trauma Registry.
+Table 2. Summary of datasets used in our experiments.
+Dataset
+# Features
+# Feature Groups
+# Classes
+# Samples
+Fluid
+224
+162
+2
+2,770
+Respiratory
+112
+35
+2
+65,515
+Bleeding
+121
+44
+2
+6,496
+Spam
+58
+–
+2
+4,601
+MiniBooNE
+51
+–
+2
+130,064
+Diabetes
+45
+–
+3
+92,062
+MNIST
+784
+–
+10
+60,000
+CIFAR-10
+1,024
+64
+10
+60,000
+B.1. MiniBooNE and spam classification
+The spam dataset includes features extracted from e-mail messages to predict whether or not a message is spam. Three
+features describes the usage of capital letters in the e-mail, and the remaining 54 features describe the frequency with which
+certain key words or characters are used. The MiniBooNE particle identification dataset involves distinguishing electron
+neutrinos from muon neutrinos based on various continuous features (Roe et al., 2005). Both datasets were obtained from
+the UCI repository (Dua & Graff, 2017).
+B.2. Diabetes classification
+The diabetes dataset was obtained from from the National Health and Nutrition Examination Survey (NHANES) (NHA,
+2018), an ongoing survey designed to assess the well-being of adults and children in the United States. We used a version
+of the data pre-processed by Kachuee et al. (2018; 2019) that includes data collected from 1999 through 2016. The input
+features include demographic information (age, gender, ethnicity, etc.), lab results (total cholesterol, triglyceride, etc.),
+examination data (weight, height, etc.), and questionnaire answers (smoking, alcohol, sleep habits, etc.). An expert was also
+asked to suggest costs for each feature based on the financial burden, patient privacy, and patient inconvenience, but we
+assume uniform feature costs in our experiments. Finally, the fasting glucose values were used to define three classes based
+on standard threshold values: normal, pre-diabetes, and diabetes.
+B.3. Image classification datasets
+The MNIST and CIFAR-10 datasets were downloaded using PyTorch (Paszke et al., 2017). We used the standard train-test
+splits, and we split the train set to obtain a validation set with the same size as the test set (10,000 examples).
+B.4. Emergency medicine datasets
+The emergency medicine datasets used in this study were gathered over a 13-year period (2007-2020) and encompass 14,463
+emergency department admissions. We excluded patients under the age of 18, and we curated 3 clinical cohorts commonly
+seen in pre-hospitalization settings. These include 1) pre-hospital fluid resuscitation, 2) emergency department respiratory
+support, and 3) bleeding after injury. These datasets are not publicly available due to patient privacy concerns.
+Pre-hospital fluid resuscitation
+We selected 224 variables that were available in the pre-hospital setting, including
+dispatch information (injury date, time, cause, and location), demographic information (age, sex), and pre-hospital vital
+signs (blood pressure, heart rate, respiratory rate). The outcome was each patient’s response to fluid resuscitation, following
+the Advanced Trauma Life Support (ATLS) definition (Subcommittee et al., 2013).
+
+Learning to Maximize Mutual Information for Dynamic Feature Selection
+Emergency department respiratory support
+In this cohort, our goal is to predict which patients require respiratory
+support upon arrival in the emergency department. Similar to the previous dataset, we selected 112 pre-hospital clinical
+features including dispatch information (injury date, time, cause, and location), demographic information (age, sex), and
+pre-hospital vital signs (blood pressure, heart rate, respiratory rate). The outcome is defined based on whether a patient
+received respiratory support, including both invasive (intubation) and non-invasive (BiPap) approaches.
+Bleeding
+In this cohort, we only included patients whose fibrinogen levels were measured, as this provides an indicator for
+bleeding or fibrinolysis (Mosesson, 2005). As with the previous datasets, demographic information, dispatch information,
+and pre-hospital observations were used as input features. The outcome, based on experts’ opinion, was defined by whether
+an individual’s fibrinogen level is below 200 mg/dL, which represents higher risk of bleeding after injury.
+C. Baselines
+This section provides more details on the baseline methods used in our experiments (Section 6).
+C.1. Global feature importance methods
+Two of our static feature selection baselines, permutation tests and SAGE, are global feature importance methods that rank
+features based on their role in improving model accuracy (Covert et al., 2021). In our experiments, we ran each method
+using a single classifier trained on the entire dataset, and we then selected the top k features depending on the budget.
+When running the permutation test, we calculated the validation AUROC while replacing values in the corresponding feature
+column with random draws from the training set. When running SAGE, we used the authors’ implementation with automatic
+convergence detection (Covert et al., 2020). To handle held-out features, we averaged across 128 sampled values for the six
+tabular datasets, and for MNIST we used a zeros baseline to achieve faster convergence.
+C.2. Local feature importance methods
+Two of our static feature selection baselines, DeepLift and Integrated Gradients, are local feature importance methods that
+rank features based on their importance to a single prediction. In our experiments, we generated feature importance scores
+for the true class using all examples in the validation set. We then selected the top k features based on their mean absolute
+importance. We used a mean baseline for Integrated Gradients (Sundararajan et al., 2017), and both methods were run using
+the Captum package (Kokhlikyan et al., 2020).
+C.3. CMI estimation
+Our experiments use two versions of the CMI estimation approach described in Section 3.2. Both are inspired by the
+EDDI method introduced by Ma et al. (2019), but a key difference is that we do not jointly model (x, y) within the same
+conditional generative model: we instead separately model the response with a classifier f(xs) ≈ p(y | xs) and the features
+with a generative model of p(xi | xs). This partially mitigates one challenge with this approach, which is working with
+mixed continuous/categorical data (i.e., we do not need to jointly model categorical response variables).
+For the first version of this approach, we train a PVAE as a generative model (Ma et al., 2019). The encoder and decoder both
+have two hidden layers, the latent dimension is set to 16, and we use 128 samples from the latent posterior to approximate
+p(xi | xs) =
+�
+p(xi | z)p(z | xs). We use Gaussian distributions for both the latent and decoder spaces, and we generate
+samples using the decoder mean, similar to the original approach (Ma et al., 2019). In the second version, we bypass the
+need for a generative model with a simple approximation: we sample features from their marginal distribution, which is
+equivalent to assuming feature independence.
+C.4. Opportunistic learning
+Kachuee et al. (2018) proposed Opportunistic Learning (OL), an approach to solve DFS using RL. The model consists
+of two networks analogous to our policy and predictor: a Q-network that estimates the value associated with each action,
+where actions correspond to features, and a P-network responsible for making predictions. When using OL, we use the same
+architectures as our approach, and OL shares network parameters between the P- and Q-networks.
+The authors introduce a utility function for their reward, shown in eq. (6), which calculates the difference in prediction
+
+Learning to Maximize Mutual Information for Dynamic Feature Selection
+uncertainty as approximated by MC dropout (Gal & Ghahramani, 2016). The reward also accounts for feature costs, but we
+set all feature costs to ci = 1:
+ri = ||Cert(xs) − Cert(xs ∪ xi)||
+ci
+(6)
+To provide a fair comparison with the remaining methods, we made several modifications to the authors’ implementation.
+These include 1) preventing the prediction action until the pre-specified budget is met, 2) setting all feature costs to be
+identical, and 3) supporting pre-defined feature groups as described in Appendix D.3. When training, we update the P-,
+Q-, and target Q-networks every 1 +
+d
+100 experiences, where d is the number of features in a dataset. In addition, the
+replay buffer is set to store the 1000d most recent experiences, and the random exploration probability is decayed so that it
+eventually reaches a value of 0.1.
+D. Training approach and hyperparameters
+This section provides more details on our training approach and hyperparameter choices.
+D.1. Training pseudocode
+Algorithm 1 summarizes our training approach. Briefly, we select features by drawing a Concrete sample using policy
+network’s logits, we calculate the loss based on the subsequent prediction, and we then update the mask for the next step
+using a discrete sample from the policy’s distribution. We implemented this approach using PyTorch (Paszke et al., 2017)
+and PyTorch Lightning2.
+Algorithm 1: Training pseudocode
+Input: Data distribution p(x, y), budget k > 0, learning rate γ > 0, temperature τ > 0
+Output: Predictor model f(x; θ), policy model π(x; φ)
+initialize f(x; θ), π(x; φ)
+while not converged do
+sample x, y ∼ p(x, y)
+initialize L = 0, m = [0, . . . , 0]
+for j = 1 to k do
+calculate logits α = π(x ⊙ m; φ), sample Gi ∼ Gumbel for i ∈ [d]
+set ˜m = max
+�
+m, softmax(G + α, τ)
+�
+// update with Concrete
+set m = max
+�
+m, softmax(G + α, 0)
+�
+// update with one-hot
+update L ← L + ℓ
+�
+f(x ⊙ ˜m; θ), y
+�
+end
+update θ ← θ − γ∇θL, φ ← φ − γ∇φL
+end
+return f(x; θ), π(x; φ)
+One notable difference between Algorithm 1 and our objective L(θ, φ) in the main text is the use of the policy π(x; φ) for
+generating feature subsets. This differs from eq. (5), which generates feature subsets using a subset distribution p(s). The
+key shared factor between both approaches is that there are separate optimization problems over each feature set that are
+effectively treated independently. For each feature set xs, the problem is the one-step-ahead loss, and it incorporates both
+the policy and predictor as follows:
+Ei∼π(xs;φ)
+�
+ℓ
+�
+f(xs ∪ xi; θ), y
+��
+.
+(7)
+The problems for each subset do not interact: during optimization, the selection given xs is based only on the immediate
+change in the loss, and gradients are not propagated through multiple selections as they would be for an RL-based solution.
+In solving these multiple problems, the difference is simply that eq. (5) weights them according to p(s), whereas Algorithm 1
+weights them according to the current policy π(x, φ).
+2https://www.pytorchlightning.ai
+
+Learning to Maximize Mutual Information for Dynamic Feature Selection
+D.2. Hyperparameters
+Our experiments with the six tabular datasets all used fully connected architectures with dropout in all layers (Srivastava
+et al., 2014). The dropout probability is set to 0.3, the networks have two hidden layers of width 128, and we performed
+early stopping using the validation loss. For our method, the predictor and policy were separate networks with identical
+architectures. When training models with the features selected by static methods, we reported results using the best model
+from multiple training runs based on the validation loss. We did not perform any additional hyperparameter tuning due to
+the large number of models being trained.
+For MNIST, we used fully connected architectures with two layers of width 512 and the dropout probability set to 0.3.
+Again, our method used separate networks with identical architectures. For CIFAR-10, we used a shared ResNet backbone
+(He et al., 2016b) consisting of several residually connected convolutional layers. The classification head consists of global
+average pooling and a linear layer, and the selection head consisted of a transposed convolution layer followed by a 1 × 1
+convolution, which output a grid of logits with size 8 × 8. Our CIFAR-10 networks are trained using random crops and
+random horizontal flips as augmentations.
+D.3. Feature grouping
+All of the methods used in our experiments were designed to select individual features, but this is undesirable when using
+categorical features with one-hot encodings. Each of our three emergency medicine tasks involve such features, so we
+extended each method to support feature grouping.
+SAGE and permutation tests are trivial to extend to feature groups: we simply removed groups of features rather than
+individual features when calculating importance scores. For DeepLift and Integrated Gradients, we used the summed
+importance within each group, which preserves each method’s additivity property. For the method based on Concrete
+Autoencoders, we implemented a generalized version of the selection layer that operates on feature groups. We also extended
+OL to operate on feature groups by having actions map to groups rather than individual features.
+Finally, for our method, we parameterized the policy network π(x; φ) so that the number of outputs is the number of groups
+g rather than the total number of features d (where g < d). When applying masking, we first generate a binary mask
+m ∈ [0, 1]g, and we then project the mask into [0, 1]d using a binary group matrix G ∈ {0, 1}d×g, where Gij = 1 if feature
+i is in group j and Gij = 0 otherwise. Thus, our masked input vector is given by x ⊙ (Gm).
+E. Additional results
+This section provides several additional experimental results. First, Figure 4 and Figure 5 show the same results as Figure 2
+but larger for improved visibility. Next, Figure 6 though Figure 11 display the feature selection frequency for each of the
+tabular datasets when using the greedy method. The heatmaps in each plot show the portion of the time that a feature (or
+feature group) is selected under a specific feature budget. These plots reveal that our method is indeed selecting different
+features for different samples.
+Finally, Figure 12 displays examples of CIFAR-10 predictions given different numbers of revealed patches. The predictions
+generally become relatively accurate after revealing only a small number of patches, reflecting a similar result as Figure 3.
+Qualitatively, we can see that the policy network learns to select vertical stripes, but the order in which it fills out each stripe
+depends on where it predicts important information may be located.
+
+Learning to Maximize Mutual Information for Dynamic Feature Selection
+0
+5
+10
+15
+20
+25
+# Selected Features
+0.55
+0.60
+0.65
+0.70
+0.75
+AUROC
+Bleeding AUROC Comparison
+0
+5
+10
+15
+20
+25
+# Selected Features
+0.65
+0.70
+0.75
+0.80
+0.85
+AUROC
+Respiratory AUROC Comparison
+2
+4
+6
+8
+10
+# Selected Features
+0.700
+0.725
+0.750
+0.775
+0.800
+0.825
+0.850
+0.875
+AUROC
+Fluid AUROC Comparison
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+IntGrad
+DeepLift
+SAGE
+Perm Test
+CAE
+Opportunistic (OL)
+CMI (Marginal)
+CMI (PVAE)
+Greedy (Ours)
+Figure 4. AUROC comparison on the three emergency medicine diagnosis tasks.
+
+Learning to Maximize Mutual Information for Dynamic Feature Selection
+0
+5
+10
+15
+20
+25
+# Selected Features
+0.70
+0.75
+0.80
+0.85
+0.90
+0.95
+AUROC
+Spam AUROC Comparison
+0
+5
+10
+15
+20
+25
+# Selected Features
+0.65
+0.70
+0.75
+0.80
+0.85
+0.90
+0.95
+AUROC
+MiniBooNE AUROC Comparison
+2
+4
+6
+8
+10
+# Selected Features
+0.75
+0.80
+0.85
+0.90
+0.95
+AUROC
+Diabetes AUROC Comparison
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+IntGrad
+DeepLift
+SAGE
+Perm Test
+CAE
+Opportunistic (OL)
+CMI (Marginal)
+CMI (PVAE)
+Greedy (Ours)
+Figure 5. AUROC comparison on the three public tabular datasets.
+
+Learning to Maximize Mutual Information for Dynamic Feature Selection
+0
+5
+10
+15
+20
+25
+30
+35
+40
+Feature Index
+0
+5
+10
+15
+20
+25
+# Selections
+Bleeding Feature Selection Frequency
+0.0
+0.2
+0.4
+0.6
+0.8
+Figure 6. Feature selection frequency for our greedy approach on the bleeding dataset.
+0
+5
+10
+15
+20
+25
+30
+Feature Index
+0
+5
+10
+15
+20
+25
+# Selections
+Respiratory Feature Selection Frequency
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Figure 7. Feature selection frequency for our greedy approach on the respiratory dataset.
+0
+20
+40
+60
+80
+100
+120
+140
+160
+Feature Index
+0
+5
+10
+15
+20
+25
+# Selections
+Fluid Feature Selection Frequency
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Figure 8. Feature selection frequency for our greedy approach on the fluid dataset.
+
+Learning to Maximize Mutual Information for Dynamic Feature Selection
+0
+10
+20
+30
+40
+50
+Feature Index
+0
+5
+10
+15
+20
+25
+# Selections
+Spam Feature Selection Frequency
+0.0
+0.2
+0.4
+0.6
+0.8
+Figure 9. Feature selection frequency for our greedy approach on the spam dataset.
+0
+10
+20
+30
+40
+Feature Index
+0
+5
+10
+15
+20
+25
+# Selections
+MiniBooNE Feature Selection Frequency
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Figure 10. Feature selection frequency for our greedy approach on the MiniBooNE dataset.
+0
+5
+10
+15
+20
+25
+30
+35
+40
+Feature Index
+0
+5
+10
+15
+20
+25
+# Selections
+Diabetes Feature Selection Frequency
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Figure 11. Feature selection frequency for our greedy approach on the diabetes dataset.
+
+Learning to Maximize Mutual Information for Dynamic Feature Selection
+Full Image
+Horse
+Automobile
+Truck
+Cat
+Dog
+Frog
+Ship
+1 Patches
+Prob = 8.04%
+Prob = 27.61%
+Prob = 12.60%
+Prob = 12.08%
+Prob = 69.06%
+Prob = 33.50%
+Prob = 40.99%
+2 Patches
+Prob = 19.55%
+Prob = 94.60%
+Prob = 3.71%
+Prob = 19.11%
+Prob = 85.20%
+Prob = 78.33%
+Prob = 52.80%
+5 Patches
+Prob = 48.14%
+Prob = 99.99%
+Prob = 16.02%
+Prob = 27.03%
+Prob = 99.98%
+Prob = 94.11%
+Prob = 94.02%
+10 Patches
+Prob = 76.57%
+Prob = 99.97%
+Prob = 94.65%
+Prob = 41.52%
+Prob = 99.99%
+Prob = 99.75%
+Prob = 82.71%
+15 Patches
+Prob = 92.00%
+Prob = 100.00%
+Prob = 88.88%
+Prob = 72.54%
+Prob = 99.97%
+Prob = 99.90%
+Prob = 98.36%
+20 Patches
+Prob = 81.35%
+Prob = 100.00%
+Prob = 96.01%
+Prob = 79.03%
+Prob = 99.93%
+Prob = 99.89%
+Prob = 99.90%
+25 Patches
+Prob = 97.02%
+Prob = 100.00%
+Prob = 96.34%
+Prob = 75.32%
+Prob = 99.91%
+Prob = 99.56%
+Prob = 99.88%
+30 Patches
+Prob = 91.91%
+Prob = 100.00%
+Prob = 96.29%
+Prob = 66.15%
+Prob = 99.78%
+Prob = 99.35%
+Prob = 99.86%
+Figure 12. CIFAR-10 predictions with different numbers of patches revealed by our approach.
+
+1

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+Magnetic anisotropy and low-energy spin dynamics in magnetic van der Waals
+compounds Mn2P2S6 and MnNiP2S6
+J. J. Abraham,1, 2, ∗ Y. Senyk,1, 2, ∗ Y. Shemerliuk,1 S. Selter,1, 2
+S. Aswartham,1 B. B¨uchner,1, 3 V. Kataev,1 and A. Alfonsov1, 3
+1Leibniz IFW Dresden, D-01069 Dresden, Germany
+2Institute for Solid State and Materials Physics, TU Dresden, D-01062 Dresden, Germany
+3Institute for Solid State and Materials Physics and W¨urzburg-Dresden
+Cluster of Excellence ct.qmat, TU Dresden, D-01062 Dresden, Germany
+(Dated: January 12, 2023)
+We report the detailed high-field and high-frequency electron spin resonance (HF-ESR) spectro-
+scopic study of the single-crystalline van der Waals compounds Mn2P2S6 and MnNiP2S6. Analysis
+of magnetic excitations shows that in comparison to Mn2P2S6 increasing the Ni content yields a
+larger magnon gap in the ordered state and a larger g-factor value and its anisotropy in the param-
+agnetic state. The studied compounds are found to be strongly anisotropic having each the unique
+ground state and type of magnetic order. Stronger deviation of the g-factor from the free electron
+value in the samples containing Ni suggests that the anisotropy of the exchange is an important
+contributor to the stabilization of a certain type of magnetic order with particular anisotropy. At
+the temperatures above the magnetic order, we have analyzed the spin-spin correlations resulting
+in a development of slowly fluctuating short-range order. They are much stronger pronounced in
+MnNiP2S6 compared to Mn2P2S6. The enhanced spin fluctuations in MnNiP2S6 are attributed to
+the competition of different types of magnetic order. Finally, the analysis of the temperature de-
+pendent critical behavior of the magnon gaps below the ordering temperature in Mn2P2S6 suggest
+that the character of the spin wave excitations in this compound undergoes a field induced crossover
+from a 3D-like towards 2D XY regime.
+I.
+INTRODUCTION
+In the past recent years magnetic van der Waals (vdW)
+materials have become increasingly attractive for the fun-
+damental investigations since they provide immense pos-
+sibility to study intrinsic magnetism in low dimensional
+limit [1–3].
+The weak vdW forces hold together the
+atomic monolayers in vdW crystals, which results in a
+poor interlayer coupling, and therefore renders these ma-
+terials intrinsically two dimensional. In addition to the
+fundamental research, these materials are very promis-
+ing as potential candidates for next-generation spintron-
+ics devices [4–7].
+Among the variety of magnetic vdW materials a par-
+ticularly interesting subclass is represented by the anti-
+ferromagnetic (TM)2P2S6 tiophosphates (TM stands for
+a transition metal ion). Here the transition metal ions
+are arranged in a graphene-like layered honeycomb lat-
+tice [8]. The high flexibility of the choice of the TM ion
+enables to control the properties. Among the tiophos-
+phates there are examples of superconductors [9], pho-
+todetectors and field effect transistors [10, 11]. They also
+can be used for ion-exchange applications [12], catalytic
+activity [13], etc. Therefore, a proper choice of TM, or of
+a mixture of magnetically inequivalent ions on the same
+crystallographic position, could lead to the possibility of
+engineering of a material with desired magnetic ground
+state, excitations and correlations.
+∗ These authors contributed equally to this work.
+In order to establish the connection between the choice
+of magnetic ion and the resulting ground state and cor-
+relations, we performed a detailed high-field and high-
+frequency electron spin resonance (HF-ESR) spectro-
+scopic study on single crystals of the van der Waals com-
+pounds Mn2P2S6 and MnNiP2S6 in a broad range of mi-
+crowave frequencies and temperatures below and above
+the magnetic order. ESR spectroscopy is a powerful tool
+that can provide insights into spin-spin correlations, mag-
+netic anisotropy and spin dynamics. This technique has
+shown to be very effective for exploration of the mag-
+netic properties of vdW systems [14–25].
+Albeit reso-
+nance studies on Mn2P2S6 were made by Okuda et al.
+[14], Joy and Vasudevan [15] and Kobets et al. [18], a
+high-frequency ESR study exploring broad range of tem-
+peratures below and above magnetic order was not yet
+performed. The MnNiP2S6 compound is barely explored
+from the point of view of spin excitations from the mag-
+netic ground state below the ordering temperature, and
+from the point of view of spin-spin correlations in the
+high temperature regime.
