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+MNRAS 000, 1–12 (2022)
+Preprint 9 January 2023
+Compiled using MNRAS LATEX style file v3.0
+A study of convective core overshooting as a function of stellar mass based
+on two-dimensional hydrodynamical simulations
+I. Baraffe,1,2 ★ J. Clarke,1 A. Morison,1 D. G. Vlaykov,1 T. Constantino,1 T. Goffrey,3 T. Guillet,1
+A. Le Saux1,2 and J. Pratt4
+1University of Exeter, Physics and Astronomy, EX4 4QL Exeter, UK
+2École Normale Supérieure, Lyon, CRAL (UMR CNRS 5574), Université de Lyon, France
+3Centre for Fusion, Space and Astrophysics, Department of Physics, University of Warwick, Coventry, CV4 7AL, UK
+4Lawrence Livermore National Laboratory, 7000 East Ave, Livermore, CA 94550, USA
+Accepted XXX. Received YYY
+ABSTRACT
+We perform two-dimensional numerical simulations of core convection for zero-age-main-sequence stars covering a mass range
+from 3 𝑀⊙ to 20 𝑀⊙. The simulations are performed with the fully compressible time-implicit code MUSIC. We study the
+efficiency of overshooting, which describes the ballistic process of convective flows crossing a convective boundary, as a function
+of stellar mass and luminosity. We also study the impact of artificially increasing the stellar luminosity for 3 𝑀⊙ models. The
+simulations cover hundreds to thousands of convective turnover timescales. Applying the framework of extreme plume events
+previously developed for convective envelopes, we derive overshooting lengths as a function of stellar masses. We find that the
+overshooting distance (𝑑ov) scales with the stellar luminosity (𝐿) and the convective core radius (𝑟conv). We derive a scaling law
+𝑑ov ∝ 𝐿1/3𝑟1/2
+conv which is implemented in a 1D stellar evolution code and the resulting stellar models are compared to observations.
+The scaling predicts values for the overshooting distance that significantly increase with stellar mass, in qualitative agreement
+with observations. Quantitatively, however, the predicted values are underestimated for masses
+>∼ 10𝑀⊙. Our 2D simulations
+show the formation of a nearly-adiabatic layer just above the Schwarzschild boundary of the convective core, as exhibited in
+recent 3D simulations of convection. The most luminous models show a growth in size with time of the nearly-adiabatic layer.
+This growth seems to slow down as the upper edge of the nearly-adiabatic layer gets closer to the maximum overshooting length
+and as the simulation time exceeds the typical thermal diffusive timescale in the overshooting layer.
+Key words: Convection – Hydrodynamics – Stars: evolution
+1 INTRODUCTION
+One of the major uncertainties in stellar evolution models is the treat-
+ment of mixing taking place at convective boundaries (see Stancliffe
+et al. 2016). Convective motions do not abruptly stop at the classical
+Schwarzschild boundary, but extend beyond it and lead to the pro-
+cess of convective boundary mixing (CBM). The complex dynamics
+resulting from convective flows penetrating in stable layers drives
+the transport of chemical species and heat, strongly affecting the
+structure and the evolution of stars. The same complex dynamics can
+also drive transport of angular momentum, impacting the rotational
+evolution of stars, the generation of magnetic field in their interior
+and their magnetic activity. CBM affects the evolution of all stars that
+develop a convective envelope, core or shell. Its treatment is one of
+the oldest unsolved problems of stellar structure and evolution theory
+(Shaviv & Salpeter 1973). This extra mixing could significantly alter
+the size of a convective core, the lifetime of major burning phases
+or the surface chemistry over a wide range of stellar masses. It can
+impact the entire evolution of massive stars (𝑀 >∼ 8𝑀⊙), determin-
+ing their structure before core-collapse supernova explosion and thus
+★ E-mail: i.baraff[email protected]
+affecting nucleosynthetic yields which are crucial for galactic evolu-
+tion studies (Arnett & Meakin 2011). There is ample observational
+evidence pointing towards the need for extra internal mixing to ex-
+plain a wide range of observations, such as eclipsing binaries (Claret
+& Torres 2016), color-magnitude diagrams (Rosenfield et al. 2017)
+or asteroseismology (Bossini et al. 2015). Rosenfield et al. (2017)
+illustrate the uncertainty due to the treatment of core overshooting on
+ages and on morphological changes in stellar evolution tracks, signif-
+icantly impacting stellar population studies. An increasing number of
+observational studies also suggests an increase of convective bound-
+ary mixing efficiency with stellar mass, using eclipsing binaries (see
+Claret & Torres 2019, and references therein) or Hertzsprung-Russell
+diagrams of massive stars (Castro et al. 2014). In a recent study, John-
+ston (2021) confirms that current stellar models with no or with little
+convective boundary mixing usually under-predict the mass of con-
+vective cores. While such comparisons between stellar models and
+observations cannot identify a mechanism responsible for mixing at
+the convective boundaries, Johnston (2021) concludes that a range of
+efficiencies for the mixing mechanism(s) should be used. In addition
+to CBM, additional mixing could be due to rotation (Zahn 1992) or
+internal gravity waves (Schatzman 1993). The latter are connected to
+CBM as they are excited at convective boundaries by turbulent con-
+© 2022 The Authors
+arXiv:2301.02604v1  [astro-ph.SR]  6 Jan 2023
+
+2
+I. Baraffe et al.
+vective motions (Press 1981; Goldreich & Kumar 1990; Lecoanet
+& Quataert 2013) and penetrating flows (Rieutord & Zahn 1995;
+Montalbán & Schatzman 2000; Pinçon et al. 2016).
+CBM is a generic term that encompasses different processes,
+namely penetration, overshooting or entrainment. The first term de-
+scribes motions that cross a convective boundary and alter the back-
+ground in such a way that the location of the convective boundary,
+defined by the Schwarzschild or the Ledoux criterion, moves inward
+or outward, resulting in the extension of the convective region. Over-
+shooting usually describes convective penetrative motions that do not
+alter the background but can still result in more or less efficient mixing
+(Zahn 1991). In the literature, the terms overshooting and penetration
+are often used interchangeably. These processes have been described
+in stellar evolution models by an overshooting distance 𝑑ov and/or a
+diffusion coefficient which remains constant or exponentially decays
+over the overshooting length (Freytag et al. 1996). These parameters
+are usually calibrated to fit observations. The temperature gradient in
+the overshooting region is either set to the radiative or to the adiabatic
+temperature gradient (see for example Michielsen et al. 2019). The
+third term entrainment is used to characterise shear-induced turbulent
+motions at the interface between the convectively stable and unstable
+regions driven by convective penetrative motions (plumes or eddies).
+Interfacial instabilities contribute to mixing fluids of different com-
+positions and/or densities, eroding the convective boundary. This one
+can then grow in time following an entrainment rate characterised
+by the bulk Richardson number (Fernando 1991; Strang & Fernando
+2001). Entrainment rates based on hydrodynamical simulations per-
+formed in a stellar context (Meakin & Arnett 2007; Jones et al. 2017;
+Cristini et al. 2019) are also implemented in stellar evolution codes to
+describe the extension of convective cores and shells (Staritsin 2013;
+Scott et al. 2021). However, as shown by Scott et al. (2021), adopting
+entrainment rates derived from existing stellar hydrodynamical sim-
+ulations to main sequence stellar models produces unrealistic growth
+of the convective cores. The parameters that control the entrainment
+rates need to be decreased by several orders of magnitude to repro-
+duce observations, questioning the reliability of the quantitative rates
+derived from existing numerical simulations and even the existence
+of an entrainment process for main sequence convective cores.
+Describing and isolating these different processes characterising
+CBM and at play at convective boundaries can be difficult in numer-
+ical simulations. Downward flows (or plumes) crossing a convective
+boundary at the bottom of an envelope are clearly observed in nu-
+merical simulations (see for example Baraffe et al. 2021). Ballistic
+plume crossings may eventually lead to a modification of the thermal
+background – the so-called penetration process. But for such modifi-
+cation to be observed, simulations must be run over many thousands
+of convective turnover timescales, as theoretically expected and re-
+cently demonstrated in simulations by Anders et al. (2022) based on
+3D simulations of convection in a Cartesian box with idealised se-
+tups. In a numerical study of solar-like convective envelopes, Baraffe
+et al. (2021) show that artificially boosting the luminosity of the
+stellar model by a factor 104 yields a significant modification of
+the thermal background below the convective boundary with an ex-
+tension of the size of the layer characterised by the penetration of
+convective flows, which could lead to a growth of the convectively
+unstable zone down to deeper levels. Whether this growth stabilises
+or whether the convective boundary continues moving downward
+indefinitely is unclear. For the solar-like model with realistic stellar
+luminosity, a slight modification of the thermal background is also
+observed in the simulations of Baraffe et al. (2021), but they show
+no trend of an extension of the Schwarzschild convective boundary
+over the simulation time.
+Following the approach developed in Pratt et al. (2017) for con-
+vective envelopes, the most vigorous plumes can be used to define
+a maximal overshooting length, which can be significantly deeper
+than the typical length reached by the bulk of the plumes (Pratt et al.
+2017; Baraffe et al. 2021; Vlaykov et al. 2022). Whether this bal-
+listic process is also observed for convective cores and can drive
+significant mixing is an open question. Arguments based on the dy-
+namics of convective motions and plumes suggest that mixing below
+a convective zone (e.g envelope overshooting) and above (e.g core
+overshooting) may indeed be different (Andrássy & Spruit 2013).
+Simple arguments based on the kinetic energy of a plume with typ-
+ical velocity and the restoring buoyancy force suggest very small
+overshooting lengths for the cores of low and intermediate mass
+zero-age-main-sequence (ZAMS) stars (Higl et al. 2021). But these
+estimates are based on typical velocities without considering possi-
+ble extreme plume events. The situation could also be different for
+convective cores on the ZAMS and on the main-sequence respec-
+tively. Indeed, the building of a molecular weight gradient at the core
+boundary due to hydrogen burning in the core can hamper the lifting
+of heavier material by ballistic processes. An entrainment process
+slowly eroding the convective boundary may thus dominate at some
+point over the ballistic process during the main sequence evolution,
+or both processes may coexist and contribute to mixing. These ques-
+tions are still unsettled. Existing numerical simulations of convective
+cores have mostly focussed on one single stellar mass model, rather
+than a range of stellar masses (Meakin & Arnett 2007; Gilet et al.
+2013; Rogers et al. 2013; Edelmann et al. 2019; Horst et al. 2020;
+Higl et al. 2021). Additionally, many of these works enhance the
+stellar luminosity of the model, to provide numerical stability, or to
+accelerate the thermal relaxation or the Mach number of the con-
+vective flow. This artefact may artificially favour one process over
+the other. At this time, it is difficult to draw any firm conclusion re-
+garding the main mechanisms that drive CBM in stars and how their
+efficiency is affected with stellar mass and with the stage of evolution
+on the main sequence.
+In this work devoted to convective cores, we study the efficiency
+for convective plumes to penetrate into the stable region as a function
+of stellar mass for ZAMS models. In the following we will refer to
+overshooting to describe this process, since we essentially describe
+the ballistic process and even if a modification of the temperature
+gradient is observed for the most luminous models (see Sect. 5),
+likely leading to penetration as defined by Zahn (1991). We perform
+two-dimensional (2D) numerical simulations of convective cores of
+ZAMS stellar models covering a range of stellar masses between 3
+𝑀⊙ and 20 𝑀⊙ (Sect. 2). Our goal is to apply the framework of ex-
+treme plume events developed for convective stellar envelopes (Pratt
+et al. 2017, 2020; Baraffe et al. 2021) to the convective cores of inter-
+mediate and massive stars. We analyse whether extreme events can
+provide overshooting lengths required for stellar models to reproduce
+observations. For this purpose, we derive a relationship between over-
+shooting length and stellar luminosity based on present numerical
+simulations (Sect. 4). We apply the relationship to one-dimensional
+stellar evolution models and test them against observations (Sect. 6).
+This is the first step for a systematic study devoted to convective core
+overshooting in intermediate mass and massive stars.
+2 NUMERICAL SIMULATIONS
+We use the fully compressible time-implicit code MUSIC. A full
+description of MUSIC and of the time-implicit integration can be
+found in Viallet et al. (2011, 2016); Goffrey et al. (2017). MUSIC
+MNRAS 000, 1–12 (2022)
+
+A study of convective core overshooting as a function of stellar mass
+3
+solves the inviscid Euler equations in the presence of external gravity
+and thermal diffusion:
+𝜕𝜌
+𝜕𝑡
+=
+−∇ · (𝜌v),
+(1)
+𝜕𝜌v
+𝜕𝑡
+=
+−∇ · (𝜌v ⊗ v) − ∇𝑝 + 𝜌g,
+(2)
+𝜕𝜌𝑒
+𝜕𝑡
+=
+−∇ · (𝜌𝑒v) − 𝑝∇ · v + ∇ · (𝜒∇𝑇) + 𝑄nuc,
+(3)
+where 𝜌 is the density, 𝑒 the specific internal energy, v the velocity,
+𝑝 the gas pressure, 𝑇 the temperature, g the gravitational accelera-
+tion, and 𝜒 the thermal conductivity. The term 𝑄nuc represents the
+nuclear energy rate. The symbol ⊗ is the outer product. All hydrody-
+namical simulations presented in this work are performed assuming
+spherically symmetric gravitational acceleration g, which is updated
+every time interval Δ𝑡1. All simulations presented in this work are
+performed with Δ𝑡 = 103 s. The typical dynamical timescale of the
+entire stellar cores analysed in this study 𝜏dyn ∼ 1/
+√︁
+(𝜌mean𝐺), with
+𝜌mean the mean density of the core and 𝐺 the gravitational constant,
+is of the order of 103 s. We have checked with a number of test
+simulations that a variation of Δ𝑡 between 102 and 105 seconds does
+not impact our results.
+In the stellar models considered, radiative transfer is the major
+heat transport that contributes to the thermal conductivity, which is
+given for photons by
+𝜒 = 16𝜎𝑇3
+3𝜅𝜌 ,
+(4)
+where 𝜅 is the Rosseland mean opacity, and 𝜎 the Stefan-Boltzmann
+constant. Realistic stellar opacities and equation of states appropriate
+for the description of stellar interiors are implemented in MUSIC.
+Opacities are interpolated from the OPAL tables (Iglesias & Rogers
+1996) for solar metallicity and the equation of state is based on the
+OPAL EOS tables of Rogers & Nayfonov (2002).
+2.1 Initial stellar models
+To provide the initial structures for the 2D simulations, we compute
+stellar models in the mass range 3-20 𝑀⊙ with the one-dimensional
+Lyon stellar evolution code (Baraffe & El Eid 1991; Baraffe et al.
+1998), using the same opacities and equation of state as MUSIC2.
+The 2D simulations require as initial input a radial profile of den-
+sity and internal energy. The 1D stellar evolution models have an
+initial helium abundance in mass fraction 𝑌=0.28 and solar metal-
+licity 𝑍=0.02 and were computed through the pre-main sequence
+and main sequence phases. All initial models for the 2D simulations
+in this study are taken at the beginning of core hydrogen burning
+and have a central abundance of helium 𝑌c=0.2838, i.e. only ∼ 1%
+of their central hydrogen has been depleted. There is thus a very
+shallow mean molecular weight gradient at the convective boundary.
+Follow-up analysis of later stages of evolution with a steeper gradient
+of molecular weight at the core boundary are in progress (Morison
+et al. in prep). Convective stability is defined by the Schwarzschild
+1 Note that Δ𝑡 is the time after which the gravitational potential is updated,
+not the numerical timestep. The numerical timestep used for these simulations
+is set by the hydrodynamical CFL number varying between 10 and 50 (see
+Viallet et al. 2011, for definitions) and corresponding to values for the timestep
+ranging between 5 s and 40 s.
+2 The 1D initial structures are available on the repository http://perso.ens-
+lyon.fr/isabelle.baraffe/2Dcore_overshooting_2023
+criterion ∇ < ∇ad, with ∇ = d log𝑇
+d log 𝑃 the temperature gradient and
+∇ad = d log𝑇
+d log 𝑃 |𝑆 the adiabatic gradient. The 1D stellar models used to
+generate the initial structures for the 2D simulations do not account
+for overshooting at the convective core boundary. In the following,
+we define the Schwarzschild boundary as the transition layer be-
+tween convective instability (∇ > ∇ad) and stability (∇ < ∇ad). The
+properties of the initial 1D stellar structures are provided in Table 1.
+Nuclear energy generated in the convective cores is accounted for in
+the internal energy equation (Eq. (3)) through the term 𝑄nuc using
+the radial profile of the nuclear energy rate from the 1D stellar model.
+Given that the simulation times are orders of magnitude smaller than
+the nuclear timescale for H burning in the cores, the nuclear energy
+is assumed to remain constant with time.
+2.2 Spherical-shell geometry and boundary conditions
+Two-dimensional simulations are performed in a spherical shell using
+spherical coordinates, namely 𝑟 the radius and 𝜃 the polar angle, and
+assuming azimuthal symmetry in the 𝜙-direction. For all models,
+the inner radius 𝑟in is defined at 0.02 𝑅star. The choice of the outer
+radius 𝑟out depends on the stellar model. Since the main motivation
+of this work is to analyse the extent of the overshooting layer for
+different stellar masses, the outer radius 𝑟out is fixed at a distance
+of ∼ 1 × 𝐻𝑃,CB for the lowest mass (3 𝑀⊙) to ∼ 3.5 × 𝐻𝑃,CB
+for the highest mass (20 𝑀⊙) away from the convective boundary
+𝑟conv. Extension of the radial domain to analyse the generation of
+internal waves at the core boundary and their propagation in the
+radiative envelope is work in progress. The angular extent ranges
+from 𝜃 = 0◦ to 𝜃 = 180◦. The grid has uniform spacing in the r and
+𝜃 coordinates. The choice for the resolution (𝑁𝑟, 𝑁𝜃) is set by the
+condition to have a good resolution of the pressure scale height at the
+Schwarzschild boundary. Effective Reynolds and Prandtl numbers
+are commonly used to set the resolution of numerical simulations.
+But given that our simulations are based on an implicit Large Eddy
+Simulation (ILES) approach, only a rough estimate can be provided
+for these numbers. They will in any case remain far away from the
+conditions prevailing in stellar interiors. We suggest that a more
+relevant resolution criterion for hydrodynamical simulations devoted
+to the study of overshooting using realistic stellar structures should be
+the number of grid cells per pressure scale height at the convective
+boundary. This should allow a more relevant comparison between
+the works of different groups devoted to the study of different stars.
+We use ∼ 110 − 140 grid cells per pressure scale height in the
+radial direction. The details of the resolution adopted in this work
+are provided in Table 2. We have also performed a few tests with
+higher resolution and analyse the impact in Sect. 4.
+The radial boundary conditions for the density correspond to a
+constant radial derivative on the density (see Pratt et al. 2016). The
+energy flux at the inner and outer radial boundaries are set to the
+value of the energy flux at that radius in the one-dimensional stellar
+evolution model. At the boundaries in 𝜃, because of the extension of
+the angular domain to the poles, reflective boundary conditions for
+the density and energy are used (i.e. the values are mirrored at the
+boundary). For the velocity, we impose reflective conditions at the
+radial and polar boundaries, corresponding to:
+• v𝑟 = 0 and 𝜕v𝜃
+𝜕𝑟 = 0 at 𝑟in and 𝑟out,
+•
+𝜕v𝑟
+𝜕𝜃 = 0 and v𝜃 = 0 at 𝜃 = 0◦ and 𝜃 = 180◦.
+We have also performed simulations for the 3 𝑀⊙ model with
+artificial enhancement of the stellar luminosity and the thermal dif-
+fusivity by factors 10, 102, 103 and 104. This covers the range of
+MNRAS 000, 1–12 (2022)
+
+4
+I. Baraffe et al.
+Table 1. Properties of the initial stellar models (all models have a central helium abundance 𝑌c=0.2838) used for the 2D hydrodynamical simulations: total mass,
+stellar luminosity, stellar radius, mass and radius of the convective core (corresponding to the location of the Schwarzschild boundary) and the pressure scale
+height at the Schwarzschild boundary.
+𝑀/𝑀⊙
+𝐿star/𝐿𝑎
+⊙
+𝑅star (cm)
+𝑀conv/𝑀⊙
+𝑟conv/𝑅star
+𝐻𝑃,CB (cm)
+3
+7.7673 × 101
+1.3855 × 1011
+0.5724
+0.1486
+1.3 × 1010
+5
+5.2186 × 102
+1.8424 × 1011
+1.212
+0.1814
+1.8 × 1010
+10
+5.5726 × 103
+2.7295 × 1011
+3.046
+0.2239
+2.7 × 1010
+15
+1.9242 × 104
+3.4255 × 1011
+5.600
+0.2580
+3.3 × 1010
+20
+4.2962 × 104
+4.0172 × 1011
+8.7947
+0.2869
+3.7 × 1010
+𝑎 We use 𝐿⊙ = 3.839 × 1033 erg/s.
+Figure 1. Evolution of the total kinetic energy (in erg; y-axis with a base-10
+log scale) as a function of time (in s) for the simulations described in Tab.
+2. Top panel: results for 3 𝑀⊙ models with various luminosity enhancement
+factors: 3L0 (black), 3L1 (blue), 3L2 (magenta), 3L3 (cyan) and 3L4 (red).
+Bottom panel: results for a range of stellar masses. The dotted line for each
+model corresponds to the value of the total kinetic energy at the beginning of
+the steady state for convection.
+luminosities of the stellar masses considered in this work (3-20 𝑀⊙).
+This choice of enhancement factor allows a comparative analysis of
+the impact of the luminosity for fixed core mass and increasing core
+mass, respectively. Note that even larger enhancement factors (up to
+107) for a 3 𝑀⊙ stellar structure can be found in previous works (e.g.
+Rogers et al. 2013; Edelmann et al. 2019). For the artificially boosted
+simulations, the energy flux (equivalently the luminosity) at the ra-
+dial boundaries is multiplied by the enhancement factor, the nuclear
+energy rate is multiplied by the same factor and the Rosseland mean
+opacities 𝜅 in MUSIC are decreased by the same factor.
+Figure 2. Radial profile of the time averaged rms velocity (solid lines) and
+rms radial velocity (dashed lines) scaled by (𝐿star/1035)1/3. Top panel: re-
+sults for 3 𝑀⊙ models with various luminosity enhancement factors: 3L0
+(black), 3L1 (blue), 3L2 (magenta), 3L3 (cyan) and 3L4 (red). Bottom panel:
+results for a range of stellar masses: 3L0 (black), 5L0 (blue), 10L0 (ma-
+genta),15L0 (red) and 20L0 (cyan). The convective boundary corresponding
+to the Schwarzschild boundary from the 1D initial model is indicated by a
+vertical solid line with the colour corresponding to each stellar mass.
+3 RESULTS: AVERAGE DYNAMICS
+The properties of all simulations are summarised in Table 2. We de-
+fine 𝑡steady as the time required to reach a steady state for convection,
+characterised by the total kinetic energy 𝐸kin of the system reaching
+a plateau. Before 𝑡steady, the initial relaxation phase is characterised
+by the propagation of strong acoustic waves and the onset of convec-
+tion. At 𝑡steady, the value of the kinetic energy starts to stabilise and
+MNRAS 000, 1–12 (2022)
+
+A study of convective core overshooting as a function of stellar mass
+5
+Table 2. Main properties of the 2D simulations.
+Model
+𝑀/𝑀⊙
+𝐿 (erg/s)
+𝑁𝑟 × 𝑁𝜃
+𝑟out/𝑅star
+𝜏𝑎conv (s)
+𝑁 𝑏
+conv
+𝑡𝑐
+steady (s)
+𝑡𝑑
+sim (s)
+3L0
+3
+2.981 ×1035
+336 x 168
+0.25
+1.9 ×106
+1442
+9.5 ×108
+3.71 ×109
+3L1
+3
+2.981 ×1036
+336 x 168
+0.25
+8 ×105
+1211
+4.6 ×108
+1.43 ×109
+3L2
+3
+2.981 ×1037
+336 x 168
+0.25
+3.9 ×105
+501
+9 ×107
+2.84 ×108
+3L2xhres
+3
+2.981 ×1037
+684 x 342
+0.25
+3.8 ×105
+514
+9 ×107
+2.84 ×108
+3L3
+3
+2.981 ×1038
+336 x 168
+0.25
+1.7 ×105
+1904
+6 ×107
+3.81×108
+3L3xhres
+3
+2.981 ×1038
+684 x 342
+0.25
+1.7 ×105
+1243
+6 ×107
+2.71 ×108
+3L4
+3
+2.981 ×1039
+336 x 168
+0.25
+8.9 ×104
+1457
+3 ×107
+1.60 ×108
+3L4xhres
+3
+2.981 ×1039
+684 x 342
+0.25
+8.7 ×104
+1400
+3 ×107
+1.52 ×108
+5L0
+5
+2.003 ×1036
+400 x 200
+0.3
+1.4 ×106
+1260
+2.45 ×108
+2.01 ×109
+10L0
+10
+2.139 ×1037
+416 x 208
+0.4
+1.2 ×106
+1260
+2.1 × 108
+1.77 ×109
+15L0
+15
+7.387 ×1037
+688 x 344
+0.5
+1.1 ×106
+875
+108
+1.14×109
+20L0
+20
+1.649 ×1038
+864 x 430
+0.6
+1.1 ×106
+800
+9 × 107
+9.99 ×108
+𝑎 Convective turnover time (see Sect. 3 for its definition).
+𝑏 Number of convective turnover times covered by the simulation once steady state convection is reached.
+𝑐Physical time to reach a steady state for convection.
+𝑑Total physical runtime of the simulation.
+from this time it remains roughly constant with time (following the
+dotted curve which corresponds to the value of 𝐸kin at 𝑡steady for each
+model). The simulations are stopped at time 𝑡sim provided in Table 2.
+None of these simulations are thermally relaxed, given that the total
+simulation times for all models are orders of magnitude smaller than
+the relevant thermal timescale ∼ 𝐺𝑀2/(𝑅star𝐿). As a consequence
+all these simulations are expected to maintain a secular drift. We
+have compared the radial profile of the internal energy, averaged in
+the angular direction, for each 2D model at time 𝑡steady and at time
+𝑡sim. We find a maximum of 0.5% relative difference for the internal
+energy at a given radius, with the largest difference found for the
+most luminous models (see Sect. 5). The above-mentioned drift is
+thus so slow that calculating statistical or averaged data during this
+very slowly changing transitional state is sensible.
+Figure 1 shows the evolution of the total kinetic energy as a func-
+tion of time for all models and the plateau characterising their steady
+state. The initial transient phase can last a relatively long time, de-
+pending on the model studied. For the model 3L0, we note a dif-
+ferent behaviour. After the peak due to strong acoustic waves, the
+kinetic energy continuously decreases until 𝑡 ∼ 2.4 × 108 s (log
+𝑡 ∼ 8.38). In this regime, convection develops in the core (within the
+1D Schwarzschild boundary) in two spatially separate regions. The
+abrupt increase of 𝐸kin observed at 𝑡 ∼ 2.4×108 s marks the merging
+of these two convective regions and the beginning of fully developed
+convection in the core. The Mach number characterising the con-
+vective velocities in model 3L0 is small, of the order of ∼ 10−4,
+which is numerically challenging. This low Mach number explains
+why several previous works artificially enhance the luminosity of the
+model (Rogers et al. 2013; Horst et al. 2020). There is no need for
+this artefact for the model 3L0 as MUSIC’s numerical scheme allows
+convection to develop and eventually reach a steady state even after a
+long transient phase. Note that this unusual transient phase observed
+for the model 3L0 will likely change with a different procedure for
+initialising the simulation. All simulations start without an imposed
+background noise (i.e. initial velocities are set to zero). Imposing
+initially a background noise for the model 3L0 may change the loca-
+tion where convection starts and thus the behaviour of the transient
+phase, which is irrelevant for the analysis performed in the following.
+A global convective turnover time 𝜏conv is estimated based on the rms
+velocity vrms(𝑟, 𝑡) at radius 𝑟 and time 𝑡, which characterises a bulk
+convective velocity. We define 𝜏conv by:
+𝜏conv =
+�∫ 𝑟conv
+𝑟in
+d𝑟
+vrms(𝑟, 𝑡)
+�
+𝑡,
+(5)
+where the rms velocity is given by
+vrms(𝑟, 𝑡) =
+√︃
+⟨v2(𝑟, 𝜃, 𝑡)⟩𝜃,
+(6)
+with v2 = v2𝑟 + v2
+𝜃, v𝑟 and v𝜃 being the radial and angular velocities,
+respectively. Time averages are denoted by ⟨⟩𝑡 and calculated between
+𝑡steady and 𝑡sim, the final time reached by the simulation (see values
+in Table 2). For any quantity 𝑋 we define:
+�
+𝑋
+�
+𝑡 =
+1
+(𝑡sim − 𝑡steady)
+∫ 𝑡sim
+𝑡steady
+𝑋d𝑡
+(7)
+The volume-weighted average in the angular direction ⟨⟩𝜃 is defined
+for any quantity X as:
+�
+𝑋(𝑟, 𝜃, 𝑡)
+�
+𝜃 =
+∫
+𝜃 𝑋(𝑟, 𝜃, 𝑡)d𝑉(𝑟, 𝜃)
+∫
+𝜃 d𝑉(𝑟, 𝜃)
+.
+(8)
+The simulations are stopped after a time 𝑡sim when convergence
+of the statistics used to determine the size of the layer penetrated
+by plumes is obtained, as explained in the next section (Sect. 4).
+Table 2 provides the values and numbers of the convective turnover
+times, respectively. Figure 2 displays the rms velocity and rms radial
+velocity for the 3 𝑀⊙ models with artificially enhanced luminosities
+(upper panel) and for the range of stellar masses investigated (lower
+panel), scaled by 𝐿1/3
+star. In the convective core, our simulations re-
+produce the expected scaling of convective velocity with luminosity
+vconv ∝ 𝐿1/3 recovered by many hydrodynamical simulations (e.g.
+Jones et al. 2017; Edelmann et al. 2019; Andrassy et al. 2020; Horst
+et al. 2020; Higl et al. 2021; Baraffe et al. 2021). This scaling is
+expected from mixing-length theory based on the argument that the
+turbulent dissipation rate of kinetic energy in a turbulent convective
+zone scales with v3 (Biermann 1932). But a general scaling of the
+total flux with v3 can also be derived for the kinetic energy and the
+enthalpy fluxes based on simple dimensional arguments (see Jones
+et al. 2017)
+MNRAS 000, 1–12 (2022)
+
+6
+I. Baraffe et al.
+The rms velocities in the stably stratified region are due to the
+penetrative flows just above the convective boundary and to the prop-
+agation of internal waves excited by the convective motions and the
+penetrating plumes. The top panel of Fig. 2 shows that these ve-
+locities also increase with the luminosity, suggesting more efficient
+overshooting of the convective motions above the convective bound-
+ary and thus larger overshooting length with increasing luminosity.
+Baraffe et al. (2021) reports similar behaviours for convective en-
+velopes of solar-like models with artificially enhanced luminosities.
+Quantitative estimate of the overshooting lengths for all models is
+performed in Sect. 4.
+4 RESULTS: EXTENT OF THE OVERSHOOTING REGION
+4.1 Determination of overshooting lengths
+To determine an overshooting length, we adopt the same approach as
+in Baraffe et al. (2021) and initially inspired by the findings of Pratt
+et al. (2017). This approach is based on the analysis of the depth of
+all convective plumes that penetrate beyond the convective boundary.
+The two criteria used to determine the depth of a penetrative plume
+at a given angle 𝜃 and time 𝑡 are based on the first zero above the
+convective boundary 𝑟conv of the vertical kinetic energy flux fk and
+vertical heat flux f𝛿T, defined by (see Pratt et al. 2017):
+fk(𝑟, 𝜃, 𝑡) = 1
+2 𝜌(𝑟, 𝜃, 𝑡)v2(𝑟, 𝜃, 𝑡)v𝑟 (𝑟, 𝜃, 𝑡),
+(9)
+f𝛿T(𝑟, 𝜃, 𝑡) = 𝜌(𝑟, 𝜃, 𝑡)𝑐𝑃(𝑟, 𝜃, 𝑡)𝛿𝑇(𝑟, 𝜃, 𝑡)v𝑟 (𝑟, 𝜃, 𝑡),
+(10)
+where 𝑐𝑃 is the specific heat at constant pressure and the temperature
+fluctuation 𝛿𝑇 is defined by:
+𝛿𝑇(𝑟, 𝜃, 𝑡) = 𝑇(𝑟, 𝜃, 𝑡) −
+��
+𝑇(𝑟, 𝜃, 𝑡)
+�
+𝜃
+�
+𝑡.
+(11)
+The method is the same as the one developed in Baraffe et al.
+(2021) for convective envelopes. At each time 𝑡, we calculate at each
+angle 𝜃 the radial positions 𝑟0(𝜃, 𝑡) of a plume corresponding to the
+first zero of fk and f𝛿T, respectively, above the convective boundary
+𝑟conv. The corresponding overshooting length 𝑙0 with respect to 𝑟conv
+is defined by
+𝑙0(𝜃, 𝑡) = 𝑟0(𝜃, 𝑡) − 𝑟conv.
+(12)
+Figure 3 illustrates the angular structure of the overshooting layer at
+an arbitrary time for the 10 𝑀⊙ stellar model.
+We then define the maximal overshooting length 𝑙max
+0
+at a given
+time by the maximum over all angles 𝜃:
+𝑙max
+0
+(𝑡) = max(𝑙0(𝜃, 𝑡)).
+(13)
+The time average 𝑙max = ⟨𝑙max
+0
+(𝑡)⟩𝑡 provides an effective width
+for the overshooting layer where the most vigorous plumes penetrate
+and which we use to characterise the extension of the mixing layer
+over the long term evolution of the star (Pratt et al. 2017; Baraffe
+et al. 2021). Table 3 displays 𝑙max based on the criterion for fk and
+f𝛿T, respectively, for all models. The distributions of overshooting
+lengths derived from fk and f𝛿T, respectively, slowly converges with
+time, as found in Pratt et al. (2017) and Baraffe et al. (2021). Several
+hundreds to thousand convective turnover times, depending on the
+stellar model, are required for the statistics to converge. Eventually,
+both criteria provide similar values for the effective overshooting
+width. The values of the overshooting width based on f𝛿T converge
+faster with time, compared to the value based on fk, as found as
+well for convective envelopes in Baraffe et al. (2021). The values
+of 𝑙max(f𝛿T) provided in Table 3 have reached a steady state for all
+Figure 3. Overshooting lengths𝑙0 defined by Eq. (12) as a function of the angle
+𝜃 at time 𝑡 = 8.3108s for the 10 𝑀⊙ model. The upper panel corresponds
+to 𝑙0 defined by fk and the lower panel to 𝑙0 defined by f𝛿T. The horizontal
+dashed line in each panel indicates the average overshooting length at this
+time.
+models after 𝑡sim. Depending on the stellar model, 𝑙max(fk) gets close
+to 𝑙max(f𝛿T) (difference of <∼ 20%) for all models but models 3L0
+and 20L0, for which 𝑙max(fk) continues slowly decreasing even after
+more than 800 ×𝜏conv. We run three simulations for the 3 𝑀⊙ models
+with enhanced luminosity with twice the resolution in both radial and
+angular directions and covering about the same simulation time as
+their lower resolution counterpart, in order to check the sensitivity
+of the values of 𝑙max to the resolution. The properties of these higher
+resolution models (labelled 2xhres) are displayed in Table 2. The
+results for the overshooting lengths are given in Table 3 and show
+similar values for lmax(f𝛿T) as found with a lower resolution. The
+values for 𝑙max(fk) of the higher resolution models are larger than the
+corresponding value for the lower resolution model, as it takes more
+time for 𝑙max(fk) in the high resolution models to decrease to the
+level of 𝑙max(f𝛿T). But the value of 𝑙max(fk) in the high resolution
+models continues decreasing with time and we expect it to eventually
+converge and thus get much closer to 𝑙max(f𝛿T) and to the value of
+𝑙max(fk) found in the lower resolution model.
+4.2 Relationship between overshooting length and stellar
+luminosity
+The variation of 𝑙max with the stellar luminosity is illustrated in
+Fig. 4 for the 3𝑀⊙ models with enhanced luminosity and for the
+set of stellar masses with realistic luminosity. As expected from
+the behaviour of the rms velocities (see Fig. 2) overshooting lengths
+increase with the stellar luminosity. To derive an approximate scaling
+relationship for the overshooting length 𝑑ov that can be implemented
+in stellar evolution codes, we use the values of 𝑙max derived from f𝛿T,
+since these values have converged with time. We derive the following
+MNRAS 000, 1–12 (2022)
+
+A study of convective core overshooting as a function of stellar mass
+7
+Table 3. Effective width 𝑙max of the overshooting layer in units of the total stellar radius and of the pressure scale height at the convective boundary, for all
+models considered in this study. The quantity 𝑙max(fk) is based on the criterion using fk (Eq. 9) and 𝑙max(f𝛿T) is based on f𝛿T (Eq. 10).
+Model
+𝑙max(fk)/𝑅star
+𝑙max(f𝛿T)/𝑅star
+𝑙max(fk)/𝐻𝑃,CB
+𝑙max(f𝛿T)/𝐻𝑃,CB
+3L0
+6.4 ×10−3
+3.7 ×10−3
+6.8 ×10−2
+3.9 ×10−2
+3L1
+4.2 ×10−3
+4.2 ×10−3
+4.5 × 10−2
+4.5 ×10−2
+3L2
+6.2 ×10−3
+6.1 ×10−3
+6.6 × 10−2
+6.5 ×10−2
+3L2xhres
+8.4 ×10−3
+6.4 ×10−3
+8.9 × 10−2
+6.8 ×10−2
+3L3
+1.8 ×10−2
+1.6 ×10−2
+1.9 ×10−1
+1.7 ×10−1
+3L3xhres
+2.2 ×10−2
+1.6 ×10−2
+2.3 ×10−1
+1.7 ×10−1
+3L4
+3.5 ×10−2
+2.8 ×10−2
+3.7 ×10−1
+3.0 ×10−1
+3L4xhres
+4.0 ×10−2
+3.0 ×10−2
+4.2 ×10−1
+3.2×10−1
+5L0
+9.3 ×10−3
+6.0 ×10−3
+9.5 ×10−2
+6.1 ×10−2
+10L0
+1.2 ×10−2
+1.1 ×10−2
+1.2 ×10−1
+1.1 ×10−1
+15L0
+1.6 ×10−2
+1.3 ×10−2
+1.66 ×10−1
+1.35 ×10−1
+20L0
+3.5 ×10−2
+2.0 ×10−2
+3.8 ×10−1
+2.17 ×10−1
+Figure 4. Overshooting length 𝑙max, in units of the pressure scale height at
+the convective boundary, as a function of the model luminosity. The 3 𝑀⊙
+models with various luminosity enhancement factors are indicated in red
+(dashed line). The results for a range of stellar masses with realistic stellar
+luminosity are indicated in blue (solid line). The dotted curve shows the fit
+for the overshooting length 𝑑𝑜𝑣/𝐻P,CB given by Eq. (14).
+expression which fits the results for the stellar mass range studied:
+𝑑ov/𝐻P,CB = 3.05 × 10−3 × (𝐿/𝐿⊙)1/3 × (𝑟conv/𝐻𝑃,CB)1/2 + 0.02
+(14)
+We find a typical scaling with the luminosity 𝑑ov ∝ 𝐿1/3 ∝ vconv.
+Numerical studies of convective envelopes report overshooting
+lengths 𝑑ov which vary with the luminosity following 𝑑ov ∝ 𝐿𝑎
+with 𝑎 varying between 0.08 and 0.31 (Hotta 2017; Käpylä 2019;
+Baraffe et al. 2021). The analytical model of Zahn (1991) for pene-
+tration, based on first order estimate of the deceleration of a plume
+in an adiabatically stratified penetration zone, predicts 𝑑ov ∝ v3/2
+conv.
+Our results also show that the overshooting lengths derived for a fixed
+stellar mass (and thus a fixed convective core size) are systematically
+smaller than the one derived for larger cores but similar luminosity.
+Interestingly, a dependence of 𝑑ov with the size of the core 𝑟conv is
+also predicted by Zahn (1991) (see their Eq. (4.5)) with the same re-
+lation of proportionality 𝑑ov ∝ (𝑟conv/𝐻𝑃,CB)1/2 as found in present
+simulations. This dependence in the Zahn model is derived from the
+strong variations with radius of various relevant quantities such as
+the gravitational acceleration 𝑔, the mass 𝑚(𝑟) enclosed in a sphere
+of radius 𝑟, the radiative conductivity 𝜒, and thus the radiative flux,
+close to the convective core boundary. In our simulations, we expect
+the radial dependence of the gravitational acceleration to have the
+main impact. We find that the larger the core (in terms of radius and
+mass), the smaller the gravitational acceleration at the core boundary
+𝑔conv ∼ 𝐺𝑀conv/𝑟2conv (see values in Table 1). Therefore, the larger
+the stellar mass, the larger the velocities at the convective boundary
+and the smaller the restoring force due to gravity, implying up-flows
+to penetrate over larger distances. This is a plausible explanation for
+the dependence of 𝑑ov on the convective core radius. We analyse
+below (Sect. 6) whether the expression provided by Eq. (14) pro-
+vides a reasonable agreement between stellar evolution models and
+observations.
+5 THERMAL BACKGROUND EVOLUTION
+The prescription used in the previous section to determine overshoot-
+ing lengths relies on two assumptions. Firstly, we consider that the
+simulations have reached a steady state for convection (i.e. a global
+dynamical steady state). This assumption is reasonable based on
+the observation that the total kinetic energy of the system reaches
+a plateau as a function of time (see Fig. 1). Secondly, we assume
+that the relevant convective boundary from which the overshooting
+lengths are defined is the 1D Schwarzschild boundary. This is directly
+useful for the purpose of implementing these overshooting lengths
+in 1D stellar evolution codes. However, we find that in all models
+a small nearly adiabatic layer just above the convective boundary
+forms rapidly once convection steady state is reached. For the most
+luminous models, we observe that this small layer slowly grows in
+size with time.
+Anders et al. (2022) also find a modification of the temperature
+gradient which becomes close to the adiabatic gradient in the pen-
+etration layer. They report that their simulations exhibit the process
+of convective penetration as defined by e.g. Zahn (1991), with con-
+vective penetrating motions mixing entropy and establishing a nearly
+adiabatic stratification above the Schwarzschild boundary (see also
+Brummell et al. 2002). Anders et al. (2022) suggest that the extent of
+convective penetration is limited and derive arguments involving the
+MNRAS 000, 1–12 (2022)
+
+8
+I. Baraffe et al.
+.
+Figure 5. Visualisation of the radial velocity v𝑟 [cm/s] (top panel) and the
+relative temperature fluctuations (𝑇 −⟨𝑇 ⟩𝜃)/⟨𝑇 ⟩𝜃 (bottom panel) in a region
+zoomed around the convective boundary (horizontal black line) for the model
+20L0 at time 𝑡 = 7 × 108 s. The x-axis represents the co-latitude (in terms of
+cos 𝜃). Note that to the better illustrate upwellings and downwellings in the
+top panel, the velocity scale is saturated, i.e. any velocity > vr,max = 5 × 103
+cm/s (< vr,min = −5×103 cm/s) are represented with the same color as vr,max
+(vr,min).
+convective flux, the viscous dissipation rate and the buoyancy work,
+providing an estimate of the penetration width. Depending on their
+setup, they find that penetration zones can take thousands of con-
+vective turnover times to saturate. They show properties of the flow
+and of the temperature fluctuations close to a convective boundary
+(see their Figure 1) which are similar to our results, as illustrated
+in Fig. 5 for the model 20L0 at a given time. As expected in con-
+vective regions, convective upflows transport hot material from the
+central regions up to the top of the convective core. Inspection of
+temperature fluctuations (i.e. the difference between the local tem-
+perature and the horizontally averaged thermal background) indeed
+indicates that upflows in the convective region are characterised by
+positive temperature fluctuations and downflows by negative tem-
+perature fluctuations. When upflows cross the convective boundary,
+at the top of the convective core, and penetrate the stably stratified
+medium, they adiabatically expand and therefore get cooler (neg-
+ative temperature fluctuation) and denser than the subadiabatically
+stratified environment.
+To understand the establishment of a nearly adiabatic layer in the
+penetration region, one needs to compare the advection timescale,
+which characterises the process of entropy mixing by penetrating
+flows (i.e. an advection process), and the thermal diffusion timescale.
+If penetrating flows, as illustrated in the top panel of Fig. 5, can drive
+efficient entropy/thermal mixing, the layer characterised by pene-
+trating up-flows will remain nearly adiabatic if thermal diffusion is
+slow enough. Table 4 provides estimates of the diffusive timescale
+Figure 6. Profile of the time and angular averages of the quantity (∇ − ∇ad)
+in the layers just above the convective core for the most luminous models.
+The 1D profile of (∇ − ∇ad) is indicated by the black dashed line and the
+1D convective core boundary by the vertical solid line. The location of 𝑙max
+derived from f𝛿T is indicated by the vertical dashed line. In both panels, the
+solid blue line corresponds to the time average between 𝑡steady and 𝑡sim. The
+curves in magenta correspond to time averages over 20×𝜏conv at a given time,
+as indicated in each panel (time 𝑡 in s).
+𝜏diff = 𝐿2/𝜅rad at the core boundary, with 𝐿 a relevant lengthscale
+and 𝜅rad = 𝜒/(𝜌𝑐𝑃) the thermal diffusivity (which is the radiative
+diffusivity for present stellar models with 𝜒 defined in Eq. (4)). Esti-
+mate of an advection timescale 𝜏adv = 𝐿/vr,rms is based on the time
+averaged rms radial velocity at the core boundary. For the charac-
+teristic lengthscales at the core boundary, we use the overshooting
+distance 𝑙max(f𝛿T) (see Table 3) and the pressure scale height ���P
+(see Table 1). As illustrated in Table 4, typical advection timescales
+are much smaller than typical thermal diffusion timescales for all
+models.
+The growth in size with time of the nearly adiabatic layer observed
+in the most luminous models is illustrated in Fig. 6 for the models
+3L3 and 3L4. This growth with time may also happen in the less
+luminous models, but their very slow evolution and less vigorous
+penetrating flows may prevent clearly exhibiting this feature over
+present simulation times. We also note that the angular averaged
+temperature gradient in the models, while getting very close to the
+adiabatic gradient, remains stable against the Schwarzschild criterion
+over the simulation times.
+For the purpose of analysing the time evolution of the nearly
+adiabatic layer, we have extended the simulation time of the models
+3L3 and 3L4 beyond the value of 𝑡sim used to determine overshooting
+depths (see Tab. 2), until 𝑡final = 5 × 108 s (∼ 2600 × 𝜏conv for 3L3
+and ∼ 5300×𝜏conv for 3L4). The aim is to reach a simulation time for
+these models close to or greater than the thermal diffusion timescale
+in the overshooting layer 𝜏diff(𝑙max). Given the smaller grid size and
+larger thermal diffusivity of these models, this is still computationally
+affordable. Figure 6 shows clearly in models 3L3 and 3L4 that the
+radial extension of the nearly adiabatic layer slows down with time
+MNRAS 000, 1–12 (2022)
+
+0.36
+4000
+0.34
+0.32
+2000
+0.30
+0
+0.28
+0.26
+-2000
+0.24
+-4000
+0.22
+-1.00 -0.75 -0.50 -0.25
+0.00
+0.25
+0.50
+0.75
+1.00
+cos θ0.36
+10-3
+0.34
+10-4
+0.32
+10-5
+10-6
+0.30
+T- (T)e)/<T)e 
+0
+0.28
+0.26
+-10-5
+0.24
+-10-4
+0.22
+-10-3
+-1.00
+-0.75 -0.50 -0.25
+0.00
+0.25
+0.50
+0.75
+1.00
+cos 0A study of convective core overshooting as a function of stellar mass
+9
+Table 4. Characteristic thermal diffusion timescales 𝜏diff = 𝐿2/𝜅rad and advection timescales 𝜏adv = 𝐿/vr,rms (in s) estimated at the core boundary for all
+models, based on two characteristic lentghscales, 𝐿 = 𝑙max(f𝛿T) and 𝐿 = 𝐻P, respectively. 𝜅rad (in cm2s−1) is the thermal diffusivity and vr,rms (in cm s−1) is
+the time averaged rms radial velocity, both estimated at the core boundary. The last two columns provide the ratio 𝜏diff
+𝜏adv for the two lentghscales.
+Model
+𝜅rad
+vr,rms
+𝜏diff (𝑙max)
+𝜏diff (𝐻P)
+𝜏adv(𝑙max)
+𝜏adv(𝐻P)
+𝜏diff
+𝜏adv (𝑙max)
+𝜏diff
+𝜏adv (𝐻P)
+3L0
+107
+3.2×101
+2.6×1010
+1.7×1013
+1.6×107
+4.1×108
+1.6×103
+4.1×104
+3L1
+108
+7.8×102
+3.3×109
+1.7×1012
+7.4×105
+1.7×107
+4.4×103
+105
+3L2
+109
+4.5×103
+7×108
+1.7×1011
+1.8×105
+2.9×106
+3.9×103
+5.8×104
+3L3
+1010
+2.1×104
+4.8×108
+1.7×1010
+105
+6.2×105
+4.8×103
+2.7×104
+3L4
+1011
+7.5×104
+1.5×108
+1.7×109
+5×104
+1.8×105
+3×103
+9.4 × 103
+5L0
+7×107
+1.7 ×102
+1.7×1010
+4.6×1012
+6.4×106
+108
+2.6×103
+4.6×104
+10L0
+7.5×108
+4.2×103
+1.2×1010
+9.7 1011
+7×105
+6.4×106
+1.7×104
+1.5×105
+15L0
+2×109
+8.5×103
+1010
+5.4×1011
+5.2×105
+3.9×106
+1.9×104
+1.4×105
+20L0
+4.5×109
+1.6×104
+1.4×1010
+3×1011
+5×105
+2.3×106
+2.8×104
+1.3×105
+as the upper edge gets closer to the location of 𝑙max. Since the change
+of the temperature gradient is driven by penetrating flows, one may
+expect that the nearly adiabatic layer would not extend beyond 𝑙max
+and that its growth may slow down when thermal diffusion starts
+to play a role. This process is likely to happen in the model 3L4,
+given that the final simulation time is significantly greater than the
+diffusive timescale over 𝑙max. It may start in the model 3L3 for which
+𝑡final ∼ 𝜏diff(𝑙max).
+We cannot exclude that the modification of the temperature gradi-
+ent is in part a transient effect in non-thermally relaxed simulations.
+The initial conditions for the simulations are based on 1D stellar
+structures relying on the mixing-length theory (MLT) to describe
+the transport of heat in the convective core. A readjustment of the
+structure inducing a change of the location of the Schwarzschild
+boundary in the 2D simulations cannot be excluded, given the uncer-
+tainty inherent to the MLT. But the only process which can readjust
+the location of the convective boundary over present simulation times
+is the penetration of convective motions across the convective bound-
+ary. A possible readjustment of the structure is thus also part of the
+process that we aim at characterising in this work (see discussion in
+Sect. 7).
+6 APPLICATION TO 1D STELLAR EVOLUTION MODELS
+AND OBSERVATIONS
+6.1 Spectroscopic Hertzsprung Russell diagram
+We implement the scaling relationship for the overshooting distance
+predicted by present simulations in stellar evolution models and com-
+pare these models to observations. For this purpose, the catalog of
+data of Castro et al. (2014) is relevant as it covers a large part of the
+stellar mass range investigated. This observational work provides the
+position of massive stars of spectral type OB in the Milky Way in the
+so-called spectroscopic Hertzsprung Russell diagram (sHRD). The
+sHRD uses a value L =𝑇4
+eff/𝑔 in place of the stellar luminosity 𝐿,
+based on spectroscopically determined effective temperature 𝑇eff and
+surface gravity 𝑔 of the star. The quantity L has the advantage that
+it can be calculated from stellar atmosphere analyses and compared
+to stellar evolution models without any knowledge of the distance or
+the extinction. Castro et al. (2014) derived an empirical location in
+the sHRD of the zero-age-main-sequence (ZAMS), for stellar masses
+above ∼ 9𝑀⊙, and terminal-age-main-sequence (TAMS) that can be
+directly compared to stellar evolution tracks. Because of the discrep-
+ancy in the main sequence width between observations and models,
+Figure 7. Evolution of massive stars in the spectroscopic Herzsprung-Russell
+diagram with different treatments of core overshooting. The symbols are
+observed stars in the Milky Way from Castro et al. (2014). The positions of
+the ZAMS and TAMS are indicated by the black solid lines. Coloured solid
+lines: Models evolved with an overshooting law given by Eq. (14). Dashed
+lines: models evolved with an arbitrary overshooting length 𝑑ov = 𝛼ov𝐻P with
+values of 𝛼ov provided in Table 5. Dotted lines: models with no overshooting.
+the main conclusion of their work is that convective core overshoot-
+ing may be mass dependent and stronger than previously thought for
+stellar masses >∼ 15𝑀⊙. We use this catalog of data to test the scaling
+relationship for 𝑑ov predicted by present numerical simulations.
+Stellar evolution models are calculated using the MESA code (Pax-
+ton et al. 2011) which provides the flexibility of easily implementing
+MNRAS 000, 1–12 (2022)
+
+3.8 -
+3.6
+3.4
+(°/)60|
+3.2
+Observed
+3.0 -
+8Mo
+9Mo
+10Mo
+2.8
+12Mo
+15Mo
+20Mo
+Best Fit
+2.6 -
+No Overshooting
+4.7
+4.6
+4.5
+4.4
+4.3
+4.2
+4.1
+log(Teff)(k)10
+I. Baraffe et al.
+Table 5. Values of 𝑑ov/𝐻P for each stellar model evolved with the scaling
+relationship given by Eq. (14) at the ZAMS and the TAMS, respectively. 𝛼ov
+is the fitted value for each stellar mass required to roughly reproduce the
+observed main sequence width.
+𝑀/𝑀⊙
+𝑑ov/𝐻P (ZAMS)
+𝑑ov/𝐻P (TAMS)
+Fitted 𝛼ov
+8
+0.09
+0.11
+0.1
+9
+0.10
+0.13
+0.2
+10
+0.11
+0.14
+0.3
+12
+0.13
+0.18
+0.35
+15
+0.16
+0.23
+0.4
+20
+0.22
+0.32
+0.45
+the scaling relation for overshooting distance given by Eq. (14).
+Instantaneous mixing is assumed over the distance 𝑑ov above the
+convective core. We have performed calculations adopting either a
+radiative or an adiabatic temperature gradient in the overshooting
+layer and find no significant impact on the evolutionary tracks. As
+done in Castro et al. (2014), we compare the data to solar metal-
+licity models. In Fig. 7 we show the evolution of massive stars in
+the mass range 8-20 𝑀⊙ with no overshooting and with the scaling
+relationship given by Eq. (14). The tracks are compared to the Castro
+et al. (2014) data and to the empirical locations of the ZAMS and
+the TAMS. Table 5 provides the values of 𝑑ov/𝐻P at the ZAMS
+and the TAMS, respectively, for the models evolved with the scaling
+relationship given by Eq. (14).
+We have also computed models with an arbitrary overshooting
+length 𝑑ov = 𝛼ov𝐻P which is fixed for a given stellar mass but
+increases with mass. The values of 𝛼ov for this set of models are
+chosen to roughly reproduce the main sequence width and are pro-
+vided in Table 5. We did not try to reproduce the ZAMS/TAMS
+empirical positions accurately. This set of models is also shown in
+Fig. 7. In agreement with the conclusions of Castro et al. (2014),
+models without overshooting are unable to reproduce the observed
+main sequence width. An increasing overshooting distance with in-
+creasing stellar mass is required to reproduce the observed width.
+The overshooting scaling law based on our present hydrodynami-
+cal simulations predict this increase with the stellar mass (see Fig.
+4). It provides a good fit to the observed main sequence width for
+𝑀 <∼ 10𝑀⊙. But it tends to under-predict the value of 𝑑ov needed for
+models of higher mass to reach the observed location of the TAMS.
+A comparison of the values of 𝑑ov given in Table 5 with the fitted
+values of 𝛼ov given in the same table suggests that values predicted
+by the hydrodynamical simulations are a factor ∼ 2 smaller than what
+is required to reach the observed location of the TAMS.