+Investigating Mn2P2S6 and MnNiP2S6 we have found
+difference in the types of magnetic order, anisotropies be-
+low the ordering temperature TN, as well as the g-factors
+and their anisotropy above TN in these compounds. In
+fact, increasing the Ni content yields a larger magnon
+gap in the ordered state (T << TN) and a larger g-
+factor value and its anisotropy in the paramagnetic state
+(T >> TN). At temperatures above the magnetic or-
+der, we have analyzed the spin-spin correlations result-
+ing in a development of slowly fluctuating short-range or-
+der. They are much stronger pronounced in MnNiP2S6
+arXiv:2301.04239v1  [cond-mat.str-el]  10 Jan 2023
+
+2
+compared to Mn2P2S6, which in our previous study has
+shown clear cut signatures of 2D correlated spin dy-
+namics [25].
+Therefore, enhanced spin fluctuations in
+MnNiP2S6 are attributed to the competition of differ-
+ent types of magnetic order. Finally, the analysis of the
+temperature dependent critical behavior of the magnon
+gaps below the ordering temperature in Mn2P2S6 sug-
+gest that the character of the spin wave excitations in
+this compound undergoes a field induced crossover from
+a 3D-like towards 2D XY regime.
+II.
+EXPERIMENTAL DETAILS
+Crystal growth of Mn2P2S6 and MnNiP2S6 samples
+investigated in this work was done using the chemical
+vapor transport technique with iodine as the transport
+agent. Details of their growth, crystallographic, composi-
+tional and static magnetic characterization are described
+in Refs. [26, 27]. Note, that the experimental value xexp
+in (Mn1−xNix)2P2S6 for the nominal MnNiP2S6 com-
+pound is found to be xexp = 0.45, considering an uncer-
+tainty of approximately 5% [27]. Both materials exhibit
+a monoclinic crystal lattice system with a C2/m space
+group [27, 28]. Each unit cell contains a [P2S6]4− cluster
+with S atoms occupying the edges of TM octahedra and
+P-P dumbbells occupying the void of each metal honey-
+comb sublattice. The crystallographic c-axis makes an
+angle of 17◦ with the normal to the ab-plane [29], which
+is known to be one of the magnetic axes and is hereafter
+called as c* [18].
+The ordering temperature of Mn2P2S6 is found to be
+TN = 77 K [27].
+In contrast, the transition tempera-
+ture of MnNiP2S6 is rather uncertain and might depend
+on the direction of the applied magnetic field. Various
+studies have reported different values of TN, for instance
+it amounts to 12 K in [30], 38 K in [27], 41 K in [31]
+and 42 K in [32].
+For the samples used in this study
+the ordering temperatures were extracted from the tem-
+perature dependence of the susceptibility χ measured at
+H = 1000 Oe (see appendix, Fig. 10 (a)). The calcula-
+tion of the maximum value of the derivative d(χ · T)/dT
+yields TN ∼ 57 K for H ∥ c* and TN ∼ 76 K for H ⊥ c*
+(hereafter called TN*).
+The antiferromagnetic resonance (AFMR) and ESR
+measurements (hereafter called HF-ESR) were performed
+on several single crystalline samples of Mn2P2S6 and
+MnNiP2S6 using a homemade HF-ESR spectrometer. A
+superconducting magnet from Oxford instruments with
+a variable temperature insert (VTI) was used to gener-
+ate magnetic fields up to 16 T allowing a continuous field
+sweep. The sample was mounted on a probe head which
+was then inserted into the VTI immersed in a 4He cryo-
+stat. A piezoelectric step-motor based sample holder was
+used for angular dependent measurements. Continuous
+He gas flow was utilized to attain stable temperatures
+in the range of 3 to 300 K. Generation and detection of
+microwaves was performed using a vector network an-
+6
+7
+8
+9 10 11 12 13
+SD (arb. u.)
+Magnetic Field (T)
+ b) MnNiP2S6, �  = 326 GHz
+300 K
+250 K
+200 K
+175 K
+150 K
+140 K
+120 K
+100 K
+90 K
+80 K
+70 K
+60 K
+50 K
+45 K
+40 K
+35 K
+30 K
+20 K
+10 K
+7.5 K
+5 K
+3 K
+2
+4
+6
+B5
+SD (arb. u.)
+Magnetic Field (T)
+70 K
+65 K
+60 K
+90 K
+80 K
+75 K
+*
+**
+*
+*
+*
+*
+****
+*
+50 K
+40 K
+30 K
+20 K
+10 K
+7.5 K
+5 K
+3 K
+293 K
+250 K
+194 K
+171 K
+150 K
+130 K
+110 K
+100 K
+B4
+a) Mn2P2S6, �  = 147 GHz
+FIG. 1. Temperature dependence of HF-ESR spectra of (a)
+Mn2P2S6 at fixed excitation frequency ν ≈ 147 GHz and (b)
+MnNiP2S6 at ν ≈ 326 GHz in H ∥ c* configuration. Spectra
+are normalized and vertically shifted for clarity. The temper-
+ature independent peaks from the impurity in the probehead
+occurring at low frequencies are marked with asterisks.
+alyzer (PNA-X) from Keysight Technologies. Equipped
+with the frequency extensions from Virginia Diodes, Inc.,
+the PNA-X can generate a frequency in the range from
+75 to 330 GHz. The measurements are performed in the
+transmission mode, where the microwaves are directed
+to the sample using oversized waveguides. All measure-
+ments were made by sweeping the field from 0 to 16 T
+and back to 0 T at constant temperature and frequency.
+HF-ESR signals generally have a Lorentzian line pro-
+file with an absorption and dispersion components. For
+such a case, the resonance field (Hres) and linewidth (full
+width at half maxima, ∆H) can be extracted by fitting
+the signal using the function:
+SD(H) = 2Amp
+π
+× (L1sinα + L2cosα)
++ Coffset + CslopeH
+(1)
+where SD(H) is the signal at the detector and Amp is
+the amplitude. Coffset represents the offset and CslopeH
+is the linear background of the spectra.
+L1 is the
+Lorentzian absorption which is defined in terms of Hres
+and ∆H. L2 is the Lorentzian dispersion which is ob-
+tained by applying the Kramers-Kronig transformation
+to L1. α is a parameter used to define the degree of in-
+strumental mixing of the absorption and dispersion com-
+ponents which is unavoidable in the used setup. Some
+of the HF-ESR signals of Mn2P2S6 could not be fitted
+using the above equation due to the development of the
+shoulders or the splitting of peaks [33]. ∆H, Hres and,
+
+3
+0
+50
+100
+150
+200
+250
+300
+-4
+-2
+0
+2
+4
+6
+� H = Hres(T) - Hres(300 K) (T)
+Temperature (K)
+ 88 GHz, H || c*
+ 88 GHz, H || c*
+ 147 GHz, H || c*
+ 147 GHz, H || c*
+ 329 GHz, H || c*
+ 329 GHz, H ^ c*
+TN
+Mn2P2S6
+B5
+B4
+B3
+0
+100
+200
+300
+0.05
+0.10
+0.15
+0.20
+0.25
+�  = 329 GHz
+ H || c*
+ H ^ c*
+∆H = Linewidth (T)
+Temperature (K)
+TN
+100
+150
+200
+250
+300
+-0.1
+0.0
+0.1
+FIG. 2. Shift of the resonance field position δH (main panel)
+and linewidth ∆H (inset) as a function of temperature. The
+horizontal dashed line represents zero shift from the room
+temperature value and the vertical dashed line (also for inset)
+represents the N´eel temperature of the material.
+therefore, δH = Hres - Hres(300 K) were then obtained
+by picking a position of the peak value and by calculating
+the full width at half maximum.
+III.
+RESULTS
+A.
+Temperature dependence of HF-ESR response
+To study the temperature evolution of the spin dynam-
+ics, the HF-ESR spectra were measured at several tem-
+peratures in the range of 3 - 300 K and at few selected
+microwave excitation frequencies ν.
+Such dependences
+measured in the H ∥ c* configuration at ν = 147 GHz
+for Mn2P2S6 and at ν = 326 GHz for MnNiP2S6 are pre-
+sented in Fig. 1. As can be seen, in the case of Mn2P2S6
+upon entering the ordered state with lowering tempera-
+ture, the single ESR line transforms into two modes B4
+and B5 (see below) at ν = 147 GHz. The temperature
+dependence of the spectra for other frequencies can be
+found in Appendix in Fig. 9.
+The shift of the obtained values of Hres from the
+resonance field position at T = 300 K, δH = Hres -
+Hres(300 K) is plotted as a function of temperature for
+Mn2P2S6 and MnNiP2S6 in Fig. 2 and Fig. 3, respec-
+tively. Hres(300 K) was calculated using the equation
+hν = gµBµ0Hres, where the g-factor is obtained from
+the frequency dependence of the resonance field at 300 K
+(see Sec. III B). In the case of Mn2P2S6, δH stays practi-
+cally constant down to T ∼ 130 − 150 K for both config-
+urations H ∥ c* and H ⊥ c*. Below this temperature it
+starts to slightly deviate (lower inset in Fig. 2), suggest-
+ing a development of the static on the ESR time scale
+0
+50
+100
+150
+200
+250
+300
+-4
+-3
+-2
+-1
+0
+ H || c*
+ H ⊥ c*
+� H = Hres (T) - Hres (300 K) (T)
+Temperature (K)
+MnNiP2S6, �  = 326 GHz
+TN TN*
+0
+100
+200
+300
+0
+1
+2
+3
+� H = Linewidth (T)
+Temperature (K)
+TN*
+TN
+FIG. 3. Temperature dependence of δH (main panel) and ∆H
+(inset) measured at ν = 325.67 GHz. The horizontal dashed
+line represents zero shift from room temperature value and
+the vertical dashed line (also for inset) represents the N´eel
+temperature of the material.
+internal fields. In contrast, the deviations of δH from
+zero value in MnNiP2S6 are larger, and are observed at
+a higher temperature T ∼ 200 K. In the vicinity of the
+ordering temperature TN* there is a strong shift of the
+ESR line, observed for both compounds. In the Mn2P2S6
+case the sign of δH below the ordering temperature de-
+pends on the particular AFMR mode, which is probed at
+the specific frequency. This is detailed in the following
+Sec. III C.
+Insets of Fig. 2 and Fig. 3 represent the evolution
+of the linewidth ∆H as a function of temperature for
+Mn2P2S6 and MnNiP2S6 compounds, respectively.
+At
+T > TN, ∆H remains practically temperature indepen-
+dent for both compounds.
+A small broadening of the
+line is observed in the vicinity of the phase transition
+temperature, and there is a drastic increase of ∆H in
+the ordered state. Note, that ∆H of MnNiP2S6 is larger
+than that of Mn2P2S6 in the whole temperature range.
+Moreover, for MnNiP2S6, ∆H increases at low tempera-
+tures by almost one order of magnitude from 0.3 to 3 T
+(inset in Fig. 3). Such extensive line broadening at low
+temperatures hampers the accurate determination of the
+linewidth and resonance field, which is accounted for in
+the error bars.
+B.
+Frequency dependence at 300 K
+The frequency dependence of the resonance field
+ν(Hres) of Mn2P2S6 and MnNiP2S6 compounds mea-
+sured in the paramagnetic state at T = 300 K is shown in
+Fig. 4. Both plots have a linear dependence which can be
+fitted with the conventional paramagnetic resonance con-
+
+4
+0
+2
+4
+6
+8
+10 12
+g|| = 2.026 ± 0.002
+g⊥ = 2.047 ± 0.004
+SD (arb. u.)
+0
+2
+4
+6
+8
+10 12
+0
+50
+100
+150
+200
+250
+300
+350
+a) Mn2P2S6
+H || c*
+H ⊥ c* 
+Frequency (GHz)
+Resonance Field (T)
+g|| = 1.992 ± 0.001
+g⊥ = 1.999 ± 0.001
+H || c*
+H ⊥ c*
+b) MnNiP2S6
+FIG. 4.
+ν(Hres) dependence measured at 300 K for (a)
+Mn2P2S6 and (b) MnNiP2S6. Blue squares represent H ∥ c*
+configuration and the red circles represent H ⊥ c* con-
+figuration.
+Solid lines show the results of the fit accord-
+ing to the resonance condition of a conventional paramagnet
+hν = gµBµ0Hres. Right vertical axis: Representative spectra
+normalized for clarity. The color of the spectra corresponds
+to the color of the data points in the ν(Hres) plot with the
+same Hres.
+dition for a gapless excitation hν = gµBµ0Hres. Here,
+h is the Plank constant, µB is the Bohr magneton, µ0 is
+the permeability of free space and g is the g-factor of res-
+onating spins. For Mn2P2S6, we obtain almost isotropic
+values of the g-factor: g∥ = 1.992 ± 0.001 (H ∥ c*) and
+g⊥ = 1.999 ± 0.001 (H ⊥ c*), which is expected for a
+Mn2+ ion [34].
+In contrast, MnNiP2S6 shows a small
+anisotropy of g-factors with g∥ = 2.026 ± 0.002 and g⊥
+= 2.047 ± 0.004. In case of Ni2+ ions (3d8, S = 1), g-
+factors are expected to be appreciably greater than free
+spin value, as is revealed in HF-ESR studies on Ni2P2S6
+[24].
+C.
+Frequency dependence at 3 K
+1.
+Mn2P2S6
+The low temperature resonance modes of Mn2P2S6 ob-
+tained at T = 3 K are plotted in Fig. 5. The measure-
+ments along the H ∥ c* configuration (Fig. 5) yield three
+branches B3, B4 and B5, two of which (B3 and B4)
+are observed below the spin-flop field, HSF = 3.62 T.
+Branches B1 and B2 are assigned to the measurements
+along a- and b-axis, respectively [35].
+Additionally, at
+the spin-flop field, a non-resonance absorption peak (full
+circles) was observed at high frequencies.
+The exact gap values are calculated by fitting the in-
+plane resonance branches B1 and B2 using the analytical
+expressions for easy-axis AFMs [36]:
+hν = [(g⊥µBµ0Hres)2 + ∆2
+1,2]1/2.
+(2)
+Here ∆1 corresponds to the magnon excitation gap for
+branch B2 (also B3), and ∆2 corresponds to B1 (also
+B4). The obtained values are ∆1 = ∆Mn2P2S6
+1
+= 101.3 ±
+0
+2
+4
+6
+8
+10
+12
+0
+50
+150
+200
+250
+300
+350
+Frequency (GHz)
+Resonance Field (T)
+Spin Flop  
+Mn2P2S6, T = 3 K
+116
+101
+AFM Branch, H || a*-axis  
+AFM Branch, H || c*-axis  
+AFM Branch, H || b*-axis  
+B1
+B2
+B3
+B4
+B5
+H || a 
+H || b 
+H || c* 
+Paramagnetic branch  
+SD (arb. u.)
+FIG. 5.
+ν(Hres) dependence of HF-ESR signals measured
+at T = 3 K (symbols).
+Solid lines are the fit to the phe-
+nomenological equations as explained in the text. The dash
+gray lines correspond to the frequencies at which temperature
+dependent measurements were performed. The dash line in
+magenta represents the paramagnetic branch. Right vertical
+scale: Normalized ESR spectra for selected frequencies. For
+clarity the spectra are shifted vertically.
+Error bars in the
+Hres are smaller than the symbol size.
+0.6 GHz and ∆2 = ∆Mn2P2S6
+2
+= 116 ± 2 GHz.
+These
+values, which agree well with previous measurements by
+Okuda et al. [14] and Kobets et al. [18], are then used
+in the theoretical description for a rhombic biaxial two-
+lattice AFM [18, 37] to match the field dependence of B3
+and B4 [38]:
+ν = gµBµ0
+2h
+×
+�
+∆2
+1 + ∆2
+2 + 2H2
+res±
+±
+�
+8H2res(∆2
+1 + ∆2
+2) + (∆2
+1 + ∆2
+2)2
+�1/2
+. (3)
+Above the spin-flop field the above model can not be
+used to describe the system. Therefore branch B5 [38]
+was simulated by the resonance condition of a conven-
+tional easy-axis AFM [36]:
+hν = [(g∥µBµ0Hres)2 − ∆2
+1]1/2.
+(4)
+The presence of the second easy-axis within the ab-
+plane is further confirmed by the angular dependence of
+Hres(θ) in the H ⊥ c* configuration (Fig. 6). It follows
+a A + Bsin2(θ) law, which suggests a 180◦ periodicity of
+Hres(θ). θ denotes the angle between the applied field
+and a-axis. For a honeycomb spin system with a N´eel
+type arrangement, a six-fold periodicity of angular de-
+pendence in the layer plane can be expected. However,
+this is absent in the case of Mn2P2S6 sample due to dom-
+inating effects of a two-fold in-plane anisotropy.
+
+5
+-120 -90 -60 -30
+0
+30
+60
+90
+120 150 180 210 240
+3.90
+3.95
+4.00
+4.05
+4.10
+4.15
+4.20
+4.25
+4.30
+4.35
+4.40
+4.45
+Resonance Field (T)
+Theta (°)
+Mn2P2S6
+H ⊥ c*, T = 3 K
+�  = 159.9 GHz
+Model
+pi_periodicity (User)
+Equation
+A + B * sin(x*pi/180 + C)^2
+Plot
+Resonance Field
+A
+3.96068 ± 0.01024
+B
+0.43992 ± 0.01675
+C
+0.03046 ± 0.01979
+Reduced Chi-Sqr
+8.35579E-4
+R-Square (COD)
+0.97185
+Adj. R-Square
+0.96903
+H || a
+H || b
+FIG. 6.
+Resonance field as a function of angle θ at T =
+3 K and ν = 160 GHz for Mn2P2S6.
+θ denotes the angle
+between the direction of the field applied along the ab-plane
+and the a-axis. Red dash line represents the result of the fit,
+as explained in the text.
+To further analyze the measured ν(Hres) dependence
+of the AFMR modes in the magnetically ordered state of
+Mn2P2S6, that correspond to the collective excitations
+of the spin lattice (spin waves), we employed a linear
+spin wave theory (LSWT) with the second quantization
+formalism [36, 39]. The details of our model are provided
+in Ref. [40]. The phenomenological Hamiltonian for the
+two-sublattice spin system, used for calculations of the
+spin waves energies, has the following form:
+H = A(M1M2)
+M 2
+0
++ Kuniax
+M1z
+2 + M2z
+2
+M 2
+0
++ Kbiax
+2
+(M 2
+1x − M 2
+1y) + (M 2
+2x − M 2
+2y)
+M 2
+0
+− (HM1) − (HM2) .
+(5)
+Here the first term represents the exchange interaction
+between the magnetic sublattices with respective magne-
+tizations M1 and M2, such that M 2
+1 = M 2
+2 = (M0)2 =
+(Ms/2)2, with M 2
+s being the square of the saturation
+magnetization. A is the mean-field antiferromagnetic ex-
+change constant. The second term in Eq. (5) is the uni-
+axial part of the magnetocrystalline anisotropy given by
+the anisotropy constant Kuniax. The third term describes
+an additional anisotropy in the xy-plane with the respec-
+tive constant Kbiax. The fourth and fifth terms are the
+Zeeman interactions for both sublattice magnetizations.
+The results of the calculation match well the measured
+data. In the calculation we assumed a full Mn satura-
+tion moment of ∼ 5µB, yielding Ms = 446 erg/(G·cm3)
+= 446 · 103 J/(T·m3), considering 4 Mn ions in the unit
+cell. The average g-factor value of 1.995 was taken from
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11
+12
+0
+50
+100
+150
+200
+250
+300
+350
+400
+450
+ AFM Branch, H || c*
+ AFM Branch, H ⊥ c*
+ Paramagnetic branch
+ H || c*
+ H ⊥ c*
+Resonance Field (T)
+Frequency (GHz)
+MnNiP2S6,
+T = 3 K
+SD (arb. u.)
+FIG. 7. ν(Hres) dependence measured at 3 K for both con-
+figurations of magnetic field. Right vertical scale: Exemplary
+spectra positioned above the resonance points. The horizontal
+dash gray line represents the frequency at which the temper-
+ature dependence was measured. The dash line in magenta
+depicts the paramagnetic resonance branch at 300 K.
+the frequency dependence measurements at T = 300 K
+(Fig. 4).
+As the result we obtain the exchange con-
+stant A = 2.53 · 108 erg/cm3 = 2.53 · 107 J/m3, uni-
+axial anisotropy constant Kuniax = −7.2 · 104 erg/cm3
+= −7.2 · 103 J/m3, and an in-plane anisotropy constant
+Kbiax = 1.9 · 104 erg/cm3 = 1.9 · 103 J/m3. Within the
+mean-field theory A is related to the Weiss constant
+Θ = A ∗ C/M 2
+0 , where C is the Curie constant.
+Θ,
+that provides an average energy scale for the exchange
+interaction in the system, amounts therefore at least
+ΘMn2P2S6 ≈ 350 K.
+2.
+MnNiP2S6
+In the case of MnNiP2S6 we observe one branch for
+H ∥ c* and another one for H ⊥ c* configuration, re-
+spectively, as shown in Fig. 7.
+HF-ESR spectra were
+also recorded at various angles for the in-plane orienta-
+tion. Within the experimental error bars of ∼ 300 mT, no
+signatures for an in-plane anisotropy were observed (see
+Fig. 10 in Appendix). Both branches follow the resonance
+condition for a hard direction of an AFM given by Eq. (2),
+which reveals that neither c*-axis nor the ab-plane are
+energetically favorable. The magnitude of the gap was
+obtained from the fit as ∆MnNiP2S6
+1
+= 115 ± 9 GHz for
+H ∥ c* and ∆MnNiP2S6
+2
+= 215 ± 1 GHz for H ⊥ c* config-
+urations, respectively.
+Unfortunately, we could not find a good matching of
+the calculated frequency dependence to the one mea-
+sured at low temperature (Fig. 7) with the AFM Hamil-
+tonian for a two sublattice model.
+Inclusion of the
+
+6
+terms describing cubic, hexagonal and symmetric ex-
+change anisotropies in addition to those given in Eq. (5)
+did not yield a good result either.
+This could be ex-
+plained by the complicated type of order of two mag-
+netically inequivalent ions Mn2+ (S =
+5
+2, g = 1.955)
+and Ni2+ (S = 1, g = 2.17), which possibly requires a
+more sophisticated model than the one used in this study.
+The analysis might be even more complicated by poten-
+tial disorder in the system due to the stochastic distribu-
+tion of these ions on the 4g Wyckoff sites. Therefore the
+full description of this system remains an open question.
+However, one could draw some conclusions by analyzing
+how the magnetization measured at low-T depends on the
+Mn/Ni ratio [27]. The reduction of the magnetization
+measured at low-T can be explained by the the reduc-
+tion of the total moment per formula unit of MnNiP2S6,
+which can be found as an average of the Mn and Ni sat-
+uration magnetizations and amounts to ∼ 7.2 µB, com-
+pared to Mn2P2S6 which has the saturation moment of
+∼ 10 µB. Additionally, an almost isotropic behavior of
+the magnetization as a function of magnetic field (inset of
+Fig. 10 (a)) suggests that the isotropic exchange energy
+is by orders of magnitude the strongest term defining the
+static magnetic properties of MnNiP2S6. In this case, the
+magnetization value, measured at the magnetic field ap-
+plied along some hard direction, should be inversely pro-
+portional to the mean-field isotropic exchange constant
+M ∼ H/A.
+The reduced magnetization in MnNiP2S6
+suggests, therefore, that Θ ∼ A should be at least as large
+in MnNiP2S6 (ΘMnNiP2S6 ≥
+∼ 350 K) as in Mn2P2S6
+(ΘMn2P2S6 ≈ 350 K, see Sec. III C 1).
+IV.
+DISCUSSION
+A.
+Spin-Spin correlations in (Mn1−xNix)2P2S6
+(T > TN*)
+As has been shown in our previous work, both the
+resonance field and the linewidth of the HF-ESR signal
+in Ni2P2S6 remain temperature independent by cooling
+the sample down to temperatures close to TN [24]. Usu-
+ally, in the quasi-2D spin systems the ESR line broad-
+ening and shift occur at T > TN due to the growth
+of the in-plane spin-spin correlations resulting in a de-
+velopment of slowly fluctuating short-range order [41].
+Specifically, the slowly fluctuating spins produce a static
+on the ESR timescale field causing a shift of the reso-
+nance line, and a distribution of these local fields and
+shortening of the spin-spin relaxation time due to the
+slowing down of the spin fluctuations increase the ESR
+linewidth.
+In the Mn2P2S6 compound these features
+are not very pronounced, only in the resonance field of
+the HF-ESR response one can detect within error bars
+small deviations starting at T ∼ 130 − 150 K. In the
+MnNiP2S6 compound, in turn, the critical broadening
+and the shift of the resonance line are observed at tem-
+perature T ∼ 200 K, which is much higher than TN.
+Even though the critical broadening and the line shift
+above TN are much stronger pronounced in MnNiP2S6,
+our previous low-frequency ESR study shows that the
+clear cut signatures of 2D correlated spin dynamics are
+present above TN only in the Mn2P2S6 compound [25].
+Interestingly, these signatures, seen in the characteristic
+angular dependence of the ESR linewidth, develop only
+at elevated temperatures, where the effect of the strong
+isotropic AFM coupling (ΘMn2P2S6 ≈ 350 K) on the
+spin fluctuations becomes gradually suppressed. Critical
+broadening and the shift of the ESR line in MnNiP2S6
+above TN could therefore be due to the stochastic dis-
+tribution of Mn and Ni ions on the 4g Wyckoff sites of
+the crystal structure causing a competition of different
+order types with contrasting magnetic anisotropies. Our
+conclusion on the drastic difference in the ground states
+is supported by the strong distinction in the energy gaps
+and magnetic field dependences of the low-T spin wave
+excitations in Mn2P2S6, MnNiP2S6 and Ni2P2S6, respec-
+tively.
+The competing types of magnetic order might
+enhance spin fluctuations seen in the HF-ESR response
+at elevated temperatures. Strong fluctuations suppress,
+in turn, the ordering temperature for MnNiP2S6 which is
+evident in the recent studies on the (Mn1−xNix)2P2S6 se-
+ries [27, 30, 32]. Moreover, in this scenario of the stochas-
+tic distribution of Mn and Ni, small deviation of the sto-
+ichiometry from sample to sample of the same nominal
+composition could vary the ordering temperature, which
+explains the broad range of TN measured in MnNiP2S6
+samples [27, 30, 32].
+B.
+Ground state and anisotropy of
+(Mn1−xNix)2P2S6 (T << TN*)
+At the lowest measurement temperature Mn2P2S6 has
+an antiferromagnetic ground state with biaxial type of
+anisotropy, and the spin wave excitations can be suc-
+cessfully modeled using LSWT. As the result we obtain
+the estimation of the exchange interaction ΘMn2P2S6 ≈
+350 K and the parameters of the anisotropy Kuniax =
+−7.2 · 104 erg/cm3 = −7.2 · 103 J/m3 and Kbiax = 1.9 ·
+104 erg/cm3 = 1.9 · 103 J/m3. There is only about four
+times difference between Kuniax and Kbiax, which sug-
+gests that the anisotropy in the ab-plane makes a sig-
+nificant contribution to the properties of the ground
+state of Mn2P2S6.
+Interestingly, the value of Kbiax =
+1.9 · 103 J/m3 ≈ 2 · 10−25 J/spin is very close to the es-
+timation of the anisotropy within the ab-plane made by
+Goossens [42], suggesting a possible dipolar nature of this
+anisotropy. In the MnNiP2S6 case we could not find an
+appropriate Hamiltonian within a two sublattice model
+which would fully describe the system, calling for a more
+sophisticated theoretical study. Interestingly, the charac-
+teristic feature of the MnNiP2S6 compound is the almost
+isotropic dependence of the magnetization as a function
+of magnetic field, measured at temperature well below TN
+[27]. The isothermal magnetization measurements made
+
+7
+on the sample used in this study, confirm the presence
+of this almost isotropic static magnetic response (see ap-
+pendix Fig. 10 (a)). Such an isotropic behavior of the
+static magnetization is related to the strong isotropic
+AFM exchange interaction (ΘMnNiP2S6 ≥
+∼ 350 K),
+which is larger than the applied magnetic field and the
+observed magnetic anisotropy in this system. However,
+the HF-ESR data reveals a substantial anisotropy in the
+magnetic field dependence of the spin waves. This seem-
+ing contradiction is actually not surprising. The magne-
+tization value at the magnetic field applied along some
+hard direction is mostly given by the mean-field exchange
+constant M ∼ H/A, whereas the magnon gap measured
+in the ESR experiment is roughly proportional to the
+square root of the product of exchange and magnetic
+anisotropy constants [36].
+Qualitatively, the evolution of the type of magnetic
+anisotropy with x in (Mn1−xNix)2P2S6 is also evident
+from our study, where, e.g., MnNiP2S6 reveals no easy-
+axis within or normal to the ab-plane. In order to quan-
+tify the change of magnetic anisotropic properties with
+the Mn/Ni content the excitation energy gaps can be
+used.
+The single gap of about 260 GHz was found in
+our previous study on Ni2P2S6 [24]. Both Mn containing
+compounds have two gaps ∆MnNiP2S6
+1
+= 115±9 GHz and
+∆MnNiP2S6
+2
+= 215 ± 1 GHz in the case of MnNiP2S6, and
+∆Mn2P2S6
+1
+= 101.3±0.6 GHz and ∆Mn2P2S6
+2
+= 116±2 GHz
+in the case of Mn2P2S6. As can be seen, there is a no-
+ticeable increase of the zero field AFM gaps in the sam-
+ples with higher Ni content, suggesting an increase of the
+magnetic anisotropy and exchange interaction. Indeed,
+the estimated energy scale of the exchange interaction in
+Mn2P2S6 is about ∼ 350 K, in MnNiP2S6 is more than
+∼ 350 K, and it is even larger in Ni2P2S6, due to the
+observation of the larger TN and as it is suggested by the
+previous investigations [25, 43–45]. Mn2+ with the half
+filled 3d electronic shell, and a small admixture of the
+excited state 4P5/2 into the ground state 6S5/2 is an ion
+with rather isotropic magnetic properties. In contrast,
+the ground state of the Ni2+ ion in the octahedral envi-
+ronment [8] is a spin triplet with the higher lying orbital
+multiplets, admixed through the spin-orbit coupling [34],
+which makes the Ni spin (S = 1) sensitive to the local
+crystal field.
+This, first, could increase a contribution
+of the local (single ion) magnetic anisotropy term in the
+Hamiltonian describing the system in the ordered and
+in the paramagnetic state, as discussed for the case of
+Ni2P2S6 in [24].
+Second, it could yield a deviation of
+the g-factor from the free electron value and also induce
+an effective g-factor anisotropy.
+The effective g-factor
+value and its anisotropy, as found in our study, increase
+with Ni content. Deviation of the g-factor from the free
+electron value (∆g) and the anisotropy of the exchange
+originate from the spin-orbit coupling effect, and there-
+fore are interrelated. In the case of symmetric anisotropic
+exchange the elements of the anisotropic exchange ten-
+sor are A ∝ (∆g/g)2J [46–48], where J is the isotropic
+exchange interaction constant. Observation of increased
+0.1
+1
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+B5, 329 GHz
+B5, 147 GHz
+B4, 147 GHz
+B4, 88 GHz
+∆(T)/∆(3 K)
+1 - T/TN
+b = 0.55
+b = 0.6
+b = 0.29
+b = 0.26
+collinear phase
+spin-flop
+phase
+H||c*
+0
+2
+4
+6
+8
+10
+12
+0
+50
+150
+200
+250
+300
+350
+100
+Frequency (GHz)
+Resonance Field (T)
+T = 3 K
+H||c*
+B1
+B2
+B3
+B4
+B5
+µ0Hsf
+collinear 
+ phase
+spin-flop 
+  phase
+FIG. 8. Main panel: Temperature dependence of the normal-
+ized energy gap ∆(T)/∆(3 K) = [1−(T/TN)]b for Mn2P2S6 at
+different field regimes. Symbol shapes and colors correspond
+to that in Fig. 2. Inset: Resonance branches at T = 3 K (solid
+lines) as in Fig. 5. Symbols (same as in the main panel) indi-
+cate the positions of the resonance modes B4 at 147 GHz and
+B5 at 147 and 329 GHz. The position of mode B4 at 88 GHz
+is not shown here since it can be detected at T ≥ 50 K only.
+The temperature dependence of these modes shown in Fig. 2
+was used to estimate that of ∆(T). (see the text)
+∆g at higher Ni content suggests that in the Ni contain-
+ing (Mn1−xNix)2P2S6 the exchange anisotropy is likely
+an important contributor to the anisotropic properties of
+the ground state at low temperatures < TN, such as, e.g.,
+increased magnon gaps.
+C.
+Critical behavior of Mn2P2S6 (T ≲ TN*)
+In the following we discuss the temperature depen-
+dence of the excitation energy gap ∆ at finite magnetic
+fields in the collinear and the spin-flop AFM ordered
+phases of Mn2P2S6, at H < Hsf and H > Hsf, re-
+spectively. This should provide useful insights onto the
+type of the critical behavior of the Mn spin lattice at
+T < TN.
+Such a dependence can be obtained by an-
+alyzing the temperature dependence of the shift of the
+resonance field positions Hres(T) of the excitation modes
+B4 and B5 for H ∥ c* (Fig. 2) with the aid of the sim-
+plified relations ∆ ≈ hν − g∥µBµ0Hres for mode B4 and
+∆ ≈ [(g∥µBµ0Hres)2 − (hν)2]1/2 for mode B5 derived
+from Eqs. (3) and (4), respectively.
+The result of this analysis is shown in Fig. 8.
+The
+∆(T) dependence can be well fitted to the power law
+∆(T) ∝ [1 − (T/TN)]b in a broad temperature range be-
+low TN with some deviations from it at lower T. The
+exponents b indicated in this Figure appear to be very
+different for modes B4 and B5. Notably, the resonance
+
+8
+field of mode B4 is always smaller than the spin-flop
+field, HB4
+res |88 GHz< HB4
+res |147 GHz< Hsf, whereas mode
+B5 occurs at larger fields with Hsf < HB5
+res |145 GHz<
+HB5
+res |329 GHz [Fig. 8(inset)]. This suggests a significant
+difference in the temperature dependence of the exci-
+tation gap in the collinear and spin-flop AFM ordered
+phases of Mn2P2S6.
+Usually, the magnetic anisotropy gap ∆(T) observed
+in quasi-2D antiferromagnets scales with the sublattice
+magnetization Msl(T) [49–51] so that the exponent b of
+the temperature dependence of ∆ can be treated as a crit-
+ical exponent β of the AFM order parameter Msl. If that
+were the case for Mn2P2S6, the value of b in the collinear
+phase would indicate the mean-field behavior of Msl(T)
+for which β = 0.5 (Fig. 8). In contrast, a strong reduction
+of b in the spin-flop phase, as seen in Fig. 8, would cor-
+respond to the critical behavior of Msl(T) in the 2D XY
+model for which β = 0.231 [52]. However, measurements
+of the temperature dependence of Msl by elastic and of
+∆ by inelastic neutron scattering in zero magnetic field
+reveal a more complex scaling between these two param-
+eters with b ≈ 3β/2 and β = 0.32 in the vicinity of TN,
+and b ≈ β with β = 0.25 at lower temperatures [53–55].
+This finding was tentatively ascribed to different tem-
+perature dependence of the competing single-ion and
+dipolar anisotropies which are both responsible for a fi-
+nite value of ∆ in the AFM ordered state of Mn2P2S6
+[53]. Theoretical analysis in Ref. [42] shows that besides
+the dipolar anisotropy which is responsible for the out-
+of-plane order of the Mn spins there is a competing, pre-
+sumably single ion anisotropy turning the spins into the
+ab plane. As argued in Ref. [54], the presence of the latter
+contribution gives rise to the 2D XY critical behavior.
+It should also be noted that the scaling b ≈ 3β/2 is
+a characteristics of a 3D antiferromagnet, as it follows
+from the theories of AFM resonance [56–58] and was con-
+firmed experimentally (see, e.g., [59, 60]). Thus, a field-
+dependent change of b indicates a kind of field-driven di-
+mensional crossover of the spin wave excitations at inter-
+mediate temperatures below TN while ramping the mag-
+netic field across the spin-flop transition. Magnetic fields
+H > Hsf push the spins into the plane, boosting the
+effective XY anisotropy, which changes the character of
+spin wave excitations observed by ESR towards the 2D
+XY scaling regime.
+V.
+CONCLUSION
+In summary, we have performed a detailed ESR spec-
+troscopic study of the single-crystalline samples of the
+van der Waals compounds Mn2P2S6 and MnNiP2S6. The
+measurements were carried out in a broad range of ex-
+citation frequencies and temperatures, and at different
+orientations of the magnetic field with respect to the sam-
+ple. Our study suggests a strong sensitivity of the type
+of magnetic order and anisotropy below TN, as well as of
+the g-factor and its anisotropy above TN to the Ni con-
+centration. Stronger deviation of the g-factor from the
+free electron value in the samples containing Ni suggests
+that the anisotropy of the exchange can be an impor-
+tant contributor to the stabilization of the certain type
+of magnetic order with particular anisotropy. Analysis of
+the spin excitations at T << TN has shown that both
+Mn2P2S6 and MnNiP2S6 are strongly anisotropic.
+In
+fact, increasing the Ni content yields a larger magnon
+gap in the ordered state (T << TN). In the Mn2P2S6
+compound we could fully describe the magnetic excita-
+tions using a two sublattice AFM Hamiltonian, which
+yielded an estimation of the uniaxial anisotropy energy,
+the anisotropy energy within the ab-plane, and the av-
+erage exchange interaction ΘMn2P2S6 ≈ 350 K. On the
+contrary, in the MnNiP2S6 compound the ground state
+and the excitations appear too complex to be described
+using two-sublattice AFM model. This could be due to a
+stochastic mixing of two magnetically inequivalent ions,
+Mn and Ni, on the 4g Wyckoff crystallographic sites.
+However, the analysis of the magnetization measured at
+low-T suggests that the exchange coupling in this com-
+pound should be comparable to or stronger than that in
+Mn2P2S6.
+We have analyzed the spin-spin correlations resulting
+in a development of slowly fluctuating short-range order,
+which, in the quasi-2D spin systems, manifest in the ESR
+line broadening and shift at T > TN. The line broaden-
+ing and shift are much stronger pronounced in MnNiP2S6
+compared to Mn2P2S6, suggesting that the critical broad-
+ening and the shift of the ESR line in MnNiP2S6 could
+be due to the enhanced spin fluctuations at the elevated
+temperatures caused by the competition of different types
+of magnetic order. Moreover, these strong spin fluctua-
+tions in the mixed Mn/Ni compounds could additionally
+lower the ordering temperature.
+Finally, the analysis of the temperature dependence of
+the spin excitation gap in Mn2P2S6 at different applied
+fields suggests a kind of field-driven dimensional crossover
+of the spin wave excitations at intermediate temperatures
+below TN.
+Strong magnetic fields push the spins into
+the plane, boosting the effective XY anisotropy, which
+changes the character of spin wave excitations observed
+by ESR from a 3D-like towards the 2D XY scaling regime.
+ACKNOWLEDGMENTS
+J.J.A. acknowledges the valuable discussions with
+Kranthi Kumar Bestha. This work was supported by the
+Deutsche Forschungsgemeinschaft (DFG) through grants
+No. KA 1694/12-1, AL 1771/8-1, AS 523/4-1, and within
+the Collaborative Research Center SFB 1143 “Correlated
+Magnetism – From Frustration to Topology” (project-id
+247310070), and the Dresden-W¨urzburg Cluster of Excel-
+lence (EXC 2147) “ct.qmat - Complexity and Topology
+in Quantum Matter” (project-id 390858490), as well as
+by the UKRATOP-project (funded by BMBF with Grant
+No. 01DK18002).
+
+9
+Appendix
+0
+1
+2
+3
+300 K
+200 K
+175 K
+150 K
+140 K
+130 K
+120 K
+110 K
+100 K
+b) Mn2P2S6, ν = 88 GHz
+10 K
+7.5  K
+5 K
+3 K
+30 K
+20 K
+65 K
+60 K
+50 K
+40 K
+90 K
+80 K
+75 K
+70 K
+11.8 12.0 12.2 12.4
+50 K
+40 K
+30 K
+20 K
+10 K
+7.5 K
+5 K
+3 K
+SD (arb. u.)
+Magnetic
+300 K
+250 K
+200 K
+175 K
+150 K
+140 K
+130 K
+120 K
+110 K
+100 K
+90 K
+80 K
+75 K
+70 K
+65 K
+60 K
+a) Mn2P2S6, ν = 326 GHz
+****
+*
+***
+*
+*
+*
+*
+6
+7
+8
+9 10 11 12 13
+Field (T)
+ d) MnNiP2S6, �  = 326 GHz
+300 K
+250 K
+200 K
+175 K
+150 K
+140 K
+120 K
+100 K
+90 K
+85 K
+80 K
+75 K
+70 K
+65 K
+60 K
+55 K
+50 K
+40 K
+30 K
+20 K
+10 K
+7.5 K
+5 K
+3 K
+11.0
+11.5
+12.0
+300 K
+250 K
+200 K
+175 K
+150 K
+140 K
+120 K
+110 K
+100 K
+95 K
+90 K
+85 K
+80 K
+   
+75 K
+70 K 
+   
+60 K
+50 K
+40 K
+30 K
+20 K
+10 K
+7.5 K
+5 K
+3 K
+c) Mn2P2S6, �  = 329 GHz
+FIG. 9. Temperature dependence of the HF-ESR spectra of (a) Mn2P2S6 at the excitation frequency, ν ≈ 326 GHz for H ∥ c*
+configuration, (b) Mn2P2S6 at ν ≈ 88 GHz for H ∥ c*. The temperature independent peaks from the impurity in the probehead
+occurring only at low frequencies are marked with asterisks. (c) Mn2P2S6 at ν ≈ 329 GHz for H ⊥ c* and (d) MnNiP2S6 at
+ν ≈ 326 GHz for H ⊥ c*. Spectra are normalized and vertically shifted for clarity.
+0
+50
+100
+150
+200
+250
+300
+1.0
+1.2
+1.4
+1.6
+1.8
+2.0
+� m (10-2 emu/ mol Oe)
+Temperature (K)
+ H || c*
+ H ⊥ c*
+H = 100 mT
+56.7 K
+75.5 K
+� � � � � �
+0
+2
+4
+6
+� � ��
+� � ��
+0.0
+0.1
+0.2
+M (µB/f.u.)
+H (T)
+T = 1.8 K
+MnNiP2S6
+a)
+0
+30
+60
+90
+120
+150
+180
+2.4
+2.6
+2.8
+3.0
+3.2
+3.4
+MnNiP2S6, H �  c*
+T = 3 K, �  = 226 GHz
+Resonance Field (T)
+Theta (° )
+b)
+FIG. 10.
+(a) Molar susceptibility at the applied field of 1000 Oe as a function of temperature measured on the sample
+of MnNiP2S6, which was used for the ESR investigations.
+The gray broken lines represent the magnetic phase transition
+temperature in both configurations. Inset: Isothermal magnetization per formula unit as a function of applied field performed
+at 1.8 K for MnNiP2S6, depicting the almost isotropic field dependence of magnetic response. (b) In-plane angular dependence
+of the resonance field at T = 3 K and ν = 226 GHz for MnNiP2S6, showing no systematic angular dependence within the
+average error bar of 0.16 T. The large linewidth values ∆H of the peaks in the ESR spectra are accounted for in the enlarged
+error bars.
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+arXiv:2301.04465v1  [cs.CV]  11 Jan 2023
+CO-TRAINING WITH HIGH-CONFIDENCE PSEUDO LABELS FOR
+SEMI-SUPERVISED MEDICAL IMAGE SEGMENTATION
+Zhiqiang Shen1,2
+Peng Cao1,2∗
+Hua Yang3
+Xiaoli Liu4
+Jinzhu Yang1,2
+Osmar R. Zaiane5
+1College of Computer Science and Engineering, Northeastern University, Shenyang, China
+2Key Laboratory of Intelligent Computing in Medical Image, Ministry of Education, Shenyang, China
+3College of Photonic and Electronic Engineering, Fujian Normal University, Fuzhou, China
+4DAMO Academy, Alibaba Group, China
+5Alberta Machine Intelligence Institute, University of Alberta, Edmonton, Alberta, Canada
+[email protected], [email protected]
+ABSTRACT
+High-quality pseudo labels are essential for semi-supervised semantic segmentation. Consistency
+regularization and pseudo labeling-based semi-supervised methods perform co-training using the
+pseudo labels from multi-view inputs. However, such co-training models tend to converge early to
+a consensus during training, so that the models degenerate to the self-training ones. Besides, the
+multi-view inputs are generated by perturbing or augmenting the original images, which inevitably
+introduces noise into the input leading to low-confidence pseudo labels. To address these issues,
+we propose an Uncertainty-guided Collaborative Mean-Teacher (UCMT) for semi-supervised se-
+mantic segmentation with the high-confidence pseudo labels. Concretely, UCMT consists of two
+main components: 1) collaborative mean-teacher (CMT) for encouraging model disagreement and
+performing co-training between the sub-networks, and 2) uncertainty-guided region mix (UMIX)
+for manipulating the input images according to the uncertainty maps of CMT and facilitating CMT
+to produce high-confidence pseudo labels. Combining the strengths of UMIX with CMT, UCMT
+can retain model disagreement and enhance the quality of pseudo labels for the co-training seg-
+mentation. Extensive experiments on four public medical image datasets including 2D and 3D
+modalities demonstrate the superiority of UCMT over the state-of-the-art. Code is available at:
+https://github.com/Senyh/UCMT.