+6.2 Massive binaries
+We also test the overshooting scaling law given by Eq. (14) against
+a selected sample of massive eccentric binaries, namely HD 152218
+(Rauw et al. 2016), HD152219 (Rosu et al. 2022b) and CPD-41◦742
+(Rosu et al. 2022a). We limit present analysis to this restricted number
+of binary systems as they belong to the same young open cluster NGC
+6231 and thus have the same metallicity, likely a solar metallicity.
+In addition, their fundamental properties are inferred using the same
+methods and tools. This selected sample thus provides a small but
+homogeneous and consistent set of data to compare to stellar models.
+Their fundamental properties are provided in Table 6.
+Figure 8 compares evolutionary tracks with different treatments of
+core overshooting with the observed properties of these binaries. For
+Table 6. Properties of the binaries used for the comparison with models.
+Binary
+𝑀/𝑀⊙
+𝑇eff(K)
+𝐿/𝐿⊙
+HD 152218a
+19.8 ±1.5
+33 400 ±1000
+7.94+2.52
+−1.77 × 104
+HD 152218b
+15.0 ±1.1
+29 900 ±1000
+4.36+1.39
+−1.48 × 104
+HD 152219a
+18.64 ±0.47
+30 900 ±1000
+(7.26±0.97) × 104
+HD 152219b
+7.70 ±0.12
+21 697 ±1000
+(2.73±0.51) × 103
+CPD-41◦742a
+17.8 ±0.5
+31 800 ±1000
+5.28+0.67
+−0.68 × 104
+CPD-41◦742b
+10.0 ±0.3
+24 098 ±1000
+5.58+0.93
+−0.94 × 103
+HD 152218 (Fig. 8, left panel), all models provide a solution within
+the error bars, but the large error bars do not provide a very stringent
+test for the treatment of overshooting. For HD 152219 (Fig. 8, middle
+panel), all models provide a solution to the secondary, while only
+models with the arbitrary overshooting width from Tab. 5 provide a
+solution for the primary. Finally, for CPD-41◦742 (Fig. 8, right panel),
+all models fall within the error bars for the secondary. For the primary,
+models with the arbitrary overshooting width provide a solution, but
+the models with the present hydrodynamical relationship provide
+solutions at the very limit of the error bars. Although this comparison
+of models with binaries is less conclusive than the one performed
+with the Castro et al. (2014) data, it suggests that larger overshooting
+widths than predicted by the hydrodynamical relationship would
+provide a better fit, particularly for primaries with masses ∼ 18 𝑀⊙.
+7 DISCUSSION AND CONCLUSION
+This work is an initial, exploratory investigation, in which we infer
+an overshooting width 𝑑ov for a broad range of ZAMS stellar models
+based on hydrodynamical simulations. The present determination of
+an effective overshooting width, characterising the extent of mixing
+on the long term evolution of the star, is based on an approach rely-
+ing on extreme events of penetrating flows previously developed for
+convective envelopes of solar-type stars (e.g. Pratt et al. 2017, 2020;
+Baraffe et al. 2021). For ZAMS stars, we find that the overshoot-
+ing distance scales with the stellar luminosity and the convective
+core radius, resulting in values of 𝑑ov which significantly increase
+with stellar mass. Obtaining this increase is an important achieve-
+ment, since such an increase is suggested by several observational
+constraints. But although the results within our framework are qual-
+itatively in agreement with the observed trends, quantitatively, they
+are unable to match the available data. Indeed, the comparison of
+stellar evolution tracks to the properties of a sample of Milky Way
+main sequence stars suggests that the predicted values of 𝑑ov are
+underestimated for 𝑀 >∼ 10𝑀⊙. The comparison to massive bina-
+ries suggests the same limitation. This points to a need for further
+computational studies, as discussed below.
+The diagnostics we have used to examine the present set of 2D
+simulations have their limitations and several physical or numeri-
+cal ingredients may increase the values of overshooting lengths. One
+limitation is our assumption that the overshooting lengths determined
+within the extreme event framework, which is based on fluxes in an
+Eulerian approach, characterise the extension of efficient chemical
+mixing above the convective core. Quantifying the extent of chemical
+mixing is the prime interest for an application to 1D stellar evolu-
+tion models. This assumption can be verified with an analysis of
+mixing based on Lagrangian tracer particles, a direction that will
+MNRAS 000, 1–12 (2022)
+
+A study of convective core overshooting as a function of stellar mass
+11
+Figure 8. Comparison of evolutionary tracks with different treatments of overshooting and observations for massive binaries in the Hertzsprung-Russell diagram.
+Green lines: Models evolved with the overshooting law given by Eq. (14). Red lines: models evolved with an arbitrary overshooting length 𝑑ov provided in
+Table 5 (Fittted 𝛼ov). Blue lines: models without overshooting. The solid lines correspond to the track for the masses provided in Table 6 and the dashed lines
+correspond to the tracks for the upper and lower masses within the errorbars. Observations are from Rauw et al. (2016) for HD 152218, Rosu et al. (2022b) for
+HD152219 and Rosu et al. (2022a) for CPD-41◦742.
+be explored in a future work. The formation of a small nearly adia-
+batic layer above the convective core of our models due to efficient
+entropy mixing by the upward penetrating flows indicates that effi-
+cient chemical mixing should also proceed between the convective
+boundary and the location of the maximal overshooting length 𝑙max.
+But the size of the layer for efficient chemical mixing and the one
+of the nearly adiabatic layer are not expected to be the same, even
+if the same initial process drives thermal and chemical mixing (i.e.
+advection by upward flows). Our results suggest that the extent of
+the nearly adiabatic layer may be limited by thermal diffusion, as
+observed for the most luminous models when the simulation time
+exceeds the typical thermal diffusive timescale in the overshooting
+layer. Thermal diffusion will not limit the extent of chemical mixing.
+Internal waves excited by convective plumes at the core boundary
+could however contribute to additional chemical mixing and extend
+the size of the chemical mixing layer beyond 𝑙max. This is also un-
+der further investigation and could provide an interesting process to
+increase the overshooting lengths derived with present approach.
+Extension to three-dimensional geometry is an obvious next step,
+since the structure and the geometry of penetrating convective flows
+are expected to be modified in 3D compared to 2D simulations (see
+Brummell et al. 2002). Despite 2D convective velocities being on
+average larger than 3D velocities (Meakin & Arnett 2007; Pratt et al.
+2020), several works have suggested that the filling factor and plume
+geometry could be smaller in 3D than in 2D (see discussion in Rogers
+et al. 2006). Simulations in 3D may thus provide larger overshooting
+lengths, as needed to reproduce stellar observations. But so far no
+conclusive study of the filling factor and plume shape using the same
+simulation framework in 2D and 3D has been performed (see Pratt
+et al. 2020).
+Further numerical studies need to be performed in order to de-
+termine the impact of rotation and whether it can provide another
+driver to increase overshooting lengths and/or to make mixing more
+efficient (see e.g. Browning et al. 2004). A limitation of the present
+simulations, and indeed many global simulations of stars, is the fact
+that they are not thermally relaxed, since this would require simu-
+lation times even greater than the values for the thermal diffusion
+timescale over a pressure scale height 𝜏diff(𝐻P) provided in Table 4.
+The direct application of the overshooting lengths predicted by these
+simulations to “real" stars must thus be taken with caution, since the
+final relaxed state for these simulations may have different properties
+from present non thermally relaxed states. This does not however pre-
+clude analysing the efficiency of overshooting as a function of stellar
+mass and luminosity during the slowly evolving transient phase dur-
+ing which convection is considered to be in steady state. One can
+speculate that even if the convective boundary moves with respect to
+the initial 1D Schwarzschild boundary after thermal relaxation, the
+overshooting lengths determined on a dynamical steady state from
+MNRAS 000, 1–12 (2022)
+
+HD152219
+5.00
++
+4.75
+4.50
+(7/7)60l
+4.25
+4.00
+3.75
+3.50
+4.60
+4.55
+4.504.454.404.354.30
+4.25
+4.20
+log(Teff) (k)5.0
+CPD-410 742
+4.8
+4.6
+(7/7)60l
+4.4
+4.2
+4.0
+3.8
+3.6
+4.60
+4.55
+4.50
+4.454.40
+4.35
+4.30
+4.25
+4.20
+log(Teff) (k)5.2
+HD152218
+5.0 -
+4.8
+(7/7)60|
+4.6
+4.4
+4.2
+4.60
+4.55
+4.50
+4.454.40
+4.35
+4.30
+4.25
+4.20
+log(Teff) (k)12
+I. Baraffe et al.
+this new boundary may still be close to the the ones determined in
+this work. Unfortunately, to verify this implies running the simula-
+tions over a thermal timescale, which is computationally not feasible.
+More extreme enhancement factors for the luminosity could allow
+reaching thermal relaxation. But as shown recently for convective en-
+velopes in Baraffe et al. (2021), large enhancement factors can push
+the simulated conditions away from the original target star, inducing
+a significant drift from the initial stellar structure.
+In addition, the scaling presented in this work is derived for ZAMS
+stars and may not apply to cores that have evolved on the main
+sequence. Indeed, the development of a molecular weight gradient
+at the core boundary due to hydrogen burning will most likely limit
+the radial penetration of upward flows above the convective core
+boundary. Numerical simulations of the convective core of main
+sequence 5 𝑀⊙ and 20 𝑀⊙ star models indicate much smaller values
+of 𝑙max compared to their ZAMS counterpart (Morison et al., in prep).
+In addition, they show no sign of entrainment which could result in
+an increase of the size of the convective core. Whether 3D, rotation
+and/or other instabilities can solve the problem of “impenetrability"
+of convective flows due to the building of a molecular weight gradient
+during the evolution on the main sequence is an open question. Other
+effects and/or improvement of present 2D simulations are needed to
+increase the overshooting lengths for both ZAMS and main sequence
+models.
+In conclusion, this work provides results which qualitatively vali-
+date the increase of overshooting lengths with stellar mass (or stellar
+luminosity) suggested by observations (e.g. Castro et al. 2014). Quan-
+titatively, however, the predicted values are underestimated for stellar
+masses >∼ 10𝑀⊙. Our present results apply only to stellar models on
+the ZAMS. Our study illustrates the challenges and the promise of
+hydrodynamical simulations. It sets the stage for broader, and more
+physically detailed studies to resolve in the future this quantitative
+discrepancy with observations.
+ACKNOWLEDGEMENTS
+This work is supported by the ERC grant No. 787361-COBOM
+and the consolidated STFC grant ST/R000395/1. We are grateful
+to Noberto Castro for providing data in a user friendly form and
+for useful advises for using the catalog. We thank our anonymous
+referee for very valuable comments and suggestions. The authors
+would like to acknowledge the use of the University of Exeter High-
+Performance Computing (HPC) facility ISCA and of the DiRAC
+Data Intensive service at Leicester, operated by the University of
+Leicester IT Services, which forms part of the STFC DiRAC HPC
+Facility. The equipment was funded by BEIS capital funding via
+STFC capital grants ST/K000373/1 and ST/R002363/1 and STFC
+DiRAC Operations grant ST/R001014/1. DiRAC is part of the Na-
+tional e-Infrastructure. Part of this work was performed under the
+auspices of the U.S. Department of Energy by Lawrence Livermore
+National Laboratory under Contract DE-AC52-07NA27344.
+DATA AVAILABILITY
+The 1D initial structures are available on the repository:
+http://perso.ens-lyon.fr/isabelle.baraffe/2Dcore_overshooting_2023.
+The other data underlying this article will be shared on reasonable
+request to the corresponding author.
+REFERENCES
+Anders E. H., Jermyn A. S., Lecoanet D., Brown B. P., 2022, ApJ, 926, 169
+Andrássy R., Spruit H. C., 2013, A&A, 559, A122
+Andrassy R., Herwig F., Woodward P., Ritter C., 2020, MNRAS, 491, 972
+Arnett W. D., Meakin C., 2011, ApJ, 733, 78
+Baraffe I., El Eid M. F., 1991, A&A, 245, 548
+Baraffe I., Chabrier G., Allard F., Hauschildt P. H., 1998, A&A, 337, 403
+Baraffe I., Pratt J., Vlaykov D. G., Guillet T., Goffrey T., Le Saux A., Con-
+stantino T., 2021, A&A, 654, A126
+Biermann L., 1932, Z. Astrophys., 5, 117
+Bossini D., et al., 2015, MNRAS, 453, 2290
+Browning M. K., Brun A. S., Toomre J., 2004, ApJ, 601, 512
+Brummell N. H., Clune T. L., Toomre J., 2002, ApJ, 570, 825
+Castro N., Fossati L., Langer N., Simón-Díaz S., Schneider F. R. N., Izzard
+R. G., 2014, A&A, 570, L13
+Claret A., Torres G., 2016, A&A, 592, A15
+Claret A., Torres G., 2019, ApJ, 876, 134
+Cristini A., Hirschi R., Meakin C., Arnett D., Georgy C., Walkington I., 2019,
+MNRAS, 484, 4645
+Edelmann P. V. F., Ratnasingam R. P., Pedersen M. G., Bowman D. M., Prat
+V., Rogers T. M., 2019, ApJ, 876, 4
+Fernando H. J. S., 1991, Annual Review of Fluid Mechanics, 23, 455
+Freytag B., Ludwig H. G., Steffen M., 1996, A&A, 313, 497
+Gilet C., Almgren A. S., Bell J. B., Nonaka A., Woosley S. E., Zingale M.,
+2013, ApJ, 773, 137
+Goffrey T., et al., 2017, A&A, 600, A7
+Goldreich P., Kumar P., 1990, ApJ, 363, 694
+Higl J., Müller E., Weiss A., 2021, A&A, 646, A133
+Horst L., Edelmann P. V. F., Andrássy R., Röpke F. K., Bowman D. M., Aerts
+C., Ratnasingam R. P., 2020, A&A, 641, A18
+Hotta H., 2017, ApJ, 843, 52
+Iglesias C. A., Rogers F. J., 1996, ApJ, 464, 943
+Johnston C., 2021, A&A, 655, A29
+Jones S., Andrassy R., Sandalski S., Davis A., Woodward P., Herwig F., 2017,
+MNRAS, 465, 2991
+Käpylä P. J., 2019, A&A, 631, A122
+Lecoanet D., Quataert E., 2013, MNRAS, 430, 2363
+Meakin C. A., Arnett D., 2007, ApJ, 667, 448
+Michielsen M., Pedersen M. G., Augustson K. C., Mathis S., Aerts C., 2019,
+A&A, 628, A76
+Montalbán J., Schatzman E., 2000, A&A, 354, 943
+Paxton B., Bildsten L., Dotter A., Herwig F., Lesaffre P., Timmes F., 2011,
+ApJS, 192, 3
+Pinçon C., Belkacem K., Goupil M. J., 2016, A&A, 588, A122
+Pratt J., et al., 2016, A&A, 593, A121
+Pratt J., Baraffe I., Goffrey T., Constantino T., Viallet M., Popov M. V., Walder
+R., Folini D., 2017, A&A, 604, A125
+Pratt J., Baraffe I., Goffrey T., Geroux C., Constantino T., Folini D., Walder
+R., 2020, A&A, 638, A15
+Press W. H., 1981, ApJ, 245, 286
+Rauw G., Rosu S., Noels A., Mahy L., Schmitt J. H. M. M., Godart M., Dupret
+M. A., Gosset E., 2016, A&A, 594, A33
+Rieutord M., Zahn J. P., 1995, A&A, 296, 127
+Rogers F. J., Nayfonov A., 2002, ApJ, 576, 1064
+Rogers T. M., Glatzmaier G. A., Jones C. A., 2006, ApJ, 653, 765
+Rogers T. M., Lin D. N. C., McElwaine J. N., Lau H. H. B., 2013, ApJ, 772,
+21
+Rosenfield P., et al., 2017, ApJ, 841, 69
+Rosu S., Rauw G., Nazé Y., Gosset E., Sterken C., 2022a, arXiv e-prints, p.
+arXiv:2205.11207
+Rosu S., Rauw G., Farnir M., Dupret M. A., Noels A., 2022b, A&A, 660,
+A120
+Schatzman E., 1993, A&A, 279, 431
+Scott L. J. A., Hirschi R., Georgy C., Arnett W. D., Meakin C., Kaiser E. A.,
+Ekström S., Yusof N., 2021, MNRAS, 503, 4208
+Shaviv G., Salpeter E. E., 1973, ApJ, 184, 191
+MNRAS 000, 1–12 (2022)
+
+A study of convective core overshooting as a function of stellar mass
+13
+Stancliffe R. J., Fossati L., Passy J. C., Schneider F. R. N., 2016, A&A, 586,
+A119
+Staritsin E. I., 2013, Astronomy Reports, 57, 380
+Strang E. J., Fernando H. J. S., 2001, Journal of Fluid Mechanics, 428, 349
+Viallet M., Baraffe I., Walder R., 2011, A&A, 531, A86
+Viallet M., Goffrey T., Baraffe I., Folini D., Geroux C., Popov M. V., Pratt J.,
+Walder R., 2016, A&A, 586, A153
+Vlaykov D. G., Baraffe I., Constantino T., Goffrey T., Guillet T., Le Saux A.,
+Morison A., Pratt J., 2022, MNRAS, 514, 715
+Zahn J. P., 1991, A&A, 252, 179
+Zahn J. P., 1992, A&A, 265, 115
+This paper has been typeset from a TEX/LATEX file prepared by the author.
+MNRAS 000, 1–12 (2022)
+

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@@ -0,0 +1,1257 @@
+arXiv:2301.03288v1  [eess.SP]  9 Jan 2023
+1
+Reconfigurable Intelligent Surfaces 2.0: Beyond
+Diagonal Phase Shift Matrices
+Hongyu Li, Student Member, IEEE, Shanpu Shen, Member, IEEE,
+Matteo Nerini, Student Member, IEEE, and Bruno Clerckx, Fellow, IEEE
+Abstract—Reconfigurable intelligent surface (RIS) has been
+envisioned as a promising technique to enable and enhance future
+wireless communications due to its potential to engineer the
+wireless channels in a cost-effective manner. Extensive research
+attention has been drawn to the use of conventional RIS 1.0
+with diagonal phase shift matrices, where each RIS element
+is connected to its own load to ground but not connected to
+other elements. However, the simple architecture of RIS 1.0
+limits its flexibility of manipulating passive beamforming. To
+fully exploit the benefits of RIS, in this paper, we introduce
+RIS 2.0 beyond diagonal phase shift matrices, namely beyond
+diagonal RIS (BD-RIS). We first explain the modeling of BD-RIS
+based on the scattering parameter network analysis and classify
+BD-RIS by the mathematical characteristics of the scattering
+matrix, supported modes, and architectures. Then, we provide
+simulations to evaluate the sum-rate performance with different
+modes/architectures of BD-RIS. We summarize the benefits of
+BD-RIS in providing high flexibility in wave manipulation,
+enlarging coverage, facilitating the deployment, and requiring low
+complexity in resolution bit and element numbers. Inspired by the
+benefits of BD-RIS, we also discuss potential applications of BD-
+RIS in various wireless systems. Finally, we list key challenges in
+modeling, designing, and implementing BD-RIS in practice and
+point to possible future research directions for BD-RIS.
+Index Terms—Beyond diagonal reconfigurable intelligent sur-
+face, full space coverage, group-connected, modes/architectures.
+I. INTRODUCTION
+Wireless networks for the first five generations have been
+operated by catering the uncontrollable wireless environ-
+ment through various sophisticated designs at the transmit-
+ter/receiver. For beyond 5G and 6G, however, wireless net-
+works are expected to have manipulations of both transmit-
+ter/receiver and wireless environment, thanks to the emergence
+of a promising technique, namely reconfigurable intelligent
+surface (RIS) [1], [2]. RIS consists of numerous passive
+reconfigurable scattering elements so that it can manipulate
+the wireless environment and thus bring the spectrum and
+energy efficiency enhancement for the wireless network. The
+advantages of RIS have been demonstrated in various wireless
+systems, such as enhancing physical layer security [3] and
+H. Li and M. Nerini are with the Department of Electrical and Electronic
+Engineering, Imperial College London, London SW7 2AZ, U.K. (email:
+{c.li21,m.nerini20}@imperial.ac.uk).
+S. Shen is with the Department of Electronic and Computer Engineering,
+The Hong Kong University of Science and Technology, Clear Water Bay,
+Kowloon, Hong Kong (email: [email protected]).
+B. Clerckx is with the Department of Electrical and Electronic Engineering,
+Imperial College London, London SW7, 2AZ, U.K. and with Silicon Austria
+Labs (SAL), Graz A-8010, Austria (email: [email protected]).
+enabling integrated sensing and communication [4]. However,
+most existing works focus on using a simple RIS model
+with diagonal phase shift matrix, here referred to as RIS
+1.0, where each RIS element is connected to its own recon-
+figurable impedance without inter-element connections. More
+specifically, there are two limitations of conventional RIS 1.0:
+1) It can only control the phase of incident signal, which
+limits capability for manipulating passive beamforming and
+thus degrades the performance. 2) It only enables the signal
+reflection towards the same side, which limits the coverage.
+To address these limitations of RIS 1.0 and further enhance
+the performance gain of RIS, in this paper, we branch out
+to RIS 2.0, whose mathematical model is not limited to be
+diagonal matrices, namely beyond diagonal RIS (BD-RIS). We
+start from the BD-RIS modeling through scattering parameter
+network analysis. Then, we classify the BD-RIS based on the
+characteristics of the BD-RIS matrix, the supported modes,
+and the architectures, and categorize the existing BD-RIS
+works accordingly. Next, we consider a BD-RIS aided multi-
+user wireless communication system and evaluate the achiev-
+able sum-rate performance with different modes/architectures
+of BD-RIS. We summarize the benefits of BD-RIS such as
+high flexibility in wave manipulation and full-space coverage.
+Inspired by the benefits of BD-RIS, we look ahead to potential
+applications of BD-RIS in future wireless networks. We also
+discuss key challenges and future work of BD-RIS. Finally,
+we conclude this paper.
+II. MODELING AND CLASSIFICATION OF BD-RIS
+In this section, we introduce the model of BD-RIS based
+on the scattering parameter network analysis, and classify BD-
+RIS based on different modes and architectures.
+A. BD-RIS Model
+In general, an M-element RIS can be modeled as M
+antennas connected to an M-port reconfigurable impedance
+network [5]. The M-port reconfigurable impedance network is
+constructed by reconfigurable passive components and mathe-
+matically characterized by the scattering matrix Φ ∈ CM×M.
+The scattering matrix describes the scattering characteristics of
+the M-port reconfigurable impedance network, which relates
+the voltage of incident waves and reflected waves from the
+M ports. As per the microwave network theory, for pas-
+sive reconfigurable impedance network, the scattering matrix
+should satisfy ΦHΦ ⪯ IM, which denotes IM − ΦHΦ is
+positive semi-definite. Particularly, when the reconfigurable
+
+Depends on Inter-Cell Architecture
+Depends on if the Scattering Matrix is Diagonal
+Depends on Supported Modes
+RIS
+Diagonal Matrix:
+Single-Connected
+Conventional RIS 
+Block Diagonal Matrix:
+Group/Fully-Connected
+Hybrid Mode
+Multi-Sector 
+Mode
+Cell-Wise 
+Single-Connected 
+Cell-Wise Group/ 
+Fully-Connected
+Cell-Wise Single-
+Connected
+Cell-Wise Group/ 
+Fully-Connected
+Group/Fully-
+Connected RIS
+STAR-RIS
+Cell-Wise Group/Fully-
+Connected BD-RIS
+Multi-Sector BD-RIS
+Permuted Block Diagonal Matrix: 
+Dynamically Group-Connected
+Non-Diagonal Phase 
+Shift Matrix
+Beyond Diagonal RIS (BD-RIS)
+Non-Diagonal 
+Matrix
+Reflective Mode
+Reflective Mode
+Reflective Mode
+Reflective Mode
+ Dynamic Grouping
+Hybrid/Multi-
+Sector Mode
+Cell-Wise 
+Dynamically 
+Group-Connected
+Depends on Supported Modes
+Depends on Inter-Cell Architecture
+Fig. 1. RIS classification tree.
+impedance network is lossless, we have a unitary constraint
+for the scattering matrix, that is the power of incident waves
+is equal to that of the reflected waves. It should be noted that
+the characteristics of the scattering matrix is associated with
+the circuit topology of the M-port reconfigurable impedance
+network. In this sense, in conventional RIS 1.0, each port is
+connected to its own reconfgurable impedance without any
+connection across ports, referred to as single-connected RIS
+in [5], which yields a diagonal scattering matrix. However, in
+RIS 2.0, part of/all the ports are connected to each other so
+that the scattering matrix is not limited to be diagonal, which is
+referred to as the BD-RIS. In the following subsection, we will
+classify BD-RIS by the characteristics of scattering matrix,
+supported modes, and architectures.
+B. BD-RIS Classification
+We establish a three-layer RIS classification tree as shown
+in Fig. 1, where each layer is explained in detail as below.
+The first layer is classified by the characteristics of the scat-
+tering matrix Φ. 1) Block Diagonal Matrix: In this category,
+the M antennas are uniformly divided into G groups and
+antennas within the same group are connected to each other
+while those across groups are not connected. We refer to this
+category as group-connected RIS [5] and the corresponding
+scattering matrix Φ is a block diagonal matrix with each
+block being unitary, which enables manipulating not only
+the phase but also the magnitude of incident waves and
+thus a better performance than the conventional RIS 1.0.
+Particularly, when there is only one group G = 1, i.e. all the M
+antennas are connected to each other, it is referred to as fully-
+connected RIS [5], which results in a unitary scattering matrix.
+Besides, the conventional RIS 1.0, i.e. single-connected RIS,
+can be regarded as a special case of group-connected RIS
+with M groups, which has a diagonal scattering matrix.
+2) Permuted Block Diagonal Matrix: In this category, the
+grouping strategy, that is how the M antennas are grouped,
+for the group-connected RIS is adaptive to the channel state
+information (CSI), which is thus referred to as dynamically
+group-connected RIS. The resulting scattering matrix is a
+permuted block diagonal matrix [6], which provides higher
+flexibility in beam control than the fixed group-connected RIS.
+3) Non-Diagonal Matrix: In this category, antennas are linked
+in pairs through phase shifters so that the signal impinging on
+one antenna is purely reflected from another antenna, which
+results in an asymmetric non-diagonal scattering matrix [7]
+and a higher power gain than conventional RIS 1.0.
+The second layer is classified by the modes supported by
+RIS, including reflective, hybrid, and multi-sector modes as
+detailed in the following. 1) Reflective Mode: In this mode,
+signals impinging on one side of the RIS are reflected toward
+the same side, yielding a half-space coverage. To support the
+reflective mode, all the M antennas of RIS are placed towards
+the same direction as shown in Fig. 2(a). Mathematically, the
+RIS with reflective mode is characterized by the matrix Φ with
+a unitary constraint. 2) Hybrid Mode: In this mode, signals
+impinging on one side of the RIS can be partially reflected
+toward the same side and partially transmitted toward the
+opposite side, yielding a whole space coverage. The RIS with
+hybrid mode is also known as simultaneous transmitting and
+reflecting RIS (STAR-RIS) or intelligent omni-surface (IOS)
+[8]. To support the hybrid mode, each two antennas with
+uni-directional radiation pattern are back to back placed to
+form one cell, and are connected to a 2-port fully-connected
+reconfigurable impedance network [9] as shown in Fig. 2(c),
+so that each antenna in one cell respectively covers half space
+to achieve full-space coverage. Mathematically, the RIS with
+hybrid mode is characterized by two matrices, Φr ∈ C
+M
+2 × M
+2
+and Φt ∈ C
+M
+2 × M
+2 , which satisfy that ΦH
+r Φr + ΦH
+t Φt = I M
+2 .
+3) Multi-Sector Mode: This mode is a generalization of hybrid
+mode. In this mode, the full space is divided into L sectors
+(L ≥ 2) and signals impinging on one sector of RIS can
+be partially reflected toward the same sector and partially
+scattered toward the other L−1 sectors. To support the multi-
+sector mode, in each cell there are L antennas placed at each
+edge of an L-sided polygon, with each antenna having a uni-
+
+2-Cell 4-Sector BD-RIS with Cell-
+Wise Single-Connected Architecture
+(Cell Size: 4)
+(f)
+Z3
+Z5
+Z5
+Z5
+Antenna 1
+Z1,3
+Z1,7
+Z7
+Z3,5
+Z5,7
+Z3,7
+Z1,5
+Antenna 3
+Antenna 5
+Antenna 7
+Z1
+Cell 1
+Z3
+Z5
+Antenna 1
+Z1,3
+Z1,7
+Z7
+Z3,5
+Z5,7
+Z3,7
+Z1,5
+Antenna 3
+Antenna 5
+Antenna 7
+Z1
+Cell 1
+Z4
+Z6
+Z6
+Z6
+Antenna 2
+Z2,4
+Z2,8
+Z8
+Z4,6
+Z6,8
+Z4,8
+Z2,6
+Antenna 4
+Antenna 6
+Antenna 8
+Z2
+Cell 2
+Z4
+Z6
+Antenna 2
+Z2,4
+Z2,8
+Z8
+Z4,6
+Z6,8
+Z4,8
+Z2,6
+Antenna 4
+Antenna 6
+Antenna 8
+Z2
+Cell 2
+2-Cell 4-Sector BD-RIS with Cell-
+Wise Single-Connected Architecture
+(Cell Size: 4)
+(f)
+Z3
+Z5
+Antenna 1
+Z1,3
+Z1,7
+Z7
+Z3,5
+Z5,7
+Z3,7
+Z1,5
+Antenna 3
+Antenna 5
+Antenna 7
+Z1
+Cell 1
+Z4
+Z6
+Antenna 2
+Z2,4
+Z2,8
+Z8
+Z4,6
+Z6,8
+Z4,8
+Z2,6
+Antenna 4
+Antenna 6
+Antenna 8
+Z2
+Cell 2
+Antenna 5
+Antenna 1
+Antenna 6
+Z5,6
+Z5
+Z1,5
+Z1
+Z2,6
+Z1,2
+Z1,6
+Z2,5
+Z2
+Z6
+Z5,6
+Z5
+Z1,5
+Z1
+Z2,6
+Z1,2
+Z1,6
+Z2,5
+Z2
+Z6
+Antenna 2
+4-Cell BD-RIS with Cell-Wise 
+Group-Connected Architecture
+(No. of Group: 2)
+Antenna 7
+Antenna 3
+Antenna 8
+Z7,8
+Z7
+Z3,7
+Z3
+Z4,8
+Z3,4
+Z3,8
+Z4,7
+Z4
+Z8
+Z7,8
+Z7
+Z3,7
+Z3
+Z4,8
+Z3,4
+Z3,8
+Z4,7
+Z4
+Z8
+Antenna 4
+Group 1
+Group 2
+Cell 1
+Cell 2
+Cell 3
+Cell 4
+(d)
+Antenna 5
+Antenna 1
+Antenna 6
+Z5,6
+Z5
+Z1,5
+Z1
+Z2,6
+Z1,2
+Z1,6
+Z2,5
+Z2
+Z6
+Antenna 2
+4-Cell BD-RIS with Cell-Wise 
+Group-Connected Architecture
+(No. of Group: 2)
+Antenna 7
+Antenna 3
+Antenna 8
+Z7,8
+Z7
+Z3,7
+Z3
+Z4,8
+Z3,4
+Z3,8
+Z4,7
+Z4
+Z8
+Antenna 4
+Group 1
+Group 2
+Cell 1
+Cell 2
+Cell 3
+Cell 4
+(d)
+8-Element RIS with Group-
+Connected Architecture
+(Group Size: 4)
+Antenna 5
+Z5,8
+Z5
+Z8
+Z5,6
+Z6
+Z7,8
+Z7
+Z6,7
+Z6,8
+Z5,7
+Antenna 6
+Antenna 7
+Antenna 8
+Group 2
+Antenna 5
+Z5,8
+Z5
+Z8
+Z5,6
+Z6
+Z7,8
+Z7
+Z6,7
+Z6,8
+Z5,7
+Antenna 6
+Antenna 7
+Antenna 8
+Group 2
+(b)
+Antenna 1
+Z1,4
+Z1
+Z4
+Z1,2
+Z2
+Z3,4
+Z3
+Z2,3
+Z2,4
+Z1,3
+Antenna 2
+Antenna 3
+Antenna 4
+Group 1
+Antenna 1
+Z1,4
+Z1
+Z4
+Z1,2
+Z2
+Z3,4
+Z3
+Z2,3
+Z2,4
+Z1,3
+Antenna 2
+Antenna 3
+Antenna 4
+Group 1
+8-Element RIS with Group-
+Connected Architecture
+(Group Size: 4)
+Antenna 5
+Z5,8
+Z5
+Z8
+Z5,6
+Z6
+Z7,8
+Z7
+Z6,7
+Z6,8
+Z5,7
+Antenna 6
+Antenna 7
+Antenna 8
+Group 2
+(b)
+Antenna 1
+Z1,4
+Z1
+Z4
+Z1,2
+Z2
+Z3,4
+Z3
+Z2,3
+Z2,4
+Z1,3
+Antenna 2
+Antenna 3
+Antenna 4
+Group 1
+Uni-Directional 
+Radiation 
+Pattern
+Uni-Directional 
+Radiation 
+Pattern
+(a)
+Uni-Directional 
+Radiation 
+Pattern
+(a)
+2-Port 
+Network
+1 Cell
+Uni-Directional 
+Radiation Pattern
+Uni-Directional 
+Radiation Pattern
+2-Port 
+Network
+1 Cell
+Uni-Directional 
+Radiation Pattern
+Uni-Directional 
+Radiation Pattern
+(c)
+2-Port 
+Network
+1 Cell
+Uni-Directional 
+Radiation Pattern
+Uni-Directional 
+Radiation Pattern
+(c)
+Uni-Directional 
+Radiation Pattern
+Uni-Directional 
+Radiation Pattern
+Sector 1
+Sector 2
+Sector L
+Sector 3
+Sector l
+Sector L-1
+Uni-Directional 
+Radiation Pattern
+Uni-Directional 
+Radiation Pattern
+Sector 1
+Sector 2
+Sector L
+Sector 3
+Sector l
+Sector L-1
+(e)
+Uni-Directional 
+Radiation Pattern
+Uni-Directional 
+Radiation Pattern
+Sector 1
+Sector 2
+Sector L
+Sector 3
+Sector l
+Sector L-1
+(e)
+2-Cell 4-Sector BD-RIS with Cell-
+Wise Single-Connected Architecture
+(Cell Size: 4)
+(f)
+Z3
+Z5
+Antenna 1
+Z1,3
+Z1,7
+Z7
+Z3,5
+Z5,7
+Z3,7
+Z1,5
+Antenna 3
+Antenna 5
+Antenna 7
+Z1
+Cell 1
+Z4
+Z6
+Antenna 2
+Z2,4
+Z2,8
+Z8
+Z4,6
+Z6,8
+Z4,8
+Z2,6
+Antenna 4
+Antenna 6
+Antenna 8
+Z2
+Cell 2
+Antenna 5
+Antenna 1
+Antenna 6
+Z5,6
+Z5
+Z1,5
+Z1
+Z2,6
+Z1,2
+Z1,6
+Z2,5
+Z2
+Z6
+Antenna 2
+4-Cell BD-RIS with Cell-Wise 
+Group-Connected Architecture
+(No. of Group: 2)
+Antenna 7
+Antenna 3
+Antenna 8
+Z7,8
+Z7
+Z3,7
+Z3
+Z4,8
+Z3,4
+Z3,8
+Z4,7
+Z4
+Z8
+Antenna 4
+Group 1
+Group 2
+Cell 1
+Cell 2
+Cell 3
+Cell 4
+(d)
+8-Element RIS with Group-
+Connected Architecture
+(Group Size: 4)
+Antenna 5
+Z5,8
+Z5
+Z8
+Z5,6
+Z6
+Z7,8
+Z7
+Z6,7
+Z6,8
+Z5,7
+Antenna 6
+Antenna 7
+Antenna 8
+Group 2
+(b)
+Antenna 1
+Z1,4
+Z1
+Z4
+Z1,2
+Z2
+Z3,4
+Z3
+Z2,3
+Z2,4
+Z1,3
+Antenna 2
+Antenna 3
+Antenna 4
+Group 1
+Uni-Directional 
+Radiation 
+Pattern
+(a)
+Port 
+Port 
+Port 
+Port 
+Port 
+Network
+Network
+Network
+Network
+Network
+1 Cell
+1 Cell
+1 Cell
+1 Cell
+Network
+Network
+Network
+Network
+Network
+Network
+Network
+Network
+2-
+Network
+Network
+Network
+Network
+Network
+-Port 
+Port 
+Port 
+Port 
+Port 
+Port 
+Port 
+Port 
+Port 
+Network
+Network
+Network
+Network
+Network
+2-Port 
+Network
+1 Cell
+Uni-Directional 
+Radiation Pattern
+Uni-Directional 
+Radiation Pattern
+(c)
+Radiation Pattern
+Radiation Pattern
+Radiation Pattern
+Radiation Pattern
+Sector 3
+Sector 3
+Sector 3
+Sector 3
+Radiation Pattern
+Radiation Pattern
+Radiation Pattern
+Radiation Pattern
+Directional 
+Directional 
+Directional 
+Directional 
+Uni-Directional 
+Radiation Pattern
+Uni-Directional 
+Radiation Pattern
+Sector 1
+Sector 2
+Sector L
+Sector 3
+Sector l
+Sector L-1
+(e)
+Fig. 2. RISs with the same circuit topologies of reconfigurable impedance network while supporting different modes. (a) RIS with reflective mode and (b)
+group-connected architecture; (c) RIS with hybrid mode and (d) cell-wise group-connected architecture; (e) RIS with multi-sector mode and (f) cell-wise
+single-connected architecture.
+directional radiation pattern covering 1/L space, and the L
+antennas are connected to an L-port fully-connected reconfig-
+urable impedance network, as shown in Fig. 2(e). Hence, the
+multi-sector mode can cover the full space while providing
+higher performance gains than the hybrid mode, thanks to the
+use of higher-gain antennas with narrower beamwidth covering
+1/L space. Mathematically, the RIS with multi-sector mode
+is characterized by L matrices, Φl ∈ C
+M
+L × M
+L , l = 1, . . . , L,
+which satisfy �L
+l=1 ΦH
+l Φl = I M
+L .
+The third layer is classified by the inter-cell architecture,
+i.e. how the cells are connected to each other, in BD-RIS with
+hybrid/multi-sector modes. Analogous to the first layer in RIS
+classification tree, here we have cell-wise single/group/fully-
+connected architectures, where the resulting Φr and Φt for
+hybrid mode or Φl ∀l for multi-sector mode are diagonal/block
+diagonal/full matrices, respectively. In [9], it is shown that
+the cell-wise group/fully connected architecture has a better
+performance than the cell-wise single connected architecture,
+i.e. the STAR-RIS/IOS. To further enhance the performance,
+we have cell-wise dynamically group-connected architecture,
+where the inter-cell grouping strategy is adaptive to CSI and
+the resulting Φr and Φt for hybrid mode or Φl ∀l for multi-
+sector mode are permuted block diagonal matrices [6].
+C. Unified Architectures and Modes
+It is worthwhile highlighting that the BD-RIS with differ-
+ent modes and architectures are realized by group-connected
+reconfigurable impedance network together with different an-
+tenna array arrangements. To get insights into the essence of
+BD-RIS with different modes/architectures, three examples are
+illustrated in Fig. 2, including 1) a BD-RIS with reflective
+mode and group-connected architecture, 2) a BD-RIS with
+hybrid mode and cell-wise group-connected architecture, and
+3) a BD-RIS with multi-sector mode and cell-wise single-
+connected architecture. From Figs. 2(b), (d), and (f), we can
+find these three BD-RISs have the same circuit topology of
+reconfigurable impedance network but different antenna array
+arrangements, which results in different modes and inter-cell
+
+TABLE I
+CIRCUIT COMPLEXITY OF BD-RIS WITH NINE MODES/ARCHITECTURES
+Mode
+Architecture
+(Inter-Cell)
+Cell-Wise
+Cell-Wise
+Cell-Wise
+Single-
+Group-
+Fully-
+Connected
+Connected
+Connected
+Reflective
+M
+( M
+G + 1) M
+2
+(M + 1) M
+2
+Hybrid
+3
+2M
+Multi-Sector
+(L + 1) M
+2
+architectures. For clarity, we summarize the circuit complexity,
+that is the required number of reconfigurable impedance com-
+ponents, of BD-RIS with nine different modes/architectures in
+Table I. Hence, appropriately designing the group-connected
+reconfigurable impedance network and arranging the antenna
+array, we can implement BD-RIS with different modes and ar-
+chitectures to enhance the performance in different scenarios.
+III. PERFORMANCE EVALUATION FOR BD-RIS
+In this section, we evaluate the performance of BD-RIS with
+different modes and architectures. To that end, we consider a
+BD-RIS aided multiuser multiple input single output (MU-
+MISO) system, where a 4-antenna transmitter serves 4 single-
+antenna users with the aid of BD-RIS. The 4 users are located
+at one side for BD-RIS with reflective mode, while they are
+located at four corners for BD-RIS with hybrid and multi-
+sector modes. The transmit precoder and BD-RIS are jointly
+optimized to maximize the sum-rate of the MU-MISO system
+as detailed in [9], [10]. Fig. 3 shows the sum-rate performance
+versus the number of BD-RIS antennas for the BD-RIS with
+nine different modes and architectures. In Fig. 3, the direct link
+between the transmitter and users is assumed to be blocked.
+The distance between the transmitter and the BD-RIS is set
+as 100 m. The distance between the BD-RIS and users is set
+as 10 m. Channels from the transmitter to the BD-RIS and
+from BD-RIS to users are modeled as a combination of small-
+scale fading and large-scale fading. Specifically, the small-
+scale fading components follow the Rician fading with Rician
+factor 5 dB. The large-scale fading components are related to
+the BD-RIS antenna gains and path loss, which are modeled
+and calculated based on [10]. Transmit power is set as P = 30
+dBm. The noise power at each user is set as −80 dBm. The
+number of groups G for all three modes is fixed to 4. We make
+the following observations.
+First, under the reflective mode, BD-RIS with group/fully-
+connected architectures always achieves better performance
+than conventional RIS 1.0 due to the more general constraint
+of the BD-RIS matrix.
+Second, with the same cell-wise architecture, the BD-RIS
+with multi-sector mode always outperforms that with hybrid
+mode, even though the number of antennas covering each user
+for the former case is reduced compared to the latter. This
+is because the BD-RIS antennas with multi-sector mode has
+narrower beamwidth compared to those with hybrid mode, and
+thus provide higher gains. More interestingly, multi-sector BD-
+RIS with cell-wise single-connected architecture outperforms
+the hybrid BD-RIS with all three inter-cell architectures. This
+finding implies that with proper antenna array arrangements
+40
+60
+80
+100
+120
+ M
+3
+4
+5
+6
+7
+8
+9
+10
+11
+12
+13
+Sum-rate(b/s/Hz)
+(a)
+Reflective, Single-Connected
+Reflective, Group-Connected
+Reflective, Fully-Connected
+40
+60
+80
+100
+120
+ M
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11
+12
+Sum-rate(b/s/Hz)
+(b)
+Hybrid, Cell-Wise Single-Connected
+Multi-Sector, Cell-Wise Single-Connected
+Hybrid, Cell-Wise Group-Connected
+Multi-Sector, Cell-Wise Group-Connected
+Hybrid, Cell-Wise Fully-Connected
+Multi-Sector, Cell-Wise Fully-Connected
+80
+90
+4
+4.5
+Fig. 3. Sum-rate versus the number of BD-RIS antennas for (a) BD-RIS with
+reflective mode and (b) BD-RIS with hybrid/multi-sector modes.
+of BD-RIS, a reduced circuit complexity can achieve both
+satisfactory performance and full-space coverage.
+Third, for all three modes, the sum-rate achieved by BD-
+RIS with (cell-wise) fully/group-connected architectures grows
+faster with M than that with single-connected architecture.
+This phenomenon can be explained by Table I, which indicates
+that the circuit complexity of BD-RIS grows linearly with M
+for single-connected architecture, but grows quadratically with
+M for group/fully-connected architectures: The higher the
+circuit complexity, the more the number of non-zero elements
+of BD-RIS matrices, and thus the higher the flexibility of
+passive beamforming.
+IV. BENEFITS AND POTENTIAL APPLICATIONS OF BD-RIS
+We have shown the pronounced benefits of BD-RIS com-
+pared to conventional RIS 1.0 in the example of MU-MISO
+system in Section III. In this section, we summarize the key
+benefits of BD-RIS and discuss potential applications of BD-
+RIS in various wireless systems as illustrated in Fig. 4.
+A. Benefits of BD-RIS
+1) High Flexibility in Wave Manipulation: Compared with
+the conventional RIS 1.0 which can only manipulate the
+phase of incident wave, the BD-RIS has higher flexibility in
+manipulating both the magnitude and phase, which further
+boosts the performance in various wireless systems such as the
+received power in single input single output (SISO) systems
+[5], [11] and sum-rate in MU-MISO systems [6], [7].
+2) Full-Space Coverage: Compared with conventional RIS
+1.0 which can only cover half-space, the BD-RIS utilizing ap-
+propriate group-connected reconfigurable impedance network
+and antenna array arrangement can support the hybrid and
+multi-sector modes to realize full-space coverage [9], [10].
+Moreover, the multi-sector mode can provide high channel
+gain and thus effectively extend the communication range for
+full-space coverage [10].
+3) Facilitating Deployments:
+BD-RIS with hybrid and
+multi-sector modes facilitates practical deployments. Benefit-
+ing from the full-space coverage, the locations of the BD-RIS
+could be more flexible than conventional RIS 1.0.
+
+BD-RIS
+Macrocell
+Picocell
+Picocell
+Picocell
+  Integrated Sensing 
+and Communication
+! Backhaul and Access
+" MmWave/THz 
+Communications
+Power Station
+Power Station
+Step-up 
+Transformer
+Step-up 
+Transformer
+Step-down 
+Transformer
+Step-down 
+Transformer
+Transmission
+Station
+# Power Grid
+BD-RIS
+$ Wireless Power 
+Transfer
+% Wireless Sensing
+BD-RIS
+& Simultaneous Wireless 
+Information and Power Transfer
+BD-RIS
+BD-RIS
+Information/Power Transfer Link
+ 
+Sensing/Communication  Link
+Fig. 4. Potential applications of BD-RIS. ① BD-RIS works as a passive relay in the power grid; ② BD-RIS insists wireless power transfer; ③ BD-RIS enables
+wireless backhaul and access; ④ BD-RIS insists millimeter wave (mmWave)/Terahertz (THz) communications; ⑤ BD-RIS insists wireless sensing; ⑥ BD-RIS
+insists simultaneous wireless information and power transfer; ⑦ BD-RIS enables integrated sensing and communication.
+4) Low Complexity in Resolution Bit Number: When con-
+sidering RIS with discrete values, BD-RIS is shown to achieve
+a better performance than conventional RIS 1.0 with fewer
+resolution bits [12], due to the high flexibility of reconfig-
+urable impedance network. Such reduction of resolution bits
+is beneficial for implementation of BD-RIS.
+5) Low Complexity in Element Number: As the BD-RIS,
+especially with multi-sector mode, greatly enhances the per-
+formance in various wireless networks, given the same perfor-
+mance requirement, the required BD-RIS element number can
+be effectively reduced compared to that of conventional RIS
+1.0, which lowers the RIS complexity, cost, and form factor.
+B. Potential Applications of BD-RIS
+1) Wireless Power Relay/Transfer: One promising applica-
+tion of BD-RIS is to deploy it in the power grid to relay
+wireless power. The power grid is an electricity system which
+is generally used to carry power from a few central generators
+to numerous users/customers/devices. Specifically, the power
+grid consists of the power generation, the transmission grid
+which moves the up-stepped power over long distances to
+substations, and the distribution grid which delivers the down-
+stepped power to serve users [13]. In Fig. 4 we provide a
+diagram of employing BD-RIS in the power distribution grid
+to relay wireless power. With proper power levels, suitable
+deployments and locations of BD-RIS, the BD-RIS could act
+as a passive energy-efficient and low-cost power relay, which
+provides better relay performance than conventional RIS 1.0
+while realizing wide coverage.
+2) Wireless Communications: Another interesting applica-
+tion of BD-RIS is to enable flexible and scalable integrated
+access and backhaul (IAB) [14]. IAB is one of the promising
+techniques for 5G networks, where the operator can use part of
+the radio resources for wireless backhauling while providing
+the existing cellular services in the same node. Fig. 4 illustrates
+the BD-RIS assisted IAB, where the BD-RIS can be flexibly
+deployed in the IAB system to not only assist the wireless
+backhauling between the macrocell and picocells, but also the
+wireless access between picocells and users. Specifically, the
+wireless backhauling usually have complicated propagation
+environments and various obstacles, e.g. trees and high build-
+ings as shown in Fig. 4. BD-RIS with full space coverage and
+high gain performance can be easily incorporated into real
+environments to bypass the obstacles and assist/enhance the
+wireless backhaul. Meanwhile, wireless access, especially in
+millimeter wave or Terahertz wireless frequencies, usually has
+sparse and highly-directional channels, suffers from high path
+loss, and is vulnerable to blockages [15]. In this case, BD-
+RIS is more appealing in providing highly-directional beams
+to align with low-rank channels, compensate for the severe
+path loss, and enlarge coverage.
+3) Wireless Sensing: BD-RIS can also be deployed to boost
+the wireless sensing performance, such as improving the target
+
+detection accuracy and reducing the parameter estimation
+error, for targets enjoying line of sight (LoS) links. More
+importantly, for those complicated propagation environments
+without LoS links between the radar and targets, BD-RIS
+enables wireless sensing and enlarges coverage by creating
+effective LoS links.
+4) Integrated Wireless Power Transfer, Communications,
+and Sensing: In addition to stand-alone wireless power trans-
+fer, communications, and sensing, BD-RIS can also be used
+to assist integrated systems, such as simultaneous wireless
+information and power transfer as shown in Fig. 4, which
+helps to increase the output power level while enhancing the
+information transfer, or integrated sensing and communication
+to enable better communication and sensing performance.
+Not limited to these applications, BD-RIS can be applied in
+all the conventional RIS 1.0 enabled systems, but with higher
+flexibility and better performance in architecture design, beam
+manipulation, and deployment than conventional RIS 1.0.
+V. CHALLENGES AND FUTURE WORK OF BD-RIS
+While the BD-RIS has benefits compared with conventional
+RIS 1.0, there exist challenges in designing and implementing
+BD-RIS for practical wireless networks, which shed light on
+future research directions for BD-RIS. In this section, we
+list four challenges and future work from the perspectives
+of hardware implementation, discrete value BD-RIS design,
+channel estimation, and wideband modeling as follows.
+A. Hardware Implementation
+The hardware implementation of BD-RIS is a fundamental
+issue. As per the model in Section II, an M-element BD-RIS
+consists of two parts, which can be implemented as follows.
+1) M-Antenna Array: For the reflective mode, we can use
+the conventional uniform linear or planar antenna array. For
+the hybrid mode, we need to place each two antennas with uni-
+directional radiation pattern (e.g. patch antenna) back to back
+to form a cell and then arrange all the cells in a uniform array.
+Furthermore, for the multi-sector mode, we need to place each
+L antennas with narrow beamwidth at each edge of an L-side
+polygon to form a cell and arrange the cells in a uniform array.
+2) M-Port Reconfigurable Impedance Network: As shown
+in Section II.C, the group-connected reconfigurable impedance
+network is the key to implement the BD-RIS with different
+modes and architecture. We can utilize tunable inductance and
+capacitance, e.g. varactors, to construct the group-connected
+reconfigurable impedance network as per the circuit topology
+shown in Section II.C, so that the continuous value BD-RIS
+can be implemented. Alternatively, we can use PIN diodes as
+switches to reconfigure the impedance network to implement
+discrete value BD-RIS. However, as the group size increases,
+the circuit complexity and cost also increases. Hence, it is
+challenging but worthwhile to achieve good trade-off between
+performance and circuit complexity/cost for BD-RIS hardware
+implementation and prototyping to verify its superiority com-
+pared to conventional RIS 1.0.
+B. Discrete Value BD-RIS Design
+When using PIN diodes to implement the discrete value
+BD-RIS, it is challenging to design discrete values of the BD-
+RIS matrix. For conventional RIS 1.0 with diagonal phase
+shift matrix, the discretization is straightforward by uniformly
+sampling the phase within 2π. However, for BD-RIS with
+unitary matrix, the discretization is difficult. Instead, we need
+to determine the discrete values of the reactance matrix for the
+reconfigurable impedance network. In the recent work [12],
+a potential direction for the codebook design of group/fully-
+connected BD-RIS with reflective mode has been provided.
+Nevertheless, investigating discrete value BD-RIS design with
+hybrid/multi-sector modes and different architectures still re-
+mains an open problem.
+C. Channel Estimation
+The pronounced performance gain brought by the BD-RIS
+requires accurate CSI. For conventional RIS 1.0, there are two
+channel estimation strategies: 1) Semi-passive channel estima-
+tion by equipping a few low-power RF chains to the RIS to
+enable the pilot transmission/reception; 2) Pure passive chan-
+nel estimation by estimating the cascaded transmitter-RIS-user
+channels with pre-defined RIS patterns, which characterize the
+variation of RIS matrix during the training period. However,
+these channel estimation strategies cannot be directly applied
+in BD-RIS. For the first strategy, we need to reconsider the
+deployment of RF chains and the pilot design in the channel
+estimation process due to the different architectures/modes of
+BD-RIS. The second strategy may not be feasible for BD-
+RIS since most existing BD-RIS designs require the CSI for
+separate transmitter-RIS and RIS-user channels. Therefore, it
+is important to develop new channel estimation strategies for
+BD-RIS in the near future.
+D. Wideband BD-RIS Modeling
+The current BD-RIS model is only for narrowband com-
+munication. When it comes to wideband communications, the
+modeling of BD-RIS should take into account the frequency
+response of the reconfigurable impedance network. Specifi-
+cally, each reconfigurable component of the impedance net-
+work is frequency dependent, where the frequency response is
+determined by the circuit designs. Consequently, the resulting
+BD-RIS matrices at different frequencies are dependent on
+each other, which will complicate the wideband BD-RIS de-
+sign. To tackle the frequency dependent BD-RIS matrices and
+simplify the wideband BD-RIS design, a possible solution is
+to 1) analyze and fit the relationship between amplitudes/phase
+shifts of BD-RIS matrices and frequencies based on practical
+and specific circuits and 2) consider the wideband BD-RIS
+design based on the fitted frequency dependent BD-RIS model.
+VI. CONCLUSION
+In this paper, we depart from conventional RIS 1.0 with
+diagonal phase shift matrices and branch out to RIS 2.0 (BD-
+RIS) with beyond diagonal scattering matrices. Specifically,
+we model and classify the BD-RIS based on fundamental
+
+circuit topologies of reconfigurable impedance network. In
+addition, we highlight the benefits of BD-RIS with different
+modes/architectures in providing high flexibility in wave ma-
+nipulation, achieving full-space coverage, flexibility in various
+deployments, and low complexity in resolution bit and element
+numbers of the impedance network. Potential applications,
+challenges, and future work of BD-RIS are also discussed
+and summarized. As BD-RIS is a brand-new advance in RIS
+technology that remains unexplored from various perspectives,
+it is hoped that this paper could offer a useful and stimulating
+guide on future research directions of BD-RIS.
+REFERENCES
+[1] M. Di Renzo, A. Zappone, M. Debbah, M.-S. Alouini, C. Yuen,
+J. De Rosny, and S. Tretyakov, “Smart radio environments empowered
+by reconfigurable intelligent surfaces: How it works, state of research,
+and the road ahead,” IEEE Journal on Selected Areas in Communica-
+tions, vol. 38, no. 11, pp. 2450–2525, 2020.
+[2] Q. Wu and R. Zhang, “Towards smart and reconfigurable environment:
+Intelligent reflecting surface aided wireless network,” IEEE Communi-
+cations Magazine, vol. 58, no. 1, pp. 106–112, 2019.
+[3] J. Zhang, H. Du, Q. Sun, B. Ai, and D. W. K. Ng, “Physical
+layer security enhancement with reconfigurable intelligent surface-aided
+networks,” IEEE Transactions on Information Forensics and Security,
+vol. 16, pp. 3480–3495, 2021.
+[4] R. Liu, M. Li, Y. Liu, Q. Wu, and Q. Liu, “Joint transmit waveform
+and passive beamforming design for RIS-aided DFRC systems,” IEEE
+Journal of Selected Topics in Signal Processing, 2022.
+[5] S. Shen, B. Clerckx, and R. Murch, “Modeling and architecture design
+of reconfigurable intelligent surfaces using scattering parameter network
+analysis,” IEEE Transactions on Wireless Communications, vol. 21,
+no. 2, pp. 1229–1243, 2021.
+[6] H. Li, S. Shen, and B. Clerckx, “A dynamic grouping strategy for beyond
+diagonal reconfigurable intelligent surfaces with hybrid transmitting and
+reflecting mode,” arXiv preprint arXiv:2210.02499, 2022.
+[7] Q. Li, M. El-Hajjar, I. A. Hemadeh, A. Shojaeifard, A. Mourad,
+B. Clerckx, and L. Hanzo, “Reconfigurable intelligent surfaces relying
+on non-diagonal phase shift matrices,” IEEE Transactions on Vehicular
+Technology, 2022.
+[8] H. Zhang and B. Di, “Intelligent omni-surfaces: Simultaneous refraction
+and reflection for full-dimensional wireless communications,” IEEE
+Communications Surveys & Tutorials, 2022.
+[9] H. Li, S. Shen, and B. Clerckx, “Beyond diagonal reconfigurable
+intelligent surfaces: From transmitting and reflecting modes to single-
+, group-, and fully-connected architectures,” IEEE Transactions on
+Wireless Communications, 2022.
+[10] ——, “Beyond diagonal reconfigurable intelligent surfaces: A multi-
+sector mode enabling highly directional full-space wireless coverage,”
+arXiv preprint arXiv:2209.00301, 2022.
+[11] M. Nerini, S. Shen, and B. Clerckx, “Optimal group and fully connected
+design for beyond diagonal reconfigurable intelligent surfaces,” arXiv
+preprint arXiv:2211.06117, 2022.
+[12] ——, “Discrete-value group and fully connected architectures for
+beyond diagonal reconfigurable intelligent surfaces,” arXiv preprint
+arXiv:2110.00077v3, 2021.
+[13] X. Fang, S. Misra, G. Xue, and D. Yang, “Smart grid–The new and
+improved power grid: A survey,” IEEE communications surveys &
+tutorials, vol. 14, no. 4, pp. 944–980, 2011.
+[14] C. Madapatha, B. Makki, C. Fang, O. Teyeb, E. Dahlman, M.-S. Alouini,
+and T. Svensson, “On integrated access and backhaul networks: Current
+status and potentials,” IEEE Open Journal of the Communications
+Society, vol. 1, pp. 1374–1389, 2020.
+[15] I. F. Akyildiz, C. Han, Z. Hu, S. Nie, and J. M. Jornet, “Terahertz band
+communication: An old problem revisited and research directions for
+the next decade,” IEEE Transactions on Communications, 2022.
+

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+Molecular and solid-state topological polaritons via optical saturation
+Sindhana Pannir-Sivajothi,1 Nathaniel P. Stern,2 and Joel Yuen-Zhou1, ∗
+1Department of Chemistry and Biochemistry, University of California San Diego, La Jolla, California 92093, USA
+2Department of Physics and Astronomy, Northwestern University, Evanston, Illinois 60208, USA
+Strong coupling between electronic excitations in materials and photon modes results in the formation of
+hybrid quasiparticles called polaritons. Polariton systems often display larger nonlinearities than their photonic
+counterparts due to their material component. In this work, we theoretically investigate how to optically control
+the topological properties of molecular and solid-state exciton-polariton systems by exploiting one such nonlin-
+earity: saturation of electronic transitions. We study an optically pumped film of porphyrin molecules strongly
+coupled to the photon modes of a perylene filled Fabry-Perot cavity. Here, optical pumping with circularly
+polarized light breaks time-reversal symmetry instead of the frequently used large magnetic fields. We can op-
+tically tune properties such as the Berry curvature and Chern numbers of the bands. Importantly, while optical
+pumping does lead to non-zero Chern invariants, unidirectional edge states do not emerge in our system as the
+bulk-boundary correspondence is not applicable. Finally, we illustrate the broad applicability of our scheme by
+computing the Berry curvature of two other systems with slightly modified level structures that lead to different
+nonlinear behavior when placed in a microcavity and pumped with circularly polarized light: (a) monolayer
+MoS2 and (b) Ce:YAG. This work demonstrates a versatile platform to control topological properties of hybrid
+light-matter systems to enrich the toolbox of optoelectronic materials.
+INTRODUCTION
+Exciton-polaritons are hybrid excitations that exist in sys-
+tems where photonic modes couple strongly with optical tran-
+sitions in materials and their coupling strength exceeds losses
+[1]. Electronic strong coupling (ESC), where the optical tran-
+sitions correspond to semiconductor excitons or molecular
+electronic transitions, has been observed in a wide variety of
+inorganic and organic materials. While some polariton sys-
+tems, such as GaAs and CdTe quantum wells in microcavi-
+ties [1, 2], often require cryogenic temperatures for operation,
+due to their small exciton binding energies, organic materials
+[3] along with others such as GaN [4], ZnO [5], perovskites
+[6, 7], and transition metal dichalcogenides (TMD) [8, 9] can
+FIG. 1. Illustration of the system under study. Porphyrin (molecules
+at the center) and perylene (green blocks) placed within a Fabry-
+Perot cavity and pumped with circularly polarized light.
+∗ [email protected]
+achieve ESC at room temperature when placed in Fabry-Perot
+cavities.
+In particular, organic exciton-polaritons have re-
+ceived attention for their ability to modify chemical reactiv-
+ity [10], demonstrate polariton condensation at room temper-
+ature [11, 12], improve photoconductivity [13], and display
+topological properties [14, 15].
+Exciton-polariton systems are versatile platforms for topo-
+logical applications as their hybrid nature provides the unique
+opportunity to take advantage of the nonlinearities and mag-
+netic response of the material component while still enjoy-
+ing benefits of the coherence properties of the photonic part
+[16–18]. In the presence of photonic lattices, they also offer
+the possibility of unidirectional transport of energy through
+edge states that are robust to disorder [19]. A few approaches
+are frequently used to achieve topological exciton-polariton
+bands. In one of the approaches, the non-trivial topology re-
+sides in the winding light-matter coupling rather than individ-
+ual photon or exciton components [19, 20]. However, it is
+limited in application due to the requirement of large mag-
+netic fields to break time-reversal symmetry (TRS) and low
+temperatures to achieve Zeeman splitting in the exciton com-
+ponent which exceeds the exciton linewidth. In another ap-
+proach, TRS is preserved and a quantum spin hall insulator
+analogue is created in a polariton system [21]. This approach
+does not require a large magnetic field, however, there, a topo-
+logical polariton system is created by coupling a topologically
+non-trivial photonic lattice with a topologically trivial exciton
+system and the interesting topology is almost entirely encoded
+in the photonic component of the polariton [21, 22]. Both the
+approaches mentioned above were experimentally realized in
+polariton lattices. More recently, polaritons in Fabry-Perot
+cavities have emerged as a viable platform for topological po-
+laritonics. Several experiments have demonstrated measure-
+ment and control of the Berry curvature of exciton-polariton
+and photon bands in these systems [23–26]. Our work will
+focus on these Fabry-Perot cavity systems.
+In this work, we theoretically propose a scheme for gener-
+ating topological polaritons that combines advantages of both
+arXiv:2301.03287v1  [physics.chem-ph]  9 Jan 2023
+
+2
+the approaches mentioned above. Here, the light-matter cou-
+pling contains the non-trivial topology instead of the individ-
+ual photon or exciton components and optical pumping with
+circularly polarized light breaks TRS instead of a large mag-
+netic field. Breaking TRS in a molecular system using the
+helicity of light is an idea that has been demonstrated in sev-
+eral other contexts; it has been used to achieve all-optical non-
+reciprocity [27, 28] and theoretical results suggest that it can
+also induce optical-activity in achiral molecules [29]. Addi-
+tionally, a similar idea that relies on breaking TRS using cir-
+cularly polarized light has been previously proposed for po-
+lariton lattices by Bleu et al. [30].
+We focus on the topological properties of polaritons formed
+by the coupling of Frenkel excitons hosted in organic semi-
+conductors with photon modes in a Fabry-Perot cavity. Here,
+optical pumping with circularly polarized light saturates cer-
+tain electronic transitions and breaks TRS in the system;
+this results in non-zero Chern numbers of polariton bands.
+We exploit the primary nonlinearity of organic exciton-
+polaritons, saturation [11], to generate topological exciton-
+polariton bands. Our scheme relies on the contraction of Rabi
+splitting due to saturation, and we find modified Berry curva-
+ture and Chern number of the bands under circularly polarized
+pumping. The Berry curvature of the more photonic sections
+of the bands computed in our work can be experimentally
+measured using pump-probe spectroscopy. Furthermore, the
+applicability of our scheme is not limited to organic polariton
+systems. It only requires certain key ingredients: transitions
+that can be selectively excited with circularly polarized light,
+saturation effects, and Rabi splitting contraction. To highlight
+this, we compute the Berry curvature of two other systems un-
+der strong coupling and optical pumping: (a) Ce:YAG and (b)
+monolayer MoS2. Our work provides a viable strategy to in-
+duce non-reciprocal behavior in standard microcavity polari-
+tons, leading to the optical tuning of isolators and circulators
+[27], as well as fabrication of elliptically-polarized lasers and
+condensates [31].
+RESULTS
+Model
+In our theoretical study, we consider a Fabry-Perot cavity
+containing a thin film of porphyrin molecules at the center and
+a bulk perylene crystal filling the rest of the volume (Fig. 1).
+The porphyrin and perylene molecules are not treated on an
+equal footing in our model; while the molecular transitions of
+porphyrin are considered explicitly in the Hamiltonian, those
+of the perylene crystal are not, and they can be accounted
+for through effective cavity modes [25]. This is a valid ap-
+proximation because we focus on photon modes with fre-
+quencies close to those of electronic transitions in porphyrin
+(∼ 3.81eV) [32, 33] and far off-resonant from the transitions
+of perylene (∼ 2.98eV) [34]. Here, the birefringent perylene
+crystal plays the role of providing anisotropy and emergent
+optical activity to the cavity modes [25].
+We model each porphyrin molecule as a three-level elec-
+1
+|G⟩
+|−!"#⟩
+|+!"#⟩
+𝝁!
+𝝁"
+a
+b
+FIG. 2. (a) Illustration of circularly polarized light exciting a met-
+alloporphyrin molecule. (b) Three-level model of porphyrin with a
+ground state |G⟩ and two degenerate excited states |+mol⟩,|−mol⟩.
+The transition dipole moment for a transition from |G⟩ to |±mol⟩ is
+µµµ± = µ0(ˆx±iˆy)/
+√
+2. The number of yellow circles at each state rep-
+resents the fraction of molecules in that state. Here, the ratio of the
+fraction of molecules in the ground, fG, and |±mol⟩ excited states,
+f±, is fG : f+ : f− = 3 : 1 : 0. Such population ratios can be achieved
+through pumping with circularly polarized light.
+tronic system with a ground state |G⟩ and two excited states
+|+mol⟩ and |−mol⟩ (see Fig. 2b) [35, 36]. In the absence of
+a magnetic field, the two excited states are degenerate and
+the energy difference between the ground and excited states is
+¯hωe = 3.81eV [37]. The transition dipole moments for transi-
+tions from |G⟩ to |+mol⟩ and |−mol⟩ are µµµ+ = µ0(ˆx+iˆy)/
+√
+2
+and µµµ− = µ0(ˆx−iˆy)/
+√
+2, respectively, with µ0 = 2.84D [37].
+Here, ˆx and ˆy are unit vectors along the x and y directions.
+Using circular polarized light, the |+mol⟩ or |−mol⟩ states can
+be selectively excited.
+In our model, we consider a thin film of metalloporphyrins
+or metallophtalocyanines arranged in a square lattice with
+nearest neighbor spacing a. The choice of lattice is irrele-
+vant because later we will take the continuum limit a → 0 as
+we are only interested in length scales much larger than the
+intermolecular spacing. Each molecule is labeled with the
+index m = (mx,my), where mx,my ∈ Z and the molecule’s
+position is given by rm = mxa��x + myaˆy. States of the mth
+molecule are then written as |m,G⟩, |m,+mol⟩ and |m,−mol⟩.
+The creation operator ˆσ†
+m,± = |m,±mol⟩⟨m,G| ⊗n̸=m In ex-
+cites the mth molecule from |m,G⟩ to |m,±mol⟩. Here, In =
+|n,G⟩⟨n,G| + |n,+mol⟩⟨n,+mol| + |n,−mol⟩⟨n,−mol| is the
+identity operator for nth molecule. These molecular operators
+satisfy commutation relations (a generalization of the commu-
+tation relations of paulion operators [38, 39]),
+�
+ˆσn,±, ˆσ†
+m,±
+�
+= δm,n(1− ˆσ†
+n,∓ ˆσn,∓ −2 ˆσ†
+n,± ˆσn,±).
+(1)
+We model the effective photon modes of a Fabry-Perot cav-
+ity filled with perylene as in Ren et al. [25] For the photon
+modes of a Fabry-Perot cavity, the component of wave vec-
+tor orthogonal to the mirrors kz = 2nπ/L is quantized, where
+L is the effective distance between the mirrors of the cav-
+ity and n is the mode index [40]. For a given n, the modes
+are labeled by the in-plane wave vector k = kxˆx + kyˆy and
+polarization α; the creation operators associated with these
+
+00003
+modes are ˆa†
+k,α and they satisfy bosonic commutation rela-
+tions
+�
+ˆak,α, ˆa†
+k′,α′
+�
+= δα,α′δk,k′. As a result of in-plane trans-
+lational invariance of a cavity, k can take any value, i.e.,
+kx,ky ∈ R. Throughout this work, we specify the cavity mode
+polarization in the circularly polarized basis α = ±.
+The Hamiltonian of the full system is
+ˆH = ˆHmol + ˆHcav + ˆHcav−mol,
+(2)
+where
+ˆHmol =∑
+m
+�
+¯hωe ˆσ†
+m,+ ˆσm,+ + ¯hωe ˆσ†
+m,− ˆσm,−
+�
+ˆHcav =∑
+k
+��
+E0 + ¯h2|k|2
+2m∗ +ζ|k|cosφ
+�
+ˆa†
+k,+ ˆak,+
++
+�
+E0 + ¯h2|k|2
+2m∗ −ζ|k|cosφ
+�
+ˆa†
+k,− ˆak,−
++
+�
+−β0 +β|k|2e−i2φ�
+ˆa†
+k,+ ˆak,−
++
+�
+−β0 +β|k|2ei2φ�
+ˆa†
+k,− ˆak,+
+�
+,
+ˆHcav−mol =∑
+m ∑
+k,α
+− ˆµµµm · ˆEk,α(rm,0)
+≈∑
+m ∑
+k
+eik·rm
+�NxNy
+�
+(µµµ+ ·Jk,+) ˆσ†
+m,+ ˆak,+
++(µµµ− ·Jk,+) ˆσ†
+m,− ˆak,+ +(µµµ+ ·Jk,−) ˆσ†
+m,+ ˆak,−
++(µµµ− ·Jk,−) ˆσ†
+m,− ˆak,−
+�
++H.c.
+(3)
+Above, ˆHmol describes the porphyrin molecules, ˆHcav the ef-
+fective cavity modes (including contributions from the pery-
+lene crystal), and ˆHcav−mol the coupling between the por-
+phyrin molecules and effective cavity modes. Here, φ is the
+angle between the in-plane wave vector and the x-axis, i.e.,
+cosφ = kx/|k|. Within ˆHcav, β specifies the TE-TM splitting,
+β0 quantifies the linear birefringence of the perylene crystal
+which splits the H-V modes, and ζ describes the emergent
+optical activity [25]. Additionally, E0 is the frequency of the
+cavity modes at |k| = 0 in the absence of the perylene crys-
+tal (β0 = 0 and ζ = 0), and m∗ is the effective mass of the
+photons in the absence of perylene (β0 = 0 and ζ = 0) and
+TE-TM splitting (β = 0). The term ˆHmol describes an Nx ×××Ny
+array of porphyrin molecules with periodic boundary condi-
+tions along both the x and y directions. We have made the
+electric dipole approximation and the rotating-wave approxi-
+mation in ˆHcav−mol. Here, ˆµµµm is the electric dipole operator
+associated with the mth molecule and ˆEk,α(r,z) is the electric
+field operator of the mode with polarization α and in-plane
+wave vector k. In addition, µµµα′ · Jk,α is the collective cou-
+pling strength of the cavity mode labeled by k,α and the |G⟩
+to
+��α′
+mol
+�
+transition of the molecules (see Supplementary sec-
+tion S1 for details).
+The photon modes of an empty cavity experience TE-TM
+splitting due to polarization dependent reflection from the mir-
+rors [41]. While the TE-TM splitting lifts the degeneracy be-
+tween photon modes at |k| ̸= 0, photon modes of both polar-
+izations remain degenerate at |k| = 0 due to rotational symme-
+try of the cavity mirrors about the z-axis. However, for Berry
+curvature and Chern invariant to be well-defined, we need the
+photon/polariton bands to be separated in energy at all k; to
+achieve this, we include the perylene crystal. The anisotropy
+and emergent optical activity of the perylene crystal lifts the
+degeneracy between the photon modes at all k [25].
+To compute the Berry curvature and Chern number, we fo-
+cus on the first excitation manifold which is spanned by states
+|m,±mol⟩ = ˆσ†
+m,± |vac⟩ and |k,±cav⟩ = ˆa†
+k,± |vac⟩. Here, |vac⟩
+is the absolute ground state of the system where the pho-
+ton modes are empty and all molecules are in their ground
+states. Rewriting the Hamiltonian with operators ˆσk,α, where
+ˆσm,α =
+1
+√
+NxNy ∑k∈BZ eik·rm ˆσk,α and restricting ourselves to
+the first excitation manifold, we find ˆH(k) = ⟨k| ˆH |k⟩ to be
+ˆH(k) = ˆHmol(k)+ ˆHcav(k)+ ˆHcav−mol(k),
+(4)
+where,
+ˆHmol(k) =¯hωe |+mol⟩⟨+mol|+ ¯hωe |−mol⟩⟨−mol|,
+ˆHcav(k) =
+�
+E0 + ¯h2|k|2
+2m∗ +ζ|k|cosφ
+�
+|+cav⟩⟨+cav|
++
+�
+E0 + ¯h2|k|2
+2m∗ −ζ|k|cosφ
+�
+|−cav⟩⟨−cav|
++
+�
+−β0 +β|k|2e−i2φ�
+|+cav⟩⟨−cav|
++
+�
+−β0 +β|k|2ei2φ�
+|−cav⟩⟨+cav|,
+ˆHcav−mol(k) =Jk,+ ·
+�
+µµµ+ |+mol⟩+ µµµ− |−mol⟩
+�
+⟨+cav|
++Jk,− ·
+�
+µµµ+ |+mol⟩+ µµµ− |−mol⟩
+�
+⟨−cav|
++H.c.
+(5)
+Here, k lies within the first Brillouin zone determined by the
+porphyrin lattice kx,ky ∈ [−π/a,π/a]. As we are only inter-
+ested in length scales much larger than a, we take the contin-
+uum limit a → 0 while keeping µ0/a a constant. Therefore,
+terms such as the collective light-matter coupling strength,
+Jk,α · µµµα′, remain constant in this limit (see Supplementary
+section S1). Moreover, upon taking the continuum limit, ˆH(k)
+does not change; only the range of k becomes infinitely large,
+kx,ky ∈ R, that is, our system acquires complete translational
+invariance in the x-y plane.
+For such continuous systems,
+since kx,ky ∈ R is unbounded, we need to map (kx,ky) onto
+a sphere which is a closed and bounded surface using stere-
+ographic projection before we compute Chern numbers [42]
+(see Supplementary section S2).
+When we diagonalize the Hamiltonian in Eq. 5, we ob-
+tain four bands which we label with l = 1,2,3,4 in increas-
+ing order of energy. In Fig. 3a we plot the Berry curvature,
+Ω1(k), of the lowest band l = 1, and in Fig. 3e we plot the
+ky = 0 slice of the band structure of the two bands lowest
+in energy, l = 1,2. As expected, in the absence of optical
+pumping, this system preserves TRS, which can be verified
+
+4
+a
+e
+f
+g
+h
+b
+c
+d
+𝑓! = 0
+𝑓" = 0
+𝑓! = 0.3
+𝑓" = 0
+𝑓! = 0
+𝑓" = 0.3
+𝑓! = 0.3
+𝑓" = 0.3
+S3
+𝐶! = 0
+𝐶" = 0
+𝐶! = 1
+𝐶" = −1
+𝑓! = 0
+𝑓" = 0
+𝑓! = 0.3
+𝑓" = 0
+𝐶! = −1
+𝐶" = 1
+𝑓! = 0
+𝑓" = 0.3
+𝐶! = 0
+𝐶" = 0
+𝑓! = 0.3
+𝑓" = 0.3
+Ω1 (𝜇m2)
+Ω1 (𝜇m2)
+Ω1 (𝜇m2)
+Ω1 (𝜇m2)
+FIG. 3. (a-d) Berry curvature of the lowest energy band, Ω1(k), and (e-h) a slice of the band structure at ky = 0 of the lower two bands,
+under different levels of optical pumping which create populations: (a,e) f+ = f− = 0, (b,f) f+ = 0.3, f− = 0, (c,g) f+ = 0, f− = 0.3, and
+(d,h) f+ = f− = 0.3. (e-h) The colors of the band indicate the value of the Stokes parameter, S3(k), which measures the degree of circular
+polarization of a mode (Eq. 8). The Chern numbers C1 and C2 of the bands are also specified and are non-zero under time-reversal symmetry
+(TRS) breaking, that is, when f+ ̸= f−. We used parameters β0 = 0.1eV, β = 9×10−4eVµm2, ζ = 2.5×10−3eVµm, m∗ = 125¯h2eV−1µm−2,
+E0 = 3.80eV and ¯hωe = 3.81eV (see Supplementary section S4 for details).
+using the condition on Berry curvature Ωl(k) = −Ωl(−k),
+and the Chern numbers of the all the bands Cl = 0. Also,
+note that, the smallest splitting between the lower two bands
+within −13µm−1 < kx,ky < 13µm−1 is ∼ 2.8meV which is
+larger than the linewidth of the transition in porphyrin at 4K
+(∼ 0.5meV) [43, 44].
+Optical pumping
+Optical pumping can saturate the electronic transitions of a
+system. This leads to reduction in the effective light-matter
+coupling strength, and, therefore, Rabi splitting contraction
+[11, 45, 46]. For instance, when the pump excites a fraction
+of molecules, fE, to the excited state and the remaining popu-
+lation stays in the ground state, fG, it results in Rabi splitting
+contraction proportional to √fG − fE = √1−2fE [47].
+In our system, when the molecules are optically pumped,
+a fraction, f+, of the molecules occupy the |+mol⟩ state, an-
+other fraction, f−, occupy the |−mol⟩ state, and the remaining
+fraction, fG, are in the ground state |G⟩. The Rabi contraction
+corresponding to the |G⟩ to |+mol⟩ transition should then be
+proportional to √fG − f+ which equals √1− f− −2 f+ since
+fG+ f++ f− = 1. Similarly, the contraction should be propor-
+tional to √1− f+ −2 f− for the |G⟩ to |−mol⟩ transition. This
+difference in light-matter coupling when f+ ̸= f− effectively
+introduces 2D chirality into the system [48].
+To derive an effective Hamiltonian under optical pumping,
+we use Heisenberg equations of motion and make a mean-
+field approximation following the approach of Ribeiro et al.
+[47] (Supplementary section S3). We then obtain the effective
+Hamiltonian,
+ˆHeff(k) = ˆHeff
+mol(k)+ ˆHeff
+cav(k)+ ˆHeff
+cav−mol(k),
+(6)
+where,
+ˆHeff
+mol(k) =¯hωe |+mol⟩′ ⟨+mol|′ + ¯hωe |−mol⟩′ ⟨−mol|′ ,
+ˆHeff
+cav(k) =
+�
+E0 + ¯h2|k|2
+2m∗ +ζ|k|cosφ
+�
+|+cav⟩′ ⟨+cav|′
++
+�
+E0 + ¯h2|k|2
+2m∗ −ζ|k|cosφ
+�
+|−cav⟩′ ⟨−cav|′
++
+�
+−β0 +β|k|2e−i2φ�
+|+cav⟩′ ⟨−cav|′
++
+�
+−β0 +β|k|2ei2φ�
+|−cav⟩′ ⟨+cav|′ ,
+ˆHeff
+cav−mol(k) =Jk,+ ·
+��
+1− f− −2 f+µµµ+ |+mol⟩′
++
+�
+1− f+ −2 f−µµµ− |−mol⟩′ �
+⟨+cav|′
++Jk,− ·
+��
+1− f− −2 f+µµµ+ |+mol⟩′
++
+�
+1− f+ −2 f−µµµ− |−mol⟩′ �
+⟨−cav|′ +H.c.
+(7)
+Here, the states |γ⟩′ are different from states |γ⟩ in eq. 5, where
+γ = ±mol,±cav. As expected, the light-matter coupling terms
+are scaled by factors √1− f∓ −2 f± which is a consequence
+of the commutation relation in eq. 1 (see Supplementary sec-
+tion S3).
+If the pump pulse is circularly polarized, f+ ̸= f−, the Rabi
+contraction factor that multiplies the light-matter coupling dif-
+fers for transitions to the |+mol⟩ and |−mol⟩ states; as a re-
+sult, time-reversal symmetry is broken. Consequently, when
+f+ > f−, we find that bands 1 and 2 have non-zero Chern
+numbers +1 and -1 (Fig. 3f). Under the opposite condition,
+f+ < f−, the Chern numbers reverse sign as seen in Fig. 3g.
+When f+ = f−, TRS is preserved, and all bands have Chern
+
+5
+a
+c
+d
+b
+𝒇! = 𝟎. 𝟑
+𝒇" = 𝟎
+𝒇! = 𝟎. 𝟑
+𝒇" = 𝟎
+S3
+Band 1
+Band 2
+𝒇! = 𝟎
+𝒇" = 𝟎. 𝟑
+𝒇! = 𝟎
+𝒇" = 𝟎. 𝟑
+Band 1
+Band 2
+S3
+S3
+S3
+FIG. 4. The Stokes parameter, S3(k), which is a measure of the
+degree of circular polarization of a mode (Eq. 8), under pumping
+with (a,c) σ+ polarized light which creates populations f+ = 0.3,
+f− = 0 and (b,d) σ− polarized light which creates populations f+ =
+0, f− = 0.3 of the two lowest energy bands (Band 1 and 2 as indicated
+in the inset). We used parameters β0 = 0.1eV, β = 9×10−4eVµm2,
+ζ = 2.5 × 10−3eVµm, m∗ = 125¯h2eV−1µm−2, E0 = 3.80eV and
+¯hωe = 3.81eV (see Supplementary section S4 for details).
+number 0 as seen in Fig. 3e and 3h. In Fig. 3b-c, we plot the
+computed Berry curvature when f+ ̸= f− and due to broken
+TRS, we find Ωl(k) ̸= −Ωl(−k).
+We also plot the Stokes parameter, S3(k), for bands 1 and
+2, under pumping with circularly polarized light, in Fig. 4.
+The Stokes parameter, S3(k), provides information on the de-
+gree of circular polarization of the photonic component of an
+exciton-polariton band and is calculated as
+S3(k) = |b+,cav(k)|2 −|b−,cav(k)|2
+|b+,cav(k)|2 +|b−,cav(k)|2
+(8)
+where
+the
+eigenvectors
+of
+the
+band
+are
+��ul,k
+�
+=
+b+,cav(k)|+cav⟩ + b−,cav(k)|−cav⟩ + b+,mol(k)|+mol⟩ +
+b−,mol(k)|−mol⟩. In the absence of pumping, we find that
+within a band, one half of the modes are predominantly
+σ+ polarized and the other half are σ− polarized (Fig.
+3e). Once TRS is broken with circularly polarized optical
+pumping, a large number of modes within each band become
+overwhelmingly of the same polarization (Fig. 3f-g and Fig.
+4).
+In experiments, the Berry curvature of photon bands in a
+Fabry-Perot cavity can be extracted from the components of
+the Stokes vector [25, 26]. However, in the case of exciton-
+polariton bands, the Berry curvature of only sections of the
+band that are predominantly photonic and have negligible
+molecular character [49] can be measured experimentally as,
+to the best of our knowledge, it is difficult to obtain the phase
+relationship between the photonic and molecular components,
+unless light-matter cross-correlation functions are measured.
+Therefore, in our case, the Berry curvature of only parts of the
+band that are mostly photonic in Fig. 3a-d can be measured
+using pump-probe spectroscopy. This measurement should
+be feasible as long as the time delay between the pump and
+probe pulses is shorter than the time the system takes to depo-
+larize and reach a state with f+ = f−. As the depolarization
+timescale for porphyrins ranges from 210 fs to 1.6 ps, this
+measurement should be viable [50].
+As the Chern numbers of bands 1 and 2 are modified
+through pumping with circularly polarized light, if we per-
+form a calculation where a region of the system is pumped
+with σ+ polarized light ( f+ ̸= 0 and f− = 0) and an adjacent
+region is pumped with σ− polarized light (f+ = 0 and f− ̸= 0),
+we expect edge states at the boundary between these regions.
+However, as our Hamiltonian does not contain couplings be-
+tween neighboring molecules, and the position of a molecule
+does not enter the Hamiltonian anywhere except through the
+phase of the light-matter coupling eik·rm, the standard bulk-
+boundary correspondence is no longer applicable and we do
+not observe edge states. We do not include plots for these cal-
+culations in this work and leave it an open question whether
+there is an analogous statement for bulk-boundary correspon-
+dence in these types of systems.
+On the other hand, for
+exciton-polariton systems where nearest-neighbor couplings
+are present, edge states have been predicted and observed
+[19, 20].
+Other systems
+To emphasize that our scheme of saturating electronic tran-
+sitions with circularly polarized light to modify topological
+properties is not limited to organic exciton-polariton systems,
+we compute the Berry curvature of two other polariton sys-
+tems where porphyrin is replaced with (i) Ce:YAG and (ii)
+MoS2 (Fig. 5a and 5d). Other materials can also be used in
+place of porphyrins, as long as they have transitions that can
+be selectively excited with circularly polarized light and these
+transitions have large enough transition dipole moments that
+they can couple strongly to the photon modes of a cavity.
+In Yttrium Aluminum garnet (YAG) doped with Cerium,
+Ce3+ ions replace some Y3+ and Ce3+ has transitions that can
+be selectively excited with circularly polarized light. Here,
+each Ce3+ has two possible ground states, one with the elec-
+tron in spin up |4 f(1) ↑⟩, and the other with it in spin down
+|4 f(1) ↓⟩. Similarly, it has a degenerate pair of excited spin
+states |5d(1) ↑⟩ and |5d(1) ↓⟩.
+The |4 f(1) ↓⟩ ↔ |5d(1) ↑⟩
+transition has ∼ 400 times larger oscillator strength for ex-
+citation with σ+ polarized light than with σ− polarized light,
+therefore, we take the transition dipole moment to be µµµ+ (Fig.
+5b) [51]. Similarly, we take the transition dipole to be µµµ− for
+the |4f(1) ↑⟩ ↔ |5d(1) ↓⟩ transition (Fig. 5b). The transitions
+in Ce:YAG do couple to photon modes, however, to the best
+of our knowledge, strong coupling has not been reported in
+the literature [52, 53]. Nevertheless, strong light-matter cou-
+pling has been achieved with a similar system: Nd3+ doped
+YSO and YVO crystals [54, 55], and based on our calcula-
+tions, with a 0.1µm thick sample of Ce:YAG at concentration
+1% Ce3+ (relative to Y3+), we should be able to attain strong
+
+6
+Ce:YAG
+a
+b
+d
+f
+e
+c
+|4f(1)↓⟩
+|5d(1)↑⟩
+|5d(1)↓⟩
+𝝁!
+𝝁"
+|4f(1)↑⟩
+𝑓↓ = 0.4
+𝑓↑ = 0.6
+𝑓# = 0.3
+𝑓#! = 0
+𝝁"
+𝝁!
+K
+K’
+Ω1 (𝜇m2)
+Ω1 (𝜇m2)
+FIG. 5. (a) Illustration of Ce:YAG (salmon block) and perylene (green blocks) within a Fabry-Perot cavity. (b) Atomic levels of Ce3+
+ions embedded in Yttrium Aluminum garnet (YAG) where the yellow circles indicate the fraction f↓ of Ce3+ ions in the |4f(1) ↓⟩ state
+and the fraction f↑ in the |4f(1) ↑⟩ state after optical pumping. The transition dipoles µµµ± = µ0(ˆx ± iˆy)/
+√
+2 are also indicated. (c) Berry
+curvature of the lowest energy band, Ω1(k), under pumping with circularly polarized which creates populations f↓ = 0.4 and f↑ = 0.6. (d)
+Illustration of monolayer MoS2 and perylene (green blocks) within a Fabry-Perot cavity. (e) Illustration of A-excitons in the K and K’ valleys
+of monolayer MoS2. (f) Berry curvature of the lowest energy band, Ω1(k), under pumping with circularly polarized which creates exciton
+populations fK = 0.3 and fK′ = 0. We used parameters β0 = 0.1eV, β = 9×10−4eVµm2, ζ = 2.5×10−3eVµm, m∗ = 125¯h2eV−1µm−2, (c)
+E0 = 2.50eV, ¯hωe = 2.53eV and (f) E0 = 1.80eV, ¯hωe = 1.855eV (see Supplementary section S4 for details).
+coupling with photon modes in a Fabry-Perot cavity (see Sup-
+plementary section S4).
+Under thermal equilibrium, the populations of the |4f(1) ↑⟩
+and |4 f(1) ↓⟩ states are equal. However, under pumping with
+pulses of σ+ polarization, in the presence of a small magnetic
+field ∼ 0.049T, the population of |4 f(1) ↑⟩ will exceed that
+of |4 f(1) ↓⟩ because population is selectively removed from
+|4f(1) ↓⟩ and added to |5d(1) ↑⟩ by the circularly polarized
+pulses, but decay from the excited |5d(1) ↑⟩ state to the two
+ground states has equal probability [56]. In principle, a mag-
+netic field is not required; however, as we do not know the spin
+relaxation time in the absence of the magnetic field, we report
+the magnetic field used in the experimental study [56]. Under
+optical pumping with circularly polarized light, the 5d states
+will have very small populations which we take to be zero,
+while the |4f(1) ↓⟩ and |4 f(1) ↑⟩ states will have unequal
+populations f↓ and f↑, respectively; here, f↓ + f↑ = 1. Op-
+tically pumped Ce:YAG can then be modeled using the effec-
+tive Hamiltonian in eq. 6 and 7, with |±mol⟩′ → |5d(1) ↑ / ↓⟩
+and √1− f∓ −2 f± → � f↓/↑. The large spin relaxation time
+of ∼ 4.5 ms makes this system particularly well-suited for
+our scheme because it maintains f↓ ̸= f↑, and hence non-zero
+Chern invariants, for an extended period of time [56]. In Fig.
+5c we plot Berry curvature of the lowest band of a perylene
+filled cavity strongly coupled with Ce:YAG, where f↓ = 0.4
+and f↑ = 0.6 (see Supplementary section S4 for values of other
+parameters).
+TMDs, such as single-layer MoS2, display optically con-
+trollable valley polarization and could also be used in place
+of porphyrins [57–59].
+Due to lack of inversion symme-
+try in these systems, the K and K’ valleys are inequivalent;
+this results in optical selection rules that allow selective cre-
+ation of excitons at K and K’ valleys with σ+ and σ− polar-
+ized light, respectively [60, 61]. Additionally, strong light-
+matter coupling has been observed when monolayer MoS2 is
+placed within a Fabry-Perot cavity [8, 9]. This system has
+depolarization times of ∼ 200fs - 5ps making it possible to
+measure Berry curvature using pump-probe spectroscopy be-
+fore depolarization occurs [62, 63]. We model this exciton-
+polariton system (Fig. 5d) using eq. 6 and eq. 7 (we focus
+on the A-exciton, see Supplementary section S4 for parame-
+ters) with |+mol⟩ → |K⟩, |−mol⟩ → |K′⟩ and √1− f∓ −2 f± →
+�1−2 fK/K′. In Fig. 5f we plot the Berry curvature of the
+lowest band when fK = 0.3 and fK′ = 0. Unfortunately, sig-
+nificant Rabi contraction upon optical pumping has not been
+experimentally observed in these systems which will make it
+challenging to observe Berry curvature as in Fig. 5f since
+our model relies on saturation effects. However, for exciton
+polaritons formed from monolayer TMDs, even if Rabi con-
+traction through resonant optical pumping may not produce
+the intended effect, off-resonant optical pumping can break
+the degeneracy of excitons in the K and K’ valleys through
+optical stark effect [64], and this may have interesting conse-
+quences for the Berry curvature. Additionally, if bilayer MoS2
+is used in place of monolayer MoS2, effects on the Berry cur-
+vature described in our work may be more pronounced as bi-
+layer MoS2 hosts interlayer excitons which possess large op-
+tical nonlinearities; specifically, they display saturation and
+
+7
+Rabi contraction under strong coupling [65, 66].
+Finally, so far we have only considered replacing porphyrin
+with a different material, such as MoS2 or Ce:YAG. In addi-
+tion to this, perylene can also be replaced with other suitable
+materials. In our work, we choose to use a cavity filled with
+perylene because we do not want degeneracy at any k within
+the photon bands. Other systems also satisfy this requirement
+and could be used instead. For instance, we could use an elec-
+trically tunable, highly anisotropic, liquid-crystal cavity with
+well separated H and V polarized photon modes [24, 67]. A
+perovskite cavity is another potential candidate due to its high
+anisotropy, and optical pumping may help lift the degeneracy
+of polariton modes in this system [49]. Additionally, other
+photonic structures can also be used instead of a cavity, as
+long as the photon bands are not degenerate at any k and have
+non-zero light-matter coupling at all k.
+CONCLUSION
+In summary, we show that TRS can be broken in organic
+exciton-polariton systems through selectively saturating elec-
+tronic transitions with a circularly polarized pump and that the
+resulting bands possess non-zero Chern invariants. In particu-
+lar, we demonstrate this theoretically for a Fabry-Perot cavity
+filled with porphyrin and perylene. The Berry curvature of
+the more photonic parts of the bands of this system can be
+measured experimentally using pump-probe spectroscopy, as
+long as the time delay is shorter than the depolarization time
+for porphyrin (210fs-1.6ps) [50], and this will reveal non-zero
+Berry curvature and Chern number under circularly polarized
+pumping. Our scheme relies on Rabi contraction from satu-
+ration of optical transitions. It is important to note that edge
+states do not emerge in our system despite non-zero Chern in-
+variants as our model does not contain sufficient positional
+information about the molecules or the unit cells. Bleu et
+al. [30] have previously proposed breaking TRS in inorganic
+exciton-polariton systems through pumping with circularly
+polarized light, however, their work relies on polariton con-
+densation and having patterned lattices. Finally, we demon-
+strate that saturating electronic transitions to modify topol-
+ogy is not limited to organic systems. To illustrate this, we
+calculate the Berry curvature and Chern numbers of exciton-
+polariton bands of two other systems under optical pumping:
+(a) Ce:YAG and (b) monolayer MoS2, and find similar results
+as the organic exciton-polariton case. In view of recent devel-
+opments on electrically tuning the Berry curvature of liquid-
+crystal and perovskite filled cavities [24, 26], our work pro-
+vides an additional control knob to optically tune the Berry
+curvature of exciton-polariton systems using circularly polar-
+ized light. Additionally, ultrafast control of topological prop-
+erties of systems with light may find use in nonreciprocal and
+nonlinear optoelectronic devices.
+ACKNOWLEDGEMENTS
+S.P.-S. acknowledges support from NSF Grant No. CA-
+REER CHE 1654732 for the development of the model and
+calculations.
+The conceptualization of the molecular and
+solid-state systems was guided by N.P.S. and J.Y.-Z. as part of
+the Center for Molecular Quantum Transduction (CMQT), an
+Energy Frontier Research Center funded by the U.S. Depart-
+ment of Energy, Office of Science, Basic Energy Sciences un-
+der Award No. DE-SC0021314. S.P.-S. thanks Kai Schwen-
+nicke and Stephan van den Wildenberg for useful discussions.
+CODE AVAILABILITY
+Code available at https://github.com/SindhanaPS/Topological_Polaritons_Submission.
+REFERENCES
+[1] Claude Weisbuch, Mr Nishioka, A Ishikawa, and Y Arakawa.
+Observation of the coupled exciton-photon mode splitting in a
+semiconductor quantum microcavity. Physical Review Letters,
+69(23):3314, 1992.
+[2] R André, D Heger, Le Si Dang, and Y Merle d’Aubigné. Spec-
+troscopy of polaritons in cdte-based microcavities. Journal of
+crystal growth, 184:758–762, 1998.
+[3] David G Lidzey, DDC Bradley, MS Skolnick, T Virgili,
+S Walker, and DM Whittaker. Strong exciton–photon coupling
+in an organic semiconductor microcavity. Nature, 395(6697):
+53–55, 1998.
+[4] R Butté, G Christmann, E Feltin, J-F Carlin, M Mosca,
+M Ilegems, and N Grandjean. Room-temperature polariton lu-
+minescence from a bulk gan microcavity. Physical Review B,
+73(3):033315, 2006.
+[5] R Shimada, J Xie, Vitaliy Avrutin, Ü Özgür, and H Morkoˇc.
+Cavity polaritons in zno-based hybrid microcavities. Applied
+Physics Letters, 92(1):011127, 2008.
+[6] Antoine Brehier, Radoslav Parashkov, Jean-Sébastien Lauret,
+and Emmanuelle Deleporte. Strong exciton-photon coupling
+in a microcavity containing layered perovskite semiconductors.
+Applied physics letters, 89(17):171110, 2006.
+[7] Rui Su, Antonio Fieramosca, Qing Zhang, Hai Son Nguyen,
+Emmanuelle Deleporte, Zhanghai Chen, Daniele Sanvitto, Tim-
+othy CH Liew, and Qihua Xiong. Perovskite semiconductors
+for room-temperature exciton-polaritonics. Nature Materials,
+20(10):1315–1324, 2021.
+[8] Xiaoze Liu, Tal Galfsky, Zheng Sun, Fengnian Xia, Erh-
+chen Lin, Yi-Hsien Lee, Stéphane Kéna-Cohen, and Vinod M
+Menon.
+Strong light–matter coupling in two-dimensional
+atomic crystals. Nature Photonics, 9(1):30–34, 2015.
+[9] Fengrui Hu and Zhe Fei. Recent progress on exciton polaritons
+in layered transition-metal dichalcogenides. Advanced Optical
+Materials, 8(5):1901003, 2020.
+[10] James A Hutchison, Tal Schwartz, Cyriaque Genet, Eloïse De-
+vaux, and Thomas W Ebbesen. Modifying chemical landscapes
+by coupling to vacuum fields.
+Angewandte Chemie Interna-
+tional Edition, 51(7):1592–1596, 2012.
+[11] KS Daskalakis, SA Maier, Ray Murray, and Stéphane Kéna-
+Cohen. Nonlinear interactions in an organic polariton conden-
+sate. Nature materials, 13(3):271–278, 2014.
+
+8
+[12] Christof P Dietrich, Anja Steude, Laura Tropf, Marcel Schu-
+bert, Nils M Kronenberg, Kai Ostermann, Sven Höfling, and
+Malte C Gather. An exciton-polariton laser based on biolog-
+ically produced fluorescent protein.
+Science advances, 2(8):
+e1600666, 2016.
+[13] Nina Krainova, Alex J Grede, Demetra Tsokkou, Natalie
+Banerji, and Noel C Giebink. Polaron photoconductivity in the
+weak and strong light-matter coupling regime. Physical review
+letters, 124(17):177401, 2020.
+[14] Qing Liao, Charly Leblanc, Jiahuan Ren, Feng Li, Yiming Li,
+Dmitry Solnyshkov, Guillaume Malpuech, Jiannian Yao, and
+Hongbing Fu.
+Experimental measurement of the divergent
+quantum metric of an exceptional point. Physical Review Let-
+ters, 127(10):107402, 2021.
+[15] Marco Dusel, Simon Betzold, Tristan H Harder, Monika Em-
+merling, Johannes Beierlein, Jürgen Ohmer, Utz Fischer, Ronny
+Thomale, Christian Schneider, Sven Hofling, et al.
+Room-
+temperature topological polariton laser in an organic lattice.
+Nano Letters, 21(15):6398–6405, 2021.
+[16] Dmitry D Solnyshkov, Guillaume Malpuech, Philippe St-Jean,
+Sylvain Ravets, Jacqueline Bloch, and Alberto Amo. Micro-
+cavity polaritons for topological photonics. Optical Materials
+Express, 11(4):1119–1142, 2021.
+[17] Charles-Edouard Bardyn, Torsten Karzig, Gil Refael, and Tim-
+othy CH Liew. Topological polaritons and excitons in garden-
+variety systems. Physical Review B, 91(16):161413, 2015.
+[18] Joel Yuen-Zhou, Semion K Saikin, Tony Zhu, Mehmet C On-
+basli, Caroline A Ross, Vladimir Bulovic, and Marc A Baldo.
+Plexciton dirac points and topological modes. Nature commu-
+nications, 7(1):1–7, 2016.
+[19] S Klembt, TH Harder, OA Egorov, K Winkler, R Ge, MA Ban-
+dres, M Emmerling, L Worschech, TCH Liew, M Segev, et al.
+Exciton-polariton topological insulator.
+Nature, 562(7728):
+552–556, 2018.
+[20] Torsten Karzig, Charles-Edouard Bardyn, Netanel H Lindner,
+and Gil Refael. Topological polaritons. Physical Review X, 5
+(3):031001, 2015.
+[21] Wenjing Liu, Zhurun Ji, Yuhui Wang, Gaurav Modi, Minsoo
+Hwang, Biyuan Zheng, Volker J Sorger, Anlian Pan, and Ritesh
+Agarwal. Generation of helical topological exciton-polaritons.
+Science, 370(6516):600–604, 2020.
+[22] Mengyao Li, Ivan Sinev, Fedor Benimetskiy, Tatyana Ivanova,
+Ekaterina Khestanova, Svetlana Kiriushechkina, Anton Vaku-
+lenko, Sriram Guddala, Maurice Skolnick, Vinod M Menon,
+et al.
+Experimental observation of topological z2 exciton-
+polaritons in transition metal dichalcogenide monolayers. Na-
+ture communications, 12(1):1–10, 2021.
+[23] A Gianfrate, O Bleu, L Dominici, V Ardizzone, M De Giorgi,
+D Ballarini, G Lerario, KW West, LN Pfeiffer, DD Solnyshkov,
+et al. Measurement of the quantum geometric tensor and of the
+anomalous hall drift. Nature, 578(7795):381–385, 2020.
+[24] Katarzyna Rechci´nska, Mateusz Król, Rafał Mazur, Prze-
+mysław Morawiak, Rafał Mirek, Karolina Łempicka, Witold
+Bardyszewski, Michał Matuszewski, Przemysław Kula, Wik-
+tor Piecek, et al. Engineering spin-orbit synthetic hamiltonians
+in liquid-crystal optical cavities. Science, 366(6466):727–730,
+2019.
+[25] Jiahuan Ren, Qing Liao, Feng Li, Yiming Li, Olivier Bleu,
+Guillaume Malpuech, Jiannian Yao, Hongbing Fu, and Dmitry
+Solnyshkov. Nontrivial band geometry in an optically active
+system. Nature communications, 12(1):1–8, 2021.
+[26] Karolina Łempicka-Mirek, Mateusz Król, Helgi Sigurdsson,
+Adam Wincukiewicz, Przemysław Morawiak, Rafał Mazur,
+Marcin Muszy´nski, Wiktor Piecek, Przemysław Kula, Tomasz
+Stefaniuk, et al. Electrically tunable berry curvature and strong
+light-matter coupling in liquid crystal microcavities with 2d
+perovskite. Science Advances, 8(40):eabq7533, 2022.
+[27] Sriram Guddala, Yuma Kawaguchi, Filipp Komissarenko, Svet-
+lana Kiriushechkina, Anton Vakulenko, Kai Chen, Andrea Alù,
+Vinod M Menon, and Alexander B Khanikaev.
+All-optical
+nonreciprocity due to valley polarization pumping in transi-
+tion metal dichalcogenides. Nature communications, 12(1):1–9,
+2021.
+[28] Erik J Lenferink, Guohua Wei, and Nathaniel P Stern. Coher-
+ent optical non-reciprocity in axisymmetric resonators. Optics
+express, 22(13):16099–16111, 2014.
+[29] Kai Schwennicke and Joel Yuen-Zhou. Optical activity from
+the exciton aharonov–bohm effect: A floquet engineering ap-
+proach. The Journal of Physical Chemistry C, 124(7):4206–
+4214, 2020.
+[30] O Bleu, DD Solnyshkov, and Guillaume Malpuech. Photonic
+versus electronic quantum anomalous hall effect. Physical Re-
+view B, 95(11):115415, 2017.
+[31] Teng Long, Xuekai Ma, Jiahuan Ren, Feng Li, Qing Liao,
+Stefan Schumacher, Guillaume Malpuech, Dmitry Solnyshkov,
+and Hongbing Fu. Helical polariton lasing from topological val-
+leys in an organic crystalline microcavity. Advanced Science, 9
+(29):2203588, 2022.
+[32] Mercedes Rubio, Björn O Roos, Luis Serrano-Andrés, and
+Manuela Merchán. Theoretical study of the electronic spectrum
+of magnesium-porphyrin. The Journal of chemical physics, 110
+(15):7202–7209, 1999.
+[33] Lawrence Edwards, David H Dolphin, and Martin Gouterman.
+Porphyrins: Xvi. vapor absorption spectra and redox reactions:
+Octalkylporphins. Journal of Molecular Spectroscopy, 35(1):
+90–109, 1970.
+[34] Tonatiuh Rangel, Andre Rinn, Sahar Sharifzadeh, Felipe H
+da Jornada, André Pick, Steven G Louie, Gregor Witte, Leeor
+Kronik, Jeffrey B Neaton, and Sangam Chatterjee. Low-lying
+excited states in crystalline perylene. Proceedings of the Na-
+tional Academy of Sciences, 115(2):284–289, 2018.
+[35] Ingo Barth, Jörn Manz, Yasuteru Shigeta, and Kiyoshi Yagi.
+Unidirectional electronic ring current driven by a few cycle cir-
+cularly polarized laser pulse: quantum model simulations for
+mg- porphyrin. Journal of the American Chemical Society, 128
+(21):7043–7049, 2006.
+[36] Joel Yuen-Zhou, Semion K Saikin, Norman Y Yao, and Alán
+Aspuru-Guzik. Topologically protected excitons in porphyrin
+thin films. Nature materials, 13(11):1026–1032, 2014.
+[37] Shichao Sun, Bing Gu, and Shaul Mukamel. Polariton ring cur-
+rents and circular dichroism of mg-porphyrin in a chiral cavity.
+Chemical science, 13(4):1037–1048, 2022.
+[38] Shaul Mukamel. Principles of nonlinear optical spectroscopy.
+Number 6. Oxford University Press on Demand, 1999.
+[39] Vladimir M Agranovich. Excitations in organic solids, volume
+142. OUP Oxford, 2009.
+[40] Alexey V Kavokin, Jeremy J Baumberg, Guillaume Malpuech,
+and Fabrice P Laussy. Microcavities, volume 21. Oxford uni-
+versity press, 2017.
+[41] Giovanna Panzarini, Lucio Claudio Andreani, A Armitage,
+D Baxter, MS Skolnick, VN Astratov, JS Roberts, Alexey V
+Kavokin, Maria R Vladimirova, and MA Kaliteevski. Exciton-
+light coupling in single and coupled semiconductor microcav-
+ities: Polariton dispersion and polarization splitting. Physical
+Review B, 59(7):5082, 1999.
+[42] Mário G Silveirinha. Chern invariants for continuous media.
+Physical Review B, 92(12):125153, 2015.
+
+9
+[43] Uzi Even, Jacob Magen, Joshua Jortner, Joel Friedman,
+and Haim Levanon.
+Isolated ultracold porphyrins in super-
+sonic expansions. i. free-base tetraphenylporphyrin and zn-
+tetraphenylporphyrin. The Journal of Chemical Physics, 77(9):
+4374–4383, 1982.
+[44] S Voelker, RM Macfarlane, AZ Genack, HP Trommsdorff, and
+JH van Der Waals. Homogeneous linewidth of the s 1← s 0
+transition of free-base porphyrin in an n-octane crystal as stud-
+ied by photochemical hole-burning. The Journal of Chemical
+Physics, 67(4):1759–1765, 1977.
+[45] Bo Xiang, Raphael F Ribeiro, Adam D Dunkelberger, Jiaxi
+Wang, Yingmin Li, Blake S Simpkins, Jeffrey C Owrutsky, Joel
+Yuen-Zhou, and Wei Xiong. Two-dimensional infrared spec-
+troscopy of vibrational polaritons. Proceedings of the National
+Academy of Sciences, 115(19):4845–4850, 2018.
+[46] Timur Yagafarov, Denis Sannikov, Anton Zasedatelev, Kyriacos
+Georgiou, Anton Baranikov, Oleksandr Kyriienko, Ivan She-
+lykh, Lizhi Gai, Zhen Shen, David Lidzey, et al. Mechanisms
+of blueshifts in organic polariton condensates. Communications
+Physics, 3(1):1–10, 2020.
+[47] Raphael F. Ribeiro, Adam D Dunkelberger, Bo Xiang, Wei
+Xiong, Blake S Simpkins, Jeffrey C Owrutsky, and Joel Yuen-
+Zhou. Theory for nonlinear spectroscopy of vibrational polari-
+tons.
+The journal of physical chemistry letters, 9(13):3766–
+3771, 2018.
+[48] Andrew H Salij, Randall H Goldsmith, and Roel Tempelaar.
+Chiral polaritons based on achiral fabry-perot cavities using ap-
+parent circular dichroism.
+arXiv preprint arXiv:2208.14461,
+2022.
+[49] Laura Polimeno, Giovanni Lerario, Milena De Giorgi, Luisa
+De Marco, Lorenzo Dominici, Francesco Todisco, Annalisa
+Coriolano, Vincenzo Ardizzone, Marco Pugliese, Carmela T
+Prontera, et al. Tuning of the berry curvature in 2d perovskite
+polaritons. Nature nanotechnology, 16(12):1349–1354, 2021.
+[50] C Galli, Klaas Wynne, Steven M LeCours, MJ Therien, and
+RM Hochstrasser. Direct measurement of electronic dephasing
+using anisotropy. Chemical physics letters, 206(5-6):493–499,
+1993.
+[51] Roman Kolesov, Kangwei Xia, Rolf Reuter, Mohammad Ja-
+mali, Rainer Stöhr, Tugrul Inal, Petr Siyushev, and Jörg
+Wrachtrup. Mapping spin coherence of a single rare-earth ion
+in a crystal onto a single photon polarization state. Physical
+review letters, 111(12):120502, 2013.
+[52] Robert J Moerland, I Gerward C Weppelman, Marijke Scotuzzi,
+and Jacob P Hoogenboom. Nanoscale imaging of light-matter
+coupling inside metal-coated cavities with a pulsed electron
+beam. Nano Letters, 18(10):6107–6112, 2018.
+[53] SRK Rodriguez, S Murai, MA Verschuuren, and J Gómez Ri-
+vas. Light-emitting waveguide-plasmon polaritons. Physical
+review letters, 109(16):166803, 2012.
+[54] Tian Zhong, Jonathan M Kindem, Evan Miyazono, and Andrei
+Faraon. Nanophotonic coherent light–matter interfaces based
+on rare-earth-doped crystals. Nature communications, 6(1):1–
+6, 2015.
+[55] Tian Zhong, Jonathan M Kindem, Jake Rochman, and An-
+drei Faraon. Interfacing broadband photonic qubits to on-chip
+cavity-protected rare-earth ensembles. Nature communications,
+8(1):1–7, 2017.
+[56] P Siyushev, K Xia, R Reuter, M Jamali, N Zhao, N Yang,
+C Duan, N Kukharchyk, AD Wieck, R Kolesov, et al. Coherent
+properties of single rare-earth spin qubits. Nature communica-
+tions, 5(1):1–6, 2014.
+[57] Kin Fai Mak, Keliang He, Jie Shan, and Tony F Heinz. Control
+of valley polarization in monolayer mos2 by optical helicity.
+Nature nanotechnology, 7(8):494–498, 2012.
+[58] Hualing Zeng, Junfeng Dai, Wang Yao, Di Xiao, and Xiaodong
+Cui. Valley polarization in mos2 monolayers by optical pump-
+ing. Nature nanotechnology, 7(8):490–493, 2012.