+1
+Introduction
+Semantic segmentation is critical for medical image analysis. Great progress has been made by deep learning-based
+segmentation models relying on a large amount of labeled data [1, 2]. However, labeling such pixel-level annotations
+is laborious and requires expert knowledge especially in medical images, resulting in that labeled data are expensive
+or simply unavailable. Unlabeled data, on the contrary, are cheap and relatively easy to obtain. Under this condition,
+semi-supervised learning (SSL) has been the dominant data-efficient strategy through exploiting information from a
+limited amount labeled data and an arbitrary amount of unlabeled data, so as to alleviate the label scarcity problem
+[3].
+Consistency regularization [4] and pseudo labeling [6] are the two main methods for semi-supervised semantic seg-
+mentation. Currently, combining consistency regularization and pseudo labeling via cross supervision between the
+sub-networks, has shown promising performance for semi-supervised segmentation [6, 7, 8, 5, 9]. One critical limita-
+tion of these approaches is that the sub-networks tend to converge early to a consensus situation causing the co-training
+model degenerating to the self-training [10]. Disagreement between the sub-networks is crucial for co-training, where
+the sub-networks initialized with different parameters or trained with different views have different biases (i.e., dis-
+agreement) ensuring that the information they provide is complementary to each other. Another key factor affecting
+∗corresponding author
+
+UCMT
+(d) Co-training disagreement
+(e) Pseudo labels uncertainty
+�
+� ��
+� �
+��
+�
+(a) MT
+�
+� ��
+� ��
+��
+��
+(b) CPS
+� �
+�
+�
+� ��
+� ��
+��
+��
+UMIX
+(c) UCMT
+(f) Semi-supervised Segmentation
+EMA
+EMA
+EMA
+Figure 1: Illustration of the architectures and curves for co-training based semi-supervised semantic segmentation.
+(a) Mean-teacher [4], (b) Cross pseudo supervision [5], (c) Uncertainty-guided collaborative mean-teacher, (d) the
+disagreement between the pseudo labels in terms of dice loss of two branches (Y 1 and Y in MT; Y 1 and Y 2 in CPS;
+Y 1 and Y 2 in UCMT) from the co-training sub-networks (w.r.t. number of iterations), (e) the uncertainty variation of
+the pseudo labels in terms of entropy w.r.t. number of iterations, and (f) the performance of MT, CPS, and UCMT on
+the semi-supervised skin lesion segmentation under different proportion of labeled data.
+the performance of these approaches is the quality of pseudo labels. More importantly, these two factors influence
+each other. Intuitively, high quality pseudo labels should have low uncertainty [11]. However, increasing the degree
+of the disagreement between the co-training sub-networks by different perturbations or augmentations could result
+in their opposite training directions, thus increasing the uncertainty of pseudo labels. To investigate the effect of the
+disagreement and the quality of pseudo labels for co-training based semi-supervised segmentation, which has not been
+studied in the literature, we conduct a pilot experiment to illustrate these correlations. As shown in Figure 1, com-
+pared with mean-teacher (MT) [4] [Figure 1 (a)], cross pseudo supervision (CPS) [5] [Figure 1 (b)] with the higher
+model disagreement [(d)] and the lower uncertainty [Figure 1 (e)] produces higher performance [Figure 1 (f)] on semi-
+supervised segmentation. Note that the dice loss of two branches are calculated to measure the disagreement. The
+question that comes to mind is: how to effectively improve the disagreement between the co-training sub-networks and
+the quality of pseudo labels jointly in a unified network for SSL.
+In this paper, we focus on two major goals: maintaining model disagreement and the high-confidence pseudo labels at
+the same time. To this end, we propose the Uncertainty-guided Collaborative Mean Teacher (UCMT) framework that
+is capable of retaining higher disagreement between the co-training segmentation sub-networks [Figure 1 (d)] based on
+the higher confidence pseudo labels [Figure 1 (e)], thus achieving better semi-supervised segmentation performance
+under the same backbone network and task settings [Figure 1 (f)]. Specifically, UCMT involves two major compo-
+nents: 1) collaborative mean-teacher (CMT), and 2) uncertainty-guided region mix (UMIX), where UMIX operates
+the input images according to the uncertainty maps of CMT while CMT performs co-training under the supervision
+of the pseudo labels derived from the UMIX images. Inspired by the co-teaching [12, 10, 5] for struggling with early
+converging to a consensus situation and degrading into self-training, we introduce a third component, the teacher
+model, into the co-training framework as a regularizer to construct CMT for more effective SSL. The teacher model
+acts as self-ensemble by averaging the student models, serving as a third part to guide the training of the two student
+models. Further, we develop UMIX to construct high-confident pseudo labels and perform regional dropout for learn-
+ing robust semi-supervised semantic segmentation models. Instead of random region erasing or swapping [13, 14],
+UMIX manipulates the original image and its corresponding pseudo labels according to the epistemic uncertainty of
+the segmentation models, which not only reduces the uncertainty of the pseudo labels but also enlarges the training
+data distribution. Finally, by combining the strengths of UMIX with CMT, the proposed approach UCMT significantly
+improves the state-of-the-art (sota) results in semi-supervised segmentation on multiple benchmark datasets. For ex-
+ample, UCMT and UCMT(U-Net) achieve 88.22% and 82.14% Dice Similarity Coefficient (DSC) on ISIC dataset
+under 5% labeled data, outperforming our baseline model CPS [5] and the state-of-the-art UGCL [15] by 1.41% and
+9.47%, respectively.
+In a nutshell, our contributions mainly include:
+2
+
+UCMT
+• We pinpoint the problem in existing co-training based semi-supervised segmentation methods: the insufficient
+disagreement among the sub-networks and the lower-confidence pseudo labels. To address the problem, we
+design an uncertainty-guidedcollaborative mean-teacher to maintain co-training with high-confidence pseudo
+labels, where we incorporate CMT and UMIX into a holistic framework for semi-supervised medical image
+segmentation.
+• To avoid introducing noise into the new samples, we propose an uncertainty-guided regional mix algorithm,
+UMIX, which encourages the segmentation model to yield high-confident pseudo labels and enlarge the
+training data distribution.
+• We conduct extensive experiments on four public medical image segmentation datasets including 2D and 3D
+scenarios to investigate the effectiveness of our method. Comprehensive results demonstrate the effectiveness
+of each component of our method and the superiority of UCMT over the state-of-the-art.
+2
+Related work
+2.1
+Semi-supervised learning
+Semi-supervised learning aims to improve performance in supervised learning by utilizing information generally asso-
+ciated with unsupervised learning, and vice versa [3]. A common form of SSL is introducing a regularization term into
+the objective function of supervised learning to leverage unlabeled data. From this perspective, SSL-based methods
+can be divided into two main lines, i.e., pseudo labeling and consistency regularization. Pseudo labeling attempts to
+generate pseudo labels similar to the ground truth, for which models are trained as in supervised learning [6]. Con-
+sistency regularization enforces the model’s outputs to be consistent for the inputs under different perturbations [4].
+Current state-of-the-art approaches have incorporated these two strategies and shown superior performance for semi-
+supervised image classification [16, 17]. Based on this line of research, we explore more effective consistency learning
+algorithms for semi-supervised semantic segmentation.
+2.2
+Semi-supervised semantic segmentation
+Compared with image classification, semantic segmentation requires much more intensively and costly labeling for
+pixel-level annotations. Semi-supervised semantic segmentation inherits the main ideas of semi-supervised image clas-
+sification. The combination of consistency regularization and pseudo labeling, mainly conducting cross supervision
+between sub-networks using pseudo labels, has become the mainstream strategy for semi-supervised semantic seg-
+mentation in both natural images [7, 5] and medical images [18, 19, 20, 21]. Specifically, these combined approaches
+enforce the consistency of the predictions under different perturbations, such as input perturbations [22, 23], feature
+perturbations [7], and network perturbations [4, 5, 20, 21]. In addition, adversarial learning-based methods, rendering
+the distribution of model predictions from labeled data to be aligned with those from unlabeled data, can also be re-
+garded as a special form of consistency regularization [24, 25]. However, such cross supervision models may converge
+early to a consensus, thus degenerating to self-training ones. We hypothesize that enlarging the disagreement for the
+co-training models based on the high-confidence pseudo labels can improve the performance of SSL. Therefore, we
+propose a novel SSL framework, i.e., UCMT, to generate more accurate pseudo labels and maintain co-training for
+semi-supervised medical image segmentation.
+2.3
+Uncertainty-guided semi-supervised semantic segmentation
+Model uncertainty (epistemic uncertainty) can guide the SSL models to capture information from the pseudo labels.
+Two critical problems for leveraging model uncertainty are how to obtain and exploit model uncertainty. Recently,
+there are mainly two strategies to estimate model uncertainty: 1) using Monte Carlo dropout [26], and 2) calculating
+the variance among different predictions [27]. For semi-supervised semantic segmentation, previous works exploit
+model uncertainty to re-weight the training loss [18] or selecting the contrastive samples [15]. However, these methods
+require manually setting a threshold to neglect the low-confidence pseudo labels, where the fixed threshold is hard to
+determine. In this paper, we obtain the epistemic uncertainty by the entropy of the predictions of CMT for the same
+input and exploit the uncertainty to guide the region mix for gradually exploring information from the unlabeled data.
+3
+Methodology
+3
+
+UCMT
+EMA
+EMA
+� �; ��
+� �; ��
+� �; �
+Collaborative mean-teacher
+UMIX
+�
+�
+�
+�
+MAA
+EMA
+EMA
+� �; ��
+� �; ��
+� �; �
+Collaborative mean-teacher
+�
+�
+�
+�
+MAA
+First step: uncertainty estimation
+Second step: training with UMIX
+� �; �
+Testing Phase
+Loss functions
+�
+�’
+��
+��
+���
+���
+����
+����
+�
+�
+���
+����
+�����
+�����
+�����
+�����
+�����
+�����
+�����
+�����
+��
+��
+Training Phase
+�� = ���� + ����� +�����
+���� = ����� + �����
+���� = ����� + �����
+�� = ���� + ����
+� = �� + ���
+Figure 2: Overview of the proposed UCMT. CMT includes three sub-networks, i.e., the teacher sub-network (f(·; θ))
+and the two student sub-networks (f(·; θ1) and f(·; θ2)). UMIX constructs each new samples X′ by replacing the top
+k most uncertain regions (red grids in V 1 and V 2) with the top k most certain regions (green grids in V 2 and V 1) in
+the original image X. Note that three sub-networks are collaboratively learning during the training stage, while only
+the teacher model is needed in the testing stage.
+3.1
+Problem definition
+Before introducing our method, we first define the semi-supervised segmentation problem with some notations used
+in this work. The training set D = {DL, DU} contains a labeled set DL = {(Xi, Yi)N
+i=1} and a unlabeled set
+DU = {(Xj)M
+j=N+1}, where Xi/Xj denotes the ith/jth labeled/unlabeled image, Yi is the ground truth of the labeled
+image, and N and M − N are the number of labeled and unlabeled samples, respectively. Given the training data D,
+the goal of semi-supervised semantic segmentation is to learn a model f(·; θ) performing well on unseen test sets.
+3.2
+Overview
+To avoid the co-training degrading to the self-training, we propose to encourage model disagreement during train-
+ing and ensure pseudo labels with low uncertainty. With this motivation, we propose uncertainty-guided collabo-
+rative mean-teacher for semi-supervised image segmentation, which includes 1) collaborative mean-teacher, and 2)
+uncertainty-guided region mix. As shown in Figure 1 (d), CMT and UCMT gradually enlarge the disagreement
+between the co-training sub-networks. Meanwhile, CMT equipped with UMIX guarantees low-uncertainty for the
+pseudo labels. With the help of these conditions, we can safely maintain the co-training status to improve the effec-
+tiveness of SSL for exploring unlabeled data. Details of CMT and UMIX are presented on Section 3.3 and Section
+3.4, respectively. Figure 2 illustrates the schematic diagram of the proposed UCMT. Generally, there are two steps in
+the training phase of UCMT. In the first step, we train CMT using the original labeled and unlabeled data to obtain the
+uncertainty maps; Then, we perform UMIX to generate the new samples based on the uncertainty maps. In the second
+step, we re-train CMT using the UMIX samples. Details of the training process of UCMT are shown in Algorithm
+1. Although UCMT includes three models, i.e., one teacher model and two student models, only the teacher model is
+required in the testing stage.
+3.3
+Collaborative mean-teacher
+Current consistency learning-based SSL algorithms, e.g., Mean-teacher [4] and CPS [5], suggest to perform consis-
+tency regularization among the pseudo labels in a multi-model architecture rather than in a single model. However,
+during the training process, the two-network SSL framework may converge early to a consensus and the co-training
+degenerate to the self-training [10]. To tackle this issue, we design the collaborative mean teacher (CMT) framework
+by introducing a "arbitrator", i.e., the teacher model, into the co-training architecture [5] to guide the training of the
+4
+
+UCMT
+Algorithm 1 UCMT algorithm
+Input: DL = {{(Xi, Yi)}N
+i=1}, DU = {{Xj}M
+j=N+1}
+Parameter: θ, θ1, θ2
+Output: f(·; θ)
+1: for T ∈ [1, numepochs] do
+2:
+for each minibatch B do
+3:
+// i/j is the index for labeled/unlabeled data
+4:
+step 1: uncertainty estimation
+5:
+ˆY 0
+i ← f(Xi ∈ B; θ), ˆY 0
+j ← f(Xj ∈ B; θ)
+6:
+ˆY 1
+i ← f(Xi ∈ B; θ1), ˆY 1
+j ← f(Xj ∈ B; θ1)
+7:
+ˆY 2
+i ← f(Xi ∈ B; θ2), ˆY 2
+j ← f(Xj ∈ B; θ2)
+8:
+L ← Ls( ˆY 0
+i , ˆY 1
+i , ˆY 2
+i , Yi) + λ(T )Lu( ˆY 0
+j , ˆY 1
+j , ˆY 2
+j )
+9:
+Update f(·; θ), f(·; θ1), f(·; θ2) using optimizer
+10:
+U 1
+i ← Uncertain(f(Xi ∈ B; θ1), f(Xi ∈ B; θ))
+11:
+U 2
+i ← Uncertain(f(Xi ∈ B; θ2), f(Xi ∈ B; θ))
+12:
+U 1
+j ← Uncertain(f(Xj ∈ B; θ1), f(Xj ∈ B; θ))
+13:
+U 2
+j ← Uncertain(f(Xj ∈ B; θ2), f(Xj ∈ B; θ))
+14:
+step 2: training with UMIX
+15:
+X′
+i/Y ′
+i ← UMIX(Xi/Yi, U 1
+i , U 2
+i ; k, 1/r)
+16:
+X′
+j/ ˆY ′0
+j ← UMIX(Xj/ ˆY 0
+j , U 1
+j , U 2
+j ; k, 1/r)
+17:
+Repeat 3-7 using X′
+i, X′
+j, Y ′
+i , and ˆ
+Y ′0
+j
+18:
+end for
+19: end for
+20: return f(·; θ)
+two student models. As shown in Figure 2, CMT consists of one teacher model and two student models, where the
+teacher model is the self-ensemble of the average of the student models. For labeled data, these models are all opti-
+mized by supervised learning. For unlabeled data, there are two critical factors: 1) co-training between the two student
+models, and 2) direct supervision from the teacher to the student models. Formally, the data flow diagram of CMT can
+be illustrated as 2,
+ր f (·; θ1) → ˆY 1
+X → f (·; θ) → ˆ
+Y 0
+ց f (·; θ2) → ˆY 2,
+(1)
+where X is an input image of the labeled or unlabeled data, ˆY 0/ ˆY 1/ ˆY 2 is the predicted segmentation map, and
+f(·; θ)/f(·; θ1)/f(·; θ2) with parameters θ, θ1 and θ2 denote the teacher model and student models, respectively.
+These models have the same architecture but initialized with different weights for network perturbations.
+To explore both the labeled and unlabeled data, the total loss L for training UCMT involves two parts, i.e., the super-
+vised loss Ls and the unsupervised loss Lu.
+L = Ls + λLu,
+(2)
+where λ is a regularization parameter to balance the supervised and unsupervised learning losses. We adopt a Gaussian
+ramp-up function to gradually increase the coefficient, i.e., λ(t) = λm × exp [−5(1 −
+t
+tm )2], where λm scales the
+maximum value of the weighted function, t denotes the current iteration, and tm is the maximum iteration in training.
+Supervised learning path. For the labeled data, the supervised loss is formulated as,
+Ls = 1
+N
+N
+�
+i=1
+Lseg (f (Xi; θ) , Yi)
++ Lseg (f (Xi; θ1) , Yi) + Lseg (f (Xi; θ2) , Yi) ,
+(3)
+where Lseg can be any supervised semantic segmentation loss, such as cross entropy loss and dice loss. Note that we
+choose dice loss in our experiments as its compelling performance in medical image segmentation.
+2We omit the image index i to indicate that X can be the labeled data or unlabeled data.
+5
+
+UCMT
+Unsupervised learning path. The unsupervised loss Lu acts as a regularization term to explore potential knowledge
+for the labeled and unlabeled data. Lu includes the cross pseudo supervision Lcps between the two student models
+and the mean-teacher supervision Lmts for guiding the student models from the teacher, as follow:
+Lu = Lcps + Lmts.
+(4)
+1) Cross pseudo supervision. The purpose of Lcps is to promote two students to learn from each other and to enforce
+the consistency between them. Lcps = Lcps1 + Lcps2 encourages bidirectional interaction for the two student sub-
+networks f(·; θ1) and f(·; θ2) as follows,
+Lcps1 =
+1
+M − N
+M−N
+�
+j=1
+Lseg
+�
+f (Xj; θ1) , ˆY 2
+j
+�
+Lcps2 =
+1
+M − N
+M−N
+�
+j=1
+Lseg
+�
+f (Xj; θ2) , ˆY 1
+j
+�
+,
+(5)
+where ˆY 1
+j and ˆY 2
+j are the pseudo labels (segmentation maps) for Xj predicted by f (·; θ1) and f (·; θ1) , respectively.
+2) Mean-teacher supervision. To avoid the two students cross supervision in the wrong direction, we introduce a
+teacher model to guide the optimization of the student models. Specifically, the teacher model is updated by the
+exponential moving average (EMA) of the average of the student models:
+θt = αθt−1 + (1 − α)θt
+1 + θt
+2
+2
+,
+(6)
+where t represents the current training iteration. α is the EMA decay that controls the parameters’ updating rate and
+we set α = 0.999 in our experiments.
+The loss of mean-teacher supervision Lmts = Lmts1 + Lmts2 is calculated from two branches:
+Lmts1 =
+1
+M − N
+M−N
+�
+j=1
+Lseg
+�
+f (Xj; θ1) , ˆY 0
+j
+�
+Lmts2 =
+1
+M − N
+M−N
+�
+j=1
+Lseg
+�
+f (Xj; θ2) , ˆY 0
+j
+�
+,
+(7)
+where ˆY 0
+j refers to the predicted segmentation map derived from f (Xj; θ).
+3.4
+Uncertainty-guided Mix
+Although CMT can promote model disagreement for co-training, it also slightly increases the uncertainty of the pseudo
+labels as depicted in Figure 1. On the other hand, random regional dropout can expand the training distribution and
+improve the generalization capability of models [13, 14]. However, such random perturbations to the input images in-
+evitably introduce noise into the new samples, thus deteriorating the quality of pseudo labels for SSL. One sub-network
+may provide some incorrect pseudo labels to the other sub-networks, degrading their performance. To overcome these
+limitations, we propose UMIX to manipulate image patches under the guidance of the uncertainty maps produced by
+CMT. The main idea of UMIX is constructing a new sample by replacing the top k most uncertain (low-confidence) re-
+gions with the top k most certain (high-confidence) regions in the input image. As illustrated in Figure 2, for example,
+we obtain the most uncertain regions (the red grids) and the most certain regions (the green grids) from an uncertainty
+map U. Then, we replace the red regions with the green regions in the input image X to construct a new sample X′.
+Formally, UMIX constructs a new sample X′ = UMIX(X, U 1, U 2; k, 1/r) by replacing the top k most uncertain
+regions (red grids in V 1 and V 2) with the top k most certain regions (green grids in V 2 and V 1) in X, where each
+region has size 1/r to the image size. To ensure the reliability of the uncertainty evaluation, we obtain the uncertain
+maps by integrating the outputs of the teacher and the student model instead of performing T stochastic forward
+passes designed by Monte Carlo Dropout estimate model [26, 18], which is equivalent to sampling predictions from
+the previous and current iterations. This process can be formulated as:
+U m = Uncertain(f(X; θm), f(X; θ)) = −
+�
+c
+Pc log(Pc),
+Pc = 1
+2(Softmax(f(X; θm)) + Softmax(f(X; θ))),
+(8)
+where m = 1, 2 denotes the index of the student models and c refers to the class index.
+6
+
+UCMT
+Table 1: Comparison with state-of-the-art methods on ISIC dataset. 5% DL and 10% DL of the labeled data are used
+for training, respectively. Results are measured by DSC.
+Method
+5% DL
+10% DL
+MT [4]
+86.67
+87.42
+CCT [7]
+83.97
+86.43
+CPS [5]
+86.81
+87.70
+UGCL(U-Net) [15]
+72.67
+79.48
+UCMT(U-Net) (ours)
+82.14
+83.33
+CMT (ours)
+87.86
+88.10
+UCMT (ours)
+88.22
+88.46
+4
+Experiments and results
+4.1
+Experiments Settings
+Datasets. We conduct extensive experiments on different medical image segmentation tasks to evaluate the proposed
+method, including skin lesion segmentation from dermoscopy images, polyp segmentation from colonoscopy images,
+and the 3D left atrium segmentation from cardiac MRI images.
+Dermoscopy. We validate our method on the ISIC dataset [28] including 2594 dermoscopy images and corresponding
+annotations. Following [15], we adopt 1815 images for training and 779 images for validation.
+Colonoscopy. We evaluate the proposed method on the two public colonoscopy datasets, including Kvasir-SEG [29]
+and CVC-ClinicDB [30]. Kvasir-SEG and CVC-ClinicDB contain 1000 and 612 colonoscopy images with correspond-
+ing annotations, respectively.