+[59] Aswini Kumar Pattanayak, Pritam Das, Devarshi Chakrabarty,
+Avijit Dhara, Shreya Paul, Satyait Maji, Maruthi Manoj Brun-
+davanam, and Sajal Dhara. Probing spin dynamics of 2d ex-
+citons with twisted light.
+ACS Photonics, 9(10):3351–3356,
+2022.
+[60] Liuyang Sun, Chun-Yuan Wang, Alex Krasnok, Junho Choi,
+Jinwei Shi, Juan Sebastian Gomez-Diaz, André Zepeda,
+Shangjr Gwo, Chih-Kang Shih, Andrea Alù, et al. Separation
+of valley excitons in a mos2 monolayer using a subwavelength
+asymmetric groove array.
+Nature Photonics, 13(3):180–184,
+2019.
+[61] Guan-Hao Peng, Oscar Javier Gomez Sanchez, Wei-Hua Li,
+Ping-Yuan Lo, and Shun-Jen Cheng. Twisted-light-induced ex-
+citon wave packets in transition-metal dichalcogenide monolay-
+ers. arXiv preprint arXiv:2203.02081, 2022.
+[62] Stefano Dal Conte, Federico Bottegoni, Eva Arianna Aurelia
+Pogna, D De Fazio, Stefano Ambrogio, Ilaria Bargigia, Cosimo
+D’Andrea, A Lombardo, M Bruna, Franco Ciccacci, et al. Ul-
+trafast valley relaxation dynamics in monolayer mos 2 probed
+by nonequilibrium optical techniques. Physical Review B, 92
+(23):235425, 2015.
+[63] Yen-Jung Chen, Jeffrey D Cain, Teodor K Stanev, Vinayak P
+Dravid, and Nathaniel P Stern.
+Valley-polarized exciton–
+polaritons in a monolayer semiconductor. Nature Photonics,
+11(7):431–435, 2017.
+[64] Trevor LaMountain, Jovan Nelson, Erik J Lenferink, Samuel H
+Amsterdam, Akshay A Murthy, Hongfei Zeng, Tobin J Marks,
+Vinayak P Dravid, Mark C Hersam, and Nathaniel P Stern.
+Valley-selective optical stark effect of exciton-polaritons in a
+monolayer semiconductor. Nature communications, 12(1):1–7,
+2021.
+[65] Biswajit
+Datta,
+Mandeep
+Khatoniar,
+Prathmesh
+Desh-
+mukh, Félix Thouin, Rezlind Bushati, Simone De Liberato,
+Stephane Kena Cohen, and Vinod M Menon.
+Highly non-
+linear dipolar exciton-polaritons in bilayer mos2.
+Nature
+communications, 13(1):1–7, 2022.
+[66] Charalambos Louca, Armando Genco, Salvatore Chiavazzo,
+Thomas P Lyons, Sam Randerson, Chiara Trovatello, Pe-
+ter Claronino, Rahul Jayaprakash, Kenji Watanabe, Takashi
+Taniguchi, et al. Nonlinear interactions of dipolar excitons and
+polaritons in mos2 bilayers. arXiv preprint arXiv:2204.00485,
+2022.
+[67] Marcin Muszy´nski, Mateusz Król, Katarzyna Rechci´nska,
+Przemysław Oliwa, Mateusz K˛edziora, Karolina Łempicka-
+Mirek, Rafał Mazur, Przemysław Morawiak, Wiktor Piecek,
+Przemysław Kula, et al.
+Realizing persistent-spin-helix las-
+ing in the regime of rashba-dresselhaus spin-orbit coupling in
+a dye-filled liquid-crystal optical microcavity. Physical Review
+Applied, 17(1):014041, 2022.
+
+Molecular and solid-state topological polaritons via optical saturation: supplemental document
+Sindhana Pannir-Sivajothi,1 Nathaniel P. Stern,2 and Joel Yuen-Zhou1, ∗
+1Department of Chemistry and Biochemistry, University of California San Diego, La Jolla, California 92093, USA
+2Department of Physics and Astronomy, Northwestern University, Evanston, Illinois 60208, USA
+S1.
+LIGHT-MATTER COUPLING
+The light-matter coupling part of the total Hamiltonian under the electric dipole approximation is,
+ˆHcav−mol =∑
+m ∑
+k,α
+− ˆµµµm · ˆEk,α(rm,0),
+=∑
+m ∑
+k,α
+−
+�
+∑
+α′=±
+(µµµα′ ˆσ†
+m,α′ + µµµ∗
+α′ ˆσm,α′)
+�
+· ˆEk,α(rm,0),
+(S1)
+where µµµα′ = µµµm,α′ =
+�
+m,α′
+mol
+�� ˆµµµ |m,G⟩ is independent of m since we assume that all porphyrin molecules lie flat in the x-y
+plane and are oriented. The electric field operator of the mode labeled by k and α is
+ˆEk,α(r,z) =
+�
+¯hωk,α
+2Vεε0
+�
+f∗
+k,α(r,z) ˆa†
+k,α +fk,α(r,z) ˆak,α
+�
+.
+(S2)
+Here, V = LxLyLz is the volume of the box we consider, where as mentioned in the main manuscript, we apply periodic boundary
+conditions along the x and y directions. From here on, we will call the in-plane area of the box A = LxLy. Here, fk,α(r,z) is the
+mode profile and it satisfies[1]
+�
+dr
+� Lz
+0
+dzf∗
+k,α(r,z)fk,α(r,z) = LzA.
+(S3)
+For the TE and TM modes[2],
+fk,TE(r,z) =eik·r√
+2sin
+�
+nzπ
+Lz
+�
+z+ Lz
+2
+��
+ˆφφφ,
+fk,TM(r,z) =eik·r
+�
+2
+|k|2 +
+� nzπ
+Lz
+�2
+��nzπ
+Lz
+�
+sin
+�
+nzπ
+Lz
+�
+z+ Lz
+2
+��
+ˆρρρ −i|k|cos
+�
+nzπ
+Lz
+�
+z+ Lz
+2
+��
+ˆz
+�
+.
+(S4)
+We make the rotating-wave approximation,
+ˆHcav−mol =∑
+m ∑
+k,α
+−
+�
+∑
+α′=±
+(µµµα′ ˆσ†
+m,α′ + µµµ∗
+α′ ˆσm,α′)
+�
+·
+��
+¯hωk,α
+2Vεε0
+�
+f∗
+k,α(rm,0) ˆa†
+k,α +fk,α(rm,0) ˆak,α
+��
+,
+≈ ∑
+m,α′ ∑
+k,α
+−
+�
+¯hωk,α
+2Vεε0
+�
+µµµα′ ·fk,α(rm,0) ˆσ†
+m,α′ ˆak,α + µµµ∗
+α′ ·f∗
+k,α(rm,0) ˆσm,α′ ˆa†
+k,α
+�
+,
+= ∑
+m,α′ ∑
+k,α
+� eik·rm
+�NxNy
+(µµµα′ ·Jk,α) ˆσ†
+m,α′ ˆak,α + e−ik·rm
+�NxNy
+(µµµ∗
+α′ ·J∗
+k,α) ˆσm,α′ ˆa†
+k,α
+�
+,
+(S5)
+where Jk,α = −�NxNy
+�
+¯hωk,α
+2Vεε0 e−ik·rfk,α(r,0) and µµµα′ ·Jk,α is the collective light-matter coupling strength.
+The annihilation operators of photon modes polarized along the horizontal (H) or x-axis and vertical (V) or y-axis are ˆak,H
+and ˆak,V, respectively. They are related to α = ± polarized modes through ˆak,± =
+1
+√
+2( ˆak,H ∓i ˆak,V)[3]. In addition, we assume
+∗ [email protected]
+arXiv:2301.03287v1  [physics.chem-ph]  9 Jan 2023
+
+2
+that they are related to the TM and TE modes through ˆak,TM = cosφ ˆak,H +sinφ ˆak,V and ˆak,TE = −sinφ ˆak,H +cosφ ˆak,V. Using
+this, we obtain the relationship between ˆak,TE, ˆak,TM and ˆak,+, ˆak,− modes to be,
+ˆak,TM = 1
+√
+2
+�
+eiφ ˆak,+ +e−iφ ˆak,−
+�
+,
+ˆak,TE = 1
+√
+2
+�
+ieiφ ˆak,+ −ie−iφ ˆak,−
+�
+.
+(S6)
+It is important to note that, based on these relationships and S4, the α =H/V modes are not completely linearly polarized and
+the α = ± modes are not completely circularly polarized when |k| becomes comparable with nzπ/Lz. We also find,
+Jk,+ = eiφ
+√
+2
+�
+Jk,TM +iJk,TE
+�
+,
+Jk,− =e−iφ
+√
+2
+�
+Jk,TM −iJk,TE
+�
+.
+(S7)
+To keep the collective coupling strength µµµα′ ·Jk,α constant while taking the a → 0 limit, we take the magnitude of the collective
+transition dipole of the bright state �NxNyµ0 over square root of the quantization area of the photon mode
+√
+A to be a constant;
+that is, we keep √ρAµ0 = µ0/a a constant, where ρA = NxNy/A is the areal density of quantum emitters.
+Jk,α =−√ρA
+�
+¯hωk,α
+2Lzεε0
+e−ik.rfk,α(r,0)
+=− 1
+a
+�
+¯hωk,α
+2Lzεε0
+e−ik.rfk,α(r,0).
+(S8)
+S2.
+CHERN NUMBER CALCULATION
+a
+b
+(kmax,kmax)
+(kmax,-kmax)
+(-kmax,-kmax)
+(-kmax,kmax)
+x
+y
+FIG. S1. (a) This is a cartoon figure that demonstrates the way Berry flux and Chern number are computed in our system. The small squares
+are the plaquettes over which Berry flux is computed. The blue arrows specify the orientation used for Berry flux computation. Note that the
+direction is opposite for the small squares and the large square. (b) Same as (a), but placed on a sphere. Here, it is more clear that the direction
+of the arrow for the large square indicates the way Berry flux is computed for the giant plaquette covering the rest of the sphere.
+For the Chern invariant to be an integer, it is important that the Berry curvature is integrated over a closed and bounded
+surface [4]. For periodic systems with a finite period, the Brillouin zone is a torus which satisfies this requirement. However,
+
+3
+for a continuous system, (kx,ky) lies on an unbounded plane; for such systems, Silveirinha[5] proposed mapping this infinitely
+large plane onto a sphere to compute the Chern number. This is the procedure we follow in our work. We discretize k-space and
+compute the Berry flux in each plaquette within a square-shaped region in k-space, −kmax ≤ kx,ky ≤ kmax [4, 6] (Fig. S1a and
+S1b). The entire region that satisfies the condition kx,ky > kmax or kx,ky < −kmax is taken as a single giant plaquette (Fig. S1b),
+and the Berry flux within this region is computed by taking the Berry phase along the boundary of the plaquette but in a direction
+opposite to that used to compute Berry flux for plaquettes within the square −kmax ≤ kx,ky ≤ kmax as indicated in Fig. S1a and
+S1b. To ensure that we obtain a converged Chern number, we calculate the Chern number for different kmax and find that, for
+our system, once kmax ≳ 100µm−1, the Chern number converges to C1 = ±1,C2 = ∓1,C3 = 0, and C4 = 0 when f+ ̸= f− with
+|f+ − f−| ≳ 0.11. Smaller differences between f+ and f−, |f+ − f−| ≲ 0.11 require larger kmax for convergence. This is not a
+problem for the f+ = f− case because the Chern invariant will always be zero due to time-reversal symmetry Ωl(k) = −Ωl(−k),
+and we can use kmax ≈ 100µm−1 to compute it.
+S3.
+OPTICAL PUMPING
+The number of excitations in the system Nex = ∑k,α a†
+k,αak,α + ∑n,α σ†
+n,ασn,α is a conserved quantity of this Hamiltonian.
+Therefore, when we have f+ fraction of molecules in the |+mol⟩ state and f− in the |−mol⟩ state, we will only have to look at
+the ( f+ + f−)Nth excitation manifold. Unfortunately, the dimensions of the Hilbert space of this manifold scale as
+�
+N
+(f++f−)N
+�
+,
+and this quickly becomes computationally intractable as the system size, N, increases. Using mean-field theory, we reduce this
+many-body problem to a one-body problem. That is, we derive an effective Hamiltonian for a single excitation in the mean-field
+of the remaining ( f+ + f−)N excitations; in this way, we reduce the dimensions of the Hilbert space to that of the first excitation
+manifold. To do this, we follow a procedure similar to that used by Ribeiro et al.[7] and write the Heisenberg equations of
+motion (EOM) for the operators ˆσm,± and ˆak,±,
+i¯hd ˆσn,±
+dt
+=
+� ˆσn,±, ˆHmol
+�
++
+� ˆσn,±, ˆHcav
+�
++
+� ˆσn,±, ˆHcav−mol
+�
+=¯hωe ˆσn,± +
+1
+�NxNy ∑
+k
+eik·rn
+�
+(1− ˆσ†
+n,∓ ˆσn,∓ −2 ˆσ†
+n,± ˆσn,±)
+�
+Jk,+ · µµµ± ˆak,+
++Jk,− · µµµ± ˆak,−
+�
+− ˆσ†
+n,∓ ˆσn,±
+�
+Jk,+ · µµµ∓ ˆak,+ +Jk,− · µµµ∓ ˆak,−
+��
+,
+i¯hd ˆak,±
+dt
+=
+�
+ˆak,±, ˆHmol
+�
++
+�
+ˆak,±, ˆHcav
+�
++
+�
+ˆak,±, ˆHcav−mol
+�
+=
+�
+E0 + ¯h2|k|2
+2m∗ ±ζ|k|cosφ
+�
+ˆak,± +
+�
+−β0 +β|k|2e∓i2φ�
+ˆa∓,k
++
+1
+�NxNy ∑
+m
+eik·rm
+�
+J∗
+k,± · µµµ∗
++ ˆσm,+ +J∗
+k,± · µµµ∗
+− ˆσm,−
+�
+.
+(S9)
+We make a mean-field approximation to linearize these EOM. For instance, we use mn ≈ ¯mn, that is,
+ˆσ†
+n,+ ˆσn,+ ˆak,+ =
+�
+⟨ ˆσ†
+n,+ ˆσn,+⟩+ ˆσ†
+n,+ ˆσn,+ −⟨ ˆσ†
+n,+ ˆσn,+⟩
+�
+ˆak,+
+=⟨ ˆσ†
+n,+ ˆσn,+⟩ ˆak,+ +( ˆσ†
+n,+ ˆσn,+ −⟨ ˆσ†
+n,+ ˆσn,+⟩)⟨ ˆak,+⟩
+≈⟨ ˆσ†
+n,+ ˆσn,+⟩ ˆak,+,
+(S10)
+where ⟨ ˆO⟩ = Tr
+� ˆρ0 ˆO
+�
+with ˆρ0 ≈ ∏m ˆρm ∏k ∏α=+,− ˆρα,k[8]. Here, we assume that after dephasing of the molecular amplitudes,
+ˆρm = fG |m,G⟩⟨m,G|+ f+ |m,+mol⟩⟨m,+mol|+ f− |m,−mol⟩⟨m,−mol|, ˆρα,k = |k,αcav,0⟩⟨k,αcav,0|, and, therefore, ⟨ ˆak,+⟩ =
+0. The EOM then become
+i¯hd ˆσn,±
+dt
+≈¯hωe ˆσn,± +
+1
+�NxNy
+(1− f∓ −2 f±)∑
+k
+eik·rn
+�
+Jk,+ · µµµ± ˆak,+
++Jk,− · µµµ± ˆak,−
+�
+,
+i¯hd ˆak,±
+dt
+=
+�
+E0 + ¯h2|k|2
+2m∗ ±ζ|k|cosφ
+�
+ˆak,± +
+�
+−β0 +β|k|2e∓i2φ�
+ˆa∓,k
++
+1
+�NxNy ∑
+m
+eik·rm
+�
+J∗
+k,± · µµµ∗
++ ˆσm,+ +J∗
+k,± · µµµ∗
+− ˆσm,−
+�
+.
+(S11)
+
+4
+We define rescaled operators ˆσ′
+n,± = ˆσn,±/√1− f∓ −2 f± and rewrite the EOM,
+i¯hd ˆσ′
+n,±
+dt
+≈¯hωe ˆσ′
+n,± +
+1
+�NxNy
+�
+1− f∓ −2f±∑
+k
+eik·rn
+�
+Jk,+ · µµµ± ˆak,+
++Jk,− · µµµ± ˆak,−
+�
+,
+i¯hd ˆak,±
+dt
+=
+�
+E0 + ¯h2|k|2
+2m∗ ±ζ|k|cosφ
+�
+ˆak,± +
+�
+−β0 +β|k|2e∓i2φ�
+ˆa∓,k
++ �NxNy ∑
+m
+eik·rm
+��
+1− f− −2f+J∗
+k,± · µµµ∗
++ ˆσ′
+m,+ +
+�
+1− f+ −2 f−J∗
+k,± · µµµ∗
+− ˆσ′
+m,−
+�
+.
+(S12)
+From these EOM, along with the fact that ˆσ′
+n,± act effectively as bosonic operators in mean-field,
+�
+ˆσ′
+n,+, ˆσ′†
+n,+
+�
+=
+1− ˆσ†
+n,− ˆσn,−−2 ˆσ†
+n,+ ˆσn,+
+1−f−−2 f+
+≈
+ˆI and
+�
+ˆσ′
+n,+, ˆσ′†
+n,−
+�
+=
+− ˆσ†
+n,− ˆσn,+
+1−f−−2 f+ ≈ ˆ0, where ˆI and ˆ0 are the identity and zero operators, we can construct an effective Hamiltonian
+ˆHeff = ˆHeff
+mol + ˆHeff
+cav + ˆHeff
+cav−mol in ˆσ′
+n,± and ˆak,±,
+ˆHeff
+mol =∑
+n
+�
+¯hωe ˆσ′†
+n,+ ˆσ′
+n,+ + ¯hωe ˆσ′†
+n,− ˆσ′
+n,−
+�
+,
+ˆHeff
+cav =∑
+k
+�
+E0 + ¯h2|k|2
+2m∗ +ζ|k|cosφ
+�
+ˆa†
+k,+ ˆak,+
++
+�
+E0 + ¯h2|k|2
+2m∗ −ζ|k|cosφ
+�
+ˆa†
+k,− ˆak,− +
+�
+−β0 +β|k|2e−i2φ�
+ˆa†
+k,+ ˆak,−
++
+�
+−β0 +β|k|2ei2φ�
+ˆa†
+k,− ˆak,+,
+ˆHeff
+cav−mol =
+1
+�NxNy ∑
+m ∑
+k
+eik·rm
+�
+�
+1− f− −2 f+
+�
+Jk,+ · µµµ+ ˆσ′†
+m,+ ˆak,+
++Jk,− · µµµ+ ˆσ′†
+m,+ ˆak,−
+�
++
+�
+1− f+ −2f−
+�
+Jk,+ · µµµ− ˆσ′†
+m,− ˆak,+
++Jk,− · µµµ− ˆσ′†
+m,− ˆak,−
+��
++H.c.,
+(S13)
+which is the mean-field Hamiltonian when the system has f+, f− excitations. Writing this effective Hamiltonian in k-space,
+ˆHeff
+mol =∑
+k
+�
+¯hωe ˆσ′†
+k,+ ˆσ′
+k,+ + ¯hωe ˆσ′†
+k,− ˆσ′
+k,−
+�
+,
+ˆHeff
+cav =∑
+k
+�
+E0 + ¯h2|k|2
+2m∗ +ζ|k|cosφ
+�
+ˆa†
+k,+ ˆak,+ +
+�
+E0 + ¯h2|k|2
+2m∗ −ζ|k|cosφ
+�
+ˆa†
+k,− ˆak,−
++
+�
+−β0 +β|k|2e−i2φ�
+ˆa†
+k,+ ˆak,− +
+�
+−β0 +β|k|2ei2φ�
+ˆa†
+k,− ˆak,+,
+ˆHeff
+cav−mol =∑
+k
+�
+�
+1− f− −2f+
+�
+Jk,+ · µµµ+ ˆσ′†
+k,+ ˆak,+
++Jk,− · µµµ+ ˆσ′†
+k,+ ˆak,−
+�
++
+�
+1− f+ −2f−
+�
+Jk,+ · µµµ− ˆσ′†
+k,− ˆak,+
++Jk,− · µµµ− ˆσ′†
+k,− ˆak,−
+��
++H.c.
+(S14)
+We define states |k,±mol⟩′ and |k,±cav⟩′ corresponding to operators ˆσ′†
+k,± and ˆa†
+k,±, respectively. Writing the Hamiltonian
+ˆHeff(k) = ⟨k| ˆHeff |k⟩ in the above basis we obtain,
+ˆHeff(k) = ˆHeff
+mol(k)+ ˆHeff
+cav(k)+ ˆHeff
+cav−mol(k),
+(S15)
+
+5
+where,
+ˆHeff
+mol(k) =¯hωe |+mol⟩′ ⟨+mol|′ + ¯hωe |−mol⟩′ ⟨−mol|′ ,
+ˆHeff
+cav(k) =
+�
+E0 + ¯h2|k|2
+2m∗ +ζ|k|cosφ
+�
+|+cav⟩′ ⟨+cav|′ +
+�
+E0 + ¯h2|k|2
+2m∗ −ζ|k|cosφ
+�
+|−cav⟩′ ⟨−cav|′
++
+�
+−β0 +β|k|2e−i2φ�
+|+cav⟩′ ⟨−cav|′ +
+�
+−β0 +β|k|2ei2φ�
+|−cav⟩′ ⟨+cav|′ ,
+ˆHeff
+cav−mol(k) =Jk,+ ·
+��
+1− f− −2 f+µµµ+ |+mol⟩′ +
+�
+1− f+ −2 f−µµµ− |−mol⟩′ �
+⟨+cav|′
++Jk,− ·
+��
+1− f− −2 f+µµµ+ |+mol⟩′ +
+�
+1− f+ −2f−µµµ− |−mol⟩′ �
+⟨−cav|′ +H.c.
+(S16)
+S4.
+PARAMETERS
+A.
+Perylene filled cavity
+We take parameters for the perylene filled cavity β0 = 0.1eV, β = 9×10−4eVµm2, ζ = 2.5×10−3eVµm, m∗ = 125¯h2eV−1µm−2,
+and Lz = 0.745µm, where these are similar to those used to model the experiments of Ren et al.[9] (Fig. 3, 4, and 5 in main
+manuscript). On the other hand, we modify E0 and nz such that they make the photon modes in our model near resonant with
+the transition that is strongly coupled to the cavity. For instance, we take E0 = 3.80eV and nz = 11 for porphyrin (Fig. 3 and 4);
+E0 = 2.50eV and nz = 9 for Ce:YAG (Fig. 5b-c); and E0 = 1.80eV and nz = 5 for MoS2 (Fig. 5e-f). We assume that perylene
+has a similar effect on these different photon modes, as it does on modes with E0 ∼ 2.27eV at k = 0 in experiments[9]. This
+may not necessarily be true, however, as we consider a perylene filled cavity only to achieve frequency separation of photon
+modes with different polarization, and this can instead be easily achieved with an electrically tunable liquid crystal cavity [10],
+replacing a perylene filled cavity with a liquid-crystal cavity will not modify the underlying physics of the phenomenon we are
+interested in, i.e., the idea of using saturation to break TRS will remain intact.
+B.
+Porphyrin, Ce:YAG, and monolayer MoS2
+We take areal density ρA = 3.55 × 105µm−2 (∼ 2000 molecules in 75nm ××× 75nm)[11], relative permittivity ε = 1.5[12],
+frequency ¯hωe = 3.8056eV and transition dipole µ0 = 1.1184au × 2.5417D/au = 2.84D [13] for the porphyrin film. Also, we
+consider 100 such porphyrin films stacked one over the other along the z direction within the cavity to achieve strong light-matter
+coupling, Nz = 100. Therefore, the effective areal density of molecules ρ′
+A = NzρA will be used instead of ρA while computing
+Jk,α. These are the parameters used to generate Fig. 3 and 4.
+Similarly, using density ρYAG = 5.11g cm−3, molar mass MYAG = 738 g mol−1, number of Y3+ per unit cell nY3+ = 3, and
+concentration of Ce3+ (relative to Y3+) 1% = 10−2 [14], we obtain the effective areal density of Ce3+ ions in a L′
+z = 0.1µm
+thick layer of Ce:YAG to be ρ′
+A = 10−2L′
+znY3+ρYAGNA/MYAG = 1.25 × 107µm−2. This will be used while computing Jk,α in
+place of ρA. We use relative permittivity ε = 12[15] and frequency ¯hωe = 2.53eV (489nm[16]) for the transition in a Ce:YAG
+crystal. Using the oscillator strength of this transition 0.286[16], we calculate the transition dipole µ0 = 5.46D. These are the
+parameters used to generate Fig. 5c.
+For monolayer MoS2, we consider A-excitons at ¯hωe = 1.855eV[17]. From Chen et al.[17], we take the Rabi splitting at
+resonance, and use µ0√ρA
+�
+¯hωe/2Lzεε0 ≈ 39meV/2 = 19.5meV in our calculations (Fig. 5f).
+SUPPLEMENTARY REFERENCES
+[1] Fabre, C. & Treps, N. Modes and states in quantum optics. Reviews of Modern Physics 92, 035005 (2020).
+[2] Zoubi, H. & La Rocca, G. Microscopic theory of anisotropic organic cavity exciton polaritons. Physical Review B 71, 235316 (2005).
+[3] Martinelli, M. & Martelli, P. Polarization, mirrors, and reciprocity: birefringence and its compensation in optical retracing circuits.
+Advances in Optics and Photonics 9, 129–168 (2017).
+[4] Asb´oth, J. K., Oroszl´any, L. & P´alyi, A. A short course on topological insulators. Lecture notes in physics 919, 166 (2016).
+[5] Silveirinha, M. G. Chern invariants for continuous media. Physical Review B 92, 125153 (2015).
+[6] Fukui, T., Hatsugai, Y. & Suzuki, H. Chern numbers in discretized brillouin zone: efficient method of computing (spin) hall conductances.
+Journal of the Physical Society of Japan 74, 1674–1677 (2005).
+[7] F. Ribeiro, R. et al. Theory for nonlinear spectroscopy of vibrational polaritons. The journal of physical chemistry letters 9, 3766–3771
+(2018).
+
+6
+[8] Fowler-Wright, P., Lovett, B. W. & Keeling, J. Efficient many-body non-markovian dynamics of organic polaritons. Physical Review
+Letters 129, 173001 (2022).
+[9] Ren, J. et al. Nontrivial band geometry in an optically active system. Nature communications 12, 1–8 (2021).
+[10] Rechci´nska, K. et al. Engineering spin-orbit synthetic hamiltonians in liquid-crystal optical cavities. Science 366, 727–730 (2019).
+[11] Hulsken, B. et al. Real-time single-molecule imaging of oxidation catalysis at a liquid–solid interface. Nature nanotechnology 2, 285–289
+(2007).
+[12] Li, D., Swanson, B. I., Robinson, J. M. & Hoffbauer, M. A. Porphyrin based self-assembled monolayer thin films: synthesis and
+characterization. Journal of the American Chemical Society 115, 6975–6980 (1993).
+[13] Sun, S., Gu, B. & Mukamel, S. Polariton ring currents and circular dichroism of mg-porphyrin in a chiral cavity. Chemical Science
+(2022).
+[14] Bachmann, V., Ronda, C. & Meijerink, A. Temperature quenching of yellow ce3+ luminescence in yag: Ce. Chemistry of Materials 21,
+2077–2084 (2009).
+[15] Ctibor, P., Sedl´aˇcek, J. & Hudec, T. Dielectric properties of ce-doped yag coatings produced by two techniques of plasma spraying.
+Bolet´ın de la Sociedad Espa˜nola de Cer´amica y Vidrio (2021).
+[16] Kolesov, R. et al. Mapping spin coherence of a single rare-earth ion in a crystal onto a single photon polarization state. Physical review
+letters 111, 120502 (2013).
+[17] Chen, Y.-J., Cain, J. D., Stanev, T. K., Dravid, V. P. & Stern, N. P. Valley-polarized exciton–polaritons in a monolayer semiconductor.
+Nature Photonics 11, 431–435 (2017).
+

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+Understanding Difficulty-based Sample Weighting with
+a Universal Difficulty Measure⋆
+Xiaoling Zhou1, Ou Wu�1, Weiyao Zhu1, and Ziyang Liang1
+Center for Applied Mathematics, Tianjin University, China.
+{xiaolingzhou,wuou}@tju.edu.cn,
+[email protected], [email protected]
+Abstract. Sample weighting is widely used in deep learning. A large number
+of weighting methods essentially utilize the learning difficulty of training sam-
+ples to calculate their weights. In this study, this scheme is called difficulty-based
+weighting. Two important issues arise when explaining this scheme. First, a uni-
+fied difficulty measure that can be theoretically guaranteed for training samples
+does not exist. The learning difficulties of the samples are determined by multiple
+factors including noise level, imbalance degree, margin, and uncertainty. Never-
+theless, existing measures only consider a single factor or in part, but not in their
+entirety. Second, a comprehensive theoretical explanation is lacking with respect
+to demonstrating why difficulty-based weighting schemes are effective in deep
+learning. In this study, we theoretically prove that the generalization error of a
+sample can be used as a universal difficulty measure. Furthermore, we provide
+formal theoretical justifications on the role of difficulty-based weighting for deep
+learning, consequently revealing its positive influences on both the optimization
+dynamics and generalization performance of deep models, which is instructive to
+existing weighting schemes.
+Keywords: Learning difficulty · Generalization error · Sample weighting · Deep
+learning interpretability.
+1
+Introduction
+Treating each training sample unequally improves the learning performance. Two cues
+are typically considered in designing the weighting schemes of training samples [1].
+The first cue is the application context of learning tasks. In applications such as medical
+diagnosis, samples with high gains/costs are assigned with high weights [2]. The second
+cue is the characteristics of the training data. For example, samples with low-confidence
+or noisy labels are assigned with low weights. Characteristic-aware weighting has at-
+tracted increasing attention owing to its effectiveness and universality [3,4,5].
+Many existing characteristic-aware weighting methods are based on an intrinsic
+property of the training samples, i.e., their learning difficulty. The measures for the
+samples’ learning difficulty can be roughly divided into five categories.
+⋆ This study is supported by NSFC 62076178, TJF 19ZXAZNGX00050, and Zhijiang Fund
+2019KB0AB03.
+Paper published at ECML PKDD 2022
+arXiv:2301.04850v1  [cs.LG]  12 Jan 2023
+
+2
+Xiaoling Zhou et al.
+– Prediction-based measures. This category directly uses the loss [3,6,7] or the pre-
+dicted probability of the ground truth [4,8] as the difficulty measures. This measure
+is simple yet effective and is widely used in various studies [3,4]. Their intention is
+that a large loss (a small probability) indicates a large learning difficulty.
+– Gradient-based measures. This category applies the loss gradient in the measure-
+ment of the samples’ learning difficulty [9,10]. Santiagoa et al. [9] uses the norm
+of the loss gradient as the difficulty measure. Their intuition is that the larger the
+norm of the gradient, the harder the sample.
+– Category proportion-based measures. This category is mainly utilized in imbal-
+anced learning [11], where the category proportion measures the samples’ diffi-
+culty. People believe that the smaller the proportion of a category, the larger the
+learning difficulty of samples in this category [11,12].
+– Margin-based measures. The term “margin” refers to the distance from the sample
+to the oracle classification boundary. The motivation is that the smaller the margin,
+the larger the difficulty of a sample [13].
+– Uncertainty-based measures. This category uses the uncertainty of a sample to mea-
+sure the difficulty. Aguilar et al. [14] identify hard samples based on epistemic un-
+certainty and leverage the Bayesian Neural Network [15] to infer it.
+Varying difficulty measures have a huge impact on a difficulty-based weighting
+strategy. The underlying factors which influence samples’ learning difficulty considered
+in the above measures include noise level [6,7], imbalance degree [11,12], margin [13],
+and uncertainty [14]. However, each measure only considers a single factor or in part,
+and comes from heuristic inspirations but not formal certifications, hindering the appli-
+cation scope of the measures. It is desirable to theoretically explore a universal measure
+capturing all of the above factors. Based on this measure, the role of difficulty-based
+sample weighting can be revealed more concretely. However, neither theoretical nor
+empirical investigations have been conducted to investigate a unified measure.
+Moreover, despite the empirical success of various difficulty-based weighting meth-
+ods, the process of how difficulty-based weighting positively influences the deep learn-
+ing models remains unclear. Two recent studies have attempted to investigate the influ-
+ence of weights in deep learning. Byrd and Lipton [16] empirically studied the train-
+ing of over-parameterized networks with sample weights and found that these sample
+weights affect deep learning by influencing the implicit bias of gradient descent-a novel
+topic in deep learning interpretability, focusing on why over-parameterized models is
+biased toward solutions that generalize well. Existing studies on this topic [17,18,19]
+reveal that the direction of the parameters (for linear predictor) and the normalized mar-
+gin (for nonlinear predictor) respectively converge to those of a max-margin solution.
+Inspired by the finding of Byrd and Lipton [16], Xu et al. [20] dedicated to studying
+how the understandings for the implicit bias of gradient descent adjust to the weighted
+empirical risk minimization (ERM) setting. They concluded that assigning high weights
+to samples with small margins may accelerate optimization. In addition, they estab-
+lished a generalization bound for models that implement learning by using sample
+weights. However, they only discussed the measurement of difficulty by using one of
+the indicators (i.e., margin), resulting in that their conclusion is limited and inaccurate
+in some cases. Furthermore, their generalization bound cannot explicitly explain why
+
+Understanding Difficulty-based Sample Weighting with a Universal Difficulty Measure
+3
+hard samples are assigned with large weights in many studies. More analyses based on
+a universal difficulty measure are in urgent demand.
+In this study, the manner of how the difficulty-based weighting affects the deep
+model training is deeply investigated. First, our analyses support that the generalization
+error of the training sample can be regarded as a universal difficulty measure for captur-
+ing all of the four factors described above. Second, based on this unified measure, we
+characterize the role of difficulty-based weighting on the implicit bias of gradient de-
+scent, especially for the convergence speed. Third, two new generalization bounds are
+constructed to demonstrate the explicit relationship between the sample weights and the
+generalization performance. The two bounds illuminate a new explanation for existing
+weighting strategies. Our study takes the first step of constructing a formal theory for
+difficulty-based sample weighting. In summary, our contributions are threefold.
+– We theoretically prove the high relevance of the generalization error with four main
+factors influencing the samples’ learning difficulty, further indicating that the gen-
+eralization error can be used as a universal difficulty measure.
+– We reveal how the difficulty-based sample weighting influences the optimization
+dynamics and the generalization performance for deep learning. Our results indi-
+cate that assigning high weights on hard samples can not only accelerate the con-
+vergence speed but also enhance the generalization performance.
+– We bring to light the characteristics of a good set of weights from multiple perspec-
+tives to illuminate the deep understanding of numerous weighting strategies.
+2
+Preliminaries
+2.1
+Description of Symbols
+Let X denote the input space and Y a set of classes. We assume that the training and
+test samples are drawn i.i.d according to some distributions Dtr and Dte over X × Y.
+The training set is denoted as T = {x, y} = {(xi, yi)}n
+i=1 that contains n training
+samples, where xi denotes the i-th sample’s feature, and yi is the associated label.
+Let di and w (di) be the learning difficulty and the difficulty-based weight of xi. The
+learning difficulty can be approximated by several values, such as loss, uncertainty and
+generalization error which will be explained in Section 3.
+The predictor is denoted by f (θ, x) and F = {f (θ, ·) |θ ∈ Θ ⊂ Rd}. For the sake
+of notation, we focus on the binary setting yi ∈ {−1, 1} with f (θ, x) ∈ R. The sign
+of the model’s output f (θ, xi) is the predicted label. However, as to be clarified later,
+our results can be easily extended to the multi-class setting where yi ∈ {1, 2, · · · , C}.
+For multi-class setting, the softmax function is used to get the probability, and the log-
+its are given by {fyj (θ, x)}C
+j=1. Given a non-negative loss ℓ and a classifier f (θ, ·),
+the empirical risk can be expressed as L(θ, w) = 1
+n
+�n
+i=1 w (di) · ℓ (yif (θ, xi)). We
+focus particularly on the exponential loss ℓ (u) = exp (−u) and logistic loss ℓ (u) =
+log (1 + exp (−u)). Let ∇l(u) be the loss gradient and f (x|T) is the trained model on
+T. The margin is denoted as γi(T) = yif (θ, xi|T) for the binary setting, where it is
+equivalently denoted as γi(T) = fyi (θ, xi|T) − maxi̸=j fyj (θ, xi|T) for the multi-
+class setting.
+
+4
+Xiaoling Zhou et al.
+2.2
+Definition of the Generalization Error
+Bias-variance tradeoff is a basic theory for the qualitative analysis of the generalization
+error [22]. This tradeoff is initially constructed via regression and mean square error,
+which is given by
+Err = Ex,yET [||y − f(x|T)||2
+2]
+≈ Ex,y[||y − f(x)||2
+2]
+�
+��
+�
+Bias
++ Ex,yET [||f(x|T) − f(x)||2
+2]
+�
+��
+�
+V ariance
+,
+(1)
+where f (x) = ET [f (x|T)]. Similarly, we define the generalization error of a single
+sample xi as
+erri = ET [ℓ (f (xi|T) , yi)] ≈ B (xi) + V (xi) ,
+(2)
+where B (xi) and V (xi) are the bias and variance of xi.
+2.3
+Conditions and Definitions
+Our theoretical analyses rely on the implicit bias of gradient descent. The gradient de-
+scent process is denoted as
+θt+1 (w) = θt (w) − ηt∇L (θt [w(d [t])]) ,
+(3)
+where ηt is the learning rate which can be a constant or step-independent, ∇L (θt [w(d [t])])
+is the gradient of L, and w(d [t]) is the difficulty-based weight of difficulty d at time
+t. The weight may be dynamic with respect to time t if difficulty measures, such as
+loss [3] and predicted probability [4], are used. To guarantee the convergence of the
+gradient descent, two conditions following the most recent study [20] are shown below.
+– The loss ℓ has an exponential tail whose definition is shown in the supplementary
+file. Thus, limu→∞ ℓ(−u) = limu→∞ ∇ℓ(−u) = 0.
+– The predictor f(θ, x) is α-homogeneous such that f(c·θ, x) = cαf(θ, x), ∀c > 0.
+It is easy to verify that losses including the exponential loss, log loss, and cross-entropy
+loss satisfy the first condition. The second condition implies that the activation functions
+are homogeneous such as ReLU and LeakyReLU, and bias terms are disallowed. In
+addition, we need certain regularities from f(θ, x) to ensure the existence of critical
+points and the convergence of gradient descent:
+– For ∀x∈X, f(θ, x) is β-smooth and l-Lipschitz on Rd.
+The third condition is a common technical assumption whose practical implications are
+discussed in the supplementary file.
+The generalization performance of deep learning models is measured by the gener-
+alization error of the test set ˆL (f) [21], defined as
+ˆL (f) = P(x,y)∼Dte[γ(f (x, y)) ≤ 0].
+(4)
+
+Understanding Difficulty-based Sample Weighting with a Universal Difficulty Measure
+5
+0
+50
+100
+150
+200
+0
+1
+2
+3
+4
+5
+Error
+Id
+0
+100
+200
+300
+400
+-2
+0
+2
+4
+6
+8
+10
+12
+14
+16
+18
+Error
+Id
+Noise
+Clean
+1 (Largest)
+2
+3
+4
+5
+6
+7
+8
+9
+10 (Smallest)
+1 (Largest)
+2
+3
+4
+5
+6
+7
+8
+9
+10 (Smallest)
+Fig. 1. (a) Generalization errors of clean and noisy samples on noisy data. The noise ratio is 10%
+(b) Generalization errors of samples in ten categories on imbalanced data. The imbalance ratio is
+10:1. CIFAR10 and ResNet32 are used. Other values of noise ratio and imbalance ratio following
+Ref. [25] are also experimented with and the same conclusions can be obtained.
+2.4
+Experiment Setup
+Demonstrated experiments are performed to support our theoretical analyses. For the
+simulated data, the linear predictor is a regular regression model, and the nonlinear pre-
+dictor is a two-layer MLP with five hidden units and ReLU as the activation function.
+Exponential loss and standard normal initialization are utilized. CIFAR10 [23] is exper-
+imented with, and ResNet32 [24] is adopted as the baseline model. For the imbalanced
+data, the imbalance setting follows Ref. [11]. For the noisy data, uniform and flip label
+noises are used and the noise setting follows Ref. [25]. The models are trained with a
+gradient descent by using 0.1 as the learning rate.
+The model uncertainty is approximated by the predictive variance of five predic-
+tions. To approximate the generalization error, we adopt the five-fold cross-validation [26]
+to calculate the average learning error for each sample.
+3
+A Universal Difficulty Measure
+As previously stated, four factors pointed out by existing studies, namely, noise, imbal-
+ance, margin, and uncertainty, greatly impact the learning difficulty of samples. Nev-
+ertheless, existing measures only consider one or part of them, and their conclusions
+are based on heuristic inspirations and empirical observations. In this section, we theo-
+retically prove that the generalization error of samples is a universal difficulty measure
+reflecting all four factors. All proofs are presented in the supplementary file. Without
+increasing the ambiguity, the generalization error of the samples is termed as error for
+brevity.
+3.1
+Noise Factor
+Noise refers to data that is inaccurate in describing the scene. Numerous studies devoted
+to reducing the influence of noisy samples in the dataset on the deep learning models
+
+6
+Xiaoling Zhou et al.
+and these literature intuitively consider noisy samples as hard ones without formal cer-
+tification [7,27]. The two kinds of noise are feature noise [31] and label noise [27]. We
+offer two propositions to reveal the relationship between the generalization error and
+the noise factor. For feature noise, we offer the following proposition:
+Proposition 1. Let ∆xi be the perturbation of sample (xi, yi), which is extremely
+small in that o(∆xi) can be omitted. Let ∠ϕ be the angle between the direction of
+∆xi and the direction of ET [f ′ (xi|T)]. If ET [f ′ (xi|T) · ∆xi] < 0 (i.e., ∠ϕ > 90◦),
+then the error of the noisy sample is increased relative to the clean one. Alternatively,
+the direction of the perturbation ∆xi and that of ET [f ′ (xi|T)] are contradictory. Oth-
+erwise, if ET [f ′ (xi|T) · ∆xi] > 0, then ∠ϕ < 90◦, and the error of the noisy sample
+is decreased.
+According to Proposition 1, feature noise can be divided into two categories, which
+increase or decrease the learning difficulty (generalization error) of the samples, respec-
+tively. In this paper, noise that increases the error is called the adversarial type, which is
+always used in the field of adversarial learning; otherwise, it is a promoted type, which
+refers to noise that decrease the learning difficulty of samples. Therefore, the variation
+of the error under feature noise is determined by the noise type. For example, as all
+feature noises are adversarial in adversarial learning [32], all of the samples’ errors are
+increased with feature noise. For label noise, we offer the following proposition:
+Proposition 2. Let π be the label corruption rate (i.e., the probability of each label
+flipping to another one). Denote the probability of correct classification for the original
+samples as p. If p > 0.5, then the errors of the noisy samples are larger than those of
+the clean ones.
+This proposition implies that the errors of the samples with label noises are larger
+than those of the clean ones on the average. Specifically, if a sample is more likely to be
+predicted correctly, its generalization error is increased due to label noise. Let L∗ be the
+global optimum of the generalization error of the clean dataset and y′ be the corrupted
+label. When the noise in Proposition 2 is added, the empirical error L′ is
+L′ = (1 − π) L∗ + πL (f (x) , y′) ,
+(5)
+where we have taken expectations over the noise. When π → 0, the noise disappears,
+and the optimal generalization is attained. Proposition 2 is consistent with the empirical
+observation shown in Fig. 1(a), where the noisy samples have larger errors than the
+clean ones on the average.
+3.2
+Imbalance Factor
+Besides noise, imbalance is another common deviation of real world datasets. The cat-
+egory distribution of the samples in the training set is non-uniform. Various methods
+solve this issue by assigning high weights on samples in tail categories which are con-
+sidered to be hard ones [4,11]. Nevertheless, a theoretical justification about why these
+samples are harder lacks. The imbalance ratio is denoted by cr =max{c1, c2, · · · , cC}:
+min{c1, c2, · · · , cC}. Then, we offer the following proposition.
+
+Understanding Difficulty-based Sample Weighting with a Universal Difficulty Measure
+7
+Fig. 2. (a) Correlation between generalization error and average margin. (b) Correlation between
+generalization error and epistemic uncertainty. CIFAR10 and ResNet32 are used in this experi-
+ment. All values are normalized.
+Proposition 3. If a predictor on an imbalanced dataset (cr > e : 1) is an approximate
+Bayesian optimal classifier (as the exponential loss is an approximation for the zero-
+one loss), which is to minimize the total risk, then the average probability of the ground
+truth of the samples in the large category is greater than that of the samples in the small
+category.
+With Proposition 3, it is easy to obtain Proposition A.1 shown in the supplemen-
+tary file that the average error of samples in the small category is larger than that of
+the samples in the large category, indicating there are more hard samples in the small
+category. This proposition is verified by the experiments, as shown in Fig. 1(b). The
+tail categories contain more samples with larger errors. To enhance the performance
+of the classification model, samples with larger errors should be assigned with higher
+weights, as most methods do [11]. Further experiments in Section 5 (Fig. 6) indicate
+that the classification performance of the small category can be improved by increasing
+its sample weights.
+3.3
+Margin Factor
+The samples’ margins measure the distances of the samples from the decision boundary.
+Some literature intuitively consider a small margin indicates a large learning difficulty
+and corresponds to a low confidence of the prediction [33,13]. However, a formal justi-
+fication is lacking. We offer the following proposition.
+Proposition 4. Let µi be the true margin of xi corresponding to the oracle decision
+boundary. The condition is that the functional margins of a sample trained on random
+datasets obey a Gaussian distribution. In other words, for sample xi, its functional
+margin γi obey a Gaussian distribution N(µi, σ2
+i ). For sample xj, γj ∼ N(µj, σ2
+j ).
+
+rwr!
+www
+hEULOL(p)WIDIDMA
+wypV
+ELLOL
+igsMELLOLELLOLbI8
+Xiaoling Zhou et al.
+Fig. 3. The distributions of samples’ margins.
+when the margin variances of the two samples are same (i.e., σ2
+i = σ2
+j ), if µi ≤ µj,
+then erri ≥errj. Similarly, when the true margins of the two samples are the same (i.e.,
+µi =µj), if σ2
+i ≥σ2
+j , then erri ≥errj.
+Proposition 5 indicates a fact that even a sample with a large true margin, as long
+as the margin variance is large, it may also have a high learning difficulty. Specifically,
+the true margin (i.e., the mean of the functional margin distribution) of a sample and
+error are negatively correlated when the margin variances of the samples are equal. By
+contrast, the margin variance and error are positively correlated when the true margins
+are equal. This illumination revises the current wisdom. The conclusion in which sam-
+ples close to the oracle decision boundary are hard ones [20] is not completely correct.
+Indeed, the relation between the margin and error of sample xi conforms with the fol-
+lowing formula:
+erri = ET [e−γi(T )] = e−µi+ 1
+2 σ2
+i ,
+(6)
+where erri, µi, and σi refer to the generalization error, the true margin, and the margin
+variance of sample xi, respectively. For the two samples xi and xj, if µi < µj and
+σ2
+i < σ2
+j , then we cannot judge whether erri is greater than errj. As shown in Fig. 2(a),
+the average margin and error are negatively correlated for most samples, but it is not
+absolute, which accords with the above analyses. Although it is intuitive that the func-
+tional margin trained on random datasets obeys a Gaussian distribution, we evaluate it
+via the Z-scores of the distributions’ Kurtosis and Skewness [34] which is shown in
+Fig 3. More margin distribution curves and all Z-score values of the distributions are
+shown in the supplementary file. As all Z-scores are in [−1.96, 1.96], under the test
+level of α = 0.05, the distribution of margin obeys the Gaussian distribution.
+3.4
+Uncertainty Factor
+Uncertainties [37] in deep learning are classified into two types. The first type is aleatoric
+uncertainty (data uncertainty), which is caused by the noise in the observation data. Its
+correlation with the error has been discussed in Section 3.1. The second type is epis-
+temic uncertainty (model uncertainty). It is used to indicate the consistency of multiple
+predictions. We give the analyses of the relationship between the generalization error
+and epistemic uncertainty.
+Let T be a training set, and let P(θ|T) be the distribution of the training models
+based on T. The predictive variance V ar(f(xi|θ1), · · · , f(xi|θK)) plus a precision
+
+'00000000
+T0000000
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+如-30000000
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+QQQQQQ0E.
+32000000
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+30-
+30-
+如-Understanding Difficulty-based Sample Weighting with a Universal Difficulty Measure
+9
+constant is a typical manner of estimating epistemic uncertainty [35,36]. Take the mean
+square loss as an example1, the epistemic uncertainty is
+�
+Var [xi] :=τ −1 +
+1
+|K|
+�
+k f(xi|θk)⊺f(xi|θk) − E[f(xi|θk)]⊺E[f(xi|θk)],
+(7)
+where τ is a constant. The second term on the right side of Eq. (7) is the second raw mo-
+ment of the predictive distribution and the third term is the square of the first moment.
+When K → ∞ and the constant term is ignored, Eq. (7) becomes
+�
+Var [xi] :=
+�
+θ
+||f(xi|θ) − f(xi)||2
+2dP(θ|T).
+(8)
+If P(θ|T) is approximated by the distribution of learned models on random training sets
+which conform to the Gaussian distribution N(T, δI), Eq. (8) is exactly the variance
+term of the error defined in Eq. (2) when the mean square loss is utilized.
+As the bias term in the error can capture the aleatoric uncertainty and the variance
+term captures the epistemic uncertainty, the overall relationship between uncertainty
+and error is positively correlated. Nevertheless, the relationship between epistemic un-
+certainty and error is not simply positively or negatively correlated. For some samples
+with heavy noises, their epistemic uncertainties will be small as their predictions remain
+erroneous. However, their errors are large due to their large bias. This phenomenon is
+consistent with the experimental results shown in Fig. 2(b). Epistemic uncertainty and
+error are positively correlated for some samples, and the two variables are negatively
+correlated for other samples.
+3.5
+Discussion about Generalization Error
+The commonly used difficulty measures, such as loss [3] and gradient norm [9], are
+mainly related to the bias term. Shin et al. [27] emphasized that only using loss as the
+measurement cannot distinguish clean and noisy samples, especially for uniform la-
+bel noise. There are also a few existing studies that use variance [28,29]. For instance,
+Agarwal et al. [30] applied the variance of gradient norms as the difficulty measure.
+Indeed, both the variance and bias terms should not be underestimated when measur-
+ing the samples’ learning difficulty. Our theoretical analyses support that generalization
+error including both the two terms can capture four main factors influencing the sam-
+ples’ learning difficulty. Thus, the error can be leveraged as a universal measure that
+is more reasonable than existing measures. Existing studies generally apply the K-fold
+cross-validation method [26] to calculate the generalization error. More efficient error
+calculation algorithms are supposed to be proposed which will be our future work.
+4
+Role of Difficulty-Based Weighting
+This section aims to solve the second issue of explaining the difficulty-based weighting
+in deep learning. Based on the universal difficulty measure, the impacts of the difficulty-
+based weighting schemes on the optimization dynamics and the generalization perfor-
+mance in deep learning are investigated. Compared with the most recent conclusions
+1 For other losses, other methods can be used to calculate the predictive variance [26].
+
+10
+Xiaoling Zhou et al.
+Fig. 4. “Cosine distance” represents the cosine of the angle between the decision boundary (at
+that epoch) and the max-margin solution. (a), (b) Cosine distance and average margin of equal
+weights and inverse margin weights using the linear predictor. (c), (d) Cosine distance and average
+margin of equal weights and inverse margin weights using the nonlinear predictor. (e), (f) Cosine
+distance and average margin of equal weights and increasing weights of noisy samples using
+the nonlinear predictor on the noisy data. (g), (h) Cosine distance and average margin of equal
+weights and increasing weights of samples in tail categories using the linear predictor on the
+imbalanced data. More results are placed in the supplementary file.
+[20] established only on the margin factor, our theoretical findings, which are based on
+our universal measure, are more applicable and precise.
+4.1
+Effects on Optimization Dynamics
+Linear Predictor We begin with the linear predictors allowing for a more refined
+analysis. Xu et al. [20] inferred an upper bound containing the term DKL(p∥w), where
+DKL is the Kullback-Leibler divergence and p is the optimal dual coefficient vector. A
+smaller value of DKL(p∥w) means that the convergence may be accelerated. There-
+fore, to accelerate the convergence, they believe that the weights w should be consistent
+with the coefficients p. Alternatively, the samples with small functional margins will
+have large coefficients and thus should be assigned with large weights. However, the
+functional margin is not the true margin that corresponds to the oracle boundary. There-
+fore, their conclusion that samples close to the oracle classification boundary should be
+assigned with large weights [20] cannot be well-drawn according to their inference. We
+offer a more precise conclusion with the unified difficulty measure (i.e., generalization
+error). As before, we assume that the functional margins of a sample xi obey a Gaus-
+sian distribution N(µi, σ2
+i ), where µi is the true margin and σ2
+i is the margin variance
+of xi. We offer the following proposition:
+Proposition 5. For two samples xi and xj, if erri ≥ errj, then we have:
+(1) When the optimal dual coefficient pi of xi on a random training set T is a linear
+function of its functional margin γi on T, if µi ≤ µj, then ET [pi] ≥ ET [pj] (i.e.,
+ET [wi] ≥ ET [wj]);
+
+S(p)Ebocj0'5 -
+0°4 -
+0.0
+8.0
+I'O -batdgisw Isupg
+0
+500
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+028.0
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+28.0
+ .0
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+2nigisM(Ol:) batdgiaw sl baoslsdmi
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+000.1
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+10
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+400
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+batdgigw Isupe
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+Ebocp
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+400
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+2nigisMUnderstanding Difficulty-based Sample Weighting with a Universal Difficulty Measure
+11
+Fig. 5. (a)-(c) Normalized margin of increasing the weights of noisy samples/samples with small
+margins/samples in tail categories. CIFAR10 data is used. Uniform label noise is adopted. The
+noise ratio and imbalance ratio are 10% and 10:1. (d) Generalization error of the test set when
+the nonlinear model is trained with different weights on simulated imbalanced data with the
+imbalance ratio as 10:1. Other noise and imbalance settings are also experimented with and the
+same conclusions can be obtained.
+(2) When the optimal dual coefficient pi of xi on a random training set T is a
+natural exponential function of its functional margin γi on T, ET [pi] ≥ ET [pj] (i.e.,
+ET [wi] ≥ ET [wj]) always holds. Notably, even when µi > µj, ET [pi] > ET [pj] may
+still hold.
+The proof is presented in the supplementary file. ET [pi] > ET [pj] implies that
+wi > wj holds on the average. The conclusion that samples with small true margins
+should be assigned with large weights may not hold on some training sets when pi is
+not a linear function of γi [17]. A sample with a small true margin may have a smaller
+weight than a sample with a large true margin yet a large error. Thus, a more general
+conclusion when pi is not a linear function of γi is that increasing the weights of hard
+samples (samples with large generalization errors) may accelerate the convergence,
+rather than just for samples with small margins. Other factors, including noise, imbal-
+ance, and uncertainty also affect samples’ learning difficulty. Notably, the weights of
+the hard samples should not be excessively increased, as to be explained in the succeed-
+ing section. We reasonably increase the weights of the hard samples shown in Figs. 4
+and A-3 in the supplementary file indicating that the optimization is accelerated.
+We also prove that difficulty-based weights do not change the convergence direction
+to the max-margin solution shown in Theorem A.1 in the supplementary file. As shown
+in Fig. 3, the cosine distance and margin value are always increasing during the training
+procedure, indicating the direction of the asymptotic margin is the max-margin solution.
+Nonlinear Predictor Analyzing the gradient dynamics of the nonlinear predictors is
+insurmountable. The main conclusion obtained by Xu et al. [20] can also be established
+for difficulty-based weights only if the bound of weights is larger than zero. However,
+their theorem has only been proven for binary cases as the employed loss is inapplicable
+in multi-class cases. Here, we extend the theory to the multi-class setting with a regu-
+larization λ||θ||r on the cross-entropy loss. Let θλ (w)∈arg min Lλ (θ, w). Formally,
+the dynamic regime for the nonlinear predictor can be described as follows:
+Theorem 1. Let w ∈ [b, B]n. Denote the optimal normalized margin as
+γ∗ =
+max
+∥θ(w)∥≤1 min
+i (fyi(θ(w), xi) − max
+j̸=i (fyj(θ(w), xi)))
+(9)
+
+Ebocp
+0
+500
+400
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+800
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+5.0
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+0
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+0
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+500
+-
+2.0
+0.1
+2.1
+0.5
+01:1 = gigw
+2:I = dgigw
+52
+Ismrronl
+2 TEbocpEbocp
+0
+J00
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+300
+400
+200
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+0
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+20
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+0.0
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+04
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+500
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+04
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+8.0
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+IpgJSUcG
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+0
+5O
+40
+80
+J00
+JSO
+J40
+2.0
+a.0
+VOSIUOA
+『.0
+WM
+8.0
+ e.0Ebocp
+0
+40
+80
+J00
+JSO
+J40
+500.0
+0°004
+00.0
+2201
+800.0
+010.0
+O'OJS
+0'014Ebocp
+0
+SO
+40
+eo
+80
+J00
+JSO
+J40
+0.0
+5.0
+nigisM
+04
+0.0
+8.0
+0.1EbocpEbocp
+0
+52
+02
+J00
+s
+J20
+r
+500
+2.0
+0.I
+ro22
+2.1
+0.5
+1:I = dgiow
+: = 
+2.5
+JOLJ
+2 TEbocJ
+0
+500
+400
+Q00
+800
+J000
+0.0
+CE
+5.0
+nigisM
+04
+a.0
+8.0
+0.IEbocp
+0
+52
+20
+J00
+cr1
+500
+0.0
+O'S
+04
+nigisM
+0.0
+8.0
+CE
+0.112
+Xiaoling Zhou et al.
+Epoch1     
+Epoch10     
+Epoch20     
+Epoch1     
+Epoch30     
+Epoch50     
+Epoch80     
+Epoch100     
+Epoch1     
+Epoch10     
+Epoch20     
+Epoch30     
+Epoch50     
+Epoch80     
+Epoch100     
+Fig. 6. Top: Equal weights of the two categories. Bottom: Samples in the small category are
+assigned with high weights, obtaining better performance for the small (red) category. The im-
+balance ratio is set to 10:1. The same conclusions can also be obtained for other imbalance ratios.
+Let θλ(w) = θλ(w)/∥θλ(w)∥. Then, it holds that (1) Denote the normalized margin
+as
+γλ(w)=min
+i (fyi(θλ (w) , xi)−max
+j̸=i fyj(θλ (w) , xi))
+(10)
+Then, γλ (w)→γ∗, as λ → 0.
+(2) There exists a λ := λ (r, a, γ∗, w). For α≤2, let θ′(w) denote a α-approximate
+minimizer of Lλ. Thus, Lλ
+�
+θ′ (w)
+�
+≤ αLλ (θλ (w)). Denote the normalized margin
+of θ′(w) by γ′ (w). Then,γ′ (w) ≥
+γ∗
+10αa/r .
+The proof is presented in the supplementary file. When λ is sufficiently small, the
+difficulty-based weighting does not affect the asymptotic margin. According to Theo-
+rem 2, the weights do affect the convergence speed. A good property is that even though
+Lλ (θλ (w)) has not yet converged but close enough to its optimum, the corresponding
+normalized margin has a reasonable lower bound. A good set of weights can help the
+deep learning model to achieve this property faster. However, the conditions in which a
+set of weights can accelerate the speed are not clearly illuminated. Notably, as shown in
+our experiments in Figs. 4 and A-3 in the supplementary file, assigning large weights for
+hard samples increases the convergence speed. The results on the multi-class cases (CI-
+FAR10) indicate that assigning large weights on hard samples increases the margin, as
+shown in Figs. 5(a-c). However, some particular occasions of difficulty-based weights,
+such as SPL [3], do not satisfy the bounding condition because the lower bounds of
+these weights are zero instead of a positive real number. The theorem requires further
+revision to accommodate this situation.
+4.2
+Effects on Generalization Performance
+Besides the role of difficulty-based weights on optimization dynamics, we are also con-
+cerned as to whether and how the difficulty-based weights affect the generalization
+performance. The generalization bound of Xu et al. [20] does not contain the sample
+weights, thus it cannot explicitly explain why hard samples are assigned with large
+weights. In addition, they assume that the source and target distributions are unequal,
+restricting the application of their conclusion. The two generalization bounds we pro-
+pose offer good solutions to these issues. They illuminate how a weighting strategies
+can be designed.
+
+Understanding Difficulty-based Sample Weighting with a Universal Difficulty Measure
+13
+Let Ps and Pt be the source (training) and target (testing) distributions, respectively,
+with the corresponding densities of ps(·) and pt(·). Assume that the two distributions
+have the same support. The training and test samples are drawn i.i.d according to dis-
+tributions Ps and Pt, respectively. Learning with sample weights w(x) is equivalent
+to learning with a new training distribution �Ps. The density of the distribution of the
+weighted training set �Ps is denoted as �ps(x) and �ps(x) ∼ w(x)ps(x). Pearson χ2-
+divergence is used to measure the difference between �Ps and Pt, i.e., Dχ2(Pt∥ �Ps) =
+�
+[(d �Ps/dPt)2−1]d �Ps. We consider depth-q (q ≥ 2) networks with the activation func-
+tion φ. The binary setting is considered, in that the network computes a real value
+f (x) := W qφ (W q−1φ (· · · φ (W 1x) · · · )) ,
+(11)
+where φ(·) is the element-wise activation function (e.g., ReLU). The training set con-
+tains n samples. Denote the generalization error for a network f as ˆL(f). The general-
+ization performance of f with weights can be described as follows.
+Theorem 2. Suppose φ is 1-Lipschitz and 1-positive-homogeneous. With a probability
+at least of 1 − δ, we have
+ˆL (f) ≤ 1
+n
+n
+�
+i=1
+pt(xi)
+�ps(xi)1(yif(xi) < γ)
+�
+��
+�
+I
++
+L ·
+�
+Dχ2
+�
+Pt∥ �Ps
+�
++ 1
+γ · q(q−1)/2√n
+�
+��
+�
+(II)
++ ϵ(γ, n, δ)
+�
+��
+�
+(III)
+,
+(12)
+where ϵ(γ, n, δ) =
+�
+log log2
+4L
+γ
+n
++
+�
+log(1/δ)
+n
+and L:=supx ∥x∥.
+The proof is presented in the supplementary file. Compared with the findings of Xu et
+al. [20], the bound of the generalization error is directly related to the sample weights
+w(x) contained in �ps(x). In view of reducing the generalization error, a natural opti-
+mization strategy can be implemented as follows: 1) an optimal weight set w(x) (in
+�ps(x)) is obtained according to decreasing the right side of Eq. (12) based on the cur-
+rent f; 2) f is then optimized under the new optimal weights w(x). In the first step,
+the reduction of generalization error can come from two aspects. One is to increase
+the weights of samples with small margins. The other is to make the test and training
+distributions close. Disappointingly, this strategy heavily relies on the current f which
+is unstable. Given a fixed training set, f depends on random variables (denoted as V)
+such as hyperparameters and initialization. To obtain a more stable weighting strategy,
+we further propose the following proposition.
+Proposition 6. Suppose φ is 1-Lipschitz and 1-positive-homogeneous. With a proba-
+bility of at least 1 − δ, we have
+EV[ ˆL (fV)] ≤ 1
+n
+n
+�
+i=1
+pt(xi)
+�ps(xi)EV[1(yifV(xi) < γ)]
+�
+��
+�
+(I)
++
+L ·
+�
+Dχ2
+�
+Pt∥ �Ps
+�
++ 1
+γ · q(q−1)/2√n
+�
+��
+�
+(II)
++(III)
+(13)
+
+14
+Xiaoling Zhou et al.
+Accordingly, increasing the �ps(xi) of the samples with large EV[1(yifV(xi) < γ)]
+will reduce (I). In fact, samples with larger generalization errors will have larger values
+of EV[1(yifV(xi) < γ)]. The proof is placed in the supplementary file. Alternatively,
+increasing the weights of the hard samples will reduce (I). However, the weights of the
+hard samples cannot be increased arbitrarily as Dχ2(Pt∥ �Ps) may be large. Therefore, a
+tradeoff between (I) and (II) should be attained to obtain a good set of weights. Alterna-
+tively, a good set of weights should increase the weights of hard samples while ensuring
+that the distributions of the training set and the test set are close.
+It is worth mentioning that our two above conclusions are still insightful when Pt =
+Ps while the conclusion of Xu et al. [20] assumes Pt ̸= Ps. Apparently, even when
+Pt =Ps, assigning weights according to the samples’ difficulties is still beneficial as the
+tradeoff between (I) and (II) still takes effect.
+5
+Discussion
+Our theoretical analyses in Sections 3 and 4 provide answers to the two concerns de-
+scribed in Section 1.
+First, the generalization error has been theoretically guaranteed as a generic diffi-
+culty measure. It is highly related to noise level, imbalance degree, margin, and uncer-
+tainty. Consequently, two directions are worth further investigating. The first direction
+pertains to investigating a more efficient and effective estimation method for the gener-
+alization error, enhancing its practicality. This will be our future work. As for the second
+direction, numerous existing and new weighting schemes can be improved or proposed
+using the generalization error as the difficulty measure. Our theoretical findings sup-
+plement or even correct the current understanding. For example, samples with large
+margins may also be hard-to-classify in some cases (e.g., with heterogeneous samples
+in their neighbors).
+Second, the existing conclusions on convergence speed have been extended. For
+the linear predictors, the existing conclusion is extended by considering our difficulty
+measure, namely, the generalization error. For the nonlinear predictors, the conclusion
+is extended into the multi-class cases. Furthermore, the explicit relationship between
+the generalization gap and sample weights has been established. Our theorem indicates
+that assigning large weights on the hard samples may be more effective even when the
+source distribution Ps and target distribution Pt are equal.
+Our theoretical findings of the generalization bounds provide better explanations to
+existing weighting schemes. For example, if heavy noise exists in the dataset, then the
+weights of the noisy samples should be decreased. As noisy samples are absent in the
+target distribution (i.e., pt(xi) = 0), the weights of the noisy samples in a data set with
+heavy noise should be decreased to better match the source and target distributions. The
+experiments on the noisy data are shown in Fig. A-5 in which decreasing the weights
+of noisy samples obtain the best performance. In imbalanced learning, samples in small
+categories have higher errors on the average. Increasing the weights of the hard samples
+will not only accelerate the optimization but also improve the performance on the tail
+categories, as shown in Figs. 5(d) and 6. These high-level intuitions justify a number
+of difficulty-based weighting methods. Easy-first schemes, such as Superloss [7] and
+
+Understanding Difficulty-based Sample Weighting with a Universal Difficulty Measure
+15
+Truncated loss [6], perform well on noisy data. Hard-first schemes, such as G-RW [12]
+and Focal Loss [4], are more suitable for imbalanced data.
+6
+Conclusion
+This study theoretically investigates difficulty-based sample weighting. First, the gen-
+eralization error is verified as a universal measure as a means of reflecting the four main
+factors influencing the learning difficulty of samples. Second, based on a universal dif-
+ficulty measure, the role of the difficulty-based weighting strategy for deep learning is
+characterized in terms of convergence dynamics and the generalization bound. Theoret-
+ical findings are also presented. Increasing the weights of the hard samples may accel-
+erate the optimization. A good set of weights should balance the tradeoff between the
+assigning of large weights on the hard samples (heavy training noises are absent) and
+keeping the test and the weighted training distributions close. These aspects enlighten
+the understanding and design of existing and future weighting schemes.
+References
+1. Zhou, X., Wu, O.: Which Samples Should be Learned First: Easy or Hard?. arXiv preprint
+arXiv:2110.05481 (2021)
+2. Khan, S.-H., Hayat, M., Bennamoun, M., Sohel, F.-A., Togneri, R.: Cost-sensitive learning of
+deep feature representations from imbalanced data. IEEE Transactions on Neural Networks
+and Learning Systems 29(8), 3573–3587 (2018)
+3. Kuma, M.-P., Packer, B., Koller, D.: Self-paced learning for latent variable models. In:
+NeurIPS, pp. 1–9 (2010)
+4. Lin, T.-Y., Goyal, P., Girshick, R., He, K., Dollar, P.: Focal Loss for Dense Object Detection.
+IEEE Transactions on Pattern Analysis and Machine Intelligence 42(2), 318–327 (2020)
+5. Bengio, Y., Louradour, J., et al.: Curriculum learning. In: ICML, pp. 41–48 (2009)
+6. Wang, W., Feng, F., He, X., Nie, L., Chua, T.-S.: Denoising Implicit Feedback for Recom-
+mendation. In: WSDM, pp. 373–381 (2021)
+7. Castells, T., Weinzaepfel, P., Revaud, J.: SuperLoss: A generic loss for robust curriculum
+learning. In: NeurIPS, pp. 1–12 (2020)
+8. Emanuel B.-B., Tal R., Nadav Z., Asaf N., Itamar F., Matan P., Lihi Z.-M.: Asymmetric Loss
+For Multi-Label Classification. arXiv preprint arXiv:2009.14119 (2020)
+9. Santiago, C., Barata, C., Sasdelli, M., et al.: LOW: Training deep neural networks by learning
+optimal sample weights. Pattern Recognition 110(1), 107585 (2021)
+10. Li, B., Liu, Y., Wang, X.: Gradient Harmonized Single-stage Detector. In: AAAI, pp. 8577–
+8584 (2019)
+11. Cui, Y., Jia, M., Lin, T.-Y., Song, Y., Belongie, S.: Class-Balanced Loss Based on Effective
+Number of Samples. In: CVPR, pp. 9260–9269 (2019)
+12. Zhang, S., Li, Z., Yan, S., He, X., Sun, J.: Distribution Alignment: A Unified Framework for
+Long-tail Visual Recognition. In: CVPR, pp. 2361–2370 (2021)
+13. Zhang, J., Zhu, J., Niu, G., Han, B., Sugiyama, M., Kankanhalli, M.: Geometry-aware
+Instance-reweighted Adversarial Training. In: ICLR, pp. 1–29 (2021)
+14. Aguilar, E., Nagarajan, B., Khatun, R., Bola˜nos, M., Radeva, P.: Uncertainty modeling and
+deep learning applied to food image analysis. In: ICBM, pp. 3–16 (2020)
+15. Xiao, Y., Wang, W.-Y. Quantifying uncertainties in natural language processing tasks. In:
+AAAI, pp. 7322–7329 (2019)
+
+16
+Xiaoling Zhou et al.
+16. Byrd, J., Lipton, Z.-C.: What is the effect of Importance Weighting in Deep Learning?. In:
+ICML, pp. 1405–1419 (2019)
+17. Soudry, D., Hoffer, E., Nacson, M.-S., Gunasekar, S., Srebro, N.: The implicit bias of gradi-
+ent descent on separable data. Journal of Machine Learning Research 19(1), 1–14 (2018)
+18. Chizat, L., Bach, F.: Implicit bias of gradient descent for wide two-layer neural networks
+trained with the logistic loss. arXiv preprint arXiv:2002.04486 (2020)
+19. Lyu, K., Li, J.: Gradient Descent Maximizes the Margin of Homogeneous Neural Networks.
+arXiv preprint arXiv:1906.05890 (2019)
+20. Xu, D., Ye, Y., Ruan, C.: Understanding the role of importance weighting for deep learning.
+In: ICLR, pp. 1–20 (2020)
+21. Goodfellow, I., Bengio, Y., Courville, A.: Deep learning (2016)
+22. Heskes, T.: Bias/Variance Decompositions for Likelihood-Based Estimators. Neural Com-
+putation 10(6), 1425–1433 (1998)
+23. Alex, K., Hinton, G.: Learning multiple layers of features from tiny images. Technical report
+(2009)
+24. He, K., Zhang, X., Ren S., Sun, J.: Deep Residual Learning for Image Recognition. In:
+CVPR, pp. 770–778 (2016)
+25. Shu, J., Xie, Q., Yi, L., Zhao, Q., Zhou, S., Xu, Z., Meng, D.: Meta-weight-net: Learning an
+explicit mapping for sample weighting. In: NeurIPS, pp. 1–23 (2019)
+26. Yang, Z., Yu, Y., You, C., Jacob, S., Yi, M.: Rethinking bias-variance trade-off for general-
+ization of neural networks. In: ICML, pp. 10767–10777 (2020)
+27. Shin, W., Ha, J.-W., Li S., Cho, Y., et al.: Which Strategies Matter for Noisy Label Classifi-
+cation? Insight into Loss and Uncertainty. arXiv preprint arXiv:2008.06218 (2020)
+28. Chang, H.-S., Erik, L.-M., McCallum A.: Active bias: Training more accurate neural net-
+works by emphasizing high variance samples. In: NeurIPS, pp. 1003–1013 (2017)
+29. Swayamdipta, S., Schwartz, R., Lourie, N., Wang, Y., Hajishirzi, H., Smith, N.-A., Choi, Y.:
+Dataset cartography: Mapping and diagnosing datasets with training dynamics. arXiv preprint
+arXiv:2009.10795 (2020)
+30. Agarwal, C., Hooker, S.: Estimating example difficulty using variance of gradients. arXiv
+preprint arXiv:2008.11600 (2020)
+31. Wolterink, J.-M., Leiner, T., et al.: Generative Adversarial Networks for Noise Reduction in
+Low-Dose CT. IEEE Transactions on Medical Imaging 36(12), 2536–2545 (2017)
+32. Lowd, D., Meek, C.: Adversarial learning. In: SIGKDD, pp. 641–647 (2005)
+33. Elsayed, G.-F., Krishnan, D., Mobahi, H., Regan, K., Bengio, S.: Large margin deep net-
+works for classification. In: NeurIPS, pp. 850–860 (2018)
+34. Ghasemi, A., Zahediasl, S.: Normality tests for statistical analysis: a guide for non-
+statisticians. International journal of endocrinology and metabolism 10(2), 486–489 (2012)
+35. Gal, Y., Ghahramani, Z.: Dropout as a bayesian approximation: Representing model uncer-
+tainty in deep learning. In: ICML, pp. 1050–1059 (2016)
+36. Abdar, M., Pourpanah, F., Hussain, S., Rezazadegan, D., Liu, L., Ghavamzadeh, M., Fieguth,
+P., Cao, X., Khosravi, A., Acharya, U.-R., Makarenkov, V., Nahavandi, S.: A review of uncer-
+tainty quantification in deep learning: Techniques, applications and challenges. Information
+Fusion 76(1), 243–297 (2021)
+37. Kendall, A., Gal, Y.: What Uncertainties Do We Need in Bayesian Deep Learning for Com-
+puter Vision?. In: NeurIPS, pp. 5575–5585 (2017)
+

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+arXiv:2301.02629v1  [math.AG]  31 Oct 2022
+Intersection theory on non-archimedean analytic spaces
+Yulin Cai
+January 9, 2023
+Abstract
+We develop the intersection theory of non-archimedean analytic spaces and prove the pro-
+jection formula and the GAGA principle. As an application, we naturally define the category
+of finite correspondences of analytic spaces.
+Contents
+1
+Introduction
+1
+2
+Preliminary
+3
+3
+Meromorphic functions and Cartier divisors
+7
+4
+Cycles, flat pull-backs and proper push-forwards
+11
+5
+Proper intersection and intersection multiplicities
+19
+6
+Projection formula
+22
+7
+GAGA
+23
+8
+The category of finite correspondences
+25
+Acknowledgements
+27
+References
+27
+1
+Introduction
+The intersection theory of non-archimedean analytic spaces has been studied in [11, Section 2] and
+[1, Section 2.2], and the author believes that some experts have concrete idea about such a theory.
+In [11], Gubler considers the Cartier divisors on rigid analytic spaces and formal schemes, and
+define their intersection with irreducible analytic subsets. This theory allows him to define the
+local height of subvarieties over non-archimedean fields.
+In [1], Ayoub develops the theory of motives on rigid analytic spaces using homotopy theory.
+He uses the presheaves on the category of affinoid spaces to construct the category of finite corre-
+spondence (for rigid analytic space) RigCor(K). Such construction avoids the intersection theory
+of analytic spaces.
+In this paper, we will develop the intersection theory of non-archimedean analytic spaces follow-
+ing the idea similar to the case of algebraic varieties. We will show the flat base change formula, the
+projection formula and the GAGA principle to relate the intersection theories of analytic spaces
+and of algebraic varieties. As an application, we will give a direct construction of RigCor(K)
+(simply denoted by CorK in this paper) like [13, Lecture 1] does. In fact, we can define the higher
+Chow groups of analytic spaces as [4] for algebraic varieties, and this definition is different from
+Ayoub’s in [1, Introduction g´en´erale].
+In Section 2, we give some basic notion in the theory of Berkovich spaces, e.g. support of
+a coherent sheaf, Zariski image and codimension. We also extend [7, Proposition 4.12] into an
+abstract form, i.e. Lemma 2.15 which is a key lemma for this paper. With this lemma, we can
+solve the compatibility problems in our theory, e.g. see Lemma 4.6 and Lemma 5.4.
+1
+
+In Section 3, we define and study the Cartier divisors on an analytic space X, which form
+a group Div(X). The group of divisors up to linear equivalence is denoted by CaCl(X). As in
+the theory of schemes, we have an injective homomorphism CaCl(X) ֒→ Pic(X), and it is an
+isomorphism if X is reduced.
+In Section 4, we give the notion of cycles, and associate a coherent sheaf with a cycle. In
+particular, we can associate a closed subspace with a cycle. As in the theory of algebraic varieties,
+the flat pull-backs and proper push-forwards of cycles are defined. We prove the following flat base
+change formula.
+Proposition 1.1 (Proposition 4.28). Let
+Y ′
+g′
+�
+f ′
+�
+Y
+f
+�
+X′
+g
+� X
+be a Cartesian diagram of separated, strictly K-analytic spaces with f proper and g flat. Then f ′
+is proper, g′ is flat and g∗ ◦ f∗ = f ′
+∗ ◦ g′∗ on Z∗(Y ).
+In Section 5, we define intersection product of proper intersection. We will give two definitions,
+meaning a local one using the scheme theory and a global using Tor formula. For a flat morphism
+f : Y → X of K-analytic spaces of pure dimension, the pull-back f ∗ : Z∗(X) → Z∗(Y ) preserves
+intersection product.
+Since we have the flat pull-backs, proper push-forwards and intersection products, the expected
+projection formula is proved in Section 6.
+Theorem 1.2 (Projection formula). Let f : Y → X be a flat, proper morphism of regular,
+separated, strictly K-analytic spaces. Let α ∈ Z∗(Y ) and β ∈ Z∗(X). Assume that α and f ∗β
+intersect properly. Then f∗(α) and β intersect properly and
+f∗(α) · β = f∗(α · f ∗β).
+In Section 7, we compare the intersection theories of algebraic varieties and of non-archimedean
+analytic spaces. We prove the GAGA principle, i.e. Proposition 7.3.
+In Section 8, we define the category of finite correspondence CorK. This category is also defined
+by Ayoub [1] using another definition.
+Notation and terminology
+Throughout this paper, we fix a complete non-archimedean field K with a non-trivial valuation.
+For a K-analytic space, we mean a Berkovich space over K, see [3, Definition 1.2.3]. The structure
+sheaf on a K-analytic space X with respect to the G-topology is denoted by OX. If it is necessary,
+we will use the notation XG for the G-topology instead of the ordinary topology on X. The (K-
+analytic) dimension of X is denoted by dimK X, or dim X when there is no confusion with the
+fields.
+Given a point x ∈ X, H (x) denotes its complete residue field and dimx X denotes the local
+dimension of X at x.
+We shall simply say ”coherent sheaf on X” for ”coherent OX-module (with respect to G-
+topology)”, and denote Pic(X) for the group of invertible sheaves on X. Assume that X is good,
+let F be a coherent sheaf on X and x ∈ X. We denote by Fx the stalk at x of F viewed as a sheaf
+of the underlying ordinary topology of X, i.e.
+Fx := lim
+−→
+U
+F(U) = lim
+−→
+V
+F(V ).
+where U runs through open neighborhoods of x, and V runs through affinoid neighborhoods of x.
+We will write Irr(X) for the set of all irreducible components of X, and write Irr(X) for the
+set of all irreducible Zariski-closed subsets of X. Notice that Irr(X) has a partial order: W ≤ Z if
+W ⊂ Z.
+2
+
+For an algebraic variety over K, we mean a separated scheme of finite type over K.
+For a commutative ring A, R(A) denotes the set of all regular elements of A and Frac(A) =
+R(A)−1A, the maximal localization containing A as a subring.
+2
+Preliminary
+For the convenience of the reader and further uses, in the section, we provide some basic concepts
+and results that are either given somewhere, or formulated easily.
+2.1
+Support of a coherent sheaf
+(cf. [8, Section 2.5])
+Definition 2.1. Let X be a K-analytic space, F be a coherent sheaf on X, and Ann(F) be the
+(coherent) annihilator ideal of F (on the site XG).
+The support of F is the closed analytic
+subspace of X defined by Ann(F), denoted by Supp(F).
+Remark 2.2.
+(1) Recall the annihilator I of F is defined as follows: for any analytic domain
+V ,
+Ann(F)(V ) := {a ∈ OX(V ) | a · F(V ) = 0},
+which is a coherent ideal. In particular, for any analytic domain V , we have Ann(F)|V =
+Ann(F|V ).
+(2) If X = M(A) is affinoid and F = �
+M for some finitely generated A-module, then it is easy
+to see that
+Ann(F) =
+�
+Ann(M).
+From the definition, we can easy deduce the following lemma.
+Lemma 2.3. Let X be a K-analytic space, F a coherent sheaf on X, and Z = Supp(F). Then
+there is a unique coherent sheaf G on Z such that F = i∗G, where i : Z ֒→ X is the canonical
+immersion.
+Proof. By uniqueness, we can glue coherent sheaf G from local parts, so we can assume that
+X = M(A). It is not hard to see the lemma in this case.
+2.2
+Zariski image of a morphism
+As in the theory of schemes, we can define Zariski image of a morphism of analytic spaces, which
+has a natural structure of analytic spaces. We follow the idea in [14, Subsection 29.6].
+Lemma 2.4. Let X be a K-analytic space, F a coherent sheaf on X, and G ⊂ F an OX-submodule.
+Then there is a unique coherent OX-submodule G′ ⊂ G with the following property: for any coherent
+OX-module H, the canonical map
+HomOX(H, G′) → HomOX(H, G)
+is bijective. In particular, G′ is the largest coherent sheaf contained in G.
+Proof. Let {Gi}i∈I be the set of coherent sheaves contained in G. We consider the morphism of
+OX-modules
+ϕ :
+�
+i∈I
+Gi → F.
+We claim its image G′ ⊂ G is coherent. Let pG′ ⊂ G be the image of ϕ as presheaves. Then G′ is
+the sheafification of pG′, and for any affinoid domain V = M(V ), pG′(V ) = �
+i
+Gi(V ) ⊂ F(V ) is a
+finitely generated A-module. By Tate acyclic theorem, we have G′(V ) = pG′(V ). So G′ is coherent.
+It is the largest coherent sheaf contained in G.
+3
+
+The map
+HomOX(H, G′) → HomOX(H, G)
+is obviously injective. For any homomorphism ψ : H → G ⊂ F, the image Im(ψ) ⊂ G is a coherent
+sheaf, so Im(ψ) ⊂ G′, so f factor thorough G′. This implies that G′ is the one we want.
+For the uniqueness, if G′′ is another coherent OX-submodule with the universal property. Then
+the bijectivity of HomOX(G′, G′′) → HomOX(G′, G) implies that we have a homomorphism G′ →
+G′′ ⊂ G, so G′ ⊂ G′′. Hence G′ = G′′.
+Proposition 2.5. Let f : Y → X be a morphism of K-analytic spaces. Then there is a closed
+analytic subspace Z of X such that
+(a) the morphism f factors through Z;
+(b) (Universal property) if f factors through a closed analytic subspace Z′ of X, then Z′ contains
+Z as a closed analytic subspace.
+The closed analytic space Z of X is called the Zariski image of f, denoted by Imzar(f).
+Proof. By (b), if Z exists, then it is unique. It remains to show the existence. Let I := Ker(OY →
+f∗OX). By Lemma 2.4, we take the largest coherent OX-submodule J ⊂ I and set Z = V (J ). It
+remains to check (a) and (b).
+(a) We have f(Y ) ⊂ Z. Indeed, for any affinoid domain V = M(A) ⊂ X and any affinoid
+domain U = M(B) ⊂ f −1(V ), we have J (V ) ⊂ I(V ) ⊂ Ker(A → B), so U → V factors through
+M(A/J (V )) = Z ∩V and f(U) ⊂ Z. Hence f(Y ) ⊂ Z. We denote the map Y → Z by f. We shall
+construct f
+# : OZ(V ∩ Z) → OX(f −1(V )) for any affinoid domain V ⊂ X. Since J (V ) ⊂ I(V ),
+the homomorphism OX(V ) → OY (f −1(V )) factor through OZ(V ∩Z) = OX(V )/J (V ), we denote
+OZ(V ∩Z) → OX(f −1(V )) by f
+# which is compatible on intersections of affinoid domains. Hence
+we have a morphism f : Y → Z and f = i ◦ f.
+(b) If f factors through a closed subspace Z′ of X with Z′ = V (J ′), then J ′ ⊂ I. By the
+choice of J , we have J ′ ⊂ J , so Z′ ⊂ Z.
+Remark 2.6.
+(1) Locally, f : M(B) → M(A) is given by ϕ : A → B, then Imzar(f) =
+M(A/ Ker(ϕ)).
+We may expect the Zariski image is exactly the usual image as sets. It is almost true if Y is
+reduced or f is quasi-compact.
+Lemma 2.7. Let f : Y → X be a morphism of K-analytic space.
+If Y is reduced, then the
+Imzar(f) = f(Y )
+Xzar with the reduce closed subspace structure.
+Proof. As a map, f factor through f(Y )
+Xzar. Since Y is reduced, so f factors through f(Y )
+Xzar
+with the reduced structure, see [7, PROPOSITION 4.2 (iii)]. It remains to show the universal
+property of Y → f(Y )
+Xzar. If f factors through a closed subspace Z of X, then f(Y )
+Xzar ⊂ Z as
+a subset. The containment is also a morphism of analytic spaces since f(Y )
+Xzar is endowed with
+the reduced structure.
+Lemma 2.8. Let f : Y → X be a morphism of K-analytic space. Assume that f is quasi-compact.
+Then the following hold.
+(1) I = Ker(OX → f∗OY ) is coherent. In particular, Imzar(f) = V (I).
+(2) f(X)
+Xzar = Imzar(f). In other word, Y → Imzar(f) is dominant.
+(3) For any analytic domain V ⊂ X, the subspace Imzar(f)∩V is the Zariski image of f|f −1(V ) :
+f −1(V ) → V .
+4
+
+Proof. (1) Suppose X = M(A) is affinoid. We take a G-covering Y =
+n�
+i=1
+Vi by affinoid domains,
+and set Y ′ =
+n�
+i=1
+Vi, π : Y ′ → Y the canonical morphism which is surjective. For any analytic
+domain V ⊂ Y , the map
+π# : OY (V ) → OY ′(π−1(V )) =
+n
+�
+i=1
+OY (V ∩ Vi)
+is injective. We consider f ′ := f ◦ π : Y ′ → X. Then
+I = Ker(OX → f ′
+∗OY ′).
+Since Y ′ is affinoid, so I = (Ker(A → OY ′(Y ′))∼ which is coherent. This implies (1).
+(3) This is from (1).
+(2) By (3), suffices to assume that X = M(A) is affinoid. We use the notations in (1). Notice
+that f(Y )
+Xzar = f ′(Y ′)
+Xzar, so we can assume that Y = M(B) is affinoid, and f is induced by
+ϕ : A → B. We have I = �
+Ker(ϕ) and Imzar(f) = M(A/ Ker(ϕ)). So the morphism X → Imzar(f)
+is induced by an injective homomorphism A/ Ker(ϕ) → B, hence it is dominant.
+2.3
+Codimension
+We recall the definition of codimension in [8, 1.5.15].
+Definition 2.9. Let X be a K-analytic space, and Y a Zariski-closed subset of X. The codimen-
+sion codim(Y, X) of Y in X is defined as follows.
+• If both Y and X are irreducible, codim(Y, X) := dimK X − dimK Y .
+• If Y is irreducible, codim(Y, X) :=
+sup
+Z∈Irr(X)
+Y ⊂Z
+codim(Y, Z).
+• In the general case, codim(Y, X) :=
+inf
+Z∈Irr(Y ) codim(Z, X).
+For x ∈ X, we define the codimension of Y in X at x as
+codimx(Y, X) :=
+
+
+
+
+
+inf
+Z∈Irr(Y )
+x∈Z
+codim(Z, X)
+if x ∈ Y ;
++∞
+if x ̸∈ Y .
+Remark 2.10.
+(1) Let W ⊂ Z ⊂ Y ⊂ X be irreducible closed analytic subspaces. Then
+codim(W, Y ) = codim(W, Z) + codim(Z, Y ),
+dimK(Z) + codim(Z, Y ) = dimK(Y ).
+Example 2.11 ([6] Proposition 1.11). Let X = M(A) be a K-affinoid space, Y = V (I) for some
+ideal I ⊂ A, and x ∈ X with image ξ ∈ Spec(A). Then
+(1) codim(Y, X) = codim(Spec(A/I), Spec(A)).
+(2) codimx(Y, X) = codimξ(Spec(A/I), Spec(A)).
+Remark 2.12.
+(1) In particular, (1) implies that
+codim(Spec(AL/IL), Spec(AL)) = codim(Spec(A/I), Spec(A))
+for any complete field extension L/K. Or we can write
+dimK X − dimK Y = codimKrull(Y, X).
+5
+
+Proposition 2.13. Let X be a K-analytic space, and Z, Y ∈ Irr(X) with Z ⊂ Y . Then
+codim(Z, Y ) = max{m | Z = Y0 ⊊ Y1 ⊊ · · · ⊊ Ym = Y },
+where Yi ∈ Irr(X). Moreover, each maximal chain has the same length, i.e. every K-analytic space
+is catenary with respect to the Zariski topology.
+Proof. Firstly, if Z ⊊ Y , then codim(Z, Y ) ≥ 1. This can be seen locally. Hence ”≥” holds.
+Conversely, it suffices to show that if codim(Z, Y ) ≥ 2, then there is W ∈ Irr(X) such that Z ⊊
+W ⊊ Y . Indeed, we take an affinoid domain V of Y are affinoid, and V = M(A), Z∩V = M(A/I).
+Then we know that
+codim(Z, Y ) = codim(Spec(A/I), Spec(A)) ≥ 2.
+So we can find a prime ideal p ∈ Spec(A) such that W := M(A/p)
+Yzar strictly contains Z. Apply
+the same method, we can see that each maximal chain has the same length (this in fact due to the
+additivity of codimension).
+Remark 2.14.
+(1) In particular, we see that the codimension is independent of the base field
+K.
+2.4
+A key lemma
+For a set S satisfying certain conditions, we can determine if S satisfies a property P or not. In
+this case, we say that the property P is well-defined on S. It is not well-defined if S does not
+satisfy these conditions at the beginning.
+The following generalized result from [7, Proposition 4.12] is crucial for extending a local result
+on irreducible closed subsets to be global.
+Lemma 2.15. Let X be a K-analytic space.
+Let P be a property on irreducible components
+satisfying the following properties:
+• there is a G-covering X = �
+i��I
+Vi by affinoid domain, the property P is well-defined (this
+means that we can determine if P is satisfied or not) on each irreducible component of Vi (or
+simply say that P is well-defined on Vi);
+• if P is well-defined on an irreducible component Z of an affinoid domain V , then P is well-
+defined on each irreducible component of W for any affinoid domain W ⊂ V . Moreover, in
+this case, for any irreducible component T of W ∩ Z, we have T satisfies P ⇐⇒ Z satisfies
+P.
+Then there exist Zariski-closed subsets X+
+P , X−
+P of X which are characterized by the following
+properties: for any affinoid domain V on which P is well-defined, we have
+X+
+P ∩ V =
+�
+T ∈Irr(V ),
+T satisfies P
+T,
+X−
+P ∩ V =
+�
+T ∈Irr(V ),
+T doesn’t satisfy P
+T.
+Notice that X = X+
+P ∪ X−
+P .
+Proof. For any affinoid domain V on which P is well-defined, set
+C+(V ) := {T ∈ Irr(V ) | T satisfies P},
+C−(V ) := {T ∈ Irr(V ) | T doesn’t satisfy P},
+E+(V ) :=
+�
+T ∈C+(V )
+T,
+E−(V ) :=
+�
+T ∈C−(V )
+T.
+6
+
+Let V be an affinoid domain on which P is well-defined, and W ⊂ V an affinoid domain. Let Z be an
+irreducible component of V and T an irreducible component of W containing Z. By our assumption,
+T ∈ C+(W)⇐⇒ Z ∈ C+(V ). By [7, COROLLAIRE 4.11], we have E+(W) = E+(V ) ∩ W and
+E−(W) = E−(V ) ∩ W.
+Let X+
+P (resp. X−
+P ) be the union of E+(V ) (resp. E−(V )) where V is an affinoid domain on
+which P is well-defined. Then for any affinoid domain V of X on which P is well-defined, we
+have X+
+P ∩ V = E+(V ) and X−
+P ∩ V = E−(V ). Since P is well-defined on Vi for some G-covering
+X = �
+i∈I
+Vi by affinoid domain, and E+(Vi), E−(Vi) ⊂ Vi are Zariski-closed, so X+
+P , X−
+P ⊂ X are
+Zariski-closed.
+3
+Meromorphic functions and Cartier divisors
+The sheaf of meromorphic functions and Cartier divisors are defined on a ringed space in [10,
+Section 20, Section 21]. On a G-ringed space, these definitions do not work since the restriction of
+a regular element is not necessarily regular. Fortunately, this can be remedied on analytic spaces
+(cf. [11, Section 2]). In this section and next section, we will following the idea in [10, Section 20,
+Section 21] to discuss meromorphic functions, Cartier divisors and cycles.
+3.1
+Meromorphic functions
+For a (commutative) ring A, denote R(A) ⊂ A the set of all regular elements, i.e. non-zero divisors,
+we know R(A) is a multiplicative set, and the corresponding localization Frac(A) := R(A)−1A is
+the maximal localization containing A as a subring.
+Definition 3.1. Let X be a K-analytic space. For any affionid domain V = M(A) ⊂ X, we
+set K′
+X(V ) := Frac(A), this will defined a presheaf on affinoid domains on X. The associated
+sheaf KX with respect to the G-topology on X is called the sheaf of meromorphic functions on
+X. An element of KX(X) is called a meromorphic function on X. The subsheaf of invertible
+elements of KX is denoted by K∗
+X.
+Remark 3.2.
+(1) For affinoid domains U = M(B) ⊂ V = M(A) of X, and f ∈ R(A), the
+restriction of f on U is in R(B), this implies that our definition of KX is well-defined.
+Proof. It is from the fact A → B is flat, or we assume that B = A{p−1
+1
+T1,··· ,p−1
+n Tn}
+(gT1−f1,··· ,gTn−fn).
+(2) For any analytic domain V ⊂ X, we have
+KX(V ) =
+
+
+
+
+
+
+
+(si)i ∈
+�
+i
+K′
+X(Vi)
+��������
+V = �
+i
+Vi is a G-covering of V with Vi affi-
+noid and si|Vijk = sj|Vijk for some G-covering
+Vi ∩ Vj = �
+k
+Vijk with Vijk affinoid
+
+
+
+
+
+
+
+�
+∼,
+where (si)i ∼ (s′
+j)j if for any i, j, there exists a G-covering Vi ∩V ′
+j = �
+k
+Vijk with Vijk affinoid
+such that si|Vijk = s′
+j|Vijk.
+If X is separated, then it can be simplified as
+KX(V ) =
+�
+(si)i ∈
+�
+i
+K′
+X(Vi)
+�����
+V = �
+i
+Vi is an G-covering of V with Vi affi-
+noid and si|Vi∩Vj = sj|Vi∩Vj
+� �
+∼,
+where (si)i ∼ (s′
+j)j if for any i, j, si|Vi∩V ′
+j = s′
+j|Vi∩V ′
+j .
+(3) For any affinoid domain V ⊂ X, the canonical map K′
+X(V ) → KX(V ) is injective.
+In
+particular, OX ⊂ KX.
+7
+
+Proof. Given an affinoid domain V and any finite G-covering V =
+n�
+i=1
+Vi by affinoid domains,
+let A = OX(V ) and Ai = OX(Vi). We consider the restriction map Frac(A) →
+n�
+i=1
+Frac(Ai).
+Let a/b ∈ Frac(A) be such that its restriction on Frac(Ai) is 0 for any i, i.e. a = 0 ∈ Ai.
+This implies that a = 0 ∈ A by Tate’s acyclic theorem. Hence K′
+X(V ) ֒→ KX(V ).
+We take a G-covering X = �
+i∈I
+Vi by affinoid domains. Then the injective map OX(Vi) ֒→
+K′
+X(Vi) will induce OX ֒→ KX.
+Definition 3.3. Keep the notion in Definition 3.1. For an OX-module F, we call F ⊗OX KX the
+sheaf of meromorphic sections of F, and we have a canonical map
+idF ⊗i : F → F ⊗OX KX.
+The sheaf F is called strictly without torsion if idF ⊗i is injective.
+A global section of F ⊗OX KX is called a meromorphic sections of F on X.
+If F is coherent on X, we say a meromorphic section s on X is defined on a Zariski-open
+subset V if s|V is in the image of F(V ) via idF ⊗i. If moreover, F is strictly without torsion, then
+there is a maximal Zariski-open subset V on which s is defined, such V is called the domain of
+definition of s, denoted by dom(s) (i.e. s ∈ F(dom(s))).
+Remark 3.4.
+(1) Notice that F → F ⊗OX KX is the sheafification of the presheaf given by
+V �→ F(V ) ⊗OX(V ) K′
+X(V )
+for any affinoid domain V . So for any analytic domain V ⊂ X, we have (F ⊗OX KX)|V ≃
+F|V ⊗OV KV . In particular, KX|V = KV .
+(2) A locally free OXG-module F is strictly without torsion. Moreover, F ⊗OX KX is a KX-
+module, here, we view (XG, KX) as a G-ringed space.
+For a good, strictly K-analytic space, the sheaf of meromorphic functions can be given in a
+similar way in [10, Section 20], and will have some good properties, i.e. properties for schemes can
+be extended to good analytic spaces.
+If X is good, strictly K-analytic, and x ∈ X is rigid, we have that
+OX,x = lim
+−→
+V
+OX(V )
+where V runs through affinoid domains containing x, see [2, Section 2.3]. In particular, it suffices
+that V runs through (strictly) affinoid neighborhoods of x in X.
+Proposition 3.5. Let X be a good, strictly K-analytic space. For any analytic domain V ⊂ X,
+set
+R(V ) := {s ∈ OX(V ) | sx ∈ R(OX,x) for any x ∈ V } ⊂ OX(V ),
+which defines a sheaf on X. Then the following statements hold:
+(1) For any affinoid domain V ⊂ X, we have R(V ) = R(OX(V )). In particular, and KX to be
+the sheafification of the following presheaf: for any analytic domain V ⊂ X,
+V �→ R(V )−1OX(V ).
+(2) For any rigid point x ∈ X, we have K′
+X,x ≃ Frac(OX,x). For any analytic domain V ⊂ X,
+the canonical homomorphism K′
+X(V ) ֒→
+�
+x∈V rigid
+K′
+X,x is injective.
+8
+
+Proof. Notice that the presheaf R is a sheaf. Since R is a subpresheaf of OX, and if V = �
+i∈I
+Vi is
+a G-covering of an analytic domain V , ai ∈ R(Vi) such that ai|Vi∩Vj = aj|Vi∩Vj then there exists
+a ∈ OX(V ) such that a|Vi = ai, then a ∈ R(V ).
+(1) For any affinoid domain V ⊂ X and a ∈ OX(V ), we have a is regular ⇐⇒ a ∈ OX,x regular
+for any x ∈ V . Indeed, ”=⇒” is from the flatness, for ”⇐=”, if a ∈ OX,x is regular, then there is
+an affinoid neighborhood Vx of x in V such that a ∈ R(OX(Vx)) (since Ker(OX(V )
+·a
+→ OX(V )) is
+finitely generated). Then a ∈ R(OX(V )) since V = �
+x∈V
+Vx is a G-covering. So R(V ) = R(OX(V )).
+Hence K′
+X(V ) = Frac(OX(V )).
+(2) By definition, we have a map
+lim
+−→
+V
+K′
+X(V ) → R−1
+x OX,x
+which is surjective, where V runs through affinoid neighborhoods of x. If a/b ∈ K′
+X(V ) with V
+affinoid neighborhood of x such that a/b = 0 ∈ R−1
+x OX,x, i.e. there is c ∈ Rx such that ac = 0.
+We can assume that c ∈ OX(V ), then a/b = 0 ∈ K′
+X(V ).
+It remains to show that Rx = R(OX,x). We have an injective map Rx ֒→ R(OX,x) by definition.
+Conversely, for a ∈ R(OX,x), we consider an affinoid neighborhood V of x with A = OX(V ) such
+that a ∈ A, then
+0
+� Ann(a)
+� A
+� A .
+Since Ann(a) is finitely generated and a ∈ R(OX,x), so we can find an affinoid neighborhood
+U ⊂ V of x with B = OX(U) such that Ann(a) ⊗A B = 0. So a ∈ R(B). By (1), we know that
+Rx = R(OX,x).
+If a/b ∈ K′
+X(V ) such that 0 = a/b ∈ K′
+X,x for any rigid x ∈ V , then there exists an affinoid
+neighborhood Vx of x such that 0 = a/b ∈ K′
+X(Vx). Since R(Vx) = R(OX(Vx)), we have 0 = a ∈
+OX(Vx) and a = 0 ∈ K′
+X(V ), a/b = 0.
+3.2
+Cartier divisors
+Definition 3.6. Let K be a complete non-archimedean field, and X a K-analytic space. We denote
+the group H0(XG, K∗
+X/O∗
+X) by Div(X). The elements of Div(X) are called Cartier divisors of
+XG.
+Let f ∈ H0(XG, K∗
+X), its image in Div(X) is called a principal Cartier divisor and denoted
+by div(f).
+We say that two Cartier divisor D1, D2 are linearly equivalent if D1 − D2 is principal, write
+D1 ∼ D2. We denote CaCl(X) the group of equivalent class of Cartier divisors.
+A Cartier divisor D is called effective if it is in the image of the canonical map H0(XG, (OX ∩
+K∗
+X)/O∗
+X) → H0(XG, K∗
+X/O∗
+X), write D ≥ 0. The set of effective Cartier divisors is denoted by
+Div+(X).
+Remark 3.7.
+(1) The exact sequence of sheaves
+0
+� O∗
+X
+� K∗
+X
+� K∗
+X/O∗
+X
+� 0
+will induce a long exact sequence
+0
+� H0(XG, O∗
+X)
+� H0(XG, K∗
+X)
+� Div(X)
+�
+H1(XG, O∗
+X)
+� H1(XG, K∗
+X)
+� · · ·
+(2) We can represent a Cartier divisor D by a system {(Ui, fi)}i∈I, where X = �
+i∈I
+Ui is a G-
+covering by affinoid domains, and fi = ai/bi ∈ K′
+X(Ui) such that fi|Ui∩Uj ∈ fj|Ui∩UjOX(Ui∩
+Uj)∗ for every i, j ∈ I.
+Two systems {(Ui, fi)}i∈I and {(Vj, gj)}j∈J represent the same
+Cartier divisor if only only if fi|Ui∩Vj ∈ gj|Ui∩VjOX(Ui ∩ Vj)∗ for any i ∈ I, j ∈ J.
+9
+
+If D1 = {(Ui, fi)}i∈I and D2 = {(Vj, gj)}j∈J, then D1 + D2 = {(Wijk, figj)}i∈I,j∈J, where
+Ui ∩ Vj = �
+k
+Wijk is a G-covering by affinoid domains.
+In particular, if X = M(A) is affinoid, let X = Spec(A), then we have an injection
+Div(X) ֒→ Div(X).
+Proposition 3.8. Keep the notion in Definition 3.6.
+(1) For any divisor D = {(Ui, fi)}i∈I ∈ CaCl(X), we can associate a subsheaf OX(D) ⊂ KX
+defined by OX(D)|Ui = f −1
+i
+OX|Ui, which is an invertible sheaf and independent of the choice
+of representative. Moreover, D ≥ 0 ⇐⇒ OX(−D) ⊂ OX.
+(2) The construction above gives a homomorphism of groups ρ : Div(X) → Pic(X),
+D �→
+OX(D).
+(3) The homomorphism ρ induces an injective homomorphism CaCl(X) → Pic(X) with image
+Im ρ = {L ∈ Pic(X) | L ֒→ KX}.
+(4) If X is affinoid and reduced, then ρ : CaCl(X) → Pic(X) is an isomorphism.
+Proof. We follow the idea of the proof of [12, Proposition 7.1.18].
+(1) Assume D = {(Vj, gj)}j∈J is another representative. Then
+OX(D)|Ui∩Vj = f −1
+i
+OX|Ui∩Vj = (gju)−1OX|Ui∩Vj = g−1
+j OX|Ui∩Vj
+where u ∈ OX(Ui ∩ Vj)∗, this implies OX(D) is independent of the choice of representative. By
+construction, OX(D) ∈ Pic(X), and D ≥ 0 if and only if OX(D) ⊂ OX.
+(2) The map is a homomorphism. Indeed, let D1 = {(fi, Ui)}i∈I and D2 = {(gi, Ui)}i∈I, then
+ρ(D1 + D2)|Ui = f −1
+i
+g−1
+i
+OX|Ui ≃ f −1
+i
+OX|Ui ⊗OX|Ui g−1
+i
+OX|Ui,
+and this isomorphism is compatible on the intersection Ui ∩ Uj.
+(3) If D = {(Ui, fi)}i∈I = div(f) is a principal divisor with f ∈ H0(XG, K∗
+X) and fi = f|Ui ∈
+K′
+X(Ui), where X = �
+i∈I
+Ui is a G-covering of X by affinoid domains. Then f −1 ∈ OX(D)(X)
+because of the following exact sequence
+0
+� OX(D)(X)
+� �
+i∈I
+f −1
+i
+OX(Ui)
+� �
+i∈I
+f −1
+i
+OX(Ui ∩ Uj) .
+So we can define the morphism OX → OX(D),
+a �→ af −1. It is an isomorphism since it is an
+isomorphism on each Ui. Hence we have a homomorphism CaCl(X) → Pic(X).
+If D = {(Ui, fi)}i∈I ∈ Div(X) such that OX(D) ≃ OX, then there is g ∈ OX(D)(X) such
+that the morphism OX
+∼
+→ OX(D),
+a �→ ag is an isomorphism. Since OX(D)|Ui ≃ f −1
+i
+OX|Ui =
+g|UiOX|Ui and f −1
+i
+∈ K′∗
+X(Ui), g|Ui = f −1
+i
+ui ∈ K′∗
+X(Ui) ⊂ K∗
+X(Ui) with ui ∈ O∗
+X(Ui), we have
+g ∈ K∗
+X(X) and D = {(Ui, fi)}i∈I = {(Ui, g−1|Ui)}i∈I is principal.
+By definition, we know that OX(D) ⊂ KX. Conversely, for L ∈ Pic(X) with L ⊂ KX, there is
+G-covering X = �
+i∈I
+Ui by affinoid domains such that OX|Ui ≃ L|Ui. We take gi ∈ L(Ui) which is
+mapped to 1. Then gi ∈ KX(Ui) and L|Ui = giOX|Ui, moreover, there is fi ∈ K∗
+X(Ui) such that
+figi = 1 because of the isomorphism. On Ui ∩ Uj, we have
+L|Ui∩Uj = f −1
+i
+OX|Ui∩Uj = f −1
+j
+OX|Ui∩Uj,
+so there is u ∈ O∗
+X(Ui ∩ Uj) such that f −1
+i
+|Ui∩Uj = uf −1
+j
+|Ui∩Uj.
+Then L = OX(D), where
+D = {(Ui, fi)}i∈I ∈ Div(X).
+10
+
+(4) Let X = Spec(OX(X)), then CaCl(X) ≃ Pic(X), see [12, Corollary 1.19]. We a commuta-
+tive diagram
+Div(X)� �
+�
+∼
+�
+Div(X)
+ρ
+�
+Pic(X)
+∼
+� Pic(X)
+,
+so our claim holds. The isomorphism Pic(X) ≃ Pic(X) is from Coh(X) ≃ Coh(X) and Tate’s
+acyclic theorem, see the proof of [3, Propostion 1.3.4 (iii)].
+Remark 3.9.
+(1) We know that H1(XG, O∗
+X) ≃ Pic(X), then ρ is the connecting map of the
+long exact sequence.
+Example 3.10. Let L be a line bundle on a normal K-analytic space X. Let s ∈ H0(X, L⊗OX KX)
+be a rational section which is non-zero on each irreducible component. Let X = �
+i∈I
+Ui be a G-
+covering of X by integral affinoid domains such that L|Ui is free and generated by an element ei.
+Then these exist fi ∈ K∗
+X(Ui) such that s|Vi = fiei. Moreover div(s) := {(Ui, fi)}i∈I is a Cartier
+divisor such that OX(div(s)) ≃ L.
+3.3
+Inverse image of a Cartier divisor
+Next we consider the restriction of Cartier divisors on a closed analytic subspace.
+Definition 3.11. Let D ∈ Div(X), and Z ∈ Irr(X) with reduced analytic space structure.We
+say D intersects Z properly if there is a G-covering X = �
+i∈I
+Ui by affinoid domains such that
+D = {(Ui, ai/bi)}i∈I with the images ai, bi ∈ R(OZ(Ui∩Z)). The set of Cartier divisor intersecting
+Z properly is a subgroup of Div(X), denoted by GZ/X.
+Remark 3.12.
+(1) There is a natural homomorphism GZ/X → Div(Z) denoted by D �→ D|Z,
+compatible with the homomorphism OX → i∗OZ. Moreover, we have a canonical isomor-
+phism OX(D)|Z ≃ OZ(D|Z).
+4
+Cycles, flat pull-backs and proper push-forwards
+4.1
+Cycles
+Definition 4.1. Let X be a K-analytic space. A prime cycle on X is an element in Irr(X). A
+cycle on X is a formal sum α =
+�
+Z∈Irr(X)
+nZ[Z] with nZ ∈ Z which is G-locally finite, i.e. the set
+{Z ∈ Irr(X) | Z ∩ V ̸= ∅, nZ ̸= 0}
+is finite for any affinoid domain V . The coefficient nZ is called the multiplicity of α at Z,
+denoted by multZ(α). We say that a cycle α is positive if multZ(α) ≥ 0 for any Z ∈ Irr(X). The
+set of cycles (resp. positive cycles) is denoted by Z(X) (resp. Z+(X)).
+The union of the Z such that nZ ̸= 0 is called the support of α, denoted by Supp(α). It is a
+Zariski-closed subset of X. By convention, Supp(0) = ∅.
+A cycle α is (purely) of codimension r (resp. of dimension r) if any Z ∈ Irr(X) with
+nZ ̸= 0 has codimension r (resp. dimension r). The cycles of codimension r (resp. of dimension
+r) form a subgroup Zr(X) (resp. Zr(X)) of the group of cycles on X.
+Remark 4.2.
+(1) For a positive cycle α =
+�
+Z∈Irr(X)
+nZ[Z] and any Z ∈ Irr(X) with nZ ≥ 1, we
+can endow Z with the reduced subscheme structure, then Z = V (IZ) is an integral closed
+analytic subspace of X, where IZ is the coherent sheaf of ideal defining Z. We view α as
+a closed analytic subspace defined by the sheaf of ideal Iα :=
+�
+Z∈Irr(X)
+InZ
+Z
+and we have a
+11
+
+canonical closed immersion j : α = V (Iα) ֒→ X. This induces a homomorphisms of semi-
+groups
+Z+(X) → {closed analytic subspace of X} = {coherent sheaves of ideals on X}.
+(2) By Proposition 2.13, we know that Zr(X) is not dependent of the base field K, but Zr(X) is.
+Example 4.3. Let X = M(A) be a K-affinoid space. Set X = Spec(A). Then
+Div(X) ֒→ Div(X),
+Z∗(X) ≃ Z∗(X).