+Cardiac MRI. We evaluate our method on the 3D left atrial (LA) segmentation challenge dataset, which consists
+of 100 3D gadolinium-enhanced magnetic resonance images and LA segmentation masks for training and validation.
+Following [18], we split the 100 scans into 80 samples for training and 20 samples for evaluation.
+4.1.1
+Implementation details
+We use DeepLabv3+ [1] equipped with ResNet50 as the baseline architecture for 2D image segmentation, whereas
+adopt VNet [31] as the baseline in the 3D scenario. All images are resized to 256×256 for inference, while the outputs
+are recovered to the original size for evaluation, in the 2D scenario. For 3D image segmentation, we randomly crop
+80 × 112 × 112(Depth × Height × Width) patches for training and iteratively crop patches using a sliding window
+strategy to obtain the final segmentation mask for testing. We implement our method using PyTorch framework on a
+NVIDIA Quadro RTX 6000 GPU. We adopt AdamW as an optimizer with the fixed learning rate of le-4. The batchsize
+is set to 16, including 8 labeled samples and 8 unlabeled samples. All 2D models are trained for 50 epochs, while the
+3D models are trained for 1000 epochs 3. We empirically set k = 2 and r = 16 for our method in the experiment.
+4.2
+Comparison with state of the arts
+We compare the proposed method with state-of-the art on the four public medical image segmentation datasets. We
+re-implement MT [4], CCT [7], and CPS [5] by adopting implementations from [5]. For other approaches, we directly
+use the results reported in their original papers.
+Results on Dermoscopy. In Table 1, we report the results of our methods on ISIC and compare them with other state-
+of-the-art approaches. UCMT substantially outperforms all previous methods and sets new state-of-the-art of 88.22%
+DSC and 88.46 DSC under 5% and 10% labeled data. For fair comparison with UGCL [15], replace the backbone
+of UCMT with U-Net. The results indicate that our UCMT(U-Net) exceeds UGCL by a large margin. Moreover, our
+CMT version also outperforms other approaches under the two labeled data rates. For example, CMT surpasses MT
+and CPS by 1.19% and 1.08% on 5% DL labeled data, showing the superiority of collaborative mean-teacher against
+the current consistency learning framework. By introducing UMIX, UCMT consistently increases the performance
+under different labeled data rates, which implies that promoting model disagreement and guaranteeing high-confident
+pseudo labels are beneficial for semi-supervised segmentation.
+3Since UCMT performs the two-step training within one iteration, it is trained for half of the epochs.
+7
+
+UCMT
+Table 2: Comparison with state-of-the-art methods on Kvasir-SEG and CVC-ClinicDB datasets. 15% DL and 30%
+DL of the labeled data are individually used for training. Results are measured by DSC.
+Method
+Kvasir-SEG
+CVC-ClinicDB
+15% DL
+30% DL
+15% DL
+30% DL
+AdvSemSeg [24]
+56.88
+76.09
+68.39
+75.93
+ColAdv [32]
+76.76
+80.95
+82.18
+89.29
+MT [4]
+87.44
+88.72
+84.19
+84.40
+CCT [7]
+81.14
+84.67
+74.20
+78.46
+CPS [5]
+86.44
+88.71
+85.34
+86.69
+CMT (ours)
+88.08
+88.61
+85.88
+86.83
+UCMT (ours)
+88.68
+89.06
+87.30
+87.51
+Results on Colonoscopy.
+We further conduct a comparative experiment on the polyp segmentation task from
+colonoscopy images. Table 2 reports the quantitative results on Kvasir-SEG and CVC-ClinicDB datasets. Com-
+pared with the adversarial learning-based [24, 32] and consistency learning-based [4, 7, 5] algorithms, the proposed
+methods achieve the state-of-the-art performance. For example, both CMT and UCMT outperform AdvSemSeg [24]
+and ColAdv [32] by large margins on Kvasir-SEG and CVC-ClinicDB, except that ColAdv shows the better perfor-
+mance of 89.29% on CVC-ClinicDB under 30% labeled data. These results demonstrate that our uncertainty-guided
+collaborative mean-teacher scheme is superior to the adversarial learning and consistency learning schemes used in
+the compared approaches. In addition, CMT and UCMT show better performance on the low-data regime, i.e., 15%
+DL, and the performance between 15% DL and 30% DL labeled data is close. This phenomenon reflects the capacity
+of our method to produce high-quality pseudo labels from unlabeled data for semi-supervised learning, even with less
+labeled data.
+Results on Cardiac MRI. We further evaluate the proposed method in the 3D medical image segmentation task. Table
+3 shows the comparison results on the 3D left atrium segmentation from cardiac MRI. The compared approaches are
+based on consistency learning and pseudo labeling, including uncertainty-aware [18, 33], shape-aware [25], structure-
+aware [34], dual-task [19], and mutual training [20] consistency. It can be observed that UCMT achieves the best
+performance under 10% and 20% DL in terms of DSC and Jaccard over the state-of-the-art methods. For example,
+compared with UA-MT [18] and MC-Net [20], UCMT shows 3.88% DSC and 0.43% DSC improvements on the 10%
+labeled data. The results demonstrate the superiority of our UCMT for 3D medical image segmentation.
+Table 3: Comparison with state-of-the-art methods on the 3D left atrial segmentation challenge dataset. 10% DL and
+20% DL of the labeled data are used for training.
+Method
+10% DL
+20% DL
+DSC
+Jaccard
+95HD
+ASD
+DSC
+Jaccard
+95HD
+ASD
+UA-MT [18]
+84.25
+73.48
+13.84
+3.36
+88.88
+80.21
+7.32
+2.26
+SASSNet [25]
+87.32
+77.72
+9.62
+2.55
+89.54
+81.24
+8.24
+2.20
+LG-ER-MT [34]
+85.54
+75.12
+13.29
+3.77
+89.62
+81.31
+7.16
+2.06
+DUWM [33]
+85.91
+75.75
+12.67
+3.31
+89.65
+81.35
+7.04
+2.03
+DTC [19]
+86.57
+76.55
+14.47
+3.74
+89.42
+80.98
+7.32
+2.10
+MC-Net [20]
+87.71
+78.31
+9.36
+2.18
+90.34
+82.48
+6.00
+1.77
+MT [4]
+86.15
+76.16
+11.37
+3.60
+89.81
+81.85
+6.08
+1.96
+CPS [5]
+86.23
+76.22
+11.68
+3.65
+88.72
+80.01
+7.49
+1.91
+CMT (ours)
+87.23
+77.83
+7.83
+2.23
+89.88
+81.74
+6.07
+1.94
+UCMT (ours)
+88.13
+79.18
+9.14
+3.06
+90.41
+82.54
+6.31
+1.70
+4.3
+Ablation study
+We conduct an ablation study in terms of network architectures, loss functions, and region mix to investigate the
+effectiveness of each component and analyze the hyperparameters of the proposed method. There are three types of
+network architectures: 1) teacher-student (TS) in MT [4], 2) student-student (SS) in CPS [5], and 3) student-teacher-
+student in the proposed CMT.
+8
+
+UCMT
+Effectiveness of each component. Table 4 reports the performance improvements over the baseline. It shows a trend
+that the segmentation performance improves when the components, including the STS (student-teacher-student), Lcps,
+Lmts, and UMIX are introduced into the baseline, and again confirms the necessity of encouraging model disagreement
+and enhancing the quality of pseudo labels for semi-supervised segmentation. The semi-supervised segmentation
+model is boosted for two reasons: 1) Lcps, Lmts and the STS architecture that force the model disagreement in CMT
+for co-training, and 2) UMIX facilitating the model to produce high-confidence pseudo labels. All the components
+contribute to UCMT to achieve 88.22% DSC. These results demonstrate their effectiveness and complementarity for
+semi-supervised medical image segmentation. On the other hand, the two groups of comparisons between "TS (teacher-
+student) + Lmts" (i.e., MT) and STS + Lmts (i.e., CMTv1), and between "SS (student-student) + Lcps" (i.e., CPS) and
+"STS + Lcps" (i.e., CMTv2) show that the STS-based approaches yield the improvements of 0.17% and 0.64%, which
+verifies the effectiveness of our STS for SSL. The performance gaps are not significant because the STS architecture
+increases the co-training disagreement but decreases the confidences of pseudo labels. However, It can be easily found
+that the results are improved to 87.86% by "STS + Lcps + Lmts" (i.e., CMTv3) and the relative improvements of 1.55%
+and 1.41% DSC have been achieved by "STS + Lcps + Lmts + UMIX" (i.e., UCMT) compared with MT and CPS.
+The results demonstrate our hypothesis that maintaining co-training with high-confidence pseudo labels can improve
+the performance of semi-supervised learning.
+Table 4: Ablation study of the different components combinations on ISIC dataset. All models are trained for 5%
+labeled data. TS: teacher-student; SS: student-student; STS: student-teacher-student; Lcps: cross pseudo supervision;
+Lmts: mean-teacher supervision; U: UMIX;
+Method
+TS
+SS
+STS
+Lcps
+Lmts
+U
+DSC
+Baseline
+83.31
+MT
+√
+√
+86.67
+CPS
+√
+√
+86.81
+CMTv1
+√
+√
+86.84
+CMTv2
+√
+√
+87.48
+CMTv3
+√
+√
+√
+87.86
+UCMT
+√
+√
+√
+√
+88.22
+Comparison with CutMix. We further compare the proposed UMIX, component of our UCMT, with CutMix [14]
+on ISIC and LA datasets with different labeled data to investigate their effects in semi-supervised medical image
+segmentation. As illustrates in Figure 3, UMIX outperforms CutMix, especial in the low-data regime, i.e., 2% labeled
+data. The reason for this phenomenon is that CutMix performs random region mix that inevitably introduces noise into
+the new samples, which reduces the quality of the pseudo labels, while UMIX processes the image regions according
+to the uncertainty of the model, which facilitates the model to generate more confident pseudo labels from the new
+samples.
+(a) ISIC
+(b) LA
+CMT+CutMix
+CMT+UMIX
+CMT
+CMT+CutMix
+CMT+UMIX
+CMT
+Figure 3: Comparison UCMT (UMIX) with CutMix on ISIC (a) and LA (b) dataset under 2%, 5%, 10%, and 20%
+DL.
+Parameter sensitivity analysis. UMIX has two hyperparamters, i.e., the top k regions for mix and the size of the
+regions (patches) 1/r to the image size. We study the influence of these factors to UCMT on ISIC dataset with 5% DL.
+It can be observed in Table 5 that reducing the patch size leads to a slight increase in performance. Moreover, varying
+the number of k does not bring us any improvement. The reason for this phenomenon is that we can choose any value
+9
+
+UCMT
+of K to eliminate outliers, thus bringing high-confidence pseudo labels for semi-supervised learning, indicating the
+robustness of UMIX.
+Table 5: Investigation on how the top k and region size affect the capacity of UMIX. All results are evaluated on ISIC
+dataset with 5% labeled data.
+1/r
+k
+1
+2
+3
+4
+5
+1/16
+87.95
+88.22
+88.12
+87.96
+88.08
+1/4
+87.65
+88.15
+87.80
+87.54
+87.90
+1/8
+87.86
+87.92
+88.03
+88.01
+87.87
+4.4
+Qualitative results
+Figure 4 visualizes some example results of polyp segmentation, skin lesion segmentation, and left atrial segmentation.
+As shown in Figure 4 (a), the supervised baseline insufficiently segments some lesion regions, mainly due to the limited
+number of labeled data. Moreover, MT [Figure 4 (b)] and CPS [Figure 4 (c)] typically under-segment certain objects,
+which can be attributed to the limited generalization capability. On the contrary, our CMT [Figure 4 (e)] corrects these
+errors and produces smoother segment boundaries by gaining more effective supervision from unlabeled data. Besides,
+our complete method UCMT [Figure 4 (f)] further generates more accurate results by recovering finer segmentation
+details through more efficient training. These examples qualitatively verify the robustness of the proposed UCMT. In
+addition, to clearly give an insight into the procedure of the pseudo label generation and utilization in the co-training
+SSL method, we illustrate the uncertainty maps for two samples during the training in Figure 5. As shown, UCMT
+generates the uncertainty maps with high uncertainty [Figure 5 (a)/(c)] in the early training stage whereas our model
+produces relative higher confidence maps [Figure 5 (b)/(d)] from the UMIX images. During training, UCMT gradually
+improves the confidence for the input images. These results prove that UMIX can facilitate SSL models to generate
+high-confidence pseudo labels during training, guaranteeing that UCMT is able to maintain co-training in a more
+proper way.
+5
+Conclusion
+We present an uncertainty-guided collaborative mean-teacher for semi-supervised medical image segmentation. Our
+main ideas lies in maintaining co-training with high-confidence pseudo labels to improve the capability of the SSL
+models to explore information from unlabeled data. Extensive experiments on four public datasets demonstrate the
+effectiveness of this idea and show that the proposed UCMT can achieve state-of-the-art performance. In the future,
+we will investigate more deeply the underlying mechanisms of co-training for more effective semi-supervised image
+segmentation.
+Acknowledgments
+This work was supported by the National Natural Science Foundation of China under Grant 62076059 and the Natural
+Science Foundation of Liaoning Province under Grant 2021-MS-105.
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+

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+Vibronic Effects on the Quantum Tunnelling of Magnetisation in Single-Molecule Magnets
+Andrea Mattioni,1, ∗ Jakob K. Staab,1 William J. A. Blackmore,1 Daniel
+Reta,1, 2 Jake Iles-Smith,3, 4 Ahsan Nazir,3 and Nicholas F. Chilton1, †
+1Department of Chemistry, School of Natural Sciences,
+The University of Manchester, Oxford Road, Manchester, M13 9PL, UK
+2Faculty of Chemistry, UPV/EHU & Donostia International Physics Center DIPC,
+Ikerbasque, Basque Foundation for Science, Bilbao, Spain
+3Department of Physics and Astronomy, School of Natural Sciences,
+The University of Manchester, Oxford Road, Manchester M13 9PL, UK
+4Department of Electrical and Electronic Engineering, School of Engineering,
+The University of Manchester, Sackville Street Building, Manchester M1 3BB, UK
+Single-molecule magnets are among the most promising platforms for achieving molecular-scale data stor-
+age and processing. Their magnetisation dynamics are determined by the interplay between electronic and
+vibrational degrees of freedom, which can couple coherently, leading to complex vibronic dynamics. Building
+on an ab initio description of the electronic and vibrational Hamiltonians, we formulate a non-perturbative vi-
+bronic model of the low-energy magnetic degrees of freedom in a single-molecule magnet, which we benchmark
+against field-dependent magnetisation measurements. Describing the low-temperature magnetism of the com-
+plex in terms of magnetic polarons, we are able to quantify the vibronic contribution to the quantum tunnelling
+of the magnetisation. Despite collectively enhancing magnetic relaxation, we observe that specific vibrations
+suppress quantum tunnelling by enhancing the magnetic axiality of the complex. Finally, we discuss how this
+observation might impact the current paradigm to chemical design of new high-performance single-molecule
+magnets, promoting vibrations to an active role rather than just regarding them as sources of noise and decoher-
+ence.
+I.
+INTRODUCTION
+Single-molecule magnets (SMMs) hold the potential for
+realising high-density data storage and quantum informa-
+tion processing [1–4].
+These molecules exhibit a doubly-
+degenerate ground state, comprising two states supporting a
+large magnetic moment with opposite orientation, which rep-
+resents an ideal platform for storing digital data. Slow reori-
+entation of this magnetic moment results in magnetic hystere-
+sis at the single-molecule level at sufficiently low tempera-
+tures [5]. The main obstacle to extending this behaviour to
+room temperature is the coupling of the magnetic degrees of
+freedom to molecular and lattice vibrations, often referred to
+as spin-phonon coupling. Thermal excitation of the molec-
+ular vibrations cause transitions between different magnetic
+states, ultimately leading to a complete loss of magnetisation.
+Advances in design, synthesis and characterisation of SMMs
+have shed light on the microscopic mechanisms underlying
+their desirable magnetic properties, extending this behaviour
+to increasingly higher temperatures [6–8].
+The mechanism responsible for magnetic relaxation in
+SMMs strongly depends on temperature. At higher temper-
+atures, relaxation is driven by one (Orbach) and two (Raman)
+phonon transitions between magnetic sublevels [9].
+When
+temperatures approach absolute zero, all vibrations are pre-
+dominantly found in their ground state.
+Thus, both Or-
+bach and Raman transitions become negligible and the dom-
+inant mechanism is quantum tunnelling of the magnetisation
+∗ [email protected]
+† [email protected]
+(QTM) between the two degenerate ground states [10, 11].
+This process relies on the presence of a coherent coupling
+mixing the two otherwise degenerate ground states, opening
+a tunnelling gap, and allowing population to redistribute be-
+tween them, thus leading to facile magnetic reorientation.
+While the role of vibrations in high-temperature magnetic
+relaxation is well understood in terms of weak-coupling rate
+equations for the electronic populations [12–15], the connec-
+tion between QTM and spin-phonon coupling is still unclear.
+Some analyses have looked at the influence of vibrations on
+QTM in integer-spin SMMs, where a model spin system was
+used to show that spin-phonon coupling could open a tunnel-
+ing gap [16, 17]. However, QTM remains more elusive to
+grasp in half-integer spin complexes, such as monometallic
+Dy(III) SMMs, since it is observed experimentally despite
+being forbidden by Kramers theorem [18].
+In this case, a
+magnetic field is needed to break the time-reversal symmetry
+of the molecular Hamiltonian and lift the degeneracy of the
+ground doublet. This magnetic field can be provided by hy-
+perfine interaction with nuclear spins or by dipolar coupling
+to other SMMs; both these effects have been shown to af-
+fect tunnelling behaviour [19–25]. Once the tunnelling gap
+is opened by a magnetic field, molecular vibrations can in
+principle affect its magnitude in a nontrivial way. In a re-
+cent work, Ortu et al. analysed the magnetic hysteresis of a
+series of Dy(III) SMMs, suggesting that QTM efficiency cor-
+relates with molecular flexibility [22]. In another work, hyper-
+fine coupling was proposed to assists QTM by facilitating the
+interaction between molecular vibrations and spin sublevels
+[26]. However, a clear and unambiguous demonstration of the
+influence of the spin-phonon coupling on QTM beyond toy-
+model approaches is still lacking to this date.
+In this work we present a theoretical analysis of the effect of
+arXiv:2301.05557v1  [quant-ph]  13 Jan 2023
+
+2
+molecular vibrations on the tunnelling dynamics in a Dy(III)
+SMM. In contrast to previous treatments, our approach is
+based on a fully ab initio description of the SMM vibrational
+environment and accounts for the spin-phonon coupling in a
+non perturbative way, overcoming the standard weak-coupling
+master equation approach commonly used to determine the
+high-temperature magnetisation dynamics. After deriving an
+effective low-energy model for the relevant vibronic degrees
+of freedom based on a polaron approach [27], we demon-
+strate that vibrations can either enhance or reduce the quantum
+tunnelling gap, depending on the orientation of the magnetic
+field relative to the main anisotropy axis of the SMM. More-
+over, we validate our vibronic model against frozen solution,
+field-dependent magnetisation measurements and show that
+vibronic effects on QTM survive the orientational averaging
+imposed by amorphous samples, leading, on average, to a sig-
+nificant enhancement of the tunnelling probability. Lastly, we
+argue that not all vibrations lead to faster QTM; depending on
+how strongly vibrations impact the axiality of the lowest en-
+ergy magnetic doublet, we show that they can play a benign
+role by suppressing tunnelling, and discuss first steps in that
+direction.
+II.
+MODEL
+The compound investigated in this work is [Dy(Cpttt)2]+,
+shown in Fig. 1a [6]. The complex consists of a dyspro-
+sium ion Dy(III) enclosed between two negatively charged
+cyclopentadienyl rings with tert-butyl groups at positions 1, 2
+and 4 (Cpttt). The crystal field generated by the axial ligands
+makes the states with larger angular momentum energetically
+favourable, resulting in the energy level diagram sketched in
+Fig. 1b. The energy barrier separating the two degenerate
+ground states results in magnetic hysteresis, which was ob-
+served up to T = 60 K [6]. Magnetic hysteresis is hindered
+by QTM, which leads to a characteristic sudden drop of the
+magnetisation at zero magnetic field.
+To single out the contribution of molecular vibrations, we
+focus on a magnetically diluted sample in a frozen solution
+of dichloromethane (DCM). Thus, our computational model
+consists of a solvated [Dy(Cpttt)2]+ cation (see Section S1 for
+details; Fig. 1a), which provides a realistic description of the
+low-frequency vibrational environment, comprised of pseudo-
+acoustic vibrational modes (Fig. 1c). These constitute the ba-
+sis to consider further contributions of dipolar and hyperfine
+interactions to QTM (Fig. 1b).
+Once the equilibrium geometry and vibrational modes of
+the solvated SMM (which are in general combinations of
+molecular and solvent vibrations) are obtained at the density-
+functional level of theory (see Section S1), we proceed to de-
+termine the equilibrium electronic structure via complete ac-
+tive space self-consistent field spin-orbit (CASSCF-SO) cal-
+culations. The electronic structure is projected onto an effec-
+tive crystal-field Hamiltonian, parametrised in terms of crys-
+tal field parameters. The spin-phonon couplings are obtained
+from a single CASSCF calculation, by computing the analytic
+derivatives of the molecular Hamiltonian with respect to the
+nuclear coordinates [14] (see Section S1 for more details).
+The
+lowest-energy
+angular
+momentum
+multiplet
+of
+[Dy(Cpttt)2]+ (J = 15/2) can thus be described by the ab ini-
+tio vibronic Hamiltonian
+ˆH = ∑
+m
+Em|m⟩⟨m|+∑
+j
+ˆVj ⊗(ˆbj + ˆb†
+j)+∑
+j
+ωj ˆb†
+j ˆbj,
+(1)
+where Em denotes the energy associated with the electronic
+state |m⟩ and ˆVj represent the spin-phonon coupling opera-
+tors. The harmonic vibrational modes of the DCM-solvated
+[Dy(Cpttt)2]+ are described in terms of their bosonic annihi-
+lation (creation) operators ˆbj (ˆb†
+j) and frequencies ωj.