+The first arrow is also an isomorphism if X is regular, see Proposition 4.13.
+Lemma 4.4. Let X be a K-analytic space. Let α ∈ Z+(X) with associated sheaf of ideal Iα. Then
+V (Iα) = Supp(α) with Irr(V (Iα)) = {maximal elements in α}.
+Proof. This is local, and we can deduce this lemma from the example above.
+The following lemma is obvious.
+Lemma 4.5. Let X = �
+i∈I
+V be a G-covering of by affinoid domains, and α, β ∈ Div(X) (resp.
+Z∗(X)). Then α = β ⇐⇒ α|Vi = β|Vi for any i ∈ I.
+Proof. It suffices to show the ”if” part. If α, β ∈ Div(X), then the result holds from the expression
+of Cartier divisors. If α = �
+Z
+nZ[Z], β = �
+Z
+mZ[Z] ∈ Zk(X) such that α|Vi = β|Vi for any i ∈ I,
+then nZ[Z ∩ Vi] = mZ[Z ∩ Vi] for any Z ∈ Irr(X) with Z ∩ Vi ̸= ∅, so nZ = mZ.
+4.2
+Cycle associated to a coherent sheaf
+We will construct a homomorphism Div(X) → Z1(X) as we do in algebraic geometry. Recall,
+for a Noetherian affine scheme X = Spec(A), a coherent sheaf F = �
+M on X, and an irreducible
+component Z of Supp(F), we set multZ(F) := lengthAp(Mp), called the multiplicity of Z in F,
+where p ∈ X is the prime ideal corresponding to Z. For a divisor D ∈ Div(X) and a codimension
+one prime cycle Z = {z} ∈ Z1(X), we set multZ(D) := multOX,z(Dz) the multiplicity of Z in D.
+For an affinoid space M(A), we have similar notation.
+Lemma 4.6. Let X be a K-analytic space. Let F be a coherent sheaf on X. For any irreducible
+component Z of Supp(F) with reduced analytic space structure, and an affinoid domain V ⊂ X
+with Z ∩ V ̸= ∅, we set
+multZ(F) := multT (F|V )
+where T is an irreducible component of Z ∩ V with T
+Supp(F)Zar = Z. Then multZ(F) is a positive
+integer which is independent of the choice of T and V . We call multZ(F) the multiplicity of Z
+in F.
+Proof. For a fixed irreducible component Z of Supp(F), and any affinoid domain V, W ⊂ X with
+W ⊂ V , Z ∩ W ̸= ∅, we claim that
+multT (F|V ) = multT ′(F|W )
+where T ∈ Irr(Z ∩ V ) (resp.T ′ ∈ Irr(Z ∩ W)) with T ′VZar = T , T
+XZar = Z. Indeed, let V =
+M(A), W = M(B) and F|V = �
+M. We shall show that
+lengthAp(Mp) = lengthBq(Mp ⊗Ap Bq)
+where p ⊂ A (resp. q ⊂ B) is the prime ideal corresponding to T (resp. T ′). Notice that the kernel
+W → Spec(B) is surjective, we can find a y ∈ W such that Ker(| · |x) = q. Let x ∈ V be the image
+of y, then Ker(| · |x) = p. We have H (x) = H (y) and
+lengthAp(Mp) = dimk(p)(M ⊗A k(p)) = dimH (x)(M ⊗A H (x)),
+12
+
+it is similar for lengthBq(Mp ⊗Ap Bq). Hence our claim holds.
+To show the lemma, we apply Lemma 2.15.
+Let Z ∈ Z1(X) be a prime cycle, and m =
+multT (F|V ) for some affinoid domain V ⊂ X with Z ∩ V ̸= ∅, where T ∈ Irr(Z ∩ V ) with
+T
+XZar = Z. For V given as before, we say an irreducible component T ∈ Irr(Z ∩ V ) satisfies P if
+multT (F|V ) = m. After replacing X by Z, from our claim, we see that P satisfies the hypothesis
+in Lemma 2.15. Then there are Zariski-closed subsets Z+
+P , Z−
+P of Z such that
+Z+
+P ∩ V =
+�
+T ∈Irr(Z∩V ),
+T satisfies P
+T,
+Z−
+P ∩ V =
+�
+T ∈Irr(Z∩V ),
+T doesn’t satisfy P
+T,
+and Z = Z+
+P ∪ Z−
+P . Since Z is irreducible and there is some T ⊂ Z+
+P , we have Z = Z+
+P . This
+implies the lemma.
+Definition 4.7. Keep the notion in Lemma 4.6. For a coherent sheaf F on X with codim(Supp(F), X) ≥
+k, we set
+[F]k :=
+�
+Z∈Irr(Supp(F))k
+multZ(F)[Z] ∈ Zk(X),
+called the cycle associated to F with codimension k.
+Remark 4.8.
+(1) By Lemma 4.6, it is hard to have the following result. Let V = M(A) ⊂ X
+is an affinoid domain, and F a coherent sheaf on X. Set V = Spec(A) and Fal
+V the coherent
+sheaf on V corresponding to F|V . Then
+[F|V ]k = [Fal
+V ]k,
+here we identify Z∗(V ) ≃ Z∗(V).
+Definition 4.9. Keep the notion in Lemma 4.6.
+For a closed analytic subspace Y of X with
+codim(Y, X) ≥ k, we set
+multZ(Y ) := multZ(OY ),
+for any Z ∈ Irr(Y ), called the multiplicity of Z in Y , and set
+[Y ]k :=
+�
+Z∈Irr(Y )
+Z∈Zk(X)
+multZ(Y )[Z] ∈ Zk(X),
+called the cycle associated to Y with codimension k.
+4.3
+Weil divisors
+Definition 4.10. Let X be a K-analytic spaces. An element in Z1(X) is called a Weil divisor
+on X.
+Lemma 4.11. Let X be a K-analytic space. Let D ∈ Div(X). For any prime cycle Z ∈ Z1(X),
+and any affinoid domain V ⊂ X with Z ∩ V ̸= ∅, D|V ∈ K′
+X(V ), we set
+multZ(D) := multT (D|V )
+where T ∈ Irr(Z ∩ V ) with T
+XZar = Z. Then multZ(D) is independent of the choice of T and V .
+We call multZ(D) the multiplicity of Z for D.
+13
+
+Proof. The proof is similar with the one of Lemma 4.6.
+For any prime cycle Z ∈ Z1(X) and any affinoid domain V, W ⊂ X with W ⊂ V , Z ∩ W ̸= ∅,
+D|V ∈ K′
+X(V ), we claim that
+multT (D|V ) = multT ′(D|W ),
+where T ∈ Irr(Z ∩V ) (resp.T ′ ∈ Irr(Z ∩W)) with T ′VZar = T , T
+XZar = Z. Indeed, since both sides
+are additive, we can assume that D|V = f ∈ R(OX(V )). Let Y ⊂ V be a closed analytic subspace
+determined by f ∈ OX(V ), then our claim is from Lemma 4.6.
+To show the lemma, we apply Lemma 2.15. Let m = multT (D|V ) for some affinoid domain
+V ⊂ X with Z ∩ V ̸= ∅, D|V ∈ K′
+X(V ), where T ∈ Irr(Z ∩ V ) with T
+XZar = Z. For V given
+as before, we say an irreducible component T ∈ Irr(Z ∩ V ) satisfies P if multT (D|V ) = m. After
+replacing X by Z, from our claim, we see that P satisfies the hypothesis in Lemma 2.15. Then
+there are Zariski-closed subset Z+
+P , Z−
+P of Z such that
+Z+
+P ∩ V =
+�
+T ∈Irr(Z∩V ),
+T satisfies P
+T,
+Z−
+P ∩ V =
+�
+T ∈Irr(Z∩V ),
+T doesn’t satisfy P
+T,
+and Z = Z+
+P ∪ Z−
+P . Since Z is irreducible, and there is some T ⊂ Z+
+P , so Z = Z+
+P . This implies
+the lemma.
+Definition 4.12. Let X be a K-analytic space. For any D ∈ Div(X), we set
+[D] :=
+�
+Z⊂Irr(X)
+codim(Z,X)=1
+multZ(D)[Z] ∈ Z1(X),
+called the Weil divisor associated to D. In particular, for any f ∈ K∗(X), we denote (f) :=
+[div(f)] ∈ Z1(X). Such a divisor (f) is called a principal divisor. The set of principal divisors
+Rat1(X) form a subgroup of Z1(X). We denote the quotient of Z1(X) by the subgroup of principal
+divisors by Cl(X) := Z1(X)/Rat1(X), called the class group of X. We say that two divisors
+Z, Z′ are rationally equivalent and write Z ∼rat Z′ if they have the same class in Cl(X).
+Recall, a K-analytic space X is regular at x ∈ X if there is a good analytic domain V of X
+containing x such that OV,x is regular. We say X is regular if X is regular at every point x ∈ X.
+This is equivalent to that for any affinoid domain V ≃ M(A) ⊂ X, we have that A is regular, see
+[8, Lemma-Definition 2.4.1, Lemma 2.4.5].
+Proposition 4.13. The map [·] : Div(X) → Z1(X) a homomorphism which sends effective divisors
+to positive cycles. This induces a homomorphism
+[·] : CaCl(X) → Cl(X).
+If X is normal (resp. regular), then these two map are injective (resp. isomorphic).
+Proof. It is easy to see that [·] : Div(X) → Z1(X) is a homomorphism and induces [·] : CaCl(X) →
+Cl(X). If X is normal, by Lemma 4.5, to show [·] : Div(X) → Z1(X) is injective, we can assume X
+is affinoid. For D ∈ Div(X) such that multZ(D) = 0 for any Z ∈ Z1(X), we take affinoid domain
+V ⊂ X with Z ∩ V ̸= ∅ and D|V ∈ K′
+X(V ). Then D|V ∈ O∗
+X(V ) since multT (D|V ) = 0 for any
+Q ∈ Z1(V ). This implies that D = 0. As for the quotient, if [D] = (f) for some f ∈ K∗
+X(X), then
+D = div(f), this implies that [·] : CaCl(X) → Cl(X) is injective.
+Assume that X is regular. To show that [·] : Div(X) → Z1(X) is surjective, we firstly assume
+that X = M(A) is affinoid and set X = Spec(A).
+In this case, Div(X) ≃ Z1(X), see [12,
+Proposition 7.2.16]. Hence, we have a commutative diagram
+Div(X)� �
+�
+∼
+�
+Div(X)
+ρ
+�
+Z1(X)
+∼
+� Z1(X)
+,
+14
+
+so our claim holds for affinoid spaces. We can glue Cartier divisors on affinoid domains together
+by injectivity of [·]. Hence [·] : Div(X) → Z1(X) is surjective.
+4.4
+Rational equivalence of cycles
+As in the classical definition of Chow group of an algebraic variety, we can extend the class group
+for any codimension.
+Definition 4.14. Let X be a K-analytic space. For any (k + 1)-dimensional irreducible closed
+analytic subspace W of X and any f ∈ K∗
+W (W), we have a k-cycle [div(f)] ∈ Zk(W) ⊂ Zk(X)
+given in Definition 4.12. A k-cycle α is rationally equivalent to zero, write α ∼ 0, if there are
+a finite number of (k + 1)-dimensional subvarieties Wi of X, and fi ∈ K∗
+Wi(Wi) such that
+α =
+�
+i
+[div(fi)].
+Since [div(f −1)] = −[div(f)], the cycles rationally equivalent to zero form a subgroup Ratk(X) ⊂
+Zk(X).
+The group of k-cycles modulo rational equivalence on X is the quotient
+Ak(X) := Zk(X)/Ratk(X).
+Define Z∗(X) (resp. A∗(X)) to be the direct sum of the Zk(X) (resp. Ak(X)) for k ∈ Z. A cycle
+class on X is an element of A∗(X).
+A cycle class is positive if it can be represented by a positive cycle.
+Remark 4.15.
+(1) The subgroup Ratk(X) ⊂ Zk(X) is well-defined by Lemma 2.13.
+(2) Ak(X) = Ak(Xred) for any k ∈ Z.
+(3) If X is of pure dimension n, then An(X) = Zn(X) is the free abelian group generated by the
+irreducible components of X.
+4.5
+Flat pull-backs
+We have introduced Cartier divisors, cycles. Next we consider their pull-backs via flat morphisms.
+Recall the definition of flatness in sense of [8, Definition 4.1.8], a morphism f : Y → X of K-
+analytic spaces is naively flat if for any y ∈ Y , there exist a good analytic domain V ⊂ Y containing
+y and a good analytic domain U ⊂ X containing f(V ) such that OV,y is flat over OU,f(y). We say
+f is flat if moreover Y ′ := Y ×X X′ → X′ is naively flat for any morphism X′ → X. If f is flat,
+then OY (V ) is flat over OX(U) for any affinoid domains V ⊂ Y and U ⊂ X with f(V ) ⊂ U. The
+converse is not true in general unless f is locally finite. Notice that for any analytic domain V of
+X, the natural morphism V ֒→ X is flat.
+Definition 4.16. A morphism f : Y → X of K-analytic spaces has relative dimension r if for
+any Z ∈ Irr(X), f −1(Z) = ∅ or any irreducible component Z′ of f −1(Z) has dimK Z′ = dimK Z+r.
+Remark 4.17.
+(1) The notion of relative dimension r is an analogue of the one in algebraic
+geometry, see [9, B.2.5]. Our definition is different from the one in [8, 1.4.13]. We don’t
+assume that such morphisms are surjective.
+Lemma 4.18. Let f : Y → X be a flat morphism of K-analytic spaces. Then f has relative
+dimension r if and only if Yx = ∅ or Yx is of equidimension r for any x ∈ X. In particular, if
+f : Y → X is flat with X, Y equidimensional, then f has relative dimension dimK Y − dimK X.
+Proof. We apply [8, Lemma 4.5.11] saying that dimy Y = dimy Yx + dimx X for any y ∈ Yx.
+Assume that f has relative dimension r. If x ∈ X such that Yx ̸= ∅, then for any Z ∈ Irr(X)
+containing x, we have dimK f −1(Z)−dimK Z = r. This implies that dimy Yx = dimy Y −dimx X =
+r for any y ∈ Yx since dimx X =
+max
+x∈Z∈Irr(X){dimK Z}.
+15
+
+Conversely, for any Z ∈ Irr(X) with f −1(Z) ̸= ∅, without loss of generality, we can assume
+that Z = X. We take y ∈ Y and x = f(y). Then dimy Y = dimx X + dimy Yx = dimK X + r. This
+implies that f has relative dimension r.
+If X, Y are equidimensional, then dimy Yx = dimy Y − dimx X implies that Yx is of equidimen-
+sion for any y ∈ Y, x = f(x).
+Definition 4.19. Let f : Y → X be a flat morphism of K-analytic spaces.
+(1) The canonical morphism f # : OX → f∗OY extends to a morphism f # : K∗
+X/O∗
+X →
+f∗(K∗
+Y /O∗
+X), then we have a homomorphism
+f ∗ : Div(X) → Div(Y ).
+This will induce a homomorphism f ∗ : CaCl(X) → CaCl(Y ).
+(2) Assume that X, Y are of equidimension. For any integral closed subspace Z ⊂ X of pure
+codimension k, we set
+f ∗[Z] := [f −1(Z)] ∈ Zk(Y ).
+This extends by linearity to a pull-back homomorphism f ∗ : Zk(X) → Zk(Y ).
+Remark 4.20.
+(1) The flat pull-backs are functorial and we have a commutative diagram
+Div(X)
+f ∗
+�
+[·]
+�
+Div(Y )
+[·]
+�
+Z1(X)
+f ∗
+� Z1(Y )
+.
+Proposition 4.21. Let f : Y → X be a flat morphism of K-analytic spaces of pure dimension.
+For a coherent sheaf F on X with codim(Supp(F), X) ≥ k, we have codim(Supp(f ∗F), X) ≥ k
+and
+[f ∗F]k = f ∗[F]k.
+In particular, if Z is a closed analytic subspace of X of pure codimension k, then f ∗[Z] = [f −1(Z)].
+Proof. We can reduce the statement to the case of affinoid spaces by Lemma 4.5, then the proposi-
+tion from the analogue result in scheme theory by Remark 4.8 (1). For the result in scheme theory,
+see proof of [14, Lemma 42.14.4 (2)].
+4.6
+Proper push-forward of cycles
+For an affinoid space X = M(A), it may happen that dimKrull A < dimK X. In order to avoid
+this dimension problem, we assume that all K-analytic spaces (including affinoid domains) in this
+subsection are strict. In this case dimKrull A = dimK X.
+Recall a theorem of Kiehl.
+Theorem 4.22 ([2] Proposition 3.3.5). Let f : Y → X be a proper morphism of K-analytic spaces,
+and F a coherent OY -module. Then Rnf∗F, n ≥ 0, are coherent OX-modules. In particular, we
+have Remmert’s mapping theorem, saying that f(Y ) is an Zariski-closed subset of X.
+A similar result of the following lemma is given in [11, 2.6].
+Lemma 4.23. Let f : Y → X be a surjective finite morphism of integral, strictly K-analytic
+spaces. For any (strictly) affinoid domain V ⊂ X and T ∈ Irr(V ), we set
+deg(Y/X) :=
+�
+Q∈Irr(f −1(V ))
+f(Q)=T
+[Frac(AQ) : Frac(AT )],
+where AT , AQ are the affinoid algebras corresponding to T, Q with reduced structure. Then deg(Y/X)
+is independent of the choice of V and T , called the degree of f.
+16
+
+Proof. Apply the usual technique with Lemma 2.15, it is sufficient to show that for any affinoid
+domain V, W ⊂ X with W ⊂ V , and any T ∈ Irr(V ), T ′ ∈ Irr(W), we have
+�
+Q∈Irr(f −1(V ))
+f(Q)=T
+[Frac(AQ) : Frac(AT )] =
+�
+Q′∈Irr(f −1(W))
+f(Q′)=T ′
+[Frac(AQ′) : Frac(AT ′)].
+This is in fact from Lemma 4.6 and Proposition 4.21 for affinoid case. Let V = M(A), f −1(V ) =
+M(B) and W = M(A′), then f −1(W) = M(B′), where B′ = A′⊗AB. Let F be the corresponding
+coherent sheaf associated to B as an A-module on V , and i : W → V the canonical morphism,
+then
+[F]0 =
+�
+T ∈Irr(V )
+(
+�
+Q∈Irr(f −1(V ))
+f(Q)=T
+[Frac(AQ) : Frac(AT )])[T ],
+and we know that
+�
+Q∈Irr(f −1(V ))
+f(Q)=T
+[Frac(AQ) : Frac(AT )] is independent of the choice of T by Lemma 4.6.
+We also have
+i∗[F]0 =
+�
+T ∈Irr(V )
+(
+�
+Q∈Irr(f −1(V ))
+f(Q)=T
+[Frac(AQ) : Frac(AT )])
+�
+T ′∈Irr(T ∩W)
+[T ′],
+[i∗F]0 =
+�
+T ′∈Irr(W)
+(
+�
+Q′∈Irr(f −1(W))
+f(Q′)=T ′
+[Frac(AQ′) : Frac(AT ′)])[T ′].
+By Proposition 4.21, we compare the coefficient of some for any irreducible component T ′, we can
+see that our claim holds.
+We have the following equivalent conditions.
+Lemma 4.24. Let f : Y → X be a morphism of integral, separated, strictly K-analytic spaces.
+Then the following are equivalent.
+(i) f is surjective and finite.
+(ii) f is surjective, proper, and dimK Y = dimK X.
+(iii.a) f is proper, and for any x ∈ X, dimH (x) f −1(x) = 0.
+(iii.b) f is proper, and for any rigid point x ∈ X, f −1(x) ̸= ∅ has finite rigid points as an H (x)-
+analytic space.
+(iv.a) f is surjective and proper, and there is a point x ∈ X such that dimH (x) f −1(x) = 0.
+(iv.b) f is surjective and proper, and there is a rigid point x ∈ X such that dimH (x) f −1(x) = 0,
+i.e. f −1(x) ̸= ∅ and has finite rigid points.
+Proof. Obviously, (i) =⇒ (iii.a), (iii.b) =⇒ (iv.b) =⇒ (iv.a).
+(iii.a) =⇒ (ii). This is from [8, 1.4.14 (3)].
+(ii) =⇒ (iii.b). Since f is quasi-compact, after taking irreducible components of affinoid domain
+of X, Y , we can assume that X = M(A), Y = M(B) are affinoid, integral and dim A = dim B.
+Moreover, since the original morphism is surjective, we know that the corresponding morphism
+ϕ : Spec(B) → Spec(A) is dominant. For any closed point x ∈ Spec(A) with ϕ−1(x) ̸= ∅, by basic
+property of strict affinoid algebras, we know that codim(x, Spec(A)) = dim A. Since ϕ is dominant,
+then dim B ≥ codim(x, Spec(A)) + dim ϕ−1(x). So dim ϕ−1(x) = 0. Notice that K → A → H (x)
+is finite, then H (x) is the residue field of Spec(A) at x, and B ⊗A H (x) = B �⊗AH (x). Hence
+the rigid points of f −1(x) is exactly the closed points of ϕ−1(x) which are finite since B �⊗AH (x)
+Noetherian.
+17
+
+(iii.b) =⇒ (i). The separatedness ensure that X, Y are also rigid K-analytic spaces, see [3,
+Theorem 1.6.1]. Then the result is from [5, Corollary 9.6.6] and [2, Proposition 3.3.2].
+(iv.a) =⇒ (ii). Notice that we have proved the equivalence (i) ⇐⇒ (ii) ⇐⇒ (iii.a) ⇐⇒ (iii.b).
+By [6, TH´EOR`EME 4.9], the set
+{y ∈ Y | dimy f ≥ 1}
+is Zariski-closed in Y . So
+{x ∈ X | dimH (x) f −1(x) ≥ 1} = f({y ∈ Y | dimy f ≥ 1})
+is Zariski-closed in X, i.e. U := {x ∈ X | dimH (x) f −1(x) ≤ 0} is Zariski-open in X. Then
+dimK f −1(U) = dimK U by the equivalence of (iii.a) and (ii). Since dimK Y = dimK f −1(U), dimK X =
+dimK U, we have (ii).
+With the lemmas above, we have the following definition.
+Definition 4.25. Let f : Y → X be a proper morphism of separated, strictly K-analytic spaces.
+For any irreducible closed subspace Z of Y , the image f(Z) is a Zariski-closed subset of Y . We set
+deg(Z/f(Z)) :=
+�
+the degree of f : Z → f(Z)
+if dimK f(Z) = dimK Z;
+0
+if dimK f(Z) < dimK Z
+(notice that dimK f(Z) = dimK Z is equivalent to f : Z → f(Z) is finite).
+Define f∗[Z] :=
+deg(Z/f(Z))[f(Z)], then extends linearly to a homomorphism (of gradding groups)
+f∗ : Z∗(Y ) → Z∗(X).
+Remark 4.26.
+(1) For Z above, we know that f(Z) with the reduced subspace structure is the
+Zariski image of Z → X by Lemma 2.7.
+We can easily prove the following lemma.
+Lemma 4.27. Let f : Y → X and g : Z → Y be proper morphism of separated strictly K-analytic
+spaces. Then g∗ ◦ f∗ = (g ◦ f)∗.
+Proposition 4.28. Let
+Y ′
+g′
+�
+f ′
+�
+Y
+f
+�
+X′
+g
+� X
+be a Cartesian diagram of separated, strictly K-analytic spaces with f proper and g flat. Then f ′
+is proper, g′ is flat and g∗ ◦ f∗ = f ′
+∗ ◦ g′∗ on Z∗(Y ).
+Proof. The morphism f ′ is proper by [3], and g′ is flat by definition.
+For the equality, notice that it holds if f is a closed immersion. In general, To show g∗(f∗α) =
+f ′
+∗(g′∗(α)), we can assume that α = [Y ] and it is irreducible.
+Moreover, we can assume that
+X = f(Y ).
+If dimK X < dimK Y , then left-handed side is 0. For any x′ ∈ X′, let x = g(x′). We have
+(f ′)−1(x′) = M(H (x′)) ×X′ Y ′ = M(H (x′)) ×X Y = M(H (x′)) ×H (x) f −1(x).
+Since f is not finite, by Lemma 4.24 (iv.a), we have dimH (x′)(f ′)−1(x) = dimH (x) f −1(x) > 0.
+This means that f ′ is not finite, and f ′∗([Y ′]) = 0.
+If dimK X = dimK Y , then f : Y → X is finite. By Lemma 4.5, it suffices to consider the affine
+case. Then the result is from Proposition 4.21, and can be proved similarly as Lemma 4.23.
+With the proposition above, we can always assume that the base space is affinoid. We can
+use this to deduce the following result to the scheme case, see [14, Lemma 42.12.4] for the scheme
+version.
+18
+
+Proposition 4.29. Let f : Y → X be a proper morphism of separated strictly K-analytic spaces.
+(1) Let Z ⊂ Y be a closed subspace with dimK Z ≤ k. Then
+f∗[Z]k = [f∗OZ]k.
+(2) Let F be a coherent sheaf on X such that dimK(Supp(F)) ≤ k. Then
+f∗[F]k = [f∗F]k.
+Proof. Obviously, it suffices to show (2). By Lemma 2.3, there is a coherent sheaf G on Z :=
+Supp(F) such that F = i∗G. Let Z′ be the Zariski image of Z → X. Notice that f(Z) = Z′ by
+Lemma 2.8 and properness of f. So we have the following commutative diagram
+Z� �
+�
+f|Z �
+Y
+f
+�
+Z′� �
+� X
+.
+By functorial property of push-forward, it suffices to show (f|Z)∗[G] = [(f|Z)∗G]. So we can assume
+that dimK X = k and f : X → Y is proper and dominant. Moreover, we can assume that Y is
+affinoid. So dimK Y ≤ k.
+We write
+f∗[F]k =
+�
+W
+nW [W]
+and
+[f∗F]k =
+�
+W
+mW [W]
+where W runs through irreducible component of X of dimension k. For a fixed irreducible com-
+ponent W, to show nW = mW , it suffices to show that (f∗[F]k)|V = ([f∗F]k)|V for some affinoid
+domain V ⊂ X with V ∩ W ̸= ∅. We can take Zariski-open subsets U ⊂ X such that U ∩ W ′ = ∅
+and U ∩ f(T ) = for any irreducible component W ′ of X which is distinct from W, and any ir-
+reducible component T of Y which doesn’t dominate W. We can take an affinoid domain of U.
+So we can assume X = M(A) is equidimensional and each irreducible component of Y dominates
+some irreducible component of X. By [2, Corollary 3.3.8], we know that Y is finite over X. So we
+reduce to the case where Y, X is affinoid and f is finite. This is an algebraic result, see the last
+part of the proof of [14, Lemma 41.13.3].
+5
+Proper intersection and intersection multiplicities
+5.1
+Proper intersection
+Lemma 5.1. Let X be a regular K-analytic space of pure dimension, and Y, �Y ∈ Irr(X). Then
+for every irreducible component Z of Y ∩ �Y , we have
+codim(Z, X) ≤ codim(Y, X) + codim(�Y , X).
+Proof. The proof is based on the corresponding result in scheme theory. We can assume that X
+is irreducible. For any affinoid domain V ⊂ X, we have codim(T, V ) = codim(Y, X), where T is a
+irreducible component of V ∩Y . Then we can apply the corresponding result in scheme theory.
+Definition 5.2. Let X be a regular K-analytic space of pure dimension.
+(1) Let Y, �Y ∈ Irr(X). We say that Y and �Y intersect properly if codim(Z, X) ≥ codim(Y, X)+
+codim(�Y , X).
+(2) Let α = �
+i∈I
+ni[Yi] ∈ Zs(X) and β = �
+j∈J
+mj[�Yj] ∈ Zr(X). We say that α and β intersect
+properly if Yi and �Yj intersect properly for all i and j.
+19
+
+Lemma 5.3. Let X be a regular K-analytic space of pure dimension, and Y, �Y ∈ Irr(X). Then
+the following statements are equivalent:
+(i) Y, �Y intersect properly;
+(ii) For any x ∈ Y ∩ �Y , there is an affinoid domain V containing x such that any Q ∈ Irr(Y ∩
+V ), �Q ∈ Irr(�Y ∩ V ) intersect properly on V ;
+(iii) For any affinoid domain V with Y ∩ V , �Y ∩ V ̸= ∅ and any Q ∈ Irr(Y ∩ V), �Q ∈ Irr(�Y ∩ V ),
+we have Q and �Q intersect properly.
+Proof. For any affinoid domain V ⊂ X with Y ∩ V = ∅ and any Q ∈ Irr(Y ∩ V ), we have
+codim(Q, V ) = codim(Y, X). Then the lemma follows.
+5.2
+Multiplicities and intersect products
+In this subsection, we will apply the intersection theory on a regular catenary Noetherian scheme
+to define multiplicities. Another definition using Tor formula will be given in the next subsection.
+Recall, on a regular, catenary Noetherian scheme X, let Q, �Q be irreducible closed subschemes
+with codim(Q, X) = s, codim( �Q, X) = t. Then intersection product of Q, �Q is defined by
+Q · �Q =
+�
+T
+eT[T ] :=
+�
+i
+(−1)i[TorOX
+i
+(OQ, O �
+Q)]s+t ∈ Zs+t(X),
+i.e.
+eT = e(X, Q · �Q, T ) =
+�
+i
+(−1)ilengthOX,T (TorOX,T
+i
+(OQ,T , O �
+Q,T ))
+where T runs through Irr(Q ∩ �Q) with codim(T, X) = s + t, and OX ,T (resp. OQ,T , resp. O �
+Q,T )
+denotes the local ring of X (resp. Q, resp. �Q) at the generic point of T .
+Lemma 5.4. Let X be a regular K-analytic space of pure dimension, and Y, �Y ⊂ X irreducible
+Zariski-closed subspaces with codim(Y, X) = s, codim(�Y , X) = t. Assume that Y and �Y intersect
+properly. For any irreducible component Z of Y ∩ �Y with codim(Z, X) = s + t, and any affinoid
+domain V ⊂ X with Z ∩ V ̸= ∅, we set
+e(X, Y · �Y , Z) :=
+�
+Q, �
+Q
+e(V, Q · �Q, T )
+where T ∈ Irr(Z ∩ V ) and (Q, �Q) runs through Irr(Y ∩ V ) × Irr(�Y ∩ V ) such that T ∈ Irr(Q ∩ �Q).
+Then e(X, Y, �Y , Z) is a positive integer which is independent of the choice of V and T . We call
+e(X, Y, �Y , Z) the multiplicity of Z on Y ∩ �Y .
+Proof. The idea of proof is similar with the proof of Lemma 4.6 and Lemma 4.11. It is sufficient
+to show that for any affinoid domain V, W ⊂ X with W ⊂ V , Z ∩ W ̸= ∅, we have that
+�
+Q, �
+Q
+e(V, Q · �Q, T ) =
+�
+Q′, �
+Q′
+e(W, Q′ · �Q′, T ′)
+where T ∈ Irr(Z ∩ V ), (Q, �Q) runs through Irr(Y ∩ V ) × Irr(�Y ∩ V ) such that T ∈ Irr(Q ∩ �Q),
+and T ′, Q′, �Q′ is given similarly with T ′VZar = T , T
+XZar = Z. Let V = M(A), W = M(B) and
+f : Spec(B) → Spec(A) is the morphism of schemes given by W ⊂ V . In the following, we view
+every irreducible subset is in the corresponding affine schemes. We fix a pair (Q, �Q). Let f ∗[Q] =
+m
+�
+i=1
+[Q′
+i], f ∗[ �Q] =
+�
+m
+�
+j=1
+[ �Q′
+j], [Q] · [Q] =
+k�
+p=1
+e(V, Q · �Q, Tp)[Tp] with T1 = T , and f ∗[Tp] =
+lq�
+q=1
+[T ′
+pq] with
+T ′
+11 = T ′. Notice that each coefficient of [Q′
+i] in f ∗[Q] is 1 by Lemma 4.6, similar for f ∗[ �Q] and
+f ∗[Tp]. We have
+f ∗[Q] · f ∗[ �Q] = f ∗([Q] · [ �Q]),
+20
+
+i.e.
+�
+i,j
+[Qi] · [ �Qj] =
+�
+i,j,p,q
+e(W, Qi · �Qj, Tpq)[Tpq] =
+�
+p,q
+e(V, Q, �Q, Tp)[Tpq],
+where e(W, Qi · �Qj, Tpq) = 0 if Tpq ̸∈ Irr(Qi ∩ �Qj). Comparing the coefficient of [T11], we have
+e(V, Q · �Q, T ) = �
+i,j
+e(W, Qi · �Qj, T ′). When (Q, �Q) runs through Irr(Y ∩ V ) × Irr(�Y ∩ V ) such that
+T ∈ Irr(Q ∩ �Q), we have the equality we want.
+Definition 5.5. Keep the notion in Lemma 5.4. We define the intersection product of Y and
+�Y as
+Y · �Y =
+�
+Z
+eZ[Z] ∈ Zs+t(X),
+where Z runs through the set Irr(Y ∩ �Y ) with codim(Z, X) = s + t, and eZ = e(X, Y · �Y , Z).
+In general, let α = �
+i∈I
+ni[Yi] ∈ Zs(X) and β = �
+j∈J
+mj[�Yj] ∈ Zr(X). Assume that α and β
+intersect properly. We define
+α · β :=
+�
+i,j
+nimjYi · �Yj.
+From the associativity of intersections in scheme theory, we have the associativity for our
+definition.
+Corollary 5.6. Keep the notion in Lemma 5.4.
+Let Y, �Y , ��Y be irreducible Zariski-closed sub-
+spaces of X. Assume that Y, �Y , ��Y intersect properly pairwise and that codim(Y ∩ �Y ∩ ��Y , X) =
+codim(Y, X) + codim(�Y , X) + codim(��Y , X). Then
+Y · (�Y · ��Y ) = (Y · �Y ) · ��Y
+as cycles on X.
+Proof. This is from Lemma 4.5 and the corresponding algebraic result, see [14, Lemma 43.20.1].
+Lemma 5.7. Let f : X → Y be flat morphism of regular K-analytic spaces. Let F, G be co-
+herent sheaves on Y with codim(Supp(F), X) ≤ r, codim(Supp(G), X) ≤ s, and codim(Supp(F) ∩
+Supp(G), X) ≥ r+s+dim(Y )−dim(X). In this case, the cycle [f ∗F]r and [f ∗G]s intersect properly
+and
+f ∗([F]r · [G]s) = [f ∗F]r · [f ∗G]s.
+Proof. This is from Lemma 4.5 and [14, Lemma 43.21.1] for regular, catenary Noetherian schemes.
+The lemma implies the following corollary directly.
+Corollary 5.8. Let f : X → Y be flat morphism of regular K-analytic spaces. Let α ∈ Zr(Y ), β ∈
+Zs(Y ). Assume that α and β intersect properly. Then f ∗α and f ∗β intersect properly and f ∗(α ·
+β) = f ∗α · f ∗β.
+5.3
+Intersection multiplicities using Tor formula
+We could define the multiplicities following the idea in [14, Section 43] by using TorOX
+i
+(F, G).
+Firstly, it is not hard to see that TorOX
+i
+(F, G) is a coherent sheaf on X. Indeed, if X = M(A)
+is affinoid, then Coh(X) ≃ Coh(Spec(A)). Since A is Noetherian, so we see that TorOX
+i
+(F, G) is a
+coherent sheaf on X. For general case,
+We show the following results.
+Proposition 5.9. Let X be a regular, strictly K-analytic space.
+21
+
+(1) Let Y, �Y be irreducible Zariski-closed subspaces of X with codim(Y, X) = s, codim(�Y , X) = t.
+Assume that Y, �Y intersect properly. Then
+Y · �Y =
+�
+i
+(−1)i[TorOX
+i
+(OY , O�Y )]s+t.
+(2) Let F, G be coherent sheaves on X with codim(F, X) ≥ s, codim(F, X) ≥ t. Assume that
+[F]s, [G]t intersecting properly. Then
+[F]s · [G]t =
+�
+i
+(−1)i[TorOX
+i
+(F, G)]s+t.
+Proof. Obviously, (2) implies (1). By Lemma 4.5, Lemma 5.3 and Lemma 5.7, we can assume
+that X is strictly affinoid.
+Then this is [14, Lemma 43.19.4] for regular, catenary Noetherian
+schemes.
+6
+Projection formula
+For a K-analytic spaces X, we denote D(Coh(X)) the derived category of Coh(X). We have the
+derived tensor product ⊗L in D(Coh(X)), see [14, Definition 20.26.14]. If f : Y → X is a morphism
+of K-analytic spaces, then we have a left derived functor
+Lf ∗ : D(Coh(X)) → D(Coh(Y ))
+see [14, Section 21.18]. If f is proper, we have a right derived functor
+Rf∗ : D(Coh(Y )) → D(Coh(X)),
+see [14, Section 21.19]. By adjointness of (Lf ∗, Rf∗), we have a morphism
+Rf∗(E) ⊗L
+OX F → Rf∗(E ⊗L
+OY Lf ∗F),
+see [14, Section 21.50]. As [14, Lemma 36.22.1], we have a similar result for K-analytic spaces.
+Lemma 6.1. Let f : Y → X be a proper morphism of strictly K-analytic spaces. Then for any F
+in D(Coh(X)) and E in D(Coh(Y )), the canonical morphism
+Rf∗(E) ⊗L
+OX F → Rf∗(E ⊗L
+OY Lf ∗F)
+is an isomorphism.
+Proof. The proof is similar with the proof of [14, Lemma 36.22.1]. We can assume that X = M(A)
+is affinoid. In this case, D(Coh(Y )) is the derived category of finitely generated A-modules, which
+is a subcategory of D(A), the derived category of A-modules. We fix a coherent sheaf E on Y . For
+an object M in D(A), we say that T (M) holds if the morphism
+Rf∗(E) ⊗L
+OX �
+M → Rf∗(E ⊗L
+OY Lf ∗ �
+M)
+is an isomorphism, where �
+M is the corresponding sheaf of M on X.
+If M = �
+i
+Mi and T (Mi) holds, then so does T (M).
+Let N → L → M → N[1] be a
+distinguished triangle in D(A). If T holds for two of N, L, M, then it holds for the third. Also
+T (A[n]) for any shifts of A in D(A).
+Hence T (M) holds for any object M in D(A), see [14,
+Remark 15.59.11].
+Theorem 6.2 (Projection formula). Let f : Y → X be a flat, proper morphism of regular,
+separated, strictly K-analytic spaces. Let α ∈ Z∗(Y ) and β ∈ Z∗(X). Assume that α and f ∗β
+intersect properly. Then f∗(α) and β intersect properly and
+f∗(α) · β = f∗(α · f ∗β).
+22
+
+Proof. Our proof is an analytic version of the proof of [14, Lemma 43.22.1]
+By Lemma 5.3, Corollary 5.8 and Lemma 4.5, we can assume that X = M(A) is affinoid and
+integral. Moreover, we assume α = [Z], β = [W] for some closed subspaces of dimension r and s.
+If dimK f(Z) ̸= dimK Z, then f∗[Z] = 0, so f∗[Z] and [W] intersect properly. It sufficient to
+show that f∗([Z] · f ∗[W]) = 0. We consider the morphism Z → f(Z), where f(Z) is endowed
+with the reduced subspace structure. By Lemma 4.24, every fiber of Z → f(Z) has dimension
+≥ 1. This implies that every fiber of the morphism Z ∩ f −1(W) → f(Z) ∩ W has dimension ≥ 1,
+and dimK(Z ∩ f −1(W)) > dimK(f(Z) ∩ W). Since every irreducible component T of Z ∩ f −1(W)
+has dimension dimK(Z ∩ f −1(W)), we conclude that dimK T > dimK f(T ). This implies what we
+want.
+If dimK f(Z) = dimK Z = r, then Z → f(Z) is finite. Let T ⊂ f(Z)∩W, and Ti ⊂ Z∩f −1(W),
+i = 1, · · · , t be the irreducible components of Z ∩ f −1(W) dominating T . Since Z ∩ f −1(W) →
+f(Z) ∩ W is finite, f is flat and Z, f −1(W) intersect properly, so
+dimK T = dimK Ti = dimK Y − (dimK Y − r + dimK X − s) = r + s − dimK X,
+Then f(Z) and W intersect properly. To show the equality, we follow the same idea of the proof
+of [14, Lemma 42.23.1]. Since f is flat, by Lemma 6.1, we have
+Rf∗(OZ) ⊗L
+OX OW ≃ Rf∗(OZ ⊗L
+OY f ∗OW ).
+So for any generic point ξ ∈ Spec(A) corresponding to an irreducible component of f(Z) ∩ W, we
+have
+(f∗TorOY
+i
+(OZ, f ∗OW ))ξ = (TorOX
+i
+(f∗OZ, OW ))ξ.
+(1)
+On the other hand, by Proposition 5.9 and Proposition 4.29, we have
+f∗([Z] · f ∗[W]) =
+�
+i
+(−1)if∗[TorOY
+i
+(OZ, f ∗OW )]r+s−dimK Y
+=
+�
+i
+(−1)i[f∗TorOY
+i
+(OZ, f ∗OW )]r+s−dimK Y ,
+f∗[Z] · [W] = [f∗OZ] · [W]
+=
+�
+i
+(−1)i[TorOX
+i
+(f∗OZ, OW )]r+s−dimK Y .
+Then f∗([Z] · f ∗[W]) = f∗[Z] · [W] by Eq. (1).
+7
+GAGA
+It is natural to expect that our definitions of cycles, flat pull-backs, proper push-forwards and
+intersection products, for algebraic variety will be coincide with the ones in the intersection theory
+of algebraic varieties.
+Proposition 7.1. Let X be an algebraic variety over K. Then we have an isomorphism Z∗(X) ≃
+Z∗(Xan),
+[Y ] �→ [Y an]. For a cycle α ∈ Z∗(X), we will denote its image in Z∗(Xan) by αan.
+Moreover, the following properties hold.
+(1) For any affinoid domain V contained in some affine open subset of Xan, the diagram diagram
+commutes:
+Z∗(X)
+�
+� Z∗(V)
+�
+Z∗(Xan)
+� Z∗(V )
+,
+where V = Spec(OXan(V )).
+23
+
+(2) Let α, β ∈ Z∗(X). Then α = β ∈ Z∗(X) (or αan = βan ∈ Z∗(Xan)) if and only if i∗α =
+i∗β ∈ Z∗(V) for any any affinoid domain V contained in some affine open subset of Xan,
+where V = Spec(OXan(V )) and i : V → X is the canonical morphism.
+Proof. The map is obviously injective. It is suffices to show that every integral closed subspace
+Z of Xan is algebraic. If X is proper over K, by GAGA result, see [2, Proposition 3.4.11], we
+know that Z is algebraic. In general case, by Nagata’s compactification theorem, there is a proper
+variety X over K such that X ⊂ X is an open immersion. We take the Zariski-closure Z of Z in
+X
+an, which is algebraic, i.e. there is an integral subvariety T ⊂ X such that T an = Z. We claim
+that (T ∩ X)an = Z. By construction of analytification, we have (T ∩ X)an = T an ∩ Xan. We also
+have Z ∩ Xan = Z. Then T an = Z implies that (T ∩ X)an = Z.
+(1) The diagram is directly from the definition of [Y an] and Remark 4.8 (1).
+(2) This is from the isomorphism Z∗(X) ≃ Z∗(Xan), the commutative diagram in (1) and
+Lemma 4.5.
+Remark 7.2.
+(1) We have a surjection CH∗(X) ։ A∗(Xan).
+Proposition 7.3. Let f : Y → X be a morphism of algebraic varieties over K. We have the
+following hold.
+(1) Let F be a coherent sheaf on X. Then [F]an = [Fan].
+(2) We have a canonical homomorphism Div(X) → Div(Xan),
+D �→ Dan such that for any
+D ∈ Div(X), we have [D]an = [Dan].
+(3) If ϕ is flat and α ∈ Z∗(X), then (ϕ∗(α))an = (ϕan)∗(αan).
+(4) If ϕ is proper and β ∈ Z∗(Y ), then (ϕ∗(β))an = (ϕan)∗(βan).
+(5) Let α, β ∈ Z∗(X). Then α, β intersect properly if and only if αan, βan ∈ Z∗(Xan) intersect
+properly, and in this case, we have (α · β)an = αan · βan.
+Proof. (1) Let V = M(B) ⊂ Xan be an affinoid domain contained in some affine open subsets of
+Xan. Then we have a canonical morphism ϕ : Spec(A) → X which is flat by [7, TH´EOR`EM 3.3].
+It is sufficient to show that [F]an|V = [Fan]|V .
+By the commutative diagram in (1), we have
+[F]an|V = [ϕ∗F]; by Remark 4.8 (1), we have [Fan]|V = [Fan|V ] = [ϕ∗F]. So our claim holds.
+(2) The homomorphism is given by the fact that V → X is flat for any an affinoid domain
+V = M(A) ⊂ Xan contained in some affine open subsets of Xan, where V = Spec(A). Then the
+compatibleness on such affinoid domains will induce a divisor on X. The equality can be proved
+as (1).
+(3) We take any affinoid domains V = M(A) ⊂ Xan and W = M(B) ⊂ Y an such that
+ϕan(W) ⊂ V and V , W are contained in some affine open subsets of Xan, Y an respectively. Let
+V = Spec(A), W = Spec(B). We have the following commutative diagram
+W
+j
+�
+�ϕ
+� V
+i
+�
+Y
+ϕ
+� X
+Then
+(ϕ∗(α))an|W = j∗ϕ∗(α) = �ϕ∗i∗(α) = (ϕan|W )∗(αan|V ) = (ϕan)∗(αan)|W .
+here we identify the canonical isomorphisms Z∗(V ) ≃ Z∗(V) and Z∗(W) ≃ Z∗(W). By Lemma 4.5,
+(3) follows.
+(4) Since ϕ is proper, we have ϕan is proper.
+We may assume that β is prime, moreover,
+assume that X, Y are integral and β = [X], ϕ is finite, surjective. Hence we can assume that
+X = Spec(A) and Y = Spec(B) are affine. Let V = M(A′) ⊂ Xan be an affinoid domain, and
+24
+
+U = (ϕan)−1(V ) = M(A′ ⊗A B). Notice that Frac(B) = B ⊗A Frac(A). We consider the following
+diagram
+Frac(A) ⊗A A′
+� Frac(B) ⊗A A′
+Frac(A)
+�
+�
+Frac(B)
+�
+.
+Notice that Frac(A) → Frac(B) is finite, so Frac(A) ⊗A A′ → Frac(B) ⊗A A′ is finite and flat. We
+have that
+[Frac(B) : Frac(A)] =
+�
+q,ϕ(q)=p
+[(Frac(B) ⊗A A′)q : (Frac(A) ⊗A A′)p]
+where q runs through the minimal ideal of Frac(B)⊗A A′, and we view ϕ : Spec(Frac(B)⊗A A′) →
+Spec(Frac(A) ⊗A A′). The right-handed side is exactly deg(Y an/Xan) defined in Lemma 4.23, so
+(4) holds.
+(5) We can assume that α, β are prime.
+Since flat pull-backs preserve proper intersection,
+by Lemma 5.3, we know that α, β intersect properly if and only if αan, βan ∈ Z∗(Xan) intersect
+properly. The proof of the equality is similar with the proof of (3).
+8
+The category of finite correspondences
+In this section, we will define the additive category CorK of finite correspondences of K-analytic
+spaces. We will follow the notation in [1] and the idea in [13, Lecture 1].
+For the K-analytic spaces in this section, we always mean separated, quasi-paracompact,
+strictly K-analytic spaces, the category of such spaces is exactly the category of separated, quasi-
+paracompact, K-rigid spaces by [3, Theorem 1.6.1].
+A K-analytic space is said to be quasi-smooth if it is geometrically regular at each point, see
+[8, Corollary 5.3.5]. In particular, a quasi-smooth space is regular.
+Definition 8.1. Let X be a quasi-smooth, connected K-analytic space, and Y any K-analytic
+space. An elementary correspondence from X to Y is an irreducible closed subset W of X ×Y
+whose associated integral subspace is finite and surjective over X.
+By an elementary corresponding from a quasi-smooth non-connected K-analytic space X to Y ,
+we mean an elementary correspondence from a connected component of X to Y .
+The group CorK(X, Y ) is the free abelian group generated by the elementary correspondences
+from X to Y . The element of CorK(X, Y ) are called finite correspondences.
+Remark 8.2.
+(1) If X is quasi-smooth, K-analytic space, one important example of elementary
+correspondence from X to Y is the graph Γf of a morphism f : X → Y . If X is not connected,
+the Γf is a finite correspondence from X to Y . Notice that Γf is closed in X × Y since Y is
+separated and Γf is a section of X × Y → X.
+(2) If X is not connected and X = � Xi is the decomposition into its connected components, we
+have CorK(X, Y ) = �
+i
+CorK(Xi, Y ).
+(3) Every closed subspace Z of X × Y which is finite and surjective over X determines a finite
+correspondence [Z] from X to Y .
+Proof. We only consider the case where X is connected. We can write [Z] = �
+i
+ni[Zi], where
+Zi are irreducible component of Z such that Zi → X is surjective, and ni is the geometric
+multiplicity of Zi of Z.
+To define the composition of morphism in the category CorK, we need the following lemmas.
+Lemma 8.3. Let f : T → T ′ be a morphism of K-analytic spaces over another K-analytic space
+S. Let W be an irreducible Zariski-closed subset of T which is finite and surjective over S. Then
+f(W) is irreducible, Zariski-closed in T ′ and finite, surjective over S.
+25
+
+Proof. Since T ′ → S is separated, W → S is finite, hence proper by [2, Corollary 3.3.8], we know
+that W → T ′ is proper, see [5, Proposition 9.6.4]. So f(X) is irreducible Zariski-closed in T ′.
+We replace T, T ′ by W, f(W) respectively, so we assume that T is finite and surjective over
+S, and surjective on T ′. By [2, Corollary 3.3.8], it remains to show that T ′ is proper over S.
+Obviously T ′ → S is quasi-compact since T → T ′ is surjective and T ′ → S quasi-compact. By [2,
+Proposition 2.5.8 (iii)], we have
+T = Int(T/S) = Int(T/T ′) ∩ f −1(Int(T ′/S)) = f −1(Int(T ′/S)),
+this implies that Int(T ′/S) = T ′, i.e. ∂(T ′/S) = ∅. So T ′ is proper over S.
+Lemma 8.4. Let Z be an integral K-analytic space, finite and surjective over a normal K-analytic
+space S. Then for every morphism S′ → S with S′ connected (resp. irreducible), every connected
+(resp. irreducible) component of Z ×S S′ is finite and surjective over S′.
+Proof. This is in fact an algebraic result from [15, Proposition 2.17]. We can assume that S =
+M(A), Z = M(B) and S′ = M(A′) are affinoid. Since B is finite over A, so B′ := B �⊗AA′ =
+B ⊗A A′.
+By [15, Proposition 2.17 (3)], we know that Spec(B) → Spec(A) is universally equidimensional,
+hence universally open.
+Then Spec(B′) → Spec(A′) is open.
+For every connected component
+T = M(C) of M(B′), the morphism Spec(C) → Spec(B′) is open. So M(C) → M(B′) has image
+that is closed and Zariski-open, which is exactly M(B′) since it is connected.
+For the irreducible case, since Spec(B′) → Spec(A′) is equidimensional. Then the image of each
+irreducible component Spec(C) of Spec(B′) is Spec(A′). Since the image of M(C) is a Zariski-
+closed subspace of M(A), it must be M(A).
+Lemma 8.5. Let X, Y, Z be K-analytic spaces. Let V ⊂ X × Y and W ⊂ Y × Z be integral closed
+subspace which are finite and surjective over X and Y respectively. Assume that Y is normal.
+Then V × Z and X × W intersect properly in X × Y × Z, and each component of the push-forward
+of the cycle [V × Z] · [X × W] on X × Z is finite and surjective over X.
+Proof. Notice that V ×Y W ֒→ X × Y ×Y Y × Z ≃ X × Y × Z is the intersection of V × Z and
+X × W in X × Y × Z, see the explanation in the remark. Then we have the following diagram
+V ×Y W
+�
+�
+W
+�
+�
+Z
+V
+�
+�
+Y
+X
+.
+By Lemma 8.4, each component of V ×Y W is finite and surjective over V , so it is also finite and
+surjective over X, and it is of dimension dim X. This implies that V × Z and X × W intersect
+properly in X × Y × Z. By Lemma 8.3, the image of each component of V ×Y W in X × Z is finite
+and surjective over X.
+Definition 8.6. Let CorK be the category defined as follows:
+• Objects: the quasi-smooth K-analytic spaces;
+• Morphisms: the finite correspondences CorK(X, Y ).
+Given V ∈ CorK(X, Y ), W ∈ CorK(Y, Z), we define W ◦V as the push-forward of [V ×Z]·[X ×W]
+on X × Z, which is an element in CorK(X, Z).
+Remark 8.7.
+(1) The composition is associative and bilinear, and the diagonal ∆X is the iden-
+tity for a quasi-smooth K-analytic space X.
+Proof. This is from Proposition 4.28 and Theorem 6.2, see the proof of [9, Proposition 16.1.1]
+for the details.
+26
+
+(2) It is not hard to show that the category QSmK of quasi-smooth K-analytic spaces is fully
+faithful subcategory of CorK.
+(3) By [1, Proposition 2.2.35] and a few work, we can see our definition of CorK coincide with
+[1, Definition 2.2.29].
+Following the idea in [4], we can define higher Chow groups CHn(X, s) for quasi-smooth K-
+analytic spaces.
+By GAGA principle, such definition will coincide with the one for algebraic
+varieties. On the other hand, the higher Chow groups is also defined in [1, Introduction g´en´erale]
+using motives of analytic spaces. It is natural to expect there is a close connection between these
+two and higher Chow groups have similar properties as in the case of algebraic varieties.
+Acknowledgements
+The author would like to thank my host professor, Yigeng Zhao for his encouragement, support
+and valuable suggestions. He would also like to thank Antoine Ducros, Walter Gubler and Michael
+Temkin for their patience and answering questions during his study of Berkovich spaces. This
+research is supported by postdoctoral research grant.
+References
+[1] Ayoub, J. (2015). Motifs des vari´et´es analytiques rigides. M´em. Soc. Math. Fr. (N.S.), (140-
+141):vi+386.
+[2] Berkovich, V. G. (1990). Spectral theory and analytic geometry over non-Archimedean fields,
+volume 33 of Mathematical Surveys and Monographs. American Mathematical Society, Provi-
+dence, RI.
+[3] Berkovich, V. G. (1993). ´Etale cohomology for non-Archimedean analytic spaces. Inst. Hautes
+´Etudes Sci. Publ. Math., (78):5–161 (1994).
+[4] Bloch, S. (1986). Algebraic cycles and higher K-theory. Adv. in Math., 61(3):267–304.
+[5] Bosch, S., G¨untzer, U., and Remmert, R. (1984).
+Non-Archimedean analysis, volume 261
+of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical
+Sciences]. Springer-Verlag, Berlin. A systematic approach to rigid analytic geometry.
+[6] Ducros, A. (2007). Variation de la dimension relative en g´eom´etrie analytique p-adique. Compos.
+Math., 143(6):1511–1532.
+[7] Ducros, A. (2009). Les espaces de Berkovich sont excellents. Ann. Inst. Fourier (Grenoble),
+59(4):1443–1552.
+[8] Ducros, A. (2018). Families of Berkovich spaces. Ast´erisque, (400):vii+262.
+[9] Fulton, W. (1998).
+Intersection theory, volume 2 of Ergebnisse der Mathematik und ihrer
+Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics
+and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics]. Springer-Verlag,
+Berlin, second edition.
+[10] Grothendieck, A. (1967). ´El´ements de g´eom´etrie alg´ebrique. IV. ´Etude locale des sch´emas et
+des morphismes de sch´emas IV. Inst. Hautes ´Etudes Sci. Publ. Math., (32):361.
+[11] Gubler, W. (1998). Local heights of subvarieties over non-Archimedean fields. J. Reine Angew.
+Math., 498:61–113.
+[12] Liu, Q. (2002). Algebraic geometry and arithmetic curves, volume 6 of Oxford Graduate Texts
+in Mathematics. Oxford University Press, Oxford. Translated from the French by Reinie Ern´e,
+Oxford Science Publications.
+27
+
+[13] Mazza, C., Voevodsky, V., and Weibel, C. (2006).
+Lecture notes on motivic cohomology,
+volume 2 of Clay Mathematics Monographs. American Mathematical Society, Providence, RI;
+Clay Mathematics Institute, Cambridge, MA.