+In the absence of magnetic fields, the Hamiltonian (1) is
+symmetric under time reversal. This symmetry results in a
+two-fold degeneracy of the energy levels Em, whose corre-
+sponding eigenstates |m⟩ and | ¯m⟩ form a time-reversal conju-
+gate Kramers doublet. The degeneracy is lifted by introducing
+a magnetic field B, which couples to the electronic degrees of
+freedom via the Zeeman interaction ˆHZee = µBgJB · ˆJ, where
+gJ is the Landé g-factor and ˆJ is the total angular momentum
+operator. To linear order in the magnetic field, each Kramers
+doublet splits into two energy levels Em±∆m/2 corresponding
+to the states
+|m+⟩ = cos θm
+2 |m⟩+eiφm sin θm
+2 | ¯m⟩
+(2)
+|m−⟩ = −sin θm
+2 |m⟩+eiφm cos θm
+2 | ¯m⟩
+(3)
+where the energy splitting ∆m and the mixing angles θm and
+φm are determined by the matrix elements of the Zeeman
+Hamiltonian on the subspace {|m⟩,| ¯m⟩}. In addition to the
+intra-doublet mixing described by Eqs. (2) and (3), the Zee-
+man interaction also mixes Kramers doublets at different ener-
+gies. The ground doublet acquires contributions from higher-
+lying states
+|1′
+±⟩ = |1±⟩+ ∑
+m̸=1,¯1
+|m⟩⟨m| ˆHZee|1±⟩
+E1 −Em
++O(B2).
+(4)
+These states no longer form a time-reversal conjugate doublet,
+meaning that the spin-phonon coupling can now contribute to
+transitions between them.
+Since QTM is typically observed at much lower tempera-
+tures than the energy gap between the lowest and first excited
+doublets (which here is ∼ 660 K [6]), we focus on the per-
+turbed ground doublet |1′
+±⟩. Within this subspace, the Hamil-
+tonian ˆH + ˆHZee takes the form
+ˆHeff = E1 + ∆1
+2 σ′
+z +∑
+j
+ωj ˆb†
+j ˆbj
+(5)
++ ∑
+j
+�
+⟨1| ˆVj|1⟩−wz
+jσ′
+z
+��
+ˆbj + ˆb†
+j
+�
+− ∑
+j
+�
+wx
+jσ′
+x +wy
+jσ′
+y
+��
+ˆbj + ˆb†
+j
+�
+.
+This Hamiltonian describes the interaction between vi-
+brational
+modes
+and
+an
+effective
+spin
+one-half
+rep-
+resented
+by
+the
+Pauli
+matrices
+σ′ = (σ′
+x,σ′
+y,σ′
+z),
+
+3
+QTM
+electronic
+vibronic
+b)
+a)
+Energy
+[Dy(Cpttt)2]+
+c)
+vibrational DOS
+DCM
+z
+d)
+polarons
+FIG. 1.
+Quantum tunnelling in single-molecule magnets. (a) Molecular structure of a Dy(III) single-molecule magnet surrounded by a
+dichloromethane bath. (b) Equilibrium energy level diagram of the lowest-energy angular momentum multiplet with J = 15/2. The second-
+lowest doublet at E2 is 524 cm−1 higher than the ground doublet at E1, while the highest doublet is 1523 cm−1 above E1. Dipolar and hyperfine
+magnetic fields (Bint) can lift the degeneracy of the doublets and cause quantum tunnelling, which results in avoided crossings when sweeping
+an external magnetic field Bext. Molecular vibrations can influence the magnitude of the avoided crossing. (c) Spin-phonon coupling for the
+solvated complex shown above, as a function of the vibrational frequency (vibrations with ωj > 1500 cm−1 not shown), calculated as the
+Frobenius norm of the operator ˆVj. The grey dashed line represents the vibrational density of states, obtained by assigning to each molecular
+vibration a (anti-symmetrised) Lorentzian lineshape with full width at half-maximum 10 cm−1 (corresponding to a typical timescale of ∼ 1 ps).
+(d) Idea behind the polaron transformation of Eq. (6). Each spin state |1′±⟩ is accompanied by a vibrational distortion (greatly exaggerated
+for visualisation), thus forming a magnetic polaron. Vibrational states |ν⟩ are now described in terms of harmonic displacements around the
+deformed structure, which depends on the state of the spin. Polarons provide an accurate physical picture when the spin-phonon coupling is
+strong and mostly modulates the energy of different spin states but not the coupling between them.
+where σ′
+z = |1′
++⟩⟨1′
++| − |1′
+−⟩⟨1′
+−|.
+The vector w j =
+(ℜ⟨1−| ˆWj|1+⟩,ℑ⟨1−| ˆWj|1+⟩,⟨1+| ˆWj|1+⟩) is defined in terms
+of the operator ˆWj = ∑m̸=1,¯1 ˆVj|m⟩⟨m| ˆHZee/(Em −E1)+ h.c.,
+describing the effect of the Zeeman interaction on the
+spin-phonon coupling. Due to the strong magnetic axiality of
+the complex considered here, the longitudinal component of
+the spin-phonon coupling wz
+j dominates over the transverse
+part wx
+j, wy
+j (see Section S3).
+In this case, we can get a
+better physical picture of the system by transforming the
+Hamiltonian (5) to the polaron frame defined by the unitary
+operator
+ˆS = exp
+�
+∑
+s=±
+|1′
+s⟩⟨1′
+s| ∑
+j
+ξ s
+j
+�
+ˆb†
+j − ˆbj
+��
+,
+(6)
+which mixes electronic and vibrational degrees of freedom by
+displacing the mode operators by ξ ±
+j = (⟨1| ˆVj|1⟩ ∓ wz
+j)/ωj
+depending on the state of the effective spin one-half [27].
+The idea behind this transformation is to allow nuclei to re-
+lax around a new equilibrium geometry, which may be differ-
+ent for every spin state. This lowers the energy of the system
+and provides a good description of the vibronic eigenstates
+when the spin-phonon coupling is approximately diagonal in
+the spin basis (Fig. 1d). In the polaron frame, the longitu-
+dinal spin-phonon coupling is fully absorbed into the purely
+electronic part of the Hamiltonian, while the transverse com-
+ponents can be approximated by their thermal average over
+vibrations, neglecting their vanishingly small quantum fluc-
+tuations (see Section S2).
+After transforming back to the
+original frame, we are left with an effective spin one-half
+Hamiltonian with no residual spin-phonon coupling Heff ≈
+ˆH(pol)
+eff
++∑j ωj ˆb†
+j ˆbj, where
+ˆH(pol)
+eff
+= E1 + ∆1
+2 σ′′
+z +2∑
+j
+⟨1| ˆVj|1��
+ωj
+w j ·σ′′.
+(7)
+The set of Pauli matrices σ′′ = ˆS†(σ′ ⊗ 1lvib) ˆS describe the
+two-level system formed by the magnetic polarons of the
+form ˆS†|1′
+±⟩|{νj}⟩vib, where {νj} is a set of occupation num-
+bers for the vibrational modes of the solvent-SMM system.
+These magnetic polarons can be thought as magnetic elec-
+tronic states strongly coupled to a distortion of the molecular
+geometry. They inherit the magnetic properties of the cor-
+responding electronic states, and can be seen as the molecu-
+
+4
+lar equivalent of the magnetic polarons observed in a range
+of magnetic materials [28–30].
+Polaron representations of
+vibronic systems have been employed in a wide variety of
+settings, ranging from spin-boson models [27, 31] to photo-
+synthetic complexes [32–34], to quantum dots [35–37], pro-
+viding a convenient basis to describe the dynamics of quan-
+tum systems strongly coupled to a vibrational environment.
+These methods are particularly well suited for condensed
+matter systems where the electron-phonon coupling is strong
+but causes very slow transitions between different electronic
+states, allowing exact treatment of the pure-dephasing part
+of the electron-phonon coupling and renormalising the elec-
+tronic parameters. For this reason, the polaron transformation
+is especially effective for describing our system (as detailed in
+Section S3). The most striking advantage of this approach is
+that the average effect of the spin-phonon coupling is included
+non-perturbatively into the electronic part of the Hamiltonian,
+leaving behind a vanishingly small residual spin-phonon cou-
+pling.
+As a last step, we bring the Hamiltonian in Eq. (7) into a
+more familiar form by expressing it in terms of an effective g-
+matrix. We recall that the quantities ∆1 and w j depend linearly
+on the magnetic field B via the Zeeman Hamiltonian ˆHZee. An
+additional dependence on the orientation of the magnetic field
+comes from the mixing angles θ1 and φ1 introduced in Eqs.
+(2) and (3), appearing in the states |1±⟩ used in the definition
+of w j. This further dependence is removed by transforming
+the Pauli operators back to the basis {|1⟩,|¯1⟩} via a three-
+dimensional rotation σ = Rθ1,φ1 ·σ′′. Finally, we obtain
+ˆH(pol)
+eff
+= E1 + µBB·
+�
+gel +∑
+j
+gvib
+j
+�
+· σ
+2 ,
+(8)
+for appropriately defined electronic and single-mode vibronic
+g-matrices gel and gvib
+j . These are directly related to the elec-
+tronic splitting term ∆1 and to the vibronic corrections de-
+scribed by w j in Eq. (7), respectively (see Section S2 for
+a thorough derivation). The main advantage of representing
+the ground Kramers doublet with an effective spin one-half
+Hamiltonian is that it provides a conceptually simple founda-
+tion for studying low-temperature magnetic behaviour of the
+complex, confining all microscopic details, including vibronic
+effects, to an effective g-matrix.
+III.
+RESULTS
+We begin by considering the influence of vibrations on the
+Zeeman splitting of the lowest doublet. The Zeeman splitting
+in absence of vibrations is simply given by ∆1 = µB|B · gel|.
+In the presence of vibrations, the electronic g-matrix gel is
+modified by adding the vibronic correction ∑j gvib
+j , resulting
+in the Zeeman splitting ∆vib
+1 . In Fig. 2 we show the Zee-
+man splittings as a function of the orientation of the mag-
+netic field B, parametrised in terms of the polar angles (θ,φ).
+Depending on the field orientation, vibrations can lead to ei-
+ther an increase or decrease of the Zeeman splitting. These
+changes seem rather small when compared to the largest elec-
+tronic splitting, obtained when B is oriented along the z-axis
+(Fig. 1a), as expected for a complex with easy-axis anisotropy.
+However, they become quite significant for field orientations
+close to the xy-plane, where the purely electronic splitting ∆1
+becomes vanishingly small and ∆vib
+1
+can be dominated by the
+vibronic contribution. This is clearly shown in Fig. 2b and
+2c, where we decompose the total field B = Bint + Bext in a
+fixed internal component Bint originating from dipolar and hy-
+perfine interactions, responsible for opening a tunnelling gap,
+and an external part Bext which we sweep along a fixed direc-
+tion across zero. We note that this effect is specific to states
+with easy-axis magnetic anisotropy, however this is the defin-
+ing feature of SMMs, such that our results should be generally
+applicable to all Kramers SMMs. A more in-depth discussion
+on the origin and magnitude of the internal field can be found
+in Section S5. When these fields lie in the plane perpendicu-
+lar to the purely electronic easy axis, i.e. the hard plane, the
+vibronic splitting can be four orders of magnitude larger than
+the electronic one (Fig. 2b). The situation is reversed when
+the fields lie in the hard plane of the vibronic g-matrix (Fig.
+2c).
+So far we have seen that spin-phonon coupling can either
+enhance or reduce the tunnelling gap in the presence of a mag-
+netic field depending on its orientation. For this reason, it is
+not immediately clear whether its effects survive ensemble av-
+eraging in a collection of randomly oriented SMMs, such as
+the frozen solutions considered in magnetometry experiments.
+In order to check this, let us consider an ideal field-dependent
+magnetisation measurement. When sweeping a magnetic field
+Bext at a constant rate from positive to negative values along
+a given direction, QTM is typically observed as a sharp step
+in the magnetisation of the sample when crossing the region
+around Bext = 0 [10]. This sudden change of the magnetisa-
+tion is due to a non-adiabatic spin-flip transition between the
+two lowest energy spin states, that occurs when traversing an
+avoided crossing (see diagram in Fig. 1b, right). The spin-flip
+probability is given by the celebrated Landau-Zener expres-
+sion [38–43], which in our case takes the form
+PLZ = 1−exp
+�
+−π|∆⊥|2
+2|v|
+�
+,
+(9)
+where we have defined v = µBdBext/dt ·g, and ∆⊥ is the com-
+ponent of ∆ = µBBint · g perpendicular to v, while g denotes
+the total electronic-vibrational g-matrix appearing in Eq. (8)
+(see Section S2 for a derivation of Eq. (9)). We account for
+orientational disorder by averaging Eq. (9) over all possible
+orientations of internal and external magnetic fields, yielding
+the ensemble average ⟨PLZ⟩.
+The effect of spin-phonon coupling on the spin-flip dynam-
+ics of an ensemble of SMMs can be clearly seen in Fig. 3. In-
+cluding the vibronic correction to the ground doublet g-matrix
+leads to enhanced spin-flip probabilities across a wide range
+of internal field strengths and field sweep rates. This is in line
+with previous results suggesting that molecular flexibility cor-
+relates with QTM [22]. To further corroborate our model, we
+test its predictions against experimental data. We extracted the
+average spin-flip probability from published hysteresis data
+
+5
+a)
+b)
+c)
+FIG. 2.
+Zeeman splitting of the ground Kramers doublet.
+(a)
+Electronic ground doublet splitting (∆1, top) and vibronic correction
+(∆vib
+1
+− ∆1, bottom) as a function of the orientation of the magnetic
+field B = (sinθ cosφ,sinθ sinφ,cosθ), with magnitude fixed to 1 T.
+The dashed (solid) line corresponds to the electronic (vibronic) hard
+plane. (b–c) Electronic (dashed) and vibronic (solid) Zeeman split-
+ting of the ground doublet as a function of the external field magni-
+tude Bext in the presence of a transverse internal field Bint = 1 mT.
+External and internal fields are perpendicular to each other and were
+both chosen to lie in the hard plane of either the electronic (b) or
+vibronic (c) g-matrix. The orientation of the external (internal) field
+is shown for both cases as circles (crosses) in the inset in (a), with
+colors matching the ones in (b) and (c).
+of [Dy(Cpttt)2][B(C6F5)4] in DCM with sweep rates ranging
+between 10–20 Oe/s [6], yielding a value of ⟨PLZ⟩ = 0.27,
+indicated by the pink line in Fig. 3. We then checked what
+strength of the internal field Bint is required to reproduce such
+spin-flip probability based on Eq. (9). In Fig. 3, we observe
+that the values of Bint required by the vibronic model to re-
+produce the observed spin-flip probability are perfectly con-
+sistent with the dipolar fields naturally occurring in the sam-
+ple, whereas the purely electronic model necessitates internal
+fields that are one order of magnitude larger. These results
+clearly demonstrate the significance of spin-phonon coupling
+for QTM in a disordered ensemble of SMMs. A detailed dis-
+cussion on the estimation of spin-flip probabilities and internal
+fields from magnetisation measurements is presented in Sec-
+FIG. 3.
+Landau-Zener spin-flip probability. Ensemble-averaged
+spin-flip probability as a function of the internal field strength Bint
+causing tunnelling within the ground Kramers doublet, shown for
+different sweep rates dBext/dt. Results for the vibronic model of Eq.
+(8) are shown as orange solid lines, together with the spin-flip prob-
+abilities predicted by a purely electronic model obtained by setting
+the spin-phonon coupling to zero, shown as blue dashed lines. The
+horizontal pink line indicates ⟨PLZ⟩ = 0.27, extracted from hysteresis
+data from Ref. [6] (Section S4). The green shaded area indicates the
+range of values for typical dipolar fields in the corresponding sample
+(Section S5).
+tions S4 and S5.
+IV.
+DISCUSSION
+As shown above, the combined effect of all vibrations in
+a randomly oriented ensemble of solvated SMMs is to en-
+hance QTM. However, not all vibrations contribute to the
+same extent. Based on the polaron model introduced above,
+vibrations with large spin-phonon coupling and low frequency
+have a larger impact on the magnetic properties of the ground
+Kramers doublet. This can be seen from Eq. (7), where the
+vibronic correction to the effective ground Kramers Hamil-
+tonian is weighted by the factor ⟨1| ˆVj|1⟩/ω j. Another prop-
+erty of vibrations that can influence QTM is their symmetry.
+In monometallic SMMs, QTM has generally been correlated
+with a reduction of the axial symmetry of the complex, either
+by the presence of flexible ligands or by transverse magnetic
+fields. Since we are interested in symmetry only as long as it
+influences magnetism, it is useful to introduce a measure of
+axiality on the g-matrix, such as
+A(g) =
+��g− 1
+3Tr g
+��
+�
+2
+3Tr g
+,
+(10)
+where ∥·∥ denotes the Frobenius norm. This measure yields 1
+for a perfect easy-axis complex, 1/2 for an easy plane system,
+and 0 for the perfectly isotropic case. The axiality of an indi-
+vidual vibrational mode can be quantified as Aj = A(gel+gvib
+j )
+by building a single-mode vibronic g-matrix, analogous to
+
+△1 (cm-1
+元
+10
+8
+6
+4
+2
+2
+0
+0
+0
+一
+元
+2
+2
+ΦAvib - △1 (cm-1)
+0.2
+0.1
+0
+2
+-0.1
+-0.2
+0
+元
+0
+一元
+2
+2(cm
+0.2
+0.1
+0
+2
+-0.1
+-0.2
+0
+一元
+2
+2(cm
+0.2
+0.1
+0
+2
+-0.1
+-0.2
+0
+一元
+2
+26
+a)
+b)
+FIG. 4.
+Single-mode contributions to tunnelling of the magnetisation. (a) Single-mode vibronic Landau-Zener probabilities plotted for
+each vibrational mode, shown as a function of the mode axiality relative to the axiality of the purely electronic g-matrix (∆Aj = Aj −Ael). The
+magnitude of the internal field is fixed to Bint = 1 mT and the external field sweep rate is 10 Oe/s. The color coding represents the spin-phonon
+coupling strength ∥ ˆVj∥. Grey dashed lines corresponds to the purely electronic model. (b) Visual representation of the displacements induced
+by the vibrational modes indicated by arrows in (a). Solvent motion is only shown for modes 2 and 6, which have negligible amplitude on the
+SMM.
+the multi-mode one introduced in Eq.
+(8).
+We might be
+tempted to intuitively conclude that vibrational motion al-
+ways decreases the axiality with respect to its electronic value
+Ael = A(gel), given that the collective effect of vibrations is to
+enhance QTM. However, when considered individually, some
+vibrations can have the opposite effect, of effectively increas-
+ing the magnetic axiality.
+In order to see how axiality correlates to QTM, we calcu-
+late the single-mode Landau-Zener probabilities ⟨Pj⟩. These
+are obtained by replacing the multi-mode vibronic g-matrix in
+Eq. (8) with the single-mode one gel +gvib
+j , and following the
+same procedure detailed in Section S2. The single-mode con-
+tribution to the spin-flip probability unambiguously correlates
+with mode axiality, as shown in Fig. 4a. Vibrational modes
+that lead to a larger QTM probability are likely to reduce the
+magnetic axiality of the complex (top-left sector). Vice versa,
+those vibrational modes that enhance axiality also suppress
+QTM (bottom-right sector).
+As a first step towards uncovering the microscopic basis
+of this unexpected behaviour, we single out the three vibra-
+tional modes that have the largest impact on axiality and spin-
+flip probability in both directions. These vibrational modes,
+labelled 1–6, represent a range of qualitatively distinct vibra-
+tions, as can be observed in Fig. 4b. Modes 4 and 5 are among
+the ones exhibiting the strongest spin-phonon coupling. Both
+of them are mainly localised on one of the Cpttt ligands and
+involve atomic displacements along the easy axis and, to a
+lesser extent, rotations of the methyl groups. Modes 1 and
+3 are among the ones with largest amplitude on the Dy ion,
+which in both cases mainly moves in the hard plane, disrupt-
+ing axial symmetry and enhancing tunnelling. Lastly, modes
+2 and 6 predominantly correspond to solvent vibrations, and
+are thus very low energy and so give a large contribution via
+the small denominator in Eq. (7).
+This analysis shows that the effect of vibrational modes on
+QTM is more nuanced than what both intuition and previous
+work would suggest. Despite leading to an overall increase
+of the spin-flip probability on average, coupling the spin to
+specific vibrations can increase the magnetic axiality of the
+complex and suppress QTM. This opens a new avenue for the
+improvement of magnetic relaxation times in SMMs, shifting
+the role of vibrations from purely antagonistic to potentially
+beneficial.
+According to the results shown above, the ideal candidates
+to observe vibronic suppression of QTM are systems exhibit-
+ing strongly axial, low frequency vibrations, strongly coupled
+to the electronic effective spin. Strong spin-phonon coupling
+and low frequency ensure a significant change in magnetic
+properties according to Eq. (7), but may not be enough to hin-
+der tunnelling. In order to be beneficial, vibrations also need
+to enhance the axiality of the ground doublet g-matrix. The
+relation between magnetic axiality and vibrational symmetry
+remains yet to be explored, and might lead to new insights
+regarding rational design of ideal ligands.
+
+17
+V.
+CONCLUSIONS
+In conclusion, we have presented a detailed description of
+the effect of molecular and solvent vibrations on the quan-
+tum tunnelling between low-energy spin states in a single-ion
+Dy(III) SMM. Our theoretical results, based on an ab initio
+approach, are complemented by a polaron treatment of the rel-
+evant vibronic degrees of freedom, which does not suffer from
+any weak spin-phonon coupling assumption and is therefore
+well-suited to other strong coupling scenarios. We have been
+able to derive a non-perturbative vibronic correction to the ef-
+fective g-matrix of the lowest-energy Kramers doublet, which
+we have used as a basis to determine the tunnelling dynamics
+in a magnetic field sweep experiment. This has allowed us to
+formulate the key observation that, vibrations collectively en-
+hance QTM, but some particular vibrational modes unexpect-
+edly suppress QTM. This behaviour correlates to the axiality
+of each mode, which can be used as a proxy for determining
+whether a specific vibration enhances or hinders tunnelling.
+The observation that individual vibrational modes can sup-
+press QTM challenges the paradigm that dismisses vibrations
+as detrimental, a mere obstacle to achieving long-lasting in-
+formation storage on SMMs, and forces us instead to recon-
+sider them under a new light, as tools that can be actively en-
+gineered to our advantage to keep tunnelling at bay and ex-
+tend relaxation timescales in molecular magnets. This idea
+suggests parallelisms with other seemingly unrelated chem-
+ical systems where electron-phonon coupling plays an im-
+portant role.