+[14] Stacks project authors, T. (2022). The stacks project. https://stacks.math.columbia.edu.
+[15] Voevodsky, V., Suslin, A., and Friedlander, E. M. (2000).
+Cycles, transfers, and motivic
+homology theories, volume 143 of Annals of Mathematics Studies. Princeton University Press,
+Princeton, NJ.
+Y. Cai, Westlake University, Dunyu Road 600, Xihu District 310024, Hangzhou, China
+E-mail address: [email protected]
+28
+

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+Draft version January 6, 2023
+Typeset using LATEX twocolumn style in AASTeX63
+Study of variability in long-term multiwavelength optical lightcurves of blazar AO 0235+164
+Abhradeep Roy
+,1 Alok C. Gupta
+,2, 3 Varsha R. Chitnis
+,1 Sergio A. Cellone
+,4, 5 Claudia M. Raiteri
+,6
+Gustavo E. Romero
+,7, 5 Paul J. Wiita
+,8 Anshu Chatterjee
+,1 Jorge A. Combi
+,5, 7, 9 Mai Liao
+,10, 11
+Arkadipta Sarkar
+,12 and Massimo Villata
+6
+1Department of High Energy Physics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai-400005, India
+2Aryabhatta Research Institute of Observational Sciences (ARIES), Manora Peak, Nainital 263001, India
+3Key Laboratory for Research in Galaxies and Cosmology, Shanghai Astronomical Observatory, Chinese Academy of Sciences, Shanghai
+200030, China
+4Complejo Astron´omico El Leoncito (CASLEO, CONICET-UNLP-UNC-UNSJ), San Juan, Argentina
+5Facultad de Ciencias Astron´omicas y Geof´ısicas, Universidad Nacional de La Plata, La Plata, Buenos Aires, Argentina
+6INAF-Osservatorio Astrofisico di Torino, Via Osservatorio 20, I-10025 Pino Torinese, Italy
+7Instituto Argentino de Radioastronom´ıa (CCT-La Plata, CONICET; CICPBA; UNLP), Buenos Aires, Argentina
+8Department of Physics, The College of New Jersey, 2000 Pennington Rd., Ewing, NJ 08628-0718, USA
+9Deptamento de Ingenier´ıa Mec´anica y Minera, Universidad de Ja´en, Campus Las Lagunillas s/n Ed. A3 Ja´en, 23071, Spain
+10CAS Key Laboratory for Researches in Galaxies and Cosmology, Department of Astronomy, University of Science and Technology of
+China, Hefei, Anhui 230026, China
+11School of Astronomy and Space Science, University of Science and Technology of China, Hefei, Anhui 230026, China
+12Deutsches Elektronen-Synchrotron, Platanenallee 6, D-15738 Zeuthen, Germany
+Submitted to ApJS
+ABSTRACT
+We present a long-term and intraday variability study on optical multiwaveband (UBVRI) data
+from the blazar AO 0235+164 collected by various telescopes for ∼44 years (1975–2019). The blazar
+was found to be significantly variable over the years in all wavebands with a variation of about six
+magnitudes between its low and active states. The variations in the different wavebands are highly
+correlated without any time-lag. We did not observe any significant trend in color variation with time,
+but we observed a bluer-when-brighter trend between the B − I color index and the R-magnitude.
+Optical BVR-band spectral energy distributions always show a convex shape. Significant intraday
+variability was frequently seen in the quasi-simultaneous observations of AO 0235+164 made on 22
+nights in R and V -bands by the CASLEO and CAHA telescopes during 1999–2019. We also estimated
+the central supermassive black-hole mass of 7.9 × 107M⊙ by analyzing the broad Mg II emission line
+in AO 0235+164’s spectrum. We briefly explore the probable physical scenarios responsible for the
+observed variability.
+Keywords: galaxies: active – BL Lacertae objects: general – quasars: individual – BL Lacertae objects:
+individual: AO 0235+164
+1. INTRODUCTION
+Blazars belong to the radio-loud (RL) class of active
+galactic nuclei (AGNs). This extremely variable class
+is the union of BL Lacertae objects (BL Lacs) and
+flat spectrum radio quasars (FSRQs).
+Blazars host a
+Corresponding author: Abhradeep Roy
+[email protected], [email protected]
+large-scale relativistic jet of plasma pointing very close
+to the observer’s line of sight (Urry & Padovani 1995).
+The jet is launched from the very near vicinity of the
+supermassive black hole (SMBH) of mass 106 – 1010
+M⊙ at the center of the AGN (e.g., Woo & Urry 2002).
+Blazars are characterized by highly variable emission
+throughout the whole electromagnetic (EM) spectrum,
+from radio to γ-rays, and their spectral energy distri-
+butions (SEDs) are characterized by two broad humps
+(Fossati et al. 1998).
+Blazars display high and vari-
+arXiv:2301.01944v1  [astro-ph.HE]  5 Jan 2023
+
+ID2
+Roy et al.
+time (JD)
+12
+13
+14
+15
+16
+17
+18
+19
+I mag
+WEBT-GASP
+CASLEO-CAHA
+12
+14
+16
+18
+20
+R mag
+WEBT-GASP
+Hagen-Thorn et al. 2008
+SMARTS
+Steward
+Takalo et al. 1998
+CASLEO-CAHA
+14
+15
+16
+17
+18
+19
+20
+V mag
+WEBT-GASP
+CASLEO-CAHA
+SMARTS
+Steward
+14
+15
+16
+17
+18
+19
+20
+21
+B mag
+WEBT-GASP
+CASLEO-CAHA
+SMARTS
+2444000
+2446000
+2448000
+2450000
+2452000
+2454000
+2456000
+2458000
+Time (JD)
+16
+17
+18
+19
+20
+21
+U mag
+WEBT-GASP
+1980
+1990
+2000
+2010
+2020
+Time (Year)
+Figure 1. Long-term multiwavelength optical (U, B, V , R, I) lightcurves of AO 0235+164 observed from multiple ground-based
+telescopes between JD 2442689 (1975 October 3) and JD 2458835 (2019 December 17).
+
+AO 0235+164 optical variability
+3
+able polarization from radio to optical bands, and emit
+predominately non-thermal emission in the entire EM
+spectrum.
+The low-energy hump is ascribed to syn-
+chrotron radiation from relativistic leptons, whereas the
+high-energy hump arises from inverse Compton (IC)
+processes and sometimes from hadronic processes (e.g.,
+Marscher 1983; M¨ucke et al. 2003; Romero et al. 2017,
+and references therein).
+Blazars display flux variability on diverse timescales
+ranging from a few minutes to several years.
+Blazar
+variability has often been divided into three categories,
+depending on the cadence of the observations: (i) mi-
+crovariability (Miller et al. 1989), or intraday variability
+(IDV) (Wagner & Witzel 1995), or intra-night variabil-
+ity (INV) (Sagar et al. 2004), focusing on the variability
+over a day or less; (ii) short-term variability (STV),
+focusing on variability over days to weeks, (iii) and
+long-term variability (LTV), focusing on timescales of
+months to years (e.g. Gupta et al. 2004).
+The BL Lac object AO 0235+164 is at redshift z =
+0.94 (Cohen et al. 1987).
+Optical spectroscopic and
+photometric observations of the object have discovered
+two foreground-absorbing systems at z = 0.524 and z =
+0.851 (Cohen et al. 1987; Nilsson et al. 1996; Raiteri
+et al. 2007).
+The flux of the source can be both ab-
+sorbed and contaminated by these foreground systems,
+and the stars in them may act as gravitational micro-
+lenses that could contribute to the observed variability.
+Abraham et al. (1993) did deep CFHT imaging of AO
+0235+164 and reported that the source is weakly am-
+plified by macrolensing / microlensing by stars in the
+foreground.
+AO 0235+164 has been extensively observed in the past
+from radio to γ-ray bands either in individual EM bands
+or quasi-simultaneously in multiple EM bands and has
+shown variations in all those bands on diverse timescales
+(e.g., Madejski et al. 1996; Rabbette et al. 1996; Takalo
+et al. 1998; Qian et al. 2000; Webb et al. 2000; Romero
+et al. 2000; Raiteri et al. 2006, 2008; Hagen-Thorn et al.
+2008; Gupta et al. 2008; Agudo et al. 2011; Ackermann
+et al. 2012; Fan et al. 2017; Kutkin et al. 2018; Wang
+& Jiang 2020, and references therein). It is one of the
+blazars which has displayed very high and variable op-
+tical/NIR polarization up to ∼45 percent (e.g., Impey
+et al. 1982; Stickel et al. 1993; Fan & Lin 1999; Cellone
+et al. 2007; Ikejiri et al. 2011; Itoh et al. 2016, and
+references therein). In the Hamburg quasar monitoring
+program (HQM) this source was observed in the optical
+R band during 1988–1993, during which a 2.36±0.25
+magnitude variation was detected; a particularly strong
+brightening in the source of ∼1.6 magnitude was re-
+ported during February 20–22, 1989 (Schramm et al.
+1994). In six nights of optical B and V bands obser-
+vations during 21–27 September 1992, the blazar was
+found in an unusually bright state and IDV was de-
+tected in both B and V bands (Rabbette et al. 1996).
+On another occasion, 6 nights of quasi-simultaneous V
+and R band observations in November 1999, revealed
+IDV with an amplitude of ∼100 percent over timescales
+of a day, while 0.5 magnitude changes were reported
+in both bands on a single night (Romero et al. 2000).
+In multicolor optical/NIR photometric (BVRIJHK)
+and R-band optical polarimetric observations of AO
+0235+164 during its 2006 December outburst, variabil-
+ity on IDV timescales was detected, with increasing
+minimum timescale of variability from optical to NIR
+wavelengths; such variations were even detected in the
+optical polarization (Hagen-Thorn et al. 2008). In three
+nights of optical observations of the blazar in January –
+March 2007, IDV and STV were detected (Gupta et al.
+2008).
+In quasi-simultaneous optical (V and R bands) and
+radio (22 GHz) observations of AO 0235+164 during
+1993–1996, the variability in optical bands showed am-
+plitudes up to 1.5 magnitudes on STV timescales; al-
+though the radio variability is less dramatic, in general,
+it followed the optical behavior (Takalo et al. 1998). For
+the 1997 AO 0235+164 outburst, quasi-simultaneous
+multi-wavelength (MW) (radio, optical, NIR, and X-
+ray) observations were carried out. It was found that
+the source varied nearly simultaneously over 6 decades
+in frequency during the outburst and this result was
+explained in terms of a microlensing event (Webb et al.
+2000).
+An analysis of this source’s variability over ∼25 years
+led to the suggestion of a ∼5.7 years quasi-periodicity
+of the main radio and optical flares (Raiteri et al. 2001);
+however, the putative next outburst, predicted to peak
+around February–March 2004, did not occur, and a
+new analysis of the optical light curves on a longer
+time span revealed a characteristic variability timescale
+of ∼8 years, which was also present in the radio data
+(Raiteri et al. 2006). Recently, optical R band photo-
+metric data taken during 1982–2019 showed 5 cycles
+of double-peaked periodicity of ∼8.13 years with a sec-
+ondary peak following the primary one by ∼(1.5–2.0)
+years (Roy et al. 2022). In another MW campaign from
+radio to UV bands in 2006–2007, a huge NIR-optical-
+UV outburst with brightness increase of ∼5 magnitudes
+
+4
+Roy et al.
+during February 19 – 21, 2007 was detected (Raiteri
+et al. 2008).
+During a major outburst seen in 2009,
+changes in radio, optical, X-ray, and γ-ray bands were
+found to be strongly associated (Agudo et al. 2011).
+In another simultaneous MW observing campaign of
+this blazar between 2008 September and 2009 February,
+γ-ray activity was found to be well correlated with a se-
+ries of NIR/optical flares, accompanied by an increase in
+the optical degree of polarization; the X-ray light curve
+showed a different 20-day high state of an unusually
+soft spectrum which did not match the extrapolation
+of the optical/UV synchrotron spectrum (Ackermann
+et al. 2012).
+AO 0235+164 is one of the sources that often used
+to be called OVV (optically violently variable). There
+are several such objects, like 3C 279, 3C 454.3, 4C
+29.45, CTA 102, BL Lacertae, etc.
+Long-term achro-
+maticity and zero lags have widely been found for these
+sources (Bonning et al. 2012; Zhang et al. 2021; Fan
+et al. 2006; Raiteri et al. 2017; Guo et al. 2015). AO
+0235+164 is peculiar because it is commonly considered
+a BL Lac, one of the furthest known, but it shares
+properties with FSRQs.
+It is also a complex source
+because its light is contaminated by the southern AGN,
+ELISA, and absorbed by an intervening galaxy. This
+paper has undertaken a detailed analysis of the source’s
+optical brightness and spectral variability over a very
+long time span (∼5 decades) as well as an investiga-
+tion of its central engine. Our aim is to shed light on
+the long and short-term behavior of an emblematic BL
+Lac object through a detailed analysis of what is likely
+the most massive data set ever assembled for an object
+of this kind.
+The paper is organized as follows.
+In
+section 2, we provide descriptions of the observations
+of AO 0235+164. The section 3 gives our data analy-
+sis methods and results. We present a discussion and
+conclusions in section 4 and section 5, respectively.
+2. OBSERVATIONS
+Most of the optical UBV RI
+observations of AO
+0235+164 we have employed in this work are taken
+from The Whole Earth Blazar Telescope1 (WEBT)
+(Villata et al. 2002; Raiteri et al. 2017) which is an in-
+ternational collaboration of optical, near-infrared, and
+radio observers. WEBT has organized several monitor-
+ing campaigns on the blazar AO 0235+164, with the
+participation of many tens of observers and telescopes
+all around the world.
+Later, this source was studied
+1 https://www.oato.inaf.it/blazars/webt
+by the WEBT and by its GLAST-AGILE Support Pro-
+gram (GASP) (Villata et al. 2008, 2009), which was
+started in 2007 to record quasi-simultaneous data of
+various blazars observed by the AGILE and Fermi (for-
+merly GLAST) satellites. WEBT/GASP data on AO
+0235+164 were published in Raiteri et al. (2001, 2005,
+2006, 2008) and Ackermann et al. (2012). Raiteri et al.
+(2005) prescribed ways to remove the contribution of
+the southern galaxy ELISA from the observed optical
+flux densities and estimated the amount of absorption
+towards the source in excess of that from our Galaxy in
+X-ray, ultraviolet, optical, and near-infrared bands.
+The WEBT and GASP data were calibrated following
+a common prescription, i.e., with the same photome-
+try for the same reference stars. For calibration of the
+AO 0235+164 observations, the adopted photometric
+sequence includes stars 1, 2, and 3 from Smith et al.
+(1985). To build a reliable lightcurve for further anal-
+ysis, clear outliers were removed and minor systematic
+offsets between various datasets were corrected.
+AO 0235+164 was also observed with the 2.2 m tele-
+scope of Calar Alto Astronomical Observatory (CAHA,
+Spain) in November – December 2005, using the CAFOS
+instrument in imaging polarimetry mode, and photo-
+metric data were obtained by adding up the ordinary
+and extraordinary fluxes from each individual image
+(Cellone et al. 2007).
+Photometric data were also
+obtained with the 2.15 m telescope at Complejo As-
+tron´omico El Leoncito (CASLEO, Argentina) along
+several runs in November 1999, December 2000, August
+2004, and January 2005. Results from these data were
+published in Romero et al. (1999, 2000, 2002) and in
+two papers by the WEBT collaboration focused on this
+blazar (Raiteri et al. 2005, 2006).
+Data from a more
+recent (December 2019) observing run with the same
+telescope were used in Roy et al. (2022).
+Magnitude
+calibration to the standard system was done using our
+own photometry of Landolt’s (2009) fields as well as
+standard stars in the field of AO 0235+164 (Smith
+et al. 1985; Gonz´alez-P´erez et al. 2001).
+We also collected the publicly available optical R and
+V -band data of AO 0235+164, taken at Steward Ob-
+servatory2, University of Arizona. These measurements
+employed the 2.3 m Bok and 1.54 m Kuiper telescopes
+between 4 October 2008 and 12 February 2018, using
+the SPOL CCD Imaging/Spectropolarimeter attached
+2 http://james.as.arizona.edu/∼psmith/Fermi/DATA/Rphotdata.
+html
+
+AO 0235+164 optical variability
+5
+Table 1. Result of flux variability on optical UBVRI long-term
+lightcurves of AO 0235+164
+Optical
+Total
+χ2
+red.
+χ2
+0.999,red.
+Status
+Variability
+filter
+Obs.
+amplitude (%)
+U
+109
+904.5
+1.47
+V
+548.8
+B
+894
+3246.7
+1.15
+V
+590.9
+V
+1403
+5968.4
+1.12
+V
+589.0
+R
+5675
+8715.5
+1.06
+V
+718.8
+I
+1173
+3555.2
+1.13
+V
+567.5
+Note—In the fourth column ’V/NV’ represents variable/non-
+variable status.
+to those two telescopes.
+Details about the instru-
+ment, observation, and data analysis are given in Smith
+et al. (2009).
+In addition, we included the optical-
+BV R data from the Small and Moderate Aperture
+Research Telescope System (SMARTS) public archive3.
+The SMARTS consortium is part of the Cerro Tololo
+Inter-American Observatory (CTIO), Chile, and has
+been observing Fermi-Large Area Telescope (LAT)-
+monitored blazars in the optical B, V , R and NIR J
+and K bands. Details about the SMARTS instruments,
+observations, and data analysis procedures are given
+in Bonning et al. (2012). These standard magnitudes
+observed by CASLEO, CAHA, SMARTS, and the Stew-
+ard observatory were further corrected for the southern
+galaxy ELISA following Raiteri et al. (2005). We also
+added other R-band optical photometric data from the
+literature (Takalo et al. 1998; Hagen-Thorn et al. 2008).
+3. DATA ANALYSIS METHODS AND RESULTS
+We combined all the optical U, B, V , R, I band data
+to plot the long term (1974–2020) MW lightcurves of
+blazar AO 0235+164 (Figure 1). We removed the ob-
+servations with errors of more than 0.1 magnitudes and
+studied long-term and intraday variability, color varia-
+tion, spectral properties, and inter-band correlations.
+3.1. Flux variability studies
+We use different tools on the observed optical magni-
+tudes to quantify the variability timescales and the cor-
+responding significance in multiple optical wavebands.
+3.1.1. The χ2test
+3 http://www.astro.yale.edu/smarts/glast/home.php#
+For a time series of flux density observations, the χ2 is
+defined as,
+χ2 =
+N
+�
+i=1
+(Mi − ¯
+M)2
+ε2
+i
+(1)
+where Mi is the magnitude obtained at the ith observa-
+tion, εi is the corresponding error in measurement, and
+¯
+M is the average magnitude. If the obtained χ2 value
+is higher than the critical χ2 value at 99.9 per cent sig-
+nificance level, we consider the source as variable. The
+critical value (χ2
+0.999,d) depends on the degrees of free-
+dom (d) of the dataset. The reduced χ2 values listed in
+Table 1 indicate that the source exhibits significant flux
+variations in all the optical wavebands.
+3.1.2. Variability amplitude
+According to the relation given by Heidt & Wagner
+(1996), we estimated the variability amplitudes (VM) in
+percentage for the lightcurves in different wavelengths
+using the following formula,
+VM = 100 ×
+�
+(Mmax − Mmin)2 − 2 ¯ε2 (%)
+(2)
+where Mmax and Mmin are the maximum and minimum
+observed magnitude in a lightcurve, respectively, while
+¯ε is the average error in magnitude measurements. We
+list the calculated variability of amplitudes in Table 1.
+3.1.3. Correlation study
+To study the inter-band correlations, we first gener-
+ated 15-minute binned optical UBVRI lightcurves, and
+plotted the average U, B, V , and I-magnitudes against
+the average R-magnitudes for the time bins when the
+source was observed at both the wavebands (Figure 2).
+The magnitude-vs-magnitude plots show very good
+linear correlations. To take the uncertainty of magni-
+tude measurements into account, we simulated 10000
+datasets assuming that each magnitude measurement
+is Gaussian distributed. Then we calculated the mean
+and standard deviation of the Pearson correlation co-
+efficients of all simulated datasets. We obtained high
+correlations (> 0.9) with small uncertainties (< 0.003)
+between all wavebands.
+Moreover, to find any time lag between the correlated
+optical lightcurves we computed the discrete correlation
+function (DCF) from the unbinned multiwavelength
+light curves, as the light curves consist of discrete data
+points.
+Following the method of Edelson & Krolik
+(1988), we computed the unbinned DCF (UDCF) be-
+
+6
+Roy et al.
+15
+16
+17
+18
+R magnitude
+17
+18
+19
+20
+U magnitude
+U-mag vs R-mag
+Pearson coeff. = 0.96±2.93e-03
+fit: Umag = 0.92*Rmag+3.24
+14
+15
+16
+17
+18
+19
+R magnitude
+15
+16
+17
+18
+19
+20
+V magnitude
+V-mag vs R-mag
+Pearson coeff. = 0.99±2.65e-04
+fit: Vmag = 1.00*Rmag+0.79
+14
+15
+16
+17
+18
+19
+R magnitude
+16
+17
+18
+19
+20
+21
+B magnitude
+B-mag vs R-mag
+Pearson coeff. = 0.99±4.30e-04
+fit: Bmag = 1.01*Rmag+1.65
+14
+15
+16
+17
+18
+19
+R magnitude
+13
+14
+15
+16
+17
+18
+19
+I magnitude
+I-mag vs R-mag
+Pearson coeff. = 0.99±2.58e-04
+fit: Imag = 0.98*Rmag-0.64
+Figure 2.
+15-minute averaged UBV I magnitudes versus R-magnitude plots for correlation study.
+U, B, V , and I-band
+observations show high linear correlation with R-band data. All the plots are fitted with straight lines.
+tween the ith data point in one waveband (a) and the
+jth data point in another (b) as
+UDCFij = (ai − ¯a)(bj − ¯b)
+σaσb
+,
+(3)
+where ¯a and ¯b are the mean of the observed magnitudes,
+and σa and σb are the standard deviations of the cor-
+responding datasets. Next, we calculated the discrete
+correlation function (DCF) at a certain time lag τ by
+averaging the UDCFijs whose corresponding time lags
+∆tij = ta
+i − tb
+j lie within the range [τ − ∆τ
+2 , τ + ∆τ
+2 ] (∆τ
+is the time lag bin width), such that,
+DCF(τ) = 1
+n
+�
+UDCFij(τ).
+(4)
+Following the suggestion of White & Peterson (1994),
+we computed the mean magnitudes (¯a and ¯b) and the
+
+AO 0235+164 optical variability
+7
+10.0
+7.5
+5.0
+2.5
+0.0
+2.5
+5.0
+7.5
+10.0
+Time lag (days)
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+1.1
+1.2
+1.3
+DCF
+U vs R
+10.0
+7.5
+5.0
+2.5
+0.0
+2.5
+5.0
+7.5
+10.0
+Time lag (days)
+0.70
+0.75
+0.80
+0.85
+0.90
+0.95
+1.00
+DCF
+B vs R
+10.0
+7.5
+5.0
+2.5
+0.0
+2.5
+5.0
+7.5
+10.0
+Time lag (days)
+0.75
+0.80
+0.85
+0.90
+0.95
+1.00
+DCF
+V vs R
+10.0
+7.5
+5.0
+2.5
+0.0
+2.5
+5.0
+7.5
+10.0
+Time lag (days)
+0.65
+0.70
+0.75
+0.80
+0.85
+0.90
+0.95
+1.00
+DCF
+I vs R
+Figure 3. Results of discrete cross-correlation analysis of U, B, V , and I-band with respect to R-band in the full time range.
+standard deviations (σa and σb) in Equation 3 using only
+those data points who fall within a given time lag bin, as
+the mean and standard deviation keep on changing for a
+time series originated from a stochastic process such as
+blazar emission. The error in the DCF(τ) computation
+in each bin is calculated as
+σDCF(τ) =
+1
+M − 1
+�
+�
+�
+�
+M
+�
+k=1
+(UDCFk − DCF(τ))2.
+(5)
+Figure 3 shows the DCFs of UBV I bands with respect
+to the R-band observations. In all cases, the DCFs peak
+at zero time lag, except the U-band vs R-band DCF
+due to poor data sampling in the U-band. This explains
+the strong linearity in Figure 2 and implies that the
+emission at all optical wavebands are coming from the
+same region in the jet and are produced from the same
+radiation mechanism.
+Table 2. Color variation with time in optical UBVRI long-
+term lightcurves of AO 0235+164
+CI
+m
+c
+ρ
+p
+U − B
+−1.52E-05
+3.74E+01
+−2.06E-01
+8.28E-02
+B − V
+6.58E-06
+−1.52E+01
+1.42E-01
+4.79E-03
+V − R
+−5.34E-06
+1.38E+01
+−9.19E-02
+1.39E-02
+R − I
+1.83E-05
+−4.40E+01
+2.85E-01
+1.74E-08
+U − I
+5.63E-05
+−1.35E+02
+4.03E-01
+1.88E-03
+B − I
+4.16E-05
+−9.92E+01
+4.50E-01
+3.41E-11
+Note—In the column headings: CI: color indices; m = slope;
+c = intercept; ρ = Pearson coefficient; p = null hypothesis
+probability for Figure 4a
+3.1.4. Color Variations
+The term ‘color’ denotes the magnitude difference be-
+tween two quasi-simultaneous observations at two dif-
+
+8
+Roy et al.
+1
+0
+1
+U-B
+0
+1
+2
+B-V
+0
+1
+2
+V-R
+0.0
+1.5
+Color
+R-I
+1.5
+3.0
+4.5
+U-I
+2444000
+2448000
+2452000
+2456000
+Time (JD)
+1.5
+3.0
+B-I
+(a)
+1
+0
+1
+U-B
+0.8
+1.6
+B-V
+0
+1
+2
+V-R
+0.0
+1.5
+Color
+R-I
+1.5
+3.0
+4.5
+U-I
+14
+15
+16
+17
+18
+19
+20
+R magnitude
+1.5
+3.0
+B-I
+(b)
+Figure 4. (a) Color variation with time. (b) Color variation with optical R magnitude. The red line in each panel represents
+the straight line fit. Fit parameters are given in Table 2 and Table 3 respectively.
+Table 3. Color variation with R-band magnitude in optical
+UBVRI long-term lightcurves of AO 0235+164
+CI
+m
+c
+ρ
+p
+U − B
+−1.36E-01
+2.37E+00
+−5.37E-01
+3.35E-05
+B − V
+1.62E-02
+7.04E-01
+1.41E-01
+7.41E-03
+V − R
+−3.54E-03
+7.98E-01
+−2.58E-02
+4.92E-01
+R − I
+1.62E-02
+7.00E-01
+1.37E-01
+7.59E-03
+U − I
+−6.47E-02
+3.85E+00
+−2.07E-01
+1.30E-01
+B − I
+6.23E-02
+1.66E+00
+3.66E-01
+1.69E-07
+Note—In the column headings: CI: color indices; m =
+slope; c = intercept; ρ = Pearson coefficient; p = null
+hypothesis probability for Figure 4b
+ferent wavebands. We plotted the variation of optical
+colors (U − B, B − V , V − R, R − I, and B − I) with
+time and R-magnitude in Figure 4. We listed the re-
+sults of a straight line (Y = mX + c) fitting to all these
+plots in Table 2 and Table 3. The linear fits of the color
+versus time plots do not show any trend, except for the
+rather sparsely sampled (B − I) color, which has a high
+slope (4.16×10−5) in Figure 4a, along with the highest
+Pearson correlation coefficient (0.45), and the lowest null
+hypothesis probability (3.41×10−11). Among the color
+versus magnitude relations, the strongest relationship is
+between (B − I) and R (Figure 4b), having a positive
+slope (6.23×10−2) with the highest Pearson coefficient
+(0.37) and the lowest p-value (1.69×10−7) (Table 3), in-
+dicates a bluer-when-brighter (BWB) trend when the
+widest range of the available colors is considered.
+3.1.5. Spectral Variations and SEDs
+We plotted the optical (BVR) spectral energy distri-
+butions for the nights where observations were taken
+at all of these three filters. Following the prescription
+of Raiteri et al. (2005), we took into account the total
+absorption by the Milky Way galaxy and the foreground
+
+AO 0235+164 optical variability
+9
+Figure 5. An example frame of the AO 0235+164 optical SED animation that is available in the HTML version of this article.
+The duration of the animation is 1 minute and it contains a total of 360 one-day averaged optical SEDs, having 6 SEDs per
+frame. The observation dates of the SEDs are given in the plot legend.
+Table 4. Spetral index variation with R-band magnitude and
+time in optical UBVRI long-term lightcurves of AO 0235+164
+Dependency
+m
+c
+ρ
+p
+αV R vs R
+−2.01E-02
+3.30E+00
+−2.58E-02
+4.92E-01
+αV R vs JD
+−3.03E-05
+7.74E+01
+−9.19E-02
+1.39E-02
+Note—In the column headings: m = slope; c = intercept; ρ =
+Pearson coefficient; p = null hypothesis probability for Figure 7.
+absorber at z = 0.524, and subtracted the extinction
+magnitudes (AU = 2.519, AB = 1.904, AV = 1.473,
+AR = 1.260, AI = 0.902) from the calibrated magni-
+tudes of respective wavebands and then converted them
+into extinction-corrected flux densities, Fν. The accom-
+panying video contains one-day averaged optical SEDs
+for those 360 nights (An example frame is shown in
+Figure 5). Figure 6 shows a few examples of SEDs of
+low, moderate, and high flux states, plotted in (νFν –
+ν) format.
+Mostly, the SEDs have a declining shape
+following a power law. However, there are evidences of
+spectral hardening on several nights (e.g., JD 2445337,
+JD 2445721, JD 2448889, JD 2452901, JD 2453230).
+From the one-day binned multiwavelength lightcurves
+we calculated the spectral indices (αV R) for all the days
+when the source was observed in both V and R bands,
+
+AO 0235+164 Optical SEDs
+10-10
+10-11
+JD 2448265
+JD 2448266
+JD 2448268
+JD 2448269
+10-12
+JD 2448889
+JD 2449601
+5 × 1014
+6 × 1014
+7 × 1014
+V (HZ)10
+Roy et al.
+5 × 1014
+6 × 1014
+7 × 1014
+8 × 1014
+ (HZ)
+10
+12
+10
+11
+10
+10
+F  (erg cm
+2 s
+1)
+JD 2449690
+JD 2452169
+JD 2445343
+JD 2451896
+JD 2457045
+JD 2450811
+JD 2454733
+JD 2446763
+JD 2453230
+Figure 6. Examples of AO 0235+164 optical intraday SEDs
+during three different states of brightness: (i) the green lines
+represent SED during quiescent states (νFν (erg cm−2 s−1)
+< 10−12), (ii) the blue lines show SED during moderately
+bright states (10−12 < νFν (erg cm−2 s−1) < 3×10−11), (iii)
+the red lines show SED during outbursts (νFν (erg cm−2
+s−1) > 5×10−11). The black lines are examples of SED with
+spectral hardening on JD 2446763 and JD 2453230.
+using the formula given by Wierzcholska et al. (2015)
+on extinction corrected magnitudes, as
+αV R = 0.4(V − R)
+log(νV /νR) ,
+(6)
+where νV and νR respectively represent the effective fre-
+quencies of V and R band filters (Bessell 2005).
+We
+plotted the variation of spectral indices with time and
+R-band magnitude (Figure 7) and listed the results of
+linear fits, Pearson coefficient, and null hypothesis prob-
+ability in Table 4. We do not find any significant long-
+term variation of the spectral index with time, nor is
+there a correlation with R-magnitude.
+3.2. Intraday Variability
+We applied four frequently used statistical tests for IDV:
+scaled C-criterion, scaled F-test, the power-enhanced F-
+test, and the nested analysis of variance (ANOVA) test
+(de Diego 2014; de Diego et al. 2015; Zibecchi et al.
+2017, 2020) to detect statistically significant intraday
+flux variability in AO 0235+164 lightcurves observed by
+CASLEO and CAHA telescopes.
+These tests mainly
+compare the variations in blazar magnitudes with the
+variations in magnitudes of one or more stars within
+the field-of-view of the blazar and have different advan-
+tages and disadvantages. We collected data from mul-
+tiple field stars along with the blazar data (Table 5).
+We applied the first three methods on the intraday dif-
+ferential lightcurves of AO 0235+164 where at least 10
+observations were recorded per night with at least one
+optical filter between 1999 November 2 to 2019 Decem-
+ber 17. We employed the nested ANOVA test only on
+lightcurves having at least 20 observations per night.
+Table 5. Equivalence between internal field star numbering in
+the CASLEO/CAHA data used in the IDV analyses and field-
+star numbering in other standard star charts during different
+observation seasons
+Season
+CASLEO/CAHA
+Heidelberga
+GKM2001b
+1999–2001
+2
+8
+10
+(CASLEO)
+4
+C1
+9
+5
+6
+11
+7
+–
+1
+8
+–
+3
+10
+–
+8
+12
+–
+16
+2004–2005
+2
+8
+10
+(CASLEO)
+4
+C1
+9
+5
+6
+11
+6
+–
+8
+7
+–
+7
+2005
+2
+8
+10
+(CAHA)
+11
+C1
+9
+12
+–
+1
+13
+–
+3
+14
+–
+7
+15
+–
+8
+16
+6
+11
+17
+–
+16
+2018–2019
+2
+8
+10
+(CASLEO)
+4
+C1
+9
+5
+6
+11
+6
+–
+8
+7
+–
+7
+8
+–
+16
+Note—a.
+https://www.lsw.uni-heidelberg.de/projects/
+extragalactic/charts/0235+164.html
+b. Gonz´alez-P´erez et al. (2001)
+3.2.1. Scaled C-criterion
+Differential photometry, where the blazar magnitudes
+are compared to one or more stars in the same field
+of view, is the usual technique for obtaining blazar
+lightcurves free from the effects of any non-astrophysical
+fluctuations.
+The simplest differential photometry in-
+volves a single comparison star, while a second star,
+whose magnitudes are measured against the same com-
+parison star, is used for a stability check. We denote B,
+S1, and S2 as the blazar, comparison, and control star,
+respectively. The variability test requires two differen-
+tial lightcurves (DLC): (blazar–comparison star) and
+(control star–comparison star). The latter is believed
+
+AO 0235+164 optical variability
+11
+Table 6. Result of scaled C-criterion and F-test for IDV on AO 0235+164 differential lightcurves from
+CASLEO and CAHA
+Date
+JD
+Band
+No. of
+S1, S2
+Γ
+CΓ
+FΓ
+F 0.005
+c
+Status
+Final
+obs.
+Status
+1999 Nov 2
+2451485
+V
+23
+2,3
+0.8886
+11.3640
+129.1405
+3.1246
+V
+V
+2,6
+1.0867
+12.9184
+166.8856
+3.1912
+V
+2,10
+1.6876
+8.1627
+66.6298
+3.1246
+V
+2,11
+0.7431
+13.4002
+179.5650
+3.1246
+V
+1999 Nov 3
+2451486
+V
+22
+2,3
+1.0707
+5.6976
+32.4624
+3.1347
+V
+V
+2,11
+0.8841
+6.0726
+36.8768
+3.1347
+V
+1999 Nov 4
+2451487
+R
+30
+2,3
+1.0059
+8.4058
+70.6582
+2.6737
+V
+V
+2,11
+0.6639
+9.8857
+97.7278
+2.6737
+V
+V
+30
+2,3
+0.9994
+8.9281
+79.7104
+2.6737
+V
+V
+2,11
+0.8286
+9.6683
+93.4770
+2.6737
+V
+1999 Nov 5
+2451488
+R
+23
+2,3
+1.4994
+1.5631
+2.4433
+3.1246
+NV
+NV
+2,11
+0.9852
+1.9303
+3.7260
+3.1246
+NV
+V
+22
+2,3
+1.4403
+3.0342
+9.2064
+3.1347
+V
+V
+1999 Nov 6
+2451489
+R
+30
+2,3
+0.8471
+17.5775
+308.9682
+2.6737
+V
+V
+2,6
+0.9769
+12.3281
+151.9824
+2.6737
+V
+2,7
+1.3573
+9.9373
+98.7501
+2.7048
+V
+2,8
+1.3805
+9.8381
+96.7876
+2.7048
+V
+2,10
+1.6936
+6.8657
+47.1376
+2.6737
+V
+2,11
+0.5616
+15.4338
+238.2019
+2.6737
+V
+V
+29
+2,3
+0.8485
+18.1892
+330.8486
+2.7233
+V
+V
+2,6
+1.0013
+11.7527
+138.1254
+2.7233
+V
+2,7
+1.3527
+12.5480
+157.4516
+2.7397
+V
+2,8
+1.4133
+13.4172
+180.0214
+2.7397
+V
+2,10
+1.5626
+17.6674
+312.1376
+2.7233
+V
+2,11
+0.7018
+17.9948
+323.8145
+2.7233
+V
+1999 Nov 7
+2451490
+R
+11
+2,3
+0.9562
+3.5930
+12.9095
+5.8479
+V
+PV
+2,4
+1.9798
+2.2801
+5.1990
+5.8479
+NV
+2,6
+1.1143
+4.3903
+19.2751
+5.8479
+V
+2,10
+1.9703
+1.7073
+2.9148
+5.8479
+NV
+2,11
+0.6197
+2.9496
+8.7003
+5.8479
+V
+V
+12
+2,3
+0.9382
+2.9304
+8.5871
+5.3191
+V
+PV
+2,4
+1.7807
+1.9342
+3.7410
+5.3191
+NV
+2,6
+1.1169
+2.8931
+8.3701
+5.3191
+V
+2,10
+1.7653
+2.1046
+4.4292
+5.3191
+NV
+2,11
+0.7772
+4.3359
+18.7997
+5.3191
+V
+Note—S1 and S2 are the comparison and control star numbers, respectively, used for the IDV tests. Star
+numbers follow the star maps shown in Table 5.
+
+12
+Roy et al.
+13
+14
+15
+16
+17
+18
+R magintude
+2
+0
+2
+4
+6
+8
+10
+VR
+-0.02*R+3.30
+(a)
+2445000
+2447500
+2450000
+2452500
+2455000
+2457500
+Time (JD)
+2
+0
+2
+4
+6
+8
+10
+VR
+-3.03e-05*Time+7.74e+01
+(b)
+Figure 7. (a) Variation of spectral index (αV R) with R-band magnitude. (b) Variation of αV R with time. The red line at each
+panel represents the linear fit.
+Table 6. Result of scaled C-test and F-test for IDV on AO 0235+164 differential lightcurves from CASLEO
+and CAHA (continued...)
+Date
+JD
+Band
+No. of
+S1, S2
+Γ
+CΓ
+FΓ
+F 0.005
+c
+Status
+Final
+obs.
+status
+2000 Dec 21
+2451900
+R
+10
+2,3
+0.9446
+2.3638
+5.5876
+6.5402
+NV
+PV
+2,6
+1.0793
+4.9877
+24.8767
+6.5402
+V
+2,7
+1.5020
+2.0187
+4.0753
+6.5402
+NV
+2,8
+1.5289
+1.9985
+3.9939
+6.5402
+NV
+2,9
+0.8790
+2.5120
+6.3100
+6.5402
+NV
+2,11
+0.6246
+7.4168
+55.0085
+6.5402
+V
+V
+10
+2,3
+0.9509
+3.4671
+12.0208
+6.5402
+V
+PV
+2,6
+1.1202
+2.4789
+6.1449
+6.5402
+NV
+2,7
+1.5357
+1.8729
+3.5079
+6.5402
+NV
+2,8
+1.6064
+2.0031
+4.0124
+6.5402
+NV
+2,9
+1.0966
+3.7299
+13.9120
+6.5402
+V
+2,11
+0.7842
+1.5920
+2.5343
+6.5402
+NV
+2000 Dec 23
+2451902
+R
+10
+2,3
+0.8588
+4.4475
+19.7803
+6.5402
+V
+V
+2,6
+0.9890
+5.1629
+26.6559
+6.5402
+V
+2,7
+1.3855
+3.5919
+12.9020
+6.5402
+V
+2,8
+1.4091
+2.8222
+7.9646
+6.5402
+V
+2,9
+0.8000
+4.6739
+21.8451
+6.5402
+V
+2,11
+0.5664
+5.3690
+28.8267
+6.5402
+V
+2,13
+1.7083
+3.0181
+9.1089
+6.5402
+V
+V
+11
+2,3
+0.8509
+6.5241
+42.5634
+5.8479
+V
+PV
+2,6
+1.0031
+5.4277
+29.4602
+5.8479
+V
+2,7
+1.3714
+5.0139
+25.1395
+5.8479
+V
+2,8
+1.4341
+5.2879
+27.9619
+5.8479
+V
+2,9
+0.9797
+1.4805
+2.1919
+5.8479
+NV
+2,11
+0.7013
+5.1765
+26.7965
+5.8479
+V
+2,13
+1.5668
+4.3770
+19.1586
+5.8479
+V
+Note—S1 and S2 are the comparison and control star numbers respectively used for the IDV tests. Star
+numbers follow the star maps shown in Table 5.
+
+AO 0235+164 optical variability
+13
+0.55
+0.60
+0.65
+0.70
+0.75
+0.80
+JD (+2451485)
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+Differential magnitude
+Date: 1999 Nov 02
+[Status: Variable]
+V band
+Blazar-S1
+(S2-S1)
+0.45
+0.50
+0.55
+0.60
+0.65
+0.70
+JD (+2453680)
+0.625
+0.650
+0.675
+0.700
+0.725
+0.750
+0.775
+0.800
+0.825
+Differential magnitude
+Date: 2005 Nov 05
+[Status: Variable]
+R band
+Blazar-S1
+(S2-S1)+0.02
+0.54
+0.56
+0.58
+0.60
+0.62
+0.64
+0.66
+0.68
+JD (+2451900)
+0.50
+0.52
+0.54
+0.56
+0.58
+0.60
+0.62
+0.64
+Differential magnitude
+Date: 2000 Dec 21
+[Status: Probably Variable]
+V band
+Blazar-S1
+(S2-S1)
+0.46
+0.48
+0.50
+0.52
+0.54
+0.56
+0.58
+JD (+2453711)
+1.40
+1.42
+1.44
+1.46
+1.48
+Differential magnitude
+Date: 2005 Dec 06
+[Status: Probably Variable]
+R band
+Blazar-S1
+(S2-S1)+0.69
+0.55
+0.60
+0.65
+0.70
+0.75
+0.80
+JD (+2452225)
+1.650
+1.675
+1.700
+1.725
+1.750
+1.775
+1.800
+Differential magnitude
+Date: 2001 Nov 11
+[Status: Non Variable]
+V band
+Blazar-S1
+(S2-S1)+1
+0.56
+0.58
+0.60
+0.62
+0.64
+0.66
+JD (+2458835)
+2.20
+2.22
+2.24
+2.26
+2.28
+2.30
+2.32
+2.34
+2.36
+Differential magnitude
+Date: 2019 Dec 17
+[Status: Non Variable]
+R band
+Blazar-S1
+(S2-S1)+1.55
+Figure 8. Some intraday lightcurves of AO 0235+164 on nights when the source showed different states of variability. S1 and
+S2 represent the comparison and control star respectively. In some panels, the differential lightcurve of the control star is shifted
+to bring it into the same frame of the blazar DLC for better visual comparison of variability.
+
+14
+Roy et al.
+Table 6. Result of scaled C-test and F-test for IDV on AO 0235+164 differential lightcurves from CASLEO
+and CAHA (continued...)
+Date
+JD
+Band
+No. of
+S1, S2
+Γ
+CΓ
+FΓ
+F 0.005
+c
+Status
+Final
+obs.
+status
+2001 Nov 9
+2452223
+R
+12
+2,11
+1.2042
+4.6476
+21.5998
+5.3191
+V
+V
+V
+12
+2,3
+1.8778
+2.5035
+6.2675
+5.3191
+NV
+NV
+2,4
+3.6191
+1.1450
+1.3111
+5.3191
+NV
+2,9
+2.2039
+1.2380
+1.5326
+5.4171
+NV
+2,10
+3.5871
+2.0056
+4.0226
+5.3191
+NV
+2,11
+1.5366
+1.7566
+3.0857
+5.3191
+NV
+2001 Nov 10
+2452224
+R
+10
+2,3
+2.3728
+1.0570
+1.1172
+6.5402
+NV
+NV
+2,9
+2.2429
+1.1058
+1.2229
+6.5402
+NV
+2,11
+1.5595
+0.9395
+0.8826
+6.5402
+NV
+V
+10
+2,3
+2.3876
+1.0788
+1.1637
+6.5402
+NV
+NV
+2,9
+2.7847
+1.3860
+1.9209
+6.5402
+NV
+2,11
+1.9713
+0.9038
+0.8168
+6.5402
+NV
+2001 Nov 11
+2452225
+R
+14
+2,3
+2.0291
+1.4125
+1.9951
+4.5724
+NV
+NV
+2,9
+1.5447
+1.2505
+1.5638
+4.6425
+NV
+2,11
+1.3171
+1.6860
+2.8427
+4.5724
+NV
+V
+14
+2,3
+2.0291
+1.4125
+1.9951
+4.5724
+NV
+NV
+2,9
+1.5447
+1.2505
+1.5638
+4.6425
+NV
+2,11
+1.3171
+1.6860
+2.8427
+4.5724
+NV
+2001 Nov 12
+2452226
+R
+12
+2,3
+1.8479
+1.5819
+2.5025
+5.3191
+NV
+PV
+2,11
+1.2074
+3.0203
+9.1222
+5.3191
+V
+V
+12
+2,3
+1.8704
+1.9230
+3.6980
+5.3191
+NV
+NV
+2,4
+3.5981
+1.0281
+1.0571
+5.3191
+NV
+2,10
+3.5672
+2.3374
+5.4634
+5.3191
+NV
+2,11
+1.5330
+1.5642
+2.4468
+5.3191
+NV
+2001 Nov 13
+2452227
+R
+11
+3,4
+2.0213
+1.1434
+1.3073
+5.8479
+NV
+NV
+V
+11
+3,4
+1.1840
+0.6858
+0.4703
+5.8479
+NV
+NV
+Note—S1 and S2 are the comparison and control star numbers respectively used for the IDV tests. Star
+numbers follow the star maps shown in Table 5.
+
+AO 0235+164 optical variability
+15
+Table 6. Result of scaled C-test and F-test for IDV on AO 0235+164 differential lightcurves from CASLEO
+and CAHA (continued...)
+Date
+JD
+Band
+No. of
+S1, S2
+Γ
+CΓ
+FΓ
+F 0.005
+c
+Status
+Final
+obs.
+status
+2005 Jan 16
+2453387
+R
+11
+2,3
+1.5238
+3.8074
+14.4962
+5.8479
+V
+V
+2,4
+3.3465
+2.7388
+7.5010
+5.8479
+V
+2,6
+3.3316
+2.7442
+7.5308
+5.8479
+V
+2,7
+1.4051
+3.4058
+11.5996
+5.8479
+V
+2005 Nov 2
+2453677
+R
+32
+2,3
+1.2848
+6.4237
+41.2636
+2.5846
+V
+V
+2,4
+0.8959
+4.5227
+20.4545
+2.5846
+V
+2,5
+0.2571
+4.3013
+18.5013
+2.5846
+V
+2,6
+0.5453
+3.9310
+15.4528
+2.5846
+V
+2,7
+0.5844
+4.9283
+24.2884
+2.5846
+V
+2,8
+0.4738
+7.0560
+49.7865
+2.5846
+V
+2,9
+0.3415
+6.5374
+42.7373
+2.5846
+V
+2,10
+0.3397
+4.0828
+16.6695
+2.5846
+V
+2005 Nov 4
+2453679
+R
+12
+2,3
+0.8534
+4.3059
+18.5409
+5.3191
+V
+V
+2,4
+0.5599
+3.7978
+14.4235
+5.3191
+V
+2,5
+0.1421
+5.0805
+25.8111
+5.3191
+V
+2,6
+0.3029
+5.3341
+28.4524
+5.3191
+V
+2,7
+0.3534
+7.6846
+59.0525
+5.3191
+V
+2,8
+0.2839
+4.3153
+18.6220
+5.3191
+V
+2,9
+0.1914
+11.0664
+122.4647
+5.3191
+V
+2,10
+0.1875
+5.4019
+29.1804
+5.3191
+V
+2005 Nov 5
+2453680
+R
+44
+2,3
+0.9749
+10.6766
+113.9894
+2.2266
+V
+V
+2,4
+0.6398
+9.3431
+87.2939
+2.2266
+V
+2,5
+0.1942
+11.0439
+121.9674
+2.2266
+V
+2,6
+0.3721
+10.7338
+115.2142
+2.2266
+V
+2,7
+0.4059
+10.2100
+104.2433
+2.2266
+V
+2,8
+0.3427
+8.3494
+69.7127
+2.2266
+V
+2,9
+0.2399
+12.1459
+147.5239
+2.2266
+V
+2,10
+0.2340
+8.5775
+73.5744
+2.2341
+V
+2005 Nov 6
+2453681
+R
+40
+2,3
+1.0022
+7.8517
+61.6495
+2.3212
+V
+V
+2,4
+0.6946
+8.8524
+78.3645
+2.3212
+V
+2,5
+0.2051
+6.6830
+44.6620
+2.3212
+V
+2,6
+0.4022
+7.6630
+58.7211
+2.3212
+V
+2,8
+0.3694
+5.9489
+35.3890
+2.3212
+V
+2,9
+0.2576
+6.8684
+47.1751
+2.3212
+V
+2,10
+0.2563
+5.1520
+26.5433
+2.3212
+V
+Note—S1 and S2 are the comparison and control star numbers respectively used for the IDV tests. Star
+numbers follow the star maps shown in Table 5.
+
+16
+Roy et al.
+Table 6. Result of scaled C-test and F-test for IDV on AO 0235+164 differential lightcurves from CASLEO
+and CAHA (continued...)
+Date
+JD
+Band
+No. of
+S1, S2
+Γ
+CΓ
+FΓ
+F 0.005
+c
+Status
+Final
+obs.
+status
+2005 Nov 8
+2453683
+R
+28
+2,3
+0.9329
+2.4256
+5.8834
+2.7940
+NV
+NV
+2,4
+0.6336
+2.3363
+5.4585
+2.7770
+NV
+2,5
+0.1788
+1.7843
+3.1836
+2.7770
+NV
+2,6
+0.3598
+2.2163
+4.9120
+2.7770
+NV
+2,7
+0.4059
+1.4945
+2.2335
+2.7770
+NV
+2,8
+0.3451
+2.1606
+4.6682
+2.9002
+NV
+2,9
+0.2318
+1.3895
+1.9307
+2.7770
+NV
+2005 Dec 5
+2453710
+R
+20
+2,3
+1.4796
+1.4053
+1.9748
+3.4317
+NV
+NV
+2,4
+1.0247
+0.7240
+0.5242
+3.4317
+NV
+2,5
+0.3133
+0.9355
+0.8752
+3.4317
+NV
+2,6
+0.6030
+1.1896
+1.4151
+3.4317
+NV
+2,8
+0.5634
+1.0332
+1.0674
+3.4317
+NV
+2,9
+0.3979
+1.0716
+1.1482
+3.4317
+NV
+2,10
+0.3915
+0.8994
+0.8089
+3.4317
+NV
+2005 Dec 6
+2453711
+R
+16
+2,3
+1.4092
+2.1709
+4.7129
+4.0698
+NV
+PV
+2,4
+0.9785
+3.7432
+14.0118
+4.0698
+V
+2,5
+0.2848
+2.1323
+4.5467
+4.0698
+NV
+2,6
+0.5570
+2.7562
+7.5967
+4.0698
+V
+2,7
+0.6157
+1.8489
+3.4186
+4.0698
+NV
+2,8
+0.5266
+1.3717
+1.8815
+4.0698
+NV
+2,8
+0.5266
+1.3717
+1.8815
+4.0698
+NV
+2,9
+0.3691
+2.4056
+5.7869
+4.0698
+NV
+2,10
+0.3688
+2.0671
+4.2727
+4.0698
+NV
+2019 Dec 17
+2458835
+R
+30
+9,10
+1.3151
+1.2773
+1.6315
+2.6740
+NV
+NV
+9,11
+0.7377
+1.3155
+1.7307
+2.6740
+NV
+9,12
+1.0425
+1.0698
+1.1445
+2.6740
+NV
+Note—S1 and S2 are the comparison and control star numbers respectively used for the IDV tests. Star
+numbers follow the star maps shown in Table 5.
+
+AO 0235+164 optical variability
+17
+Table 7. Result of power enhanced F-test and nested ANOVA test for IDV on AO 0235+164 differential lightcurves from CASLEO and CAHA
+Obs.
+Band
+No. of
+Power enhanced F-test
+Nested ANOVA test
+Status
+Variability
+doubling
+date
+Obs.
+Comp.
+amplitude(%)
+timescale
+star
+DOF(ν1,ν2)
+Fenh
+F 0.005
+c
+DOF(ν1,ν2)
+F
+F 0.005
+c
+(days)
+1999 Nov 2
+V
+23
+2
+(22, 87)
+116.132
+2.209
+(5, 17)
+58.924
+5.075
+V
+43.99
+0.103
+1999 Nov 3
+V
+22
+2
+(21, 42)
+34.529
+2.540
+(5, 16)
+10.920
+5.212
+V
+24.47
+0.145
+1999 Nov 4
+V
+30
+2
+(29, 58)
+86.046
+2.216
+(7, 22)
+38.922
+4.109
+V
+34.48
+0.106
+R
+30
+(29, 58)
+82.016
+2.216
+(7, 22)
+40.356
+4.109
+V
+32.59
+0.083
+1999 Nov 5
+V
+22
+2
+(21, 21)
+9.207
+3.216
+(5, 16)
+4.426
+5.212
+NV
+10.94
+0.140
+R
+23
+(22, 44)
+2.951
+2.487
+(5, 17)
+9.426
+5.075
+V
+9.03
+0.335
+1999 Nov 6
+V
+29
+2
+(28, 166)
+211.363
+1.960
+(7, 21)
+58.114
+4.179
+V
+36.79
+0.092
+R
+30
+(29, 170)
+107.913
+1.941
+(7, 22)
+74.686
+4.109
+V
+37.90
+0.085
+1999 Nov 7
+V
+12
+2
+(11, 55)
+6.392
+2.854
+–
+–
+–
+PV
+9.13
+0.170
+R
+11
+(10, 50)
+6.413
+2.988
+–
+–
+–
+PV
+5.36
+0.244
+2000 Dec 21
+V
+10
+2
+(9, 54)
+4.813
+3.055
+–
+–
+–
+PV
+6.95
+0.275
+R
+10
+(9, 54)
+6.73
+3.055
+–
+–
+–
+PV
+7.67
+0.428
+2000 Dec 23
+V
+11
+2
+(10, 70)
+10.314
+2.846
+–
+–
+–
+PV
+20.58
+0.200
+R
+10
+(9, 63)
+14.542
+2.989
+–
+–
+–
+PV
+14.18
+0.180
+2001 Nov 9
+V
+12
+2
+(11, 54)
+2.345
+2.863
+–
+–
+–
+NV
+12.13
+0.372
+R
+12
+(11, 22)
+5.91
+3.612
+–
+–
+–
+PV
+12.73
+0.441
+2001 Nov 10
+V
+10
+2
+(9, 27)
+1.152
+3.557
+–
+–
+–
+NV
+8.49
+0.227
+R
+10
+(9, 27)
+1.054
+3.557
+–
+–
+–
+NV
+5.64
+0.660
+2001 Nov 11
+V
+14
+2
+(13, 38)
+2.02
+2.923
+–
+–
+–
+NV
+9.63
+0.364
+R
+14
+(13, 38)
+2.02
+2.923
+–
+–
+–
+NV
+9.63
+0.364
+2001 Nov 12
+V
+12
+2
+(11, 44)
+2.212
+2.969
+–
+–
+–
+NV
+12.06
+0.539
+R
+12
+2
+(11, 22)
+3.928
+3.612
+–
+–
+–
+PV
+11.74
+0.856
+2001 Nov 13
+V
+11
+3
+(10, 10)
+0.470
+5.847
+–
+–
+–
+NV
+10.81
+0.160
+R
+11
+3
+(10, 10)
+1.307
+5.847
+–
+–
+–
+NV
+10.19
+0.178
+2005 Jan 16
+R
+11
+2
+(10, 40)
+9.842
+3.117
+–
+–
+–
+PV
+32.92
+0.095
+2005 Nov 2
+R
+32
+2
+(31, 247)
+27.709
+1.868
+(7, 24)
+37.156
+3.991
+V
+8.98
+0.189
+2005 Nov 4
+R
+12
+2
+(11, 88)
+31.995
+2.689
+–
+–
+–
+V
+6.59
+0.166
+2005 Nov 5
+R
+44
+2
+(43, 343)
+124.459
+1.713
+(10, 33)
+16.301
+3.26
+V
+13.60
+0.146
+2005 Nov 6
+R
+40
+2
+(39, 273)
+57.755
+1.767
+(9, 30)
+87.95
+3.45
+V
+9.79
+0.227
+2005 Nov 8
+R
+28
+2
+(27, 182)
+4.371
+1.965
+(6, 21)
+0.449
+4.393
+PV
+3.18
+0.365
+2005 Dec 5
+R
+20
+2
+(19, 133)
+1.067
+2.200
+(4, 15)
+14.394
+5.803
+PV
+2.61
+0.391
+2005 Dec 6
+R
+16
+2
+(15, 120)
+4.863
+2.373
+–
+–
+–
+PV
+3.53
+0.746
+2019 Dec 17
+R
+30
+9
+(29, 87)
+1.453
+2.075
+(7, 22)
+2.341
+4.109
+NV
+7.74
+0.038
+Note—Comparison star numbers follow the star maps shown in Table 5.