+For example, the study of electronic energy
+transfer across photosynthetic complexes was radically trans-
+formed by the simple observation that vibrations could play
+an active role, maintaining quantum coherence in noisy room-
+temperature environments, rather than just passively causing
+decoherence between electronic states [44]. Identifying these
+beneficial vibrations and amplifying their effect via chemical
+design of new SMMs remains an open question, whose solu-
+tion we believe could greatly benefit from the results and the
+methods introduced in this work.
+ACKNOWLEDGEMENTS
+This work was made possible thanks to the ERC
+grant 2019-STG-851504 and Royal Society fellowship
+URF191320. The authors also acknowledge support from the
+Computational Shared Facility at the University of Manch-
+ester.
+DATA AVAILABILITY
+The data that support the findings of this study are available
+at http://doi.org/10.48420/21892887.
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+ular magnets, Nature 410, 789 (2001).
+[2] R. Sessoli, Magnetic molecules back in the race, Nature 548,
+400 (2017).
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+
+1
+Supplementary Information:
+Vibronic Effects on the Quantum Tunnelling of Magnetisation
+in Single-Molecule Magnets
+Andrea Mattioni,1,∗ Jakob K. Staab,1 William J. A. Blackmore,1 Daniel Reta,1,2
+Jake Iles-Smith,3,4 Ahsan Nazir,3 and Nicholas F. Chilton1,†
+1Department of Chemistry, School of Natural Sciences,
+The University of Manchester, Oxford Road, Manchester, M13 9PL, UK
+2 Faculty of Chemistry, UPV/EHU & Donostia International Physics Center DIPC,
+Ikerbasque, Basque Foundation for Science, Bilbao, Spain
+3Department of Physics and Astronomy, School of Natural Sciences,
+The University of Manchester, Oxford Road, Manchester M13 9PL, UK
+4Department of Electrical and Electronic Engineering, School of Engineering,
+The University of Manchester, Sackville Street Building, Manchester M1 3BB, UK
+∗ [email protected]
+† [email protected]
+
+2
+S1.
+AB INITIO CALCULATIONS
+The ab initio model of the DCM-solvated [Dy(Cpttt)2]+ molecule is constructed using a multi-layer approach. During ge-
+ometry optimisation and frequency calculation the system is partitioned into two layers following the ONIOM scheme [1].
+The high-level layer, consisting of the SMM itself and the first solvation shell of 26 DCM molecules, is described by Density
+Functional Theory (DFT) while the outer bulk of the DCM ball constitutes the low-level layer modelled by the semi-empirical
+PM6 method. All DFT calculations are carried out using the pure PBE exchange-correlation functional [2] with Grimme’s D3
+dispersion correction. Dysprosium is replaced by its diamagnetic analogue yttrium for which the Stuttgart RSC 1997 ECP basis
+is employed [3]. Cp ring carbons directly coordinated to the central ion are equipped with Dunning’s correlation consistent
+triple-zeta polarised cc-pVTZ basis set and all remaining atoms with its double-zeta analogue cc-pVDZ [4]. Subsequently, the
+electronic spin states and spin-phonon coupling parameters are calculated at the CASSCF-SO level explicitly accounting for the
+strong static correlation present in the f-shell of Dy(III) ions. At this level, environmental effects are treated using an electrostatic
+point charge representation of all DCM atoms. All DFT/PM6 calculations are carried out with GAUSSIAN version 9 revision
+D.01 [5] and the CASSCF calculations are carried out with OPENMOLCAS version 21.06 [6].
+The starting [Dy(Cpttt)2]+ solvated system was obtained using the solvate program belonging to the AmberTool suite of
+packages, with box as method and CHCL3BOX as solvent model. Chloroform molecules were subsequently converted to
+DCM. From this large system, only molecules falling within 9 Å from the central metal atom are considered from now on.
+The initial disordered system of 160 DCM molecules packed around the [Dy(Cpttt)2]+ crystal structure [7] is pre-optimised
+in steps, starting by only optimising the high-level layer atoms and freezing the rest of the system. The low-layer atoms are
+pre-optimised along the same lines starting with DCM molecules closest to the SMM and working in shells towards the outside.
+Subsequently, the whole system is geometry optimised until RMS (maximum) values in force and displacement corresponding
+to 0.00045 au (0.0003 au) and 0.0018 au (0.0012 au) are reached, respectively. After adjusting the isotopic mass of yttrium to
+that of dysprosium mDy = 162.5u, vibrational normal modes and frequencies of the entire molecular aggregate are computed
+within the harmonic approximation.
+Electrostatic atomic point charge representations of the environment DCM molecules are evaluated for each isolated solvent
+molecule independently at the DFT level of theory employing the CHarges from ELectrostatic Potentials using a Grid-based
+(ChelpG) method [8], which serve as a classical model of environmental effects in the subsequent CASSCF calculations.
+The evaluation of equilibrium electronic states and spin-phonon coupling parameters is carried out at the CASSCF level
+including scalar relativistic effects using the second-order Douglas-Kroll Hamiltonian and spin-orbit coupling through the atomic
+mean field approximation implemented in the restricted active space state interaction approach [9, 10]. The dysprosium atom is
+equipped with the ANO-RCC-VTZP, the Cp ring carbons with the ANO-RCC-VDZP and the remaining atoms with the ANO-
+RCC-VDZ basis set [11]. The resolution of the identity approximation with an on-the-fly acCD auxiliary basis is employed to
+handle the two-electron integrals [12]. The active space of 9 electrons in 7 orbitals, spanned by 4f atomic orbitals, is employed
+in a state-average CASSCF calculation including the 18 lowest lying sextet roots which span the 6H and 6F atomic terms.
+We use our own implementation of spin Hamiltonian parameter projection to obtain the crystal field parameters Bq
+k entering
+the Hamiltonian
+ˆHCF = ∑
+k=2,4,6
+k
+∑
+q=−k
+θkBq
+kOq
+k(ˆJ),
+(S1)
+describing the 6H15/2 ground state multiplet. Operator equivalent factors and Stevens operators are denoted by θk and Oq
+k(ˆJ),
+where ˆJ = ( ˆJx, ˆJy, ˆJz) are the angular momentum components. Spin-phonon coupling arises from changes to the Hamiltonian
+(S1) due to slight distortions of the molecular geometry, parametrised as
+Bq
+k({Xj}) = Bq
+k +
+M
+∑
+j=1
+∂Bq
+k
+∂Xj
+Xj +...,
+(S2)
+where Xj denotes the dimensionless j-th normal coordinate of the complex under consideration. The derivatives ∂Bq
+k/∂Xj are
+calculated using the Linear Vibronic Coupling (LVC) approach described in Ref. [13] based on the state-average CASSCF
+density-fitting gradients and non-adiabatic coupling involving all 18 sextet roots.
+The final step leading to Eq. (1) in the main text is to quantise the normal modes and express them in terms of bosonic
+annihilation and creation operators satisfying [ˆbi, ˆb†
+j] = δij as
+ˆXj =
+ˆbj + ˆb†
+j
+√
+2
+.
+(S3)
+
+3
+Defining the spin-phonon coupling operators
+ˆVj = 1
+√
+2 ∑
+k,q
+θk
+∂Bq
+k
+∂Xj
+Oq
+k(ˆJ),
+(S4)
+we can finally write down the crystal field Hamiltonian including linear spin-phonon coupling as
+ˆH = ˆHCF +∑
+j
+ˆVj ⊗(ˆbj + ˆb†
+j)+∑
+j
+ωj ˆb†
+j ˆb j.
+(S5)
+
+4
+S2.
+DERIVATION OF THE EFFECTIVE VIBRONIC DOUBLET HAMILTONIAN
+A.
+Electronic perturbation Theory
+The starting point for our analysis of vibronic effects on QTM is the vibronic Hamiltonian
+ˆH = ∑
+m>0
+Em(|m⟩⟨m|+| ¯m⟩⟨ ¯m|)+ ˆHZee +∑
+j
+ˆVj ⊗(ˆbj + ˆb†
+j)+∑
+j
+ωj ˆb†
+j ˆb j,
+(S6)
+where ˆHZee = µBgJB · ˆJ. is the Zeeman interaction with a magnetic field B. The doubly degenerate eigenstates of the crystal
+field Hamiltonian HCF = ∑m>0 Em(|m⟩⟨m|+| ¯m⟩⟨ ¯m|) are related by time-reversal symmetry, i.e. ˆΘ|m⟩ ∝ | ¯m⟩ with ˆΘ2|m⟩ = −|m⟩,
+where ˆΘ is the time-reversal operator. In the case of [Dy(Cpttt)2]+, the total electronic angular momentum is J = 15/2, leading
+to 2J + 1 = 16 electronic states. We label these states in ascending energy with integers m = ±1,...,±8, using the compact
+notation |−m⟩ = | ¯m⟩.
+We momentarily neglect the spin-phonon coupling and focus on the purely electronic Hamiltonian Hel = HCF +HZee. Within
+each degenerate subspace, the Zeeman term selects a specific electronic basis and lifts its degeneracy. This can be seen by
+projecting the electronic Hamitonian onto the m-th subspace and diagonalising the 2×2 matrix
+H(m)
+el
+= Em + µBgJ
+�
+⟨m|B· ˆJ|m⟩ ⟨m|B· ˆJ| ¯m⟩
+⟨ ¯m|B· ˆJ|m⟩ ⟨ ¯m|B· ˆJ| ¯m⟩
+�
+.
+(S7)
+For each individual cartesian component of the angular momentum, we decompose the corresponding 2 × 2 matrix in terms of
+Pauli spin operators, which allows to rewrite the Hamiltonian of the m-th doublet as H(m)
+el
+= Em + µBB·g(m)
+el ·σ(m)/2, where
+g(m)
+el
+= 2gJ
+�
+�
+ℜ⟨ ¯m| ˆJx|m⟩ ℑ⟨ ¯m| ˆJx|m⟩ ⟨m| ˆJx|m⟩
+ℜ⟨ ¯m| ˆJy|m⟩ ℑ⟨ ¯m| ˆJy|m⟩ ⟨m| ˆJy|m⟩
+ℜ⟨ ¯m| ˆJz|m⟩ ℑ⟨ ¯m| ˆJz|m⟩ ⟨m| ˆJz|m⟩
+�
+�
+(S8)
+is the g-matrix for an effective spin 1/2 and σ(m) = (σ(m)
+x
+,σ(m)
+y
+,σ(m)
+z
+), with σ(m)
+z
+= |m⟩⟨m|−| ¯m⟩⟨ ¯m|. We note that in general the
+g-matrix in Eq. (S8) is not hermitean, but can be brought to such form by transforming the spin operators σ(m) to an appropriate
+basis [14]. An easier prescription to find the hermitean form af any g-matrix g is to redefine it as
+�
+gg†.
+To lowest order in the magnetic field, the Zeeman interaction lifts the two-fold degeneracy by selecting the basis
+|m+⟩ = cos θm
+2 |m⟩+eiφm sin θm
+2 | ¯m⟩
+(S9)
+|m−⟩ = −sin θm
+2 |m⟩+eiφm cos θm
+2 | ¯m⟩
+(S10)
+and shifting the energies according to Em,± = Em ±∆m/2, where the gap
+∆m = ⟨m+| ˆHZee|m+⟩−⟨m−| ˆHZee|m−⟩
+(S11)
+= 2µBgJ
+�
+⟨m|B· ˆJ|m⟩2 +|⟨m|B· ˆJ| ¯m⟩|2
+can be obtained as the norm of the vector jm = µBB·g(m)
+el
+and the phase and mixing angles are defined as
+eiφm = ⟨ ¯m|B· ˆJ|m⟩
+|⟨ ¯m|B· ˆJ|m⟩|,
+tanθm = |⟨ ¯m|B· ˆJ|m⟩|
+⟨m|B· ˆJ|m⟩ ,
+(S12)
+or equivalently as the azimuthal and polar angles determining the direction of jm.
+Besides selecting a preferred basis and lifting the degeneracy of each doublet, the Zeeman interaction also causes mixing
+between different doublets. In particular, the lowest doublet will change according to
+|1′
+±⟩ = |1±⟩+ ∑
+m̸=1,¯1
+|m⟩⟨m| ˆHZee|1±⟩
+E1 −Em
++O(B2) ≈
+�
+1− ˆQ1 ˆHZee
+�
+|1±⟩,
+(S13)
+with
+ˆQ1 = ∑
+m̸=1,¯1
+|m⟩
+1
+Em −E1
+⟨m|.
+(S14)
+
+5
+B.
+Spin-boson Hamiltonian for the ground doublet
+Now that we have an approximate expression for the relevant electronic states, we reintroduce the spin-phonon coupling into
+the picture. First, we project the vibronic Hamiltonian (S6) onto the subspace spanned by |1′
+±⟩, yielding
+ˆHeff = E1 +
+� ∆1
+2
+0
+0
+− ∆1
+2
+�
++∑
+j
+�
+⟨1′
++| ˆVj|1′
++⟩ ⟨1′
++| ˆVj|1′
+−⟩
+⟨1′
+−| ˆVj|1′
++⟩ ⟨1′
+−| ˆVj|1′
+−⟩
+�
+⊗(ˆbj + ˆb†
+j)+∑
+j
+ωj ˆb†
+j ˆb j.
+(S15)
+On this basis, the purely electronic part ˆHCF + ˆHZee is diagonal with eigenvalues E1 ± ∆1/2, and the purely vibrational part is
+trivially unaffected. On the other hand, the spin-phonon couplings can be calculated to lowest order in the magnetic field strength
+B as
+⟨1′
+±| ˆVj|1′
+±⟩ = ⟨1±|
+�
+1− ˆHZee ˆQ1
+� ˆVj
+�
+1− ˆQ1 ˆHZee
+�
+|1±⟩+O(B2)
+(S16)
+= ⟨1±| ˆVj|1±⟩−⟨1±|
+� ˆVj ˆQ1 ˆHZee + ˆHZee ˆQ1 ˆVj
+�
+|1±⟩+O(B2)
+= ⟨1| ˆVj|1⟩−⟨1±| ˆWj|1±⟩+O(B2),
+⟨1′
+∓| ˆVj|1′
+±⟩ = ⟨1∓|
+�
+1− ˆHZee ˆQ1
+� ˆVj
+�
+1− ˆQ1 ˆHZee
+�
+|1±⟩+O(B2)
+(S17)
+= ⟨1∓| ˆVj|1±⟩−⟨1∓|
+� ˆVj ˆQ1 ˆHZee + ˆHZee ˆQ1 ˆVj
+�
+|1±⟩+O(B2)
+= −⟨1∓| ˆWj|1±⟩+O(B2),
+where we have defined
+ˆWj = ˆVj ˆQ1 ˆHZee + ˆHZee ˆQ1 ˆVj
+(S18)
+and used the time-reversal invariance of the spin-phonon coupling operators to obtain ⟨1±| ˆVj|1±⟩ = ⟨1| ˆVj|1⟩ and ⟨1∓| ˆVj|1±⟩ = 0.
+The two states |1±⟩ form a conjugate pair under time reversal, meaning that ˆΘ|1±⟩ = ∓eiα|1∓⟩ for some α ∈ R. Using the
+fact that for any two states ψ, ϕ, and for any operator ˆO we have ⟨ψ| ˆO|ϕ⟩ = ⟨ ˆΘϕ| ˆΘ ˆO† ˆΘ−1| ˆΘψ⟩, and recalling that the angular
+momentum operator is odd under time reversal, i.e. ˆΘˆJ ˆΘ−1 = −ˆJ, we can show that
+⟨1−| ˆWj|1−⟩ = ⟨ ˆΘ1−| ˆΘ ˆWj ˆΘ−1| ˆΘ1−⟩ = −⟨1+| ˆWj|1+⟩.
+Keeping in mind these observations, and defining the vector
+w j =
+�
+�
+wx
+j
+wy
+j
+wz
+j
+�
+� =
+�
+�
+ℜ ⟨1−| ˆWj|1+⟩
+ℑ ⟨1−| ˆWj|1+⟩
+⟨1+| ˆWj|1+⟩
+�
+�,
+(S19)
+we can rewrite the spin-phonon coupling operators in Eq. (S15) as
+�
+⟨1′
++| ˆVj|1′
++⟩ ⟨1′
++| ˆVj|1′
+−⟩
+⟨1′
+−| ˆVj|1′
++⟩ ⟨1′
+−| ˆVj|1′
+−⟩
+�
+= ⟨1| ˆVj|1⟩−
+�
+⟨1+| ˆWj|1+⟩ ⟨1−| ˆWj|1+⟩∗
+⟨1−| ˆWj|1+⟩ −⟨1+| ˆWj|1+⟩
+�
+= ⟨1| ˆVj|1⟩−w j ·σ′
+(S20)
+where σ′ is a vector whose entries are the Pauli matrices in the basis |1′
+±⟩, i.e. σ′
+z = |1′
++⟩⟨1′
++| − |1′
+−⟩⟨1′
+−|. Plugging this back
+into Eq. (S15) and explicitly singling out the diagonal components of ˆHeff in the basis |1′
+±⟩, we obtain
+ˆHeff = |1′
++⟩⟨1′
++|
+�
+E1 + ∆1
+2 +∑
+j
+�
+⟨1| ˆVj|1⟩−wz
+j
+��
+ˆbj + ˆb†
+j
+�
++∑
+j
+ω j ˆb†
+j ˆbj
+�
+(S21)
++ |1′
+−⟩⟨1′
+−|
+�
+E1 − ∆1
+2 +∑
+j
+�
+⟨1| ˆVj|1⟩+wz
+j
+��
+ˆbj + ˆb†
+j
+�
++∑
+j
+ω j ˆb†
+j ˆbj
+�
+− ∑
+j
+�
+wx
+jσ′
+x +wy
+jσ′
+y
+��
+ˆbj + ˆb†
+j
+�
+.
+At this point, we apply a unitary polaron transformation to the Hamiltonian (S21)
+ˆS = exp
+�
+∑
+s=±
+|1′
+s⟩⟨1′
+s| ∑
+j
+1
+ωj
+�
+⟨1| ˆVj|1⟩−swz
+j
+��
+ˆb†
+j − ˆbj
+��
+(S22)
+= ∑
+s=±
+|1′
+s⟩⟨1′
+s| ∏
+j
+ˆD j(ξ s
+j)
+
+6
+where ξ s
+j =
+�
+⟨1| ˆVj|1⟩−swz
+j
+�
+/ω j and
+ˆD j(ξ s
+j) = eξ s
+j
+�
+ˆb†
+j−ˆb j
+�
+(S23)
+is the bosonic displacement operator acting on mode j, i.e. ˆD j(ξ)ˆbj ˆD†
+j(ξ) = ˆbj −ξ. The Hamiltonian thus becomes
+ˆS ˆHeff ˆS† = ∑
+s=±
+|1′
+s⟩⟨1′
+s|
+�
+E1 +s∆1
+2 −∑
+j
+ωj|ξ s
+j|2
+�
++∑
+j
+ωj ˆb†
+j ˆbj −∑
+j
+ˆS
+�
+wx
+jσ′
+x +wy
+jσ′
+y
+��
+ˆb j + ˆb†
+j
+�
+ˆS†.
+(S24)
+The polaron transformation reabsorbes the diagonal component of the spin-phonon coupling (S20) proportional to wz
+j into the
+energy shifts ωj|ξ ±
+j |2, leaving a residual off-diagonal spin-phonon coupling proportional to wx
+j and wy
+j. Note that the polaron
+transformation exactly diagonalises the Hamiltonian (S15) if wx
+j = wy
+j = 0. In Section S3, we argue in detail that in our case
+|wx
+j|,|wy
+j| ≪ |wz
+j| to a very good approximation. Based on this argument, we could decide to neglect the residual spin-phonon
+coupling in the polaron frame. The energies of the states belonging to the lowest doublet are shifted by a vibronic correction
+E1′± = E1 ± ∆1
+2 −∑
+j
+1
+ω j
+�
+⟨1| ˆVj|1⟩∓wz
+j
+�2
+(S25)
+= E1 ± ��1
+2 −∑
+j
+1
+ω j
+�
+⟨1| ˆVj|1⟩2 ∓2⟨1| ˆVj|1⟩wz
+j +O(B2)
+�
+,
+(S26)
+leading to a redefinition of the energy gap
+E1′+ −E1′− = ∆1 +4∑
+j
+⟨1| ˆVj|1⟩
+ωj
+wz
+j.
+(S27)
+Although the off-diagonal components of the spin-phonon coupling wx
+j and wy
+j are several orders of magnitude smaller than
+the diagonal one wz
+j (see Section S3), the sheer number of vibrational modes could still lead to an observable effect on the
+electronic degrees of freedom. We can estimate this effect by averaging the residual spin-phonon coupling over a thermal
+phonon distribution in the polaron frame. Making use of Eq. (S22), the off-diagonal coupling in Eq. (S24) can be written as
+ˆH(pol)
+sp-ph = −∑
+j
+ˆS
+�
+wx
+jσ′
+x +wy
+jσ′
+y
+��
+ˆbj + ˆb†
+j
+�
+ˆS†
+(S28)
+= −∑
+j
+|1′
+−⟩⟨1−| ˆWj|1+⟩⟨1′
++| ˆD j(ξ −
+j )
+�
+ˆbj + ˆb†
+j
+�
+ˆD†
+j(ξ +
+j )+h.c.
+Assuming the vibrations to be in a thermal state at temperature T in the polaron frame
+ρ(th)
+ph = ∏
+j
+ρ(th)
+j
+= ∏
+j
+e−ωj ˆb†
+j ˆb j/kBT
+Tr
+�
+e−ωj ˆb†
+j ˆb j/kBT�,
+(S29)
+obtaining the average of Eq. (S28) reduces to calculating the dimensionless quantity
+κj = −Tr
+�
+ˆD j(ξ −
+j )
+�
+ˆbj + ˆb†
+j
+�
+ˆD†
+j(ξ +
+j )ρ(th)
+j
+�
+(S30)
+=
+�
+ξ +
+j +ξ −
+j
+�
+e− 1
+2
+�
+ξ +
+j −ξ −
+j
+�2
+coth
+� ωj
+2kBT
+�
+= 2⟨1| ˆVj|1⟩
+ωj
+e
+−2
+(wz
+j)2
+ω2
+j
+coth
+� ωj
+2kBT
+�
+= 2⟨1| ˆVj|1⟩
+ωj
+�
+1+O(B2),
+�
+which appears as a multiplicative rescaling factor for the off-diagonal couplings ⟨1∓| ˆWj|1±⟩. Note that, when neglecting second
+and higher order terms in the magnetic field, κj does not show any dependence on temperature or on the magnetic field orientation
+via θ1 and φ1.
+
+7
+After thermal averaging, the effective electronic Hamiltonian for the lowest energy doublet becomes
+ˆHel = Trph
+�
+ˆS ˆHeff ˆS†ρ(th)
+ph
+�
+= E1 +δE1 +
+�
+2∑
+j
+⟨1| ˆVj|1⟩
+ωj
+wx
+j,2∑
+j
+⟨1| ˆVj|1⟩
+ωj
+wy
+j, ∆1
+2 +2∑
+j
+⟨1| ˆVj|1⟩
+ωj
+wz
+j
+�
+·
+�
+�
+σ′
+x
+σ′
+y
+σ′
+z
+�
+�
+(S31)
+where the energy of the lowest doublet is shifted by
+δE1 = −∑
+j
+⟨1| ˆVj|1⟩2
+ωj
++∑
+j
+ω j
+eωj/kBT −1
+(S32)
+due to the spin-phonon coupling and to the thermal phonon energy. Eq. (S31) thus represents a refined description of the lowest
+effective spin-1/2 doublet in the presence of spin-phonon coupling.
+We can finally recast the Hamiltonian (S31) in terms of a g-matrix for an effective spin 1/2, similarly to what we did earlier
+in the case of no spin-phonon coupling. In order to do so, we first recall from Eq. (S11) and (S19) that the quantities ∆1 and
+(wx
+j,wy
+j,wz
+j) appearing in Eq. (S31) depend on the magnetic field orientation via the states |1±⟩, and on both orientation and
+intensity via ˆHZee. We can get rid of the first dependence by expressing the Zeeman eigenstates |1±⟩ in terms of the original
+crystal field eigenstates |1⟩, |¯1⟩. For the spin-phonon coupling vector wj, we obtain
+w j =
+�
+�
+ℜ⟨1−| ˆWj|1+⟩
+ℑ⟨1−| ˆWj|1+⟩
+⟨1+| ˆWj|1+⟩
+�
+� =
+�
+�
+cosθ1 cosφ1 cosθ1 sinφ1 −sinθ1
+−sinφ1
+cosφ1
+0
+sinθ1 cosφ1
+sinθ1 sinφ1
+cosθ1
+�
+�
+�
+�
+ℜ⟨¯1| ˆWj|1⟩
+ℑ⟨¯1| ˆWj|1⟩
+⟨1| ˆWj|1⟩
+�
+� = R(θ1,φ1)· ˜w j.
+(S33)
+where R(θ1,φ1) is a rotation matrix. Similarly, the elctronic contribution ∆1 transforms as
+(0,0,∆1) = j1 ·R(θ1,φ1)T,= µBB·g(1)
+el ·R(θ1,φ1)T.
+(S34)
+The Pauli spin operators need to be changed accordingly to ˜σ = R(θ1,φ1)T · σ′. Lastly, we single out explicitly the magnetic
+field dependence of ˆWj, defined in Eq. (S18), by introducing a three-component operator ˆKj = ( ˆKx
+j, ˆKy
+j, ˆKz
+j), such that
+ˆWj = µBgJB·
+� ˆVj ˆQ1ˆJ+ ˆJ ˆQ1 ˆVj
+�
+(S35)
+= µBgJB· ˆK j.
+Thus, the effective electronic Hamiltonian in Eq. (S31) can be finally rewritten as
+ˆHel = E1 +δE1 + µBB·
+�
+g(1)
+el +gvib
+�
+· ˜σ/2
+(S36)
+where g(1)
+el is the electronic g-matrix defined in Eq. (S8), and
+gvib = 4gJ∑
+j
+⟨1| ˆVj|1⟩
+ωj
+�
+�
+ℜ⟨¯1| ˆKx
+j|1⟩ ℑ⟨¯1| ˆKx
+j|1⟩ ⟨1| ˆKx
+j|1⟩
+ℜ⟨¯1| ˆKy
+j|1⟩ ℑ⟨¯1| ˆKy
+j|1⟩ ⟨1| ˆKy
+j|1⟩
+ℜ⟨¯1| ˆKz
+j|1⟩ ℑ⟨¯1| ˆKz
+j|1⟩ ⟨1| ˆKz
+j|1⟩
+�
+�
+(S37)
+is a vibronic correction.
+Note that this correction is non-perturbative in the spin-phonon coupling, despite only containing quadratic terms in ˆVj (recall
+that ˆK j depends linearly on ˆVj). The only approximations leading to Eq. (S36) are a linear perturbative expansion in the magnetic
+field B and neglecting quantum fluctuations of the off-diagonal spin-phonon coupling in the polaron frame, which is accounted
+for only via its thermal expectation value. This approximation relies on the fact that the off-diagonal couplings are much smaller
+than the diagonal spin-phonon coupling that is treated exactly by the polaron transformation (see Section S3).
+C.
+Landau-Zener probability
+Let us consider a situation in which the magnetic field comprises a time-independent contribution arising from internal dipolar
+or hyperfine fields Bint and a time dependent external field Bext(t). Let us fix the orientation of the external field and vary its
+magnitude at a constant rate, such that the field switches direction at t = 0. Under these circumstances, the Hamiltonian of Eq.
+(S36) becomes
+ˆHel(t) = E1 +δE1 + µB
+�
+Bint + dBext
+dt
+t
+�
+·g· ˜σ
+2 ,
+(S38)
+
+8
+where g = g(1)
+el +gvib. Neglecting the constant energy shift and introducing the vectors
+∆
+= µBBext ·g,
+(S39)
+v = µBdBext/dt ·g,
+(S40)
+the Hamiltonian then becomes
+ˆHel(t) = ∆
+2 · ˜σ + vt
+2 · ˜σ = ∆⊥
+2
+· ˜σ + vt +∆∥
+2
+· ˜σ.
+(S41)
+In the second equality, we have split the vector ∆ = ∆⊥ +∆∥ into a perpendicular and a parallel component to v. Choosing an
+appropriate reference frame, we can write
+ˆHel(t′) = ∆⊥
+2 ˜σx + vt′
+2 ˜σz,
+(S42)
+in terms of the new time variable t′ = t +∆∥/v. Assuming that the spin is initialised in its ground state at t′ → −∞, the probability
+of observing a spin flip at t′ → +∞ is given by the Landau-Zener formula [15–20]
+PLZ = 1−exp
+�
+−π∆2
+⊥
+2v
+�
+.
+(S43)
+We remark that tunnelling is only made possible by the presence of ∆⊥, which stems from internal fields that have a perpen-
+dicular component to the externally applied field. We also observe that a perfectly axial system would not exhibit tunnelling
+behaviour, since in that case the direction of B · g would always point along the easy axis (i.e. along the only eigenvector of
+g with a non-vanishing eigenvalue), and therefore v and ∆ would always be parallel. Thus, deviations from axiality and the
+presence of transverse fields are both required for QTM to occur.
+
+9
+S3.
+DISTRIBUTION OF SPIN-PHONON COUPLING VECTORS
+The effective polaron Hamiltonian presented in Eq. (7) and derived in the previous section provides a good description of
+the ground doublet only if the spin-phonon coupling operators are approximately diagonal in the electronic eigenbasis. This is
+equivalent to requiring that the components of the vectors w j defined in Eq. (S19) satisfy
+|wx
+j|,|wy
+j| ≪ |wz
+j|.
+(S44)
+Fig. S1a shows the distribution of points {w j, j = 1,...,M} (where M is the number of vibrational modes) in 3D space for
+different orientations of the magnetic field. As a consequence of the strong magnetic axiality of the complex under consideration,
+we see that these points are mainly distributed along the z-axis, therefore satisfying the criterion expressed in Eq. (S44) (note
+the different scale on the xy-plane).
+b)
+a)
+FIG. S1. Distribution of spin-phonon coupling vectors wj. (a) The points w j distribute along a straight line in 3D space (units: cm−1) when
+the magnetic field is oriented along x, y, z. The magnitude is fixed to 1 T. Note that, owing to the definition of w j, a different magnitude would
+yield a uniformly rescaled distribution of points, leaving the shape unchanged. (b) Variance of the points w j in the xy-plane in units of the total
+variance, as a function of magnetic field orientation.
+In order to confirm that the points w j maintain a similar distribution regardless of the magnetic field orientation, we calculate
+their variances along different directions of the 3D space they inhabit. We define
+σ2
+α = var(wα
+j ) =
+1
+M −1
+M
+∑
+j=1
+�
+wα
+j − µα
+�2 ,
+(S45)
+where α = x,y,z and µα = 1
+M ∑M
+j=1 wα
+j . The dependence of these variances on the field orientation is made evident by recalling
+that the points w j are related via a rotation R(θ1,φ1) to the set of points ˜w j, which only depend linearly on the field B, as shown
+in Eqs. (S33) and (S35). If the points are mainly distributed along z for any field orientation, we expect the combined variance
+in the xy-plane to be much smaller than the total variance of the dataset, i.e.
+σ2
+x +σ2
+y ≪ σ2
+x +σ2
+y +σ2
+z .
+(S46)
+Fig. S1b provides a direct confirmation of this hypothesis, showing that the variance in the xy-plane is at most 6 × 10−4 times
+smaller than the total variance. Therefore, we conclude that the approach followed in Section S2 is fully justified.
+
+0.05
+0.00
+-0.0002
+-0.05
+0.0000
+0.0002
+0.0000
+0.0002
+-0.00020.05
+0.00
+-0.0002
+-0.05
+0.0000
+0.0002
+0.0000
+0.0002
+-0.00020.005
+0.000
+-0.0002
+-0.005
+0.0000
+0.0002
+0.0000
+0.0002
+-0.0002元
+0.0006
+0.0004
+元
+2
+0.0002
+0
+0
+爪
+0
+一元
+元
+2
+210
+S4.
+EXPERIMENTAL ESTIMATE OF THE SPIN-FLIP PROBABILITY
+In order to provide experimental support for our vibronic model of QTM, we compare the calculated spin-flip probabilities
+with values extracted from previously reported measurements of magnetic hysteresis. We use data from field-dependent mag-
+netisation measurements reported in Ref. [7](Fig. S35, sample 4), reproduced here in Fig. S2. The sample consisted of a 83 µL
+volume of a 170 mM solution of [Dy(Cpttt)2][B(C6F5)4] in dichloromethane (DCM). The field-dependent magnetisation was
+measured at T = 2 K while sweeping an external magnetic field Bext from +7 T to −7 T and back again to +7 T. The result-
+ing hysteresis loop is shown in Fig. S2a. The sweep rate dBext/dt is not constant throughout the hysteresis loop, as shown in
+Fig. S2b. In particular, it takes values between 10 Oe/s and 20 Oe/s across the zero field region where QTM takes place.
+QTM results in a characteristic step around the zero field region in magnetic hysteresis curves (Fig. S2a). The spin-flip
+probability across the tunnelling transition can be easily related to the height of this step via the expression [21]
+P↑→↓ = 1
+2
+� M
+Msat
+− M′
+Msat
+�
+.
+(S47)
+The value of the magnetisation before (M) and after (M′) the QTM drop is estimated by performing a linear fit of the field-
+dependent magnetisation close to the zero field region, for both Bext > 0 and Bext < 0, and extrapolating the magnetisation at
+Bext = 0 (Fig. S2a, inset). The saturation value of the magnetisation Msat is obtained by measuring the magnetisation at low
+temperature in a strong external magnetic field (T = 2 K, Bext = 7 T). Following this method, we obtain a spin-flip probability
+P↑→↓ = 0.27, which is shown as a purple horizontal line in Fig. 4 in the main text.
+b)
+a)
+FIG. S2. Magnetic hysteresis of [Dy(Cpttt)2]+ from Ref. [7]. (a) Field-dependent magnetisation was measured on a 170 mM frozen solution
+of [Dy(Cpttt)2]+ (counter ion [B(C6F5)4]−) in DCM at T = 2 K. Data presented in [7] (Fig. S35, sample 4). The loop is traversed in the
+direction indicated by the blue arrows. The sudden drop of the magnetisation from M to M′ around Bext = 0 is a characteristic signature
+of QTM. The slow magnetisation decay around the QTM step can be ascribed to other magnetic relaxation mechanisms (Raman). (b) Time
+dependence of the magnetic field Bext (top) and instantaneous sweep rate (bottom). Note that the sweep rate is not constant around the avoided
+crossing at Bext = 0, but assumes values in the range 10–20 Oe/s.
+
+11
+S5.
+ESTIMATE OF THE INTERNAL FIELDS IN A FROZEN SOLUTION
+A.
+Dipolar fields
+In this section we provide an estimate of the internal fields Bint in a disordered ensemble of SMMs, based on field-dependent
+magnetisation data introduced in Section S4.
+When a SMM with strongly axial magnetic anisotropy is placed in a strong external magnetic field Bext, it gains a non-zero
+magnetic dipole moment along its easy axis. Once the external field is removed, the SMM partially retains its magnetisation
+µ = µ ˆµ, which produces a microscopic dipolar field
+Bdip(r) = µ0µ
+4πr3 [3ˆr( ˆµ· ˆr)− ˆµ]
+(S48)
+at a point r = rˆr in space. This field can then cause a tunnelling gap to open in neighboring SMMs, depending on their relative
+distance and orientation.
+In order to estimate the strength of typical dipolar fields, we need to determine the average distance between SMMs in the
+sample, and the magnetic dipole moment associated with a single SMM. Since we know both volume V and concentration of
+Dy centres in the sample (see previous section), we can easily obtain the number of SMMs in solution N. The average distance
+between SMMs can then be obtained simply by taking the cubic root of the volume per particle, as
+r =
+�V
+N
+�1/3
+≈ 21.4 Å.
+(S49)
+The magnetic moment can be obtained from the hysteresis curve shown in Fig. S2a, by reading the value of the magnetisation
+M right before the QTM step. This amounts to an average magnetic moment per molecule
+⟨µ∥⟩ = M
+N ≈ 4.07µB
+(S50)
+along the direction of the external field Bext, where ⟨·⟩ denotes the average over the ensemble of SMMs. Since the orientation
+of SMMs in a frozen solution is random, the component of the magnetisation µ perpendicular to the applied field averages to
+zero, i.e. ⟨µ⊥⟩ = 0. However, it still contributes to the formation of the microscopic dipolar field (S48), which depends on
+µ = µ∥ +µ⊥. Since the sample consists of many randomly oriented SMMs, the average magnetisation in Eq. (S50) can also be
+expressed in terms of µ = |µ| via the orientational average
+⟨µ∥⟩ =
+� π/2
+0
+dθ sinθ µ∥(θ) = µ
+2 ,
+(S51)
+where µ∥(θ) = µ cosθ is the component of the magnetisation of a SMM along the direction of the external field Bext. Thus, the
+magnetic moment responsible for the microscopic dipolar field is twice as big as the measured value (S50).
+Based on these estimates, the magnitude of dipolar fields experienced by a Dy atom in the sample is
+Bdip = 0.77 mT×
+�
+|3( ˆµ· ˆr)2 −1|.
+(S52)
+The square root averages to 1.38 for randomly oriented µ and r and can take values between 1 and 2, represented by the green
+shaded area in Fig. 4 in the main text.
+B.
+Hyperfine coupling
+Another possible source of microscopic magnetic fields are nuclear spins. Among the different isotopes of dysprosium, only
+161Dy and 163Dy have non-zero nuclear spin (I = 5/2), making up for approximately 44 % of naturally occurring dysprosium.
+The nucear spin degrees of freedom are described by the Hamiltonian
+ˆHnuc = ˆHQ + ˆHHF = ˆI·P· ˆI+ ˆI·A· ˆJ,
+(S53)
+where the first term is the quadrupole Hamiltonian ˆHQ = ˆI·P· ˆI, accounting for the zero-field splitting of the nuclear spin states,
+and the second term ˆHHF = ˆI·A· ˆJ accounts for the hyperfine coupling between nuclear spin ˆI and electronic angular momentum
+
+12
+ˆJ operators. In analogy with the electronic Zeeman Hamiltonian ˆHZee = µBgJB· ˆJ, we define the effective nuclear magnetic field
+operator
+µBgJ ˆBnuc = AT · ˆI,
+(S54)
+so that the hyperfine coupling Hamiltonian takes the form of a Zeeman interaction ˆHHF = µBgJ ˆB†
+nuc · ˆJ. If we consider the nuclear
+spin to be in a thermal state at temperature T with respect to the quadrupole Hamiltonian ˆHQ, the resulting expectation value of
+the nuclear magnetic field vanishes, since the nuclear spin is completely unpolarised. However, the external field Bext will tend
+to polarise the nuclear spin via the nuclear Zeeman Hamiltonian
+ˆHnuc, Zee = µNg Bext · ˆI,
+(S55)
+where µN is the nuclear magneton and g is the nuclear g-factor of a Dy nucleus. In this case, the nuclear spin is described by the
+thermal state
+ρ(th)
+nuc =
+e−( ˆHQ+ ˆHnuc, Zee)/kBT
+Tr
+�
+e−( ˆHQ+ ˆHnuc, Zee)/kBT�
+(S56)
+and the effective nuclear magnetic field can be calculated as
+Bnuc = Tr
+� ˆBnucρ(th)
+nuc
+�
+.
+(S57)
+To the best of our knowledge, quadrupole and hyperfine coupling tensors for Dy in [Dy(Cpttt)2]+ have not been reported in
+the literature. However, ab initio calculations of hyperfine coupling tensors have been performed on DyPc2 [22]. Although the
+dysprosium atom in DyPc2 and [Dy(Cpttt)2]+ interacts with different ligands, the crystal field is qualitatively similar for these
+two complexes, therefore we expect the nuclear spin Hamiltonian to be sufficiently close to the one for [Dy(Cpttt)2]+, at least for
+the purpose of obtaining an approximate estimate. Using the quadrupolar and hyperfine tensors determined for DyPc2 [22] and
+the nuclear g-factors measured for 161Dy and 163Dy [23], we can compute Bnuc = |Bnuc| from Eq. (S57) for different orientations
+of the external magnetic field. As shown in Table S1, the effective nuclear magnetic fields at T = 2 K are at least one order of
+magnitude smaller than the dipolar fields calculated in the previous section, regardless of the orientation of the external field.
+161Dy
+163Dy
+Bext//ˆx 2.82×10−8 5.34×10−8
+Bext//ˆy 1.77×10−8 3.38×10−8
+Bext//ˆz 5.51×10−5 1.08×10−4
+TABLE S1. Effective Dy nuclear magnetic field Bnuc (T) at T = 2 K.
+
+13
+S6.
+RESULTS FOR A DIFFERENT SOLVENT CONFIGURATION
+In this section we show that the results presented in the main text are robust against variations of the solvent environment on
+a qualitative level. In order to show this, we consider a smaller and rounder solvent ball consisting of 111 DCM molecules, and
+reproduce the results shown in the main text, as shown in Fig. S3. It is worth noting that the vibronic spin-flip probabilities are
+significantly smaller for the smaller solvent ball, confirming the importance of the low-frequency vibrational modes associated
+to the solvent for determining QTM behaviour. The general tendency of vibrations to enhance QTM, however, is correctly
+reproduced.
+vibrational DOS
+a)
+b)
+c)
+d)
+e)
+j
+el
+j
+FIG. S3.
+Results for a different solvent configuration. (a) Alternative arrangement of 111 DCM molecules around [Dy(Cpttt)2]+. (b)
+Spin-phonon coupling strength and vibrational density of states (see Fig. 1c). (c) Vibronic correction to the energy splitting of the ground
+Kramers doublet (∆vib
+1
+− ∆1) for different orientations of the magnetic field (see Fig. 2a). (d) Ensemble-averaged spin-flip probability for
+different field sweep rates as a function of the internal field strength (see Fig. 3). (e) Orientationally averaged single-mode spin-flip probability
+⟨Pj⟩ vs change in magnetic axiality ∆Aj/Ael (see Fig. 4).
+The most evident difference between these results and the ones presented in the main text is the shape of the single-mode
+axiality distribution (Fig. S3e). In this case, single-mode spin-flip probability ⟨Pj⟩ still correlates to relative single-mode axiality
+∆A j/Ael. However, instead of taking values on a continuous range, the relative axiality seems to cluster around discrete values.
+In an attempt to clarify the origin of this strange behaviour, we looked at the composition of the vibrational modes belonging
+to the different clusters. Vibrational modes belonging to the same cluster were not found to share any evident common feature.
+Rather than in the structure of the vibrational modes, this behaviour seems to originate from the equilibrium electronic g-matrix
+gel. This can be seen by computing the single-mode axiality Aj = A(gel +gvib
+j ) for slightly different choices of gel. In particular,
+we checked how axiality of the electronic g-matrix affects the mode axiality. In order to do that, we considered the singular
+value decomposition of the electronic g-matrix
+gel = U·diag(g1,g2,g3)·V†,
+(S58)
+
+△1
+-A1
+(cm
+0.2
+0.1
+2
+0
+0.1
+-
+0.2
+0
+-元
+0
+元
+2
+2(cm
+0.2
+0.1
+0
+2
+-0.1
+-0.2
+0
+一元
+2
+2A1
+VID
+A1
+(cm
+0.2
+0.1
+2
+0
+0.1
+0.2
+-元
+0
+元
+2
+214
+the matrices U and V contain its left and right eigenvectors. The singular values are g1 = 19.99, g2 = 3.40 × 10−6, g3 =
+2.98 × 10−6, and the axiality is very close to one, i.e. 1 − Ael = 4.79 × 10−7. We artificially change the axiality of gel by
+rescaling the hard-plane g-values by a factor α and redefining the electronic g-matrix as
+gel
+α = U·diag(g1,αg2,αg3)·V†.
+(S59)
+The results are shown in Fig. S4. The three different colours distinguish the vibrational modes belonging to the three clusters
+visible in Fig. S3e (corresponding to α = 1). When α = 0, the g-matrix has perfect easy-axis anisotropy. In this case, the
+vibronic correction to the g-matrix is too small to cause significant changes in the magnetic axiality, and all the vibrational
+modes align around A j ≈ Ael. Increasing α to 0.9, clusters begin to appear. For α = 1.3, the single-mode axiality distribution
+begins to look like the one shown in Fig. 4a in the main text. The electronic g-matrix obtained for the solvent ball considered in
+the main text has a lower axiality than the one used throughout this section, i.e. 1−Ael = 1.12×10−6. Therefore, it makes sense
+that for α sufficiently larger than 1 we recover the same type of distribution as in the main text, since increasing α corresponds
+to lowering the electronic axiality A(gel
+α).
+FIG. S4. Impact of electronic axiality on single-mode axiality. Distribution of single-mode spin-flip probability ⟨Pj⟩ and g-matrix axiality
+A j = A(gelα + gvib
+j ) relative to the axiality of the modified electronic g-matrix A(gelα) defined in Eq. (S59). Vibrational modes belonging to
+different clusters in Fig. S3e (α = 1) are labelled with different colors.
+
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