+to be affected only by instrumental fluctuations as any
+known or suspected variable star can be discarded.
+Jang & Miller (1997) and Romero et al. (1999) in-
+troduced a parameter C defined as C = σB−S1/σS2−S1,
+where σB−S1 and σS2−S1 are the standard deviations in
+blazar DLC and control star DLC, respectively.
+The
+blazar is considered to be variable with 99.5 per cent
+confidence level if C is greater than a critical value of
+2.576.
+Howell et al. (1988) pointed out that it is important
+to select non-variable stars with magnitudes close to
+the blazar magnitude as comparison and control stars.
+Otherwise, even if the blazar is non-variable, there will
+be difference between σB−S1 and σS2−S1 due to dif-
+
+18
+Roy et al.
+Figure 9. Spectral fitting of AO 0235+164, where the black
+line is the original spectrum while the green line is the single
+power law for the fitted continuum.
+The inset shows Mg
+II line fitting where the blue, green, and red lines are the
+narrow, broad, and total components, respectively.
+ferences in photon statistics and other random-noise
+terms (sky, read-out noise). To use field stars with dif-
+ferent magnitude levels, Howell et al. (1988) suggests
+calculating a correction factor Γ to scale σS2−S1 to the
+instrumental level of σB−S1 for proper comparison. Γ
+can be estimated using the following formula:
+Γ2 =
+�NS2
+NB
+�2 � N 2
+S1(NB + P) + N 2
+B(NS1 + P)
+N 2
+S2(NS1 + P) + N 2
+S1(NS2 + P)
+�
+(7)
+where N is the total (sky-subtracted) counts within the
+aperture, while the sub-indices B, S1 and S2 correspond
+to N of the blazar, comparison star and control star,
+respectively. The factor P contains the common noise-
+terms, as P = npix(Nsky + N 2
+RON), where npix is the
+number of pixels within the aperture, Nsky is the sky
+level and NRON is the read-out noise. We used the me-
+dian values of N of the objects and sky for calculating
+Γ. Thus, the scaled C parameter (CΓ) is defined as
+CΓ = C
+Γ = 1
+Γ
+� σB−S1
+σS2−S1
+�
+.
+(8)
+The source is considered variable if CΓ ≥ 2.576. Even
+though the C parameter is not a proper statistic, it re-
+mains a useful indicator of stability (de Diego 2014; de
+Diego et al. 2015; Zibecchi et al. 2017, 2020).
+3.2.2. Scaled F-test
+The
+standard
+F-statistics
+parameter
+is
+F
+=
+σ2
+B−S1/σ2
+S2−S1, where σ2
+B−S1 and σ2
+S2−S1 are the vari-
+ances in blazar DLC and a control star DLC respectively.
+The scaled F-statistics FΓ is given as
+FΓ = F
+Γ2 = 1
+Γ2
+� σ2
+B−S1
+σ2
+S2−S1
+�
+.
+The F-statistic assumes that the uncertainties in the
+observations are normally distributed. If n(B−S1) and
+n(S2−S1) are the sizes of the blazar and control star
+DLC respectively, the number of degrees of freedom in
+the numerator and denominator of the F-statistic are
+ν1 = n(B−S1) − 1 and ν2 = n(S2−S1) − 1, respectively.
+We calculated FΓ and considered the blazar to be vari-
+able with 99.5 per cent confidence if FΓ was greater than
+the critical value F α
+c (ν1, ν2) at α = 0.005 (Zibecchi et al.
+2017, 2020).
+3.2.3. Power-enhanced F-test
+The power-enhanced F -test (PEF) has been used in
+various recent blazar IDV studies (Pandey et al. 2019;
+Pandey et al. 2020, and references therein). The power-
+enhanced F-statistic has the advantage of comparing the
+blazar variance to the combined variance of multiple
+field stars and is given as (de Diego 2014)
+Fenh = s2
+blz
+s2c
+,
+(9)
+where s2
+blz is the variance of the DLC of the blazar with
+respect to a reference star, and s2
+c is the combined vari-
+ance of the comparison stars’ DLCs with respect to the
+reference star. Thus, s2
+c is given as
+s2
+c =
+1
+��k
+j=1 nj
+�
+− k
+k
+�
+j=1
+nj
+�
+i=1
+s2
+j,i.
+(10)
+Here, k is the total number of available comparison stars
+in the DLC, nj is the number of observations of the jth
+comparison star, and s2
+j,i is the scaled square deviation
+of the ith observation of the jth comparison star given
+as
+s2
+j,i = Γj(mj,i − ¯
+mj)2.
+(11)
+Here Γj is the scale factor of the jth comparison star
+DLC computed following Equation 7.
+Using the data of the field stars, we first checked the
+star–star DLCs to identify any spikes due to instru-
+mental errors or improper removal of cosmic rays, and
+removed them iteratively if they were more than 3
+standard deviations from the mean magnitude.
+We
+considered a “well-behaved” star with low fluctuations
+and an average magnitude close to the blazar as the
+reference star.
+The number of degrees of freedom in
+the numerator and denominator of the F-statistics are
+
+AO 0235+164 optical variability
+19
+ν1 = nblz − 1 and ν2 =
+��k
+j=1 nj
+�
+− k, respectively.
+We calculated Fenh, and considered the blazar to be
+variable (V) with 99.5 percent confidence if Fenh was
+greater than the critical value Fc(ν1, ν2) at α = 0.005.
+3.2.4. Nested ANOVA test
+In the nested analysis of variance (ANOVA) test, DLCs
+of the blazar are generated with respect to all the com-
+parison stars used as reference stars. The details of this
+method are given in de Diego et al. (2015). The nested
+ANOVA test needs a large number of points in the light
+curves, strongly limiting its application to densely pop-
+ulated DLCs. We divided the DLCs with at least 20
+observations into groups such that each group contains
+4 observations. Equation (4) of de Diego et al. (2015)
+considers an ideal set of lightcurves where the total
+number of observations are divisible by the group size.
+In most of the DLCs in this work, the total number of
+observations was not an integral multiple of the group
+size of 4. So, in those cases, the last group contained less
+than 4 observations, and we calculated the degrees of
+freedom accordingly to compute the mean square due to
+groups (MSG) and mean square due to the nested obser-
+vations in groups (MSO(G)). The ANOVA F-statistic is
+given as, F = MSG/MSO(G). For a significance level of
+α = 0.005, if the F -statistic is greater than the critical
+value (Fc), the blazar is taken as variable (V), other-
+wise as non-variable (NV) with 99.5 per cent confidence.
+We have listed the results of the scaled C-criterion
+and scaled F-test in Table 6 and those of power en-
+hanced F-test and the nested ANOVA test in Table 7.
+In the case of scaled C-criterion and F-test, we fixed one
+particular star as the comparison star for each dataset.
+The source is declared variable with respect to one
+comparison-control star pair if both scaled C-statistics
+and F-statistics cross their respective critical values.
+We declare the final variability status of the blazar
+as variable/non-variable (V/NV) if it is variable/non-
+variable against all control stars. If the blazar is variable
+against some of the control stars, we call it probably
+variable (PV). We did not carry out the nested ANOVA
+test in a few datasets containing less than 20 obser-
+vations.
+In the case of the power-enhanced F-test in
+absence of the corresponding nested ANOVA test, we
+call the blazar probably variable (PV) even if the F-
+statistic crosses the critical value, as the F-test is more
+prone to give a false positive result (Zibecchi et al. 2017,
+2020). If nested ANOVA is present and both the tests
+cross the critical values, we call the blazar variable (V).
+Otherwise, we declare the source non-variable (NV). We
+list the summary of the IDV tests in Table 8. We give
+a final verdict on the variability status of the source
+after comparing the results of the combination of the
+C-test and F-test (C&F) from Table 6 and results of
+the combination of the power-enhance F-test and nested
+ANOVA test (P&N) from Table 7. If the results from
+both combinations were the same, we kept that result.
+If C&F declared “V” and P&N declared “PV” due to
+the absence of nested ANOVA, we finally consider the
+source variable (V). We considered variability on 2005
+November 8 as “NV” because both C-test and nested
+ANOVA resulted in non-variability. Despite being vari-
+able in nested ANOVA, we consider the 2005 December
+5 lightcurve “NV” as the F-test and PEF-test detected
+no variability. A few examples of DLCs of AO 0235+164
+having different variability characteristics (V/PV/NV)
+are shown in Figure 8.
+3.2.5. Doubling timescale
+A flux doubling/halving timescale gives an estimate of
+the variability timescale (τvar) of a source. We calcu-
+late the flux doubling/halving timescale (τd) between
+two consecutive observations and its corresponding sig-
+nificance (σ) as
+F(ti+1) = F(ti) ∗ 2(ti+1−ti)/τd
+σ = |F(ti+1) − F(ti)|/εi,
+(12)
+where F(ti) and εi are the flux observed at time ti
+and the corresponding measurement uncertainty, respec-
+tively. We consider the fastest doubling timescale (τ min
+d
+)
+with a higher significance than 3σ as an estimate for
+τvar. We obtained τ min
+d
+< 1 day for all the nights when
+the source showed significant IDV both in scaled F-test
+and nested ANOVA test. This further strengthens our
+claims for the frequent presence of IDV. Following Equa-
+tion 2 we computed the variability amplitudes on the
+same nights. All these results are listed in Table 7.
+3.2.6. Duty cycle
+We calculated the duty cycle (DC) of AO 0235+164
+using the definition of Romero et al. (1999), that was
+used later by multiple authors (e.g., Stalin et al. 2009;
+Agarwal et al. 2016). The formula for DC for a partic-
+ular waveband is given as,
+DC = 100
+�n
+i=1 Ni(1/∆ti)
+�n
+i=1(1/∆ti) %
+(13)
+where ∆ti = ∆ti,obs/(1+z) (duration of the monitoring
+session on ith night is ∆ti,obs). Thus, this formula cal-
+culates the duty cycle weighted by the cosmological red-
+shift corrected monitoring duration of each night. We
+set Ni = 1, 0.5, and 0 for the nights with variability
+
+20
+Roy et al.
+Table 8. Summary of statistical tests for IDV on AO 0235+164
+differential lightcurves from CASLEO and CAHA
+Obs.
+Band
+Combined variability status
+Final
+date
+(C & F-test)a
+(PEF &
+status
+N-ANOVA)b
+1999 Nov 2
+V
+V
+V
+V
+1999 Nov 3
+V
+V
+V
+V
+1999 Nov 4
+V
+V
+V
+V
+R
+V
+V
+V
+1999 Nov 5
+V
+V
+NV
+PV
+R
+NV
+V
+PV
+1999 Nov 6
+V
+V
+V
+V
+R
+V
+V
+V
+1999 Nov 7
+V
+PV
+PV
+PV
+R
+PV
+PV
+PV
+2000 Dec 21
+V
+PV
+PV
+PV
+R
+PV
+PV
+PV
+2000 Dec 23
+V
+PV
+PV
+PV
+R
+V
+PV
+V
+2001 Nov 9
+V
+NV
+NV
+NV
+R
+V
+PV
+V
+2001 Nov 10
+V
+NV
+NV
+NV
+R
+NV
+NV
+NV
+2001 Nov 11
+V
+NV
+NV
+NV
+R
+NV
+NV
+NV
+2001 Nov 12
+V
+NV
+NV
+NV
+R
+PV
+PV
+PV
+2001 Nov 13
+V
+NV
+NV
+NV
+R
+NV
+NV
+NV
+2005 Jan 16
+R
+V
+PV
+V
+2005 Nov 2
+R
+V
+V
+V
+2005 Nov 4
+R
+V
+V
+V
+2005 Nov 5
+R
+V
+V
+V
+2005 Nov 6
+R
+V
+V
+V
+2005 Nov 8
+R
+NV
+PV
+NV
+2005 Dec 5
+R
+NV
+PV
+NV
+2005 Dec 6
+R
+PV
+PV
+PV
+2019 Dec 17
+R
+NV
+NV
+NV
+Note—aTable 6, bTable 7, PEF=power-enhanced F-test.
+status “V”, “PV”, and “NV” respectively. We obtained
+the duty cycle of AO 0235+164 to be ∼44 percent in V -
+band, and ∼45 percent in R-band considering the nights
+where the source was observed for at least 2 hours.
+3.3. The mass of the central black hole
+We estimate the mass of the SMBH in AO 0235+164
+by using its spectrum observed using the CCD Imag-
+ing/Spectropolarimeter (SPOL) at the Steward Obser-
+vatory4 on 2011 January 8 (air mass = 1.12).
+This
+spectrum was selected since the blazar was then at its
+lowest level during the period 2008–2018, and should
+ensure the best visibility of the emission lines because
+of the lower continuum contribution from the jet. The
+observed wavelength range of the spectrum we used is
+4000–7550 ˚A, with a spectral resolution of 4 ˚A, and it is
+analyzed by following the procedure given in Liao & Gu
+(2020). Firstly, it was corrected for Galactic extinction
+with the reddening map of Schlegel et al. (1998), and
+then was shifted to the rest-frame wavelength by using
+the redshift of 0.94.
+This spectral coverage meant we could use the Mg
+II line, which is prominent on the spectrum shown in
+Figure 9 (focused on the 2400−3100 ˚A range), to es-
+timate the SMBH mass.
+We modeled the continuum
+by applying a single power law (fλ ∝ λα) (as Fe II
+emission is rather weak). A Gaussian profile was then
+used to fit the Mg II line, centered at the position of
+2800 ˚A, on the continuum-subtracted spectrum.
+The
+broad component of Mg II was fitted with a Gaussian
+with a 1000 km s−1 lower limit, while a Gaussian with
+upper limit of 1000 km s−1 was applied for the narrow
+component.
+In order to estimate the corresponding
+errors of full width at half maximum (FWHM) and
+flux, we generated 100 mock spectra by adding random
+Gaussian noise to the original spectrum using the flux
+density errors, and then took the standard deviation of
+measurements from those mock spectra as the uncer-
+tainties.
+Here, the flux density errors were the RMS
+value of the spectrum calculated over the spectral win-
+dow of (3000−3100) ˚A, after subtracting a second-order
+polynomial function.
+Figure 9 shows the resulting fit
+to the spectrum. Our best fitting results indicate that
+the line width of the broad Mg II component is FWHM
+= 3151 km s−1, with log-scale luminosity in erg s−1,
+log(LMgII) = 42.8.
+The line width and the Mg II line luminosity we find
+are consistent with the range of values FWHM=3100–
+3500 km s−1 and log(LMgII)=42.5–42.8, respectively,
+which were derived by Raiteri et al. (2007) from one
+VLT and four TNG spectra of AO 0235+164 acquired
+in 2003–2004.
+We use the FWHM and luminosity of
+the broad Mg II line, not the continuum luminosity, as
+4 http://james.as.arizona.edu/∼psmith/Fermi
+
+AO 0235+164 optical variability
+21
+1015
+1016
+ (HZ)
+10
+14
+10
+13
+10
+12
+F  (erg cm
+2 s
+1)
+Disk thermal
+U
+B
+V
+R
+I
+JD 2452169
+Figure 10. Comparison of the SED of the lowest flux state
+observed on JD 2452169 and the thermal emission from the
+accretion disk in the observer’s frame. The thermal emis-
+sion component is calculated using a multi-temperature disk
+model with the black hole mass log(MBH/M⊙) = 7.9±0.25,
+and the log-scale disk luminosity in erg s−1, log(Ldisk) =
+45.01±0.20. The shaded region indicates the uncertainties
+in the calculation of the disk thermal component.
+we are unable to exclude the jet emission contribution,
+despite the low state spectrum that we could use for
+this blazar.
+The black hole mass is derived from the
+empirical relation used for Mg II (Kong et al. 2006),
+which is based on measured broad line region sizes in
+the reverberation-mapping AGN sample of Peterson
+et al. (2004), as
+MBH
+M⊙
+= 2.9×106
+�
+LMgII
+1042 erg s−1
+�0.57±0.12 �FWHMMgII
+103 km s−1
+�2
+(14)
+Thus, the SMBH mass is log(MBH/M⊙) = 7.90 ± 0.25,
+where the uncertainty is estimated from the measure-
+ment uncertainties of the FWHM and luminosity of
+Mg II. Using optical spectroscopy data from the SDSS
+archive, Paliya et al. (2021) reported a somewhat higher
+mass, log(MBH/M⊙) = 8.58 ± 0.34, and an accretion
+disk luminosity (in erg s−1), of log(Ldisk) = 45.30 ±
+0.22. Using the method mentioned in Paliya et al. (2021)
+with log(LMgII) = 42.8, we obtained a lower disk lumi-
+nosity (in erg s−1) of log(Ldisk) = 45.01 ± 0.20 from the
+spectrum observed on 2011 January 8.
+4. DISCUSSION
+In this work, we have presented a detailed temporal
+and spectral study of the highly variable emission from
+the blazar AO 0235+164 observed at multiple optical
+wavebands (UBVRI) from October 1975 to December
+2019. The lightcurves have highly uneven data sampling
+due to gaps in observation seasons and non-uniform ob-
+servation campaigns. Although U-band data are quite
+sparsely sampled the BVRI observations have denser
+sampling when the source was highly active. Multiple
+long-term studies suggested that AO 0235+164 shows
+∼2-year long flaring episodes with multiple sub-flares
+after intervals of ∼8 years (Raiteri et al. 2006; Fan et al.
+2017; Roy et al. 2022). Figure 1 shows a difference of
+about six magnitudes between the quiescent and out-
+burst states in all optical wavebands, corresponding to
+an energy flux variation of more than two orders of
+magnitude (Figure 6). The long-term variability ampli-
+tudes at all five wavebands are quite similar (Table 1).
+Also, we found a strong correlation with zero time-lag
+between the UBVI observations and the R-band data
+(Figure 2 and Figure 3), which implies a common ra-
+diative process at a single emission zone is responsible
+for the bulk of the emission at the optical wavebands.
+Sometimes during the quiescent states of powerful
+blazars, the disk thermal emission component becomes
+visible as a big blue bump on top of the synchrotron
+emission component from the jet in the optical-UV
+wavebands (e.g., Roy et al. 2021). As the disk emission
+is bluer than the jet synchrotron emission, an increase
+in the jet activity during low flux states displays a
+redder-when-brighter trend.
+The enhanced jet activ-
+ity is observed when the charged particles inside the
+jet get accelerated to higher energies, and then radiate
+faster. Thus, the jet synchrotron component tends to
+get bluer with the increase in flux. If the jet emission
+completely outshines the disk emission, we expect to
+see a bluer-when-brighter trend (e.g., Isler et al. 2017).
+The flux increment can also be attributed to the in-
+crease in the jet Doppler factor (e.g., Papadakis et al.
+2007), which blueshifts the spectrum and produces a
+bluer-when-brighter trend because of the convexity of
+the spectrum. Such a trend is seen in the (B − I) vs R
+magnitude diagram (Figure 4b) and indicates the dom-
+ination of non-thermal jet emission over the thermal
+emission component of the accretion disk during both
+flaring and quiescent states.
+From the convex shapes
+of the optical BVR SEDs during states ranging from
+quiescent to flaring (see the accompanying SED video
+and Figure 6), we may infer that the effect of the disk
+thermal emission is not significant in optical wavebands
+even during the low flux states.
+This can be explained in terms of the nature of disk
+thermal emission given the disk luminosity and the cen-
+tral black hole mass computed in subsection 3.3. The
+primary, and most precise, black hole mass estimation
+methods are based on stellar and gas kinematics and
+reverberation mapping (e.g. Vestergaard 2004). These
+
+22
+Roy et al.
+methods need high spatial resolution spectroscopy data
+from the host galaxy and/or higher ionization emission
+lines and are not applicable to most BL Lacertae objects
+(BL Lacs). But in BL Lacs, if the weak emission lines
+are present, we can use the empirical methods (Kong
+et al. 2006) for BH mass estimation. The most common
+methods used for BH mass estimation for BL Lacs are
+the shortest variability timescales and periods of QPOs
+(Gupta et al. 2012). Since BL Lacs are highly variable
+objects, any BH mass estimation may be treated as an
+upper limit, and there are possibilities of detection of
+a shorter variability timescale or shorter QPO period.
+We obtained a log-scale BH mass of 7.90±0.25 in so-
+lar mass unit. The Steward observatory spectrum we
+used in our analysis had a narrower Mg II emission
+line (FWHM=3151 km s−1) than those of Raiteri et al.
+(2007) and Paliya et al. (2021), thus resulting in a lower
+mass estimate.
+We considered a multi-temperature
+blackbody type accretion disk model, where the temper-
+ature at any portion of the disk is a function of the disk
+luminosity and the central black hole mass, to compute
+the thermal emission component. In Figure 10 we plot-
+ted the thermal component along with the optical-UV
+SED during the lowest activity state of AO 0235+164
+observed on JD 2452169. It is evident that, as the ther-
+mal emission peaks at far UV frequencies (∼3.5×1015
+Hz) in the observer’s frame of reference, the jet emission
+always dominates in BVRI wavebands. We do not see
+any significant trend in the variation of the (V − R)
+spectral index (αV R) (Figure 7).
+The sudden rise of
+the U-band flux in Figure 10 is an indicator of a prob-
+able UV-soft X-ray bump as discussed in Raiteri et al.
+(2005, 2006).
+According to these studies, the source
+of the bump is either an additional synchrotron com-
+ponent coming from a separate emission region in the
+jet or the emission of a continuous inhomogeneous jet
+is suppressed in near UV region due to a discontinuity
+in opacity or misalignment of that particular emission
+region.
+Ackermann et al. (2012) mentioned that the
+whole optical-UV spectrum is produced by a single syn-
+chrotron emitting zone as the shape of the bump does
+not change with luminosity.
+They attributed the UV
+spectral hardening to an artifact due to the overestima-
+tion of extinction by Junkkarinen et al. (2004).
+For the detection of any statistically significant intraday
+variability in 33 lightcurves of AO 0235+164 observed
+at CASLEO/CAHA, we employed different statistical
+tests widely used in AGN variability studies. The re-
+liability of each of these tests has been disputed (e.g.
+de Diego et al. 2015; Zibecchi et al. 2017), so we here
+employed a comparative approach that could allow us
+to circumvent the limitations affecting any individual
+test. In the first place, we used the scaled C-criterion
+and the F-test. The first compares the dispersion of the
+blazar lightcurve to the dispersion of a field star (con-
+trol star), while the latter does so with the variances.
+According to Zibecchi et al. (2017) and Zibecchi et al.
+(2020), the F-test has a tendency to classify noisy non-
+variable curves as a variable (i.e., give false positives),
+while the C-criterion tends to give false negatives. Even
+though the C-criterion (Romero et al. 1999) cannot be
+considered as an actual statistical test, it may still be a
+useful parameter to detect variability with high signifi-
+cance. The F-test, on the other hand, does not always
+work as expected, because it is particularly sensitive to
+non-Gaussian errors (“red noise”), which are usually an
+issue when analyzing blazars DLCs.
+We also used the power-enhanced F-test and the nested
+ANOVA test, which involve multiple field stars. It is ex-
+pected that the power-enhanced F-test may also suffer
+from the same drawback of detecting false variability
+as the (original) F-test.
+In the nested ANOVA test,
+in turn, data grouping may lead to false results if data
+within a time span larger than the (unknown) variability
+timescale are grouped. Comparing the results of Table 6
+and Table 7, while considering the tendencies of giving
+false results by the respective tests, we can confirm that
+the source was significantly variable in 4 out of 13 V -
+band lightcurves, and 9 out of 20 R-band lightcurves.
+The source seems to be probably variable in 3 V -band
+and 4 R-band lightcurves, and non-variable in the rest.
+On 1999 November 5, the combination of C-criterion
+and F-test indicates non-variability but the combination
+of power-enhanced F-test and nested ANOVA detects
+variability in the R-band lightcurve. The results in the
+V -band lightcurve on that day are exactly the opposite.
+Similar situations were observed also on 2001 November
+9 and 2001 November 12.
+A visual inspection of the
+DLCs of these nights reveals that the blazar DLCs were
+classified as non-variable when either the control star
+DLC had higher variability (1999 November 5) or the
+measurement errors of the blazar DLCs were higher due
+to its low-flux state (2001 November 9 and 12). Higher
+measurement errors lead to a lower chance of signifi-
+cant variability detection.
+These strange results may
+be an example of the drawbacks of the applied methods
+when trying to recover low-amplitude variations from
+DLCs affected by non-Gaussian noise (part of the ob-
+servations on that night were taken at air mass > 2
+and under non-photometric conditions). Otherwise, the
+combined results of different methods seem to more or
+less agree.
+Alongside the optical SED patterns, such
+frequent IDV establishes AO 0235+164 as a low-energy
+
+AO 0235+164 optical variability
+23
+Table 9.
+Variation of duty cycle with
+the duration of observation in R-band.
+Observation
+No. of
+Duty
+duration (hours)
+nights
+cycle (%)
+> 1
+20
+52
+> 2
+19
+45
+> 3
+17
+50
+> 4
+14
+57
+> 5
+13
+64
+> 6
+8
+77
+peaked BL Lac (LBL) object. High energy peaked BL
+Lacs (HBL) show significantly less optical intraday vari-
+ability than the LBLs (Heidt & Wagner 1998; Romero
+et al. 1999).
+The differences in IDV behavior have been attributed
+to the strength of magnetic fields present in the jet of
+HBLs. A higher axial magnetic field (B) than a critical
+value (Bc) may prevent the generation of any bends and
+Kelvin-Helmhotz instabilities in the jet-base responsible
+for creating intraday microvariabilities. This indicates
+the presence of a weaker magnetic field than Bc in the
+jet of AO 0235+164. The critical magnetic field (Bc) is
+given in Romero (1995) as
+Bc =
+�
+4πnemec2(Γ2 − 1)/Γ,
+(15)
+where ne is the electron density in the emission region,
+me is the electron rest mass, and here Γ is the bulk
+Lorentz factor of the jet flow. Considering a typical set
+of parameters, ne = 429 cm−3 and Γ = 20 (Ackermann
+et al. 2012), we get Bc ≃ 0.07 G.
+From Table 7 and Figure 8, we can say that the vari-
+ability amplitudes were higher in the 1999 season when
+the source was in a fainter state (higher magnitude) than
+its brighter state in the 2005 season. Marscher (2013)
+suggested that enhancement of flux can arise from a
+more uniform flow of particles inside the jet, which in
+turn decreases the amplitude of microvariability asso-
+ciated with the turbulence inside the jet. Equation 9
+indicates that the probability of detection of significant
+variability increases with the duration of observation.
+Similar results for other blazars were found by Gupta
+& Joshi (2005), Rani et al. (2010), and Agarwal et al.
+(2016).
+From the flux doubling timescales listed in Table 7, we
+can estimate the upper limit to the size of the emission
+region (Rmax) using the light travel-time argument given
+as
+Rmax = cδtvar
+1 + z
+(16)
+where z is the cosmological redshift of 0.94, tvar is the
+variability timescale, and δ is the Doppler boost of the
+jet. Considering δ = 24 (Hovatta et al. 2009) and tvar
+to be the shortest flux doubling timescale of 0.083 days
+(when the source was significantly variable), we obtain
+an emission region size upper limit of ∼ 2.6 × 1015 cm.
+Assuming a conical jet model where the emission re-
+gion fills up the entire jet cross-section, we can estimate
+the probable maximum distance (dmax) of the emission
+region from the central black hole as, dmax = ΓRmax =
+5.2×1016 cm. To explain the observed strong variability,
+Marchesini et al. (2016) attempted to apply a swinging
+jet model that attributes the observed variability to a
+change in the viewing angle of the emission region with
+time (i.e. variation in the associated bulk Doppler fac-
+tor). They reported a high rate of change in viewing an-
+gle of about 7−10 arcmin per day, considering a mean
+viewing angle of 2.3◦, would be necessary.
+However,
+they found that this geometric wiggling-jet scenario was
+disfavored when considering the observed variation in
+color index with time.
+Several earlier studies on AO
+0235+164 associated the observed fast optical variabil-
+ity with gravitational microlensing by the foreground
+absorber at z = 0.524. Webb et al. (2000) proposed that
+the 1997 flare resulted due to microlensing because of an
+observed correlation with zero lag between radio and op-
+tical lightcurves following Stickel et al. (1988), but the
+absence of any correlated flare in the X-ray lightcurve
+makes this explanation less likely. Abraham et al. (1993)
+and Raiteri et al. (2007) explained that such microlens-
+ing events can produce small amounts of fast flux ampli-
+fication but are unlikely to dominate the high variability
+observed in AO 0235+164.
+5. CONCLUSIONS
+In this work, we conducted a study of long-term and
+short-term (intraday) variability in the optical mul-
+tiwaveband observations of the blazar AO 0235+164.
+Here we summarize our results and the probable physi-
+cal scenarios.
+1. We observed a variation of about six magnitudes
+between the quiescent and flaring episodes, or over
+two orders of magnitude variation in the SEDs.
+2. UBVI lightcurves are highly correlated with the
+R-band lightcurve with zero time lag.
+
+24
+Roy et al.
+3. A significant bluer-when-brighter trend is observed
+in the (B − I) color variation with R-magnitude.
+4. All the optical BVR-band SEDs show convexity.
+These observations indicate that the optical emis-
+sion is dominated by jet radiation.
+5. AO 0235+164 frequently shows statistically sig-
+nificant intraday variability in optical wavebands.
+This implies that AO 0235+164 is an LBL and
+probably has a weak magnetic field in the jet en-
+vironment.
+6. From the analysis of a broad Mg II emission line
+in a spectrum of AO 0235+164 taken at a low
+state, we estimate a central black-hole mass of ∼
+7.9 × 107M⊙.
+ACKNOWLEDGMENTS
+Data from the Steward Observatory spectropolari-
+metric monitoring project were used.
+This pro-
+gram is supported by Fermi Guest Investigator grants
+NNX08AW56G, NNX09AU10G, NNX12AO93G, and
+NNX15AU81G. This paper has made use of up-to-
+date SMARTS optical/near-infrared light curves that
+are available at www.astro.yale.edu/smarts/glast/home.
+php. This work is partly based on data taken and as-
+sembled by the WEBT collaboration and stored in the
+WEBT archive at the Osservatorio Astrofisico di Torino
+-
+INAF
+(https://www.oato.inaf.it/blazars/webt/).
+These data are available upon request to the WEBT
+President Massimo Villata ([email protected]).
+This work is based on data acquired at Complejo
+Astron´omico El Leoncito, operated under an agree-
+ment between the Consejo Nacional de Investigaciones
+Cient´ıficas y T´ecnicas de la Rep´ublica Argentina and
+the National Universities of La Plata, C´ordoba and San
+Juan. We thank Anabella Araudo and Ileana Andru-
+chow for help with the observations made with CASLEO
+and the data analysis.
+We thankfully acknowledge the anonymous reviewer
+for very useful comments which helped us to improve
+the manuscript.
+We acknowledge the support of the
+Department of Atomic Energy, Government of India,
+under project identification number RTI 4002.
+ACG
+is partially supported by Chinese Academy of Sciences
+(CAS) President’s International Fellowship Initiative
+(PIFI) (grant no.
+2016VMB073).
+GER acknowl-
+edges support from grants PIP 0554 (CONICET),
+PIP
+2021-1639
+(CONICET),
+and
+grant
+PID2019-
+105510GBC31 of the Spanish Ministerio de Ciencia,
+Innovaci´on y Universidades and through the Center
+of Excellence Mara de Maeztu 2020-2023 award to
+the ICCUB (CEX2019-000918-M). JAC is Mar´ıa Zam-
+brano researcher fellow funded by the European Union
+-NextGenerationEU- (UJAR02MZ), supported by PIP
+0113 (CONICET) and PICT-2017-2865 (ANPCyT).
+JAC was also supported by grant PID2019-105510GB-
+C32/AEI/10.13039/501100011033 from the Agencia Es-
+tatal de Investigaci´on of the Spanish Ministerio de
+Ciencia, Innovaci´on y Universidades, and by Consejer´ıa
+de Econom´ıa, Innovaci´on, Ciencia y Empleo of Junta
+de Andaluc´ıa as research group FQM-322, as well as
+FEDER funds.
+Facilities:
+WEBT, SMARTS, Bok,
+SO:Kuiper,
+MMT, CASLEO:JST, CAO:2.2m
+
+AO 0235+164 optical variability
+25
+Software:
+Astropy (Astropy Collaboration et al.
+2013), DAOPHOT (Stetson 1987), IRAF (Tody 1986)
+REFERENCES
+Abraham, R. G., Crawford, C. S., Merrifield, M. R.,
+Hutchings, J. B., & McHardy, I. M. 1993, ApJ, 415, 101,
+doi: 10.1086/173147
+Ackermann, M., Ajello, M., Ballet, J., et al. 2012, ApJ, 751,
+159, doi: 10.1088/0004-637X/751/2/159
+Ackermann, M., Ajello, M., Ballet, J., et al. 2012, ApJ, 751,
+doi: 10.1088/0004-637X/751/2/159
+Agarwal, A., Gupta, A. C., Bachev, R., et al. 2016,
+MNRAS, 455, 680, doi: 10.1093/mnras/stv2345
+Agudo, I., Marscher, A. P., Jorstad, S. G., et al. 2011,
+ApJL, 735, L10, doi: 10.1088/2041-8205/735/1/L10
+Astropy Collaboration, Robitaille, T. P., Tollerud, E. J.,
+et al. 2013, A&A, 558, A33,
+doi: 10.1051/0004-6361/201322068
+Bessell, M. S. 2005, ARA&A, 43, 293,
+doi: 10.1146/annurev.astro.41.082801.100251
+Bonning, E., Urry, C. M., Bailyn, C., et al. 2012, The
+Astrophysical Journal, 756, 13,
+doi: 10.1088/0004-637X/756/1/13
+Cellone, S. A., Romero, G. E., Combi, J. A., & Mart´ı, J.
+2007, MNRAS, 381, L60,
+doi: 10.1111/j.1745-3933.2007.00366.x
+Cohen, R. D., Smith, H. E., Junkkarinen, V. T., &
+Burbidge, E. M. 1987, ApJ, 318, 577, doi: 10.1086/165393
+de Diego, J. A. 2014, AJ, 148, 93,
+doi: 10.1088/0004-6256/148/5/93
+de Diego, J. A., Polednikova, J., Bongiovanni, A., et al.
+2015, AJ, 150, 44, doi: 10.1088/0004-6256/150/2/44
+Edelson, R. A., & Krolik, J. H. 1988, ApJ, 333, 646,
+doi: 10.1086/166773
+Fan, J. H., & Lin, R. G. 1999, ApJS, 121, 131,
+doi: 10.1086/313191
+Fan, J. H., Tao, J., Qian, B. C., et al. 2006, Publications of
+the Astronomical Society of Japan, 58, 797,
+doi: 10.1093/pasj/58.5.797
+Fan, J. H., Kurtanidze, O., Liu, Y., et al. 2017, ApJ, 837,
+45, doi: 10.3847/1538-4357/aa5def
+Fossati, G., Maraschi, L., Celotti, A., Comastri, A., &
+Ghisellini, G. 1998, MNRAS, 299, 433,
+doi: 10.1046/j.1365-8711.1998.01828.x
+Gonz´alez-P´erez, J. N., Kidger, M. R., & Mart´ın-Luis, F.
+2001, AJ, 122, 2055, doi: 10.1086/322129
+Guo, Y. C., Hu, S. M., Xu, C., et al. 2015, NewA, 36, 9,
+doi: 10.1016/j.newast.2014.09.011
+Gupta, A. C., Banerjee, D. P. K., Ashok, N. M., & Joshi,
+U. C. 2004, A&A, 422, 505,
+doi: 10.1051/0004-6361:20040306
+Gupta, A. C., Fan, J. H., Bai, J. M., & Wagner, S. J. 2008,
+AJ, 135, 1384, doi: 10.1088/0004-6256/135/4/1384
+Gupta, A. C., & Joshi, U. C. 2005, A&A, 440, 855,
+doi: 10.1051/0004-6361:20042370
+Gupta, S. P., Pandey, U. S., Singh, K., et al. 2012, NewA,
+17, 8, doi: 10.1016/j.newast.2011.05.005
+Hagen-Thorn, V. A., Larionov, V. M., Jorstad, S. G., et al.
+2008, ApJ, 672, 40, doi: 10.1086/523841
+Heidt, J., & Wagner, S. J. 1996, A&A, 305, 42.
+https://arxiv.org/abs/astro-ph/9506032
+—. 1998, A&A, 329, 853.
+https://arxiv.org/abs/astro-ph/9709116
+Hovatta, T., Valtaoja, E., Tornikoski, M., & L¨ahteenm¨aki,
+A. 2009, A&A, 494, 527,
+doi: 10.1051/0004-6361:200811150
+Howell, S. B., Mitchell, K. J., & Warnock, A. I. 1988, AJ,
+95, 247
+Ikejiri, Y., Uemura, M., Sasada, M., et al. 2011, PASJ, 63,
+639, doi: 10.1093/pasj/63.3.327
+Impey, C. D., Brand, P. W. J. L., & Tapia, S. 1982,
+MNRAS, 198, 1, doi: 10.1093/mnras/198.1.1
+Isler, J. C., Urry, C. M., Coppi, P., et al. 2017, The
+Astrophysical Journal, 844, 107,
+doi: 10.3847/1538-4357/aa79fc
+Itoh, R., Nalewajko, K., Fukazawa, Y., et al. 2016, ApJ,
+833, 77, doi: 10.3847/1538-4357/833/1/77
+Jang, M., & Miller, H. R. 1997, AJ, 114, 565,
+doi: 10.1086/118493
+Junkkarinen, V. T., Cohen, R. D., Beaver, E. A., et al.
+2004, ApJ, 614, 658, doi: 10.1086/423777
+Kong, M.-Z., Wu, X.-B., Wang, R., & Han, J.-L. 2006,
+ChJA&A, 6, 396, doi: 10.1088/1009-9271/6/4/02
+Kutkin, A. M., Pashchenko, I. N., Lisakov, M. M., et al.
+2018, MNRAS, 475, 4994, doi: 10.1093/mnras/sty144
+Landolt, A. U. 2009, AJ, 137, 4186,
+doi: 10.1088/0004-6256/137/5/4186
+Liao, M., & Gu, M. 2020, MNRAS, 491, 92,
+doi: 10.1093/mnras/stz2981
+Madejski, G., Takahashi, T., Tashiro, M., et al. 1996, ApJ,
+459, 156, doi: 10.1086/176877
+Marchesini, E. J., Andruchow, I., Cellone, S. A., et al. 2016,
+A&A, 591, A21, doi: 10.1051/0004-6361/201527632
+
+26
+Roy et al.
+Marscher, A. P. 1983, ApJ, 264, 296, doi: 10.1086/160597
+Marscher, A. P. 2013, The Astrophysical Journal, 780, 87,
+doi: 10.1088/0004-637x/780/1/87
+Miller, H. R., Carini, M. T., & Goodrich, B. D. 1989,
+Nature, 337, 627, doi: 10.1038/337627a0
+M¨ucke, A., Protheroe, R. J., Engel, R., Rachen, J. P., &
+Stanev, T. 2003, Astroparticle Physics, 18, 593,
+doi: 10.1016/S0927-6505(02)00185-8
+Nilsson, K., Charles, P. A., Pursimo, T., et al. 1996, A&A,
+314, 754
+Paliya, V. S., Dom´ınguez, A., Ajello, M., Olmo-Garc´ıa, A.,
+& Hartmann, D. 2021, ApJS, 253, 46,
+doi: 10.3847/1538-4365/abe135
+Pandey, A., Gupta, A. C., Wiita, P. J., & Tiwari, S. N.
+2019, ApJ, 871, 192, doi: 10.3847/1538-4357/aaf974
+Pandey, A., Gupta, A. C., Kurtanidze, S. O., et al. 2020,
+The Astrophysical Journal, 890, 72,
+doi: 10.3847/1538-4357/ab698e
+Papadakis, I. E., Villata, M., & Raiteri, C. M. 2007, A&A,
+470, 857, doi: 10.1051/0004-6361:20077516
+Peterson, B. M., Ferrarese, L., Gilbert, K. M., et al. 2004,
+ApJ, 613, 682, doi: 10.1086/423269
+Qian, S. J., Kraus, A., Witzel, A., Krichbaum, T. P., &
+Zensus, J. A. 2000, A&A, 357, 84
+Rabbette, M., McBreen, S., Steel, B., & Smith, N. 1996,
+A&A, 310, 1
+Raiteri, C. M., Villata, M., Capetti, A., et al. 2007, A&A,
+464, 871, doi: 10.1051/0004-6361:20066599
+Raiteri, C. M., Villata, M., Aller, H. D., et al. 2001, A&A,
+377, 396, doi: 10.1051/0004-6361:20011112
+Raiteri, C. M., Villata, M., Ibrahimov, M. A., et al. 2005,
+A&A, 438, 39, doi: 10.1051/0004-6361:20042567
+Raiteri, C. M., Villata, M., Kadler, M., et al. 2006, A&A,
+459, 731, doi: 10.1051/0004-6361:20065744
+Raiteri, C. M., Villata, M., Larionov, V. M., et al. 2008,
+A&A, 480, 339, doi: 10.1051/0004-6361:20079044
+Raiteri, C. M., Villata, M., Acosta-Pulido, J. A., et al.
+2017, Nature, 552, 374, doi: 10.1038/nature24623
+Rani, B., Gupta, A. C., Strigachev, A., et al. 2010, Monthly
+Notices of the Royal Astronomical Society, 404, 1992,
+doi: 10.1111/j.1365-2966.2010.16419.x
+Romero, G. E. 1995, Ap&SS, 234, 49,
+doi: 10.1007/BF00627281
+Romero, G. E., Boettcher, M., Markoff, S., & Tavecchio, F.
+2017, SSRv, 207, 5, doi: 10.1007/s11214-016-0328-2
+Romero, G. E., Cellone, S. A., & Combi, J. A. 1999,
+A&AS, 135, 477, doi: 10.1051/aas:1999184
+—. 2000, A&A, 360, L47.
+https://arxiv.org/abs/astro-ph/0007407
+Romero, G. E., Cellone, S. A., Combi, J. A., & Andruchow,
+I. 2002, A&A, 390, 431, doi: 10.1051/0004-6361:20020743
+Roy, A., Patel, S. R., Sarkar, A., Chatterjee, A., & Chitnis,
+V. R. 2021, MNRAS, 504, 1103,
+doi: 10.1093/mnras/stab975
+Roy, A., Chitnis, V. R., Gupta, A. C., et al. 2022, MNRAS,
+513, 5238, doi: 10.1093/mnras/stac1287
+Sagar, R., Stalin, C. S., Gopal-Krishna, & Wiita, P. J. 2004,
+MNRAS, 348, 176, doi: 10.1111/j.1365-2966.2004.07339.x
+Schlegel, D. J., Finkbeiner, D. P., & Davis, M. 1998, ApJ,
+500, 525, doi: 10.1086/305772
+Schramm, K. J., Borgeest, U., Kuehl, D., et al. 1994,
+A&AS, 106, 349
+Smith, P. S., Balonek, T. J., Heckert, P. A., Elston, R., &
+Schmidt, G. D. 1985, AJ, 90, 1184, doi: 10.1086/113824
+Smith, P. S., Montiel, E., Rightley, S., et al. 2009, arXiv
+e-prints, arXiv:0912.3621.
+https://arxiv.org/abs/0912.3621
+Stalin, C. S., Kawabata, K. S., Uemura, M., et al. 2009,
+Monthly Notices of the Royal Astronomical Society, 399,
+1357, doi: 10.1111/j.1365-2966.2009.15354.x
+Stetson, P. B. 1987, Publications of the Astronomical
+Society of the Pacific, 99, 191, doi: 10.1086/131977
+Stickel, M., Fried, J. W., & Kuehr, H. 1988, A&A, 198, L13
+—. 1993, A&AS, 98, 393
+Takalo, L. O., Sillanpaeae, A., Valtaoja, E., et al. 1998,
+A&AS, 129, 577, doi: 10.1051/aas:1998205
+Tody, D. 1986, in Society of Photo-Optical Instrumentation
+Engineers (SPIE) Conference Series, Vol. 627,
+Instrumentation in astronomy VI, ed. D. L. Crawford,
+733, doi: 10.1117/12.968154
+Urry, C. M., & Padovani, P. 1995, PASP, 107, 803,
+doi: 10.1086/133630
+Vestergaard, M. 2004, in Astronomical Society of the
+Pacific Conference Series, Vol. 311, AGN Physics with
+the Sloan Digital Sky Survey, ed. G. T. Richards & P. B.
+Hall, 69. https://arxiv.org/abs/astro-ph/0401436
+Villata, M., Raiteri, C. M., Kurtanidze, O. M., et al. 2002,
+A&A, 390, 407, doi: 10.1051/0004-6361:20020662
+Villata, M., Raiteri, C. M., Larionov, V. M., et al. 2008,
+A&A, 481, L79, doi: 10.1051/0004-6361:200809552
+Villata, M., Raiteri, C. M., Gurwell, M. A., et al. 2009,
+A&A, 504, L9, doi: 10.1051/0004-6361/200912732
+Wagner, S. J., & Witzel, A. 1995, ARA&A, 33, 163,
+doi: 10.1146/annurev.aa.33.090195.001115
+Wang, Y.-F., & Jiang, Y.-G. 2020, ApJ, 902, 41,
+doi: 10.3847/1538-4357/abb36c
+Webb, J. R., Howard, E., Ben´ıtez, E., et al. 2000, AJ, 120,
+41, doi: 10.1086/301432
+
+AO 0235+164 optical variability
+27
+White, R. J., & Peterson, B. M. 1994, PASP, 106, 879,
+doi: 10.1086/133456
+Wierzcholska, A., Ostrowski, M., Stawarz, �L., Wagner, S.,
+& Hauser, M. 2015, A&A, 573, A69,
+doi: 10.1051/0004-6361/201423967
+Woo, J.-H., & Urry, C. M. 2002, ApJ, 579, 530,
+doi: 10.1086/342878
+Zhang, B.-K., Jin, M., Zhao, X.-Y., Zhang, L., & Dai, B.-Z.
+2021, Research in Astronomy and Astrophysics, 21, 186,
+doi: 10.1088/1674-4527/21/8/186
+Zibecchi, L., Andruchow, I., Cellone, S. A., & Carpintero,
+D. D. 2020, MNRAS, 498, 3013,
+doi: 10.1093/mnras/staa2544
+Zibecchi, L., Andruchow, I., Cellone, S. A., et al. 2017,
+MNRAS, 467, 340, doi: 10.1093/mnras/stx054
+

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+LeaFTL: A Learning-based Flash Translation Layer
+for Solid-State Drives
+Jinghan Sun
+UIUC
+[email protected]
+Shaobo Li
+UIUC
+[email protected]
+Yunxin Sun∗
+ETH Zurich
+[email protected]
+Chao Sun
+Western Digital Research
+[email protected]
+Dejan Vucinic
+Western Digital Research
+[email protected]
+Jian Huang
+UIUC
+[email protected]
+ABSTRACT
+In modern solid-state drives (SSDs), the indexing of flash pages is a
+critical component in their storage controllers. It not only affects
+the data access performance, but also determines the efficiency
+of the precious in-device DRAM resource. A variety of address
+mapping schemes and optimizations have been proposed. However,
+most of them were developed with human-driven heuristics.
+In this paper, we present a learning-based flash translation layer
+(FTL), named LeaFTL, which learns the address mapping to tolerate
+dynamic data access patterns via linear regression at runtime. By
+grouping a large set of mapping entries into a learned segment, it
+significantly reduces the memory footprint of the address mapping
+table, which further benefits the data caching in SSD controllers.
+LeaFTL also employs various optimization techniques, including
+out-of-band metadata verification to tolerate mispredictions, opti-
+mized flash allocation, and dynamic compaction of learned index
+segments. We implement LeaFTL with both a validated SSD sim-
+ulator and a real open-channel SSD board. Our evaluation with
+various storage workloads demonstrates that LeaFTL saves the
+memory consumption of the mapping table by 2.9× and improves
+the storage performance by 1.4× on average, in comparison with
+state-of-the-art FTL schemes.
+CCS CONCEPTS
+• Hardware → External storage; • Computer systems orga-
+nization → Architectures; • Computing methodologies →
+Learning linear models.
+KEYWORDS
+Learning-Based Storage, Flash Translation Layer, Solid-State Drive
+1
+INTRODUCTION
+Flash-based SSDs have become an indispensable part in modern
+storage systems, as they outperform conventional hard-disk drives
+(HDDs) by orders of magnitude, and their cost is close to that of
+HDDs [22, 30, 51, 62]. The SSD capacity continues to boost by
+increasing the number of flash channels and chips with the rapidly
+shrinking process and manufacturing technology [22, 25, 41, 46].
+The flash translation layer (FTL) is the core component of man-
+aging flash memory in SSDs, including address translation, garbage
+collection (GC), and wear leveling [20, 66]. The FTL maintains meta-
+data structures for different functions such as address translation
+∗Work done when visiting the Systems Platform Research Group at UIUC as a research
+intern.
+and valid page tracking, and caches them in the in-device DRAM
+(SSD DRAM) for improved performance [7, 12, 25].
+Among these data structures, the address mapping table has
+the largest memory footprint. In general, the address mapping
+table can be categorized in three types: page-level mapping, block-
+level mapping, and hybrid mapping. Modern SSDs usually use the
+page-level mapping, as it offers the best performance for the flash
+page lookup, and incurs minimal GC overhead, in comparison with
+the other two mapping schemes [20, 66]. However, the page-level
+mapping table size is large, as it stores the entry for the LPA-to-PPA
+address translation for each flash page.
+The address mapping table significantly affects the performance
+of SSDs, as it not only determines the efficiency of indexing flash
+pages, but also affects the utilization of SSD DRAM. Moreover, due
+to the limitations of the cost and power budget in SSD controllers,
+it is challenging for SSD vendors to scale the in-device DRAM
+capacity [12, 41]. This challenge becomes even worse with the
+increasing flash memory capacity in an SSD, as larger capacity
+usually requires a larger address mapping table for indexing.
+To improve the address mapping and translation for SSDs, vari-
+ous optimization schemes have been developed [9, 25, 29, 38, 39, 66].
+However, most of them were developed based on human-driven
+heuristics [25], and cannot capture dynamic data access patterns
+at runtime. Employing more semantic knowledge into the FTL,
+such as GraphSSD [44], can improve the data indexing and address
+translation, however, it is application specific and complicates the
+management of address mappings [7], which does not scale for the
+development of generic SSDs. In this work, we do not expect that
+we can obtain application semantics from the host and the SSD con-
+troller. Instead, we focus on utilizing simple yet effective machine
+learning (ML) techniques to automate the address mapping table
+management in the SSDs, with the capability of learning diverse
+and dynamic data access patterns.
+To this end, we propose a learning-based FTL, named LeaFTL, by
+utilizing the piecewise linear regression technique to learn the LPA-
+PPA mappings, and automatically exploiting the data locality of
+various data access patterns at runtime. Unlike the state-of-the-art
+page-level mapping, the key idea of LeaFTL is that it can learn the
+correlation between a set of LPAs and their mapped PPAs, based
+on which it can build a space-efficient index segment, as presented
+in A in Figure 1. Since the learned index segment can be simply
+represented with (𝑆, 𝐿, 𝐾, 𝐼), where [𝑆,𝑆 + 𝐿] denotes the interval
+of LPAs, 𝐾 is the slope of the segment, and 𝐼 is the intercept of the
+segment (see the last diagram in Figure 1), each segment will take
+arXiv:2301.00072v1  [cs.OS]  30 Dec 2022
+
+Jinghan Sun, Shaobo Li, Yunxin Sun, Chao Sun, Dejan Vucinic, and Jian Huang
+30
+LPA
+PPA
+31
+32
+33
+34
+155
+156
+157
+158
+159
+60
+62
+64
+66
+68
+200
+201
+203
+204
+205
+80
+82
+83
+84
+87
+304
+305
+306
+307
+308
+Index Segment
+A
+Index Segment
+B
+Index Segment
+C
+LPA
+PPA
+A
+B
+C
+error bound
+1
+1
+1
+1
+2
+2
+2
+2
+2
+1
+1
+3
+Figure 1: An illustrative example of learning LPA-PPA mappings using piecewise linear regression in LeaFTL. It can learn
+various patterns of LPA-PPA mappings with guaranteed error bound. Each learned index segment can be represented with
+(𝑆, 𝐿, 𝐾, 𝐼), where [𝑆,𝑆 + 𝐿] denotes the interval of LPAs, 𝐾 is the slope, and 𝐼 is the intercept of the index segment.
+only 8 bytes (1 byte for 𝑆 and 𝐿, 2 bytes for 𝐾, and 4 bytes for 𝐼)
+with our optimizations (see the details in §3). Compared to the on-
+demand page-level mapping [20], the learned segment reduces the
+mapping table size by a factor of 𝑚 ∗ 𝑎𝑣𝑔(𝐿)/8, where 𝑚 is the size
+(8 bytes) of each entry in the on-demand page-level mapping table,
+and 𝑎𝑣𝑔(𝐿) is the average number of LPA-PPA mappings that can
+be represented in a learned index segment, 𝑎𝑣𝑔(𝐿) is 20.3 according
+to our study of various storage workloads.
+Beyond learning contiguous LPA-PPA mappings, LeaFTL also
+learns different correlation patterns, such as regular and irregular
+strided data accesses as shown in B and C , respectively. Unlike
+existing indexing optimizations based on human-driven heuristics,
+LeaFTL can learn more irregular patterns of LPA-PPA mappings
+with guaranteed error bound, as shown in C . This enables LeaFTL
+to further condense the address mapping table. Therefore, given a
+limited DRAM capacity in the SSD controller, LeaFTL can maximally
+utilize the DRAM caching and improve the storage performance.
+For the worst case like random I/O accesses, LeaFTL will transfer
+the mapping into single-point linear segments (𝐿 = 0, 𝐾 = 0, and
+𝐼 = 𝑃𝑃𝐴 in Figure 1), and its memory consumption will be no more
+than that of the page-level mapping.
+With the learned index segments, LeaFTL may occasionally re-
+turn an inaccurate PPA (i.e., address misprediction), which incurs
+additional flash accesses until the correct PPA is identified. To over-
+come this challenge, we develop an error-tolerant mechanism in
+LeaFTL. For each flash page access, we use the reverse mapping
+stored in the out-of-band (OOB) metadata of each flash page to
+verify the correctness of the data access. Since the OOB usually has
+64–256 bytes [20, 23], we use it to store the accurate LPAs mapped
+to the neighbor PPAs. Thus, upon an address misprediction, we use
+the stored reverse mappings to find the correct PPA, avoiding addi-
+tional flash accesses. LeaFTL leverages the intrinsic OOB structure
+to handle address mispredictions and make SSD perfectly-suited
+for practical learned indexing.
+Due to the intrinsic out-of-place write property of SSDs (see
+§2), the learned index segments will be disrupted by writes and
+GC, and the segments need to be relearned with new LPA-PPA
+mappings. To tolerate these disruptions, the learned segments are
+organized within multiple levels to maintain the temporal order
+in a log-structured manner: the topmost level has the most recent
+segments, and the lower level stores older segments. The segments
+at the same level are sorted without overlapping. If the new segment
+has a conflict with an existing segment, the old segment will be
+moved to the lower level. Therefore, LeaFTL can always identify
+the latest version of the corresponding LPA-PPA mapping in a top
+level of learned index segments. LeaFTL will compact the learned
+segments periodically to reduce its memory footprint.
+To further maximize the efficiency of LeaFTL, we coordinate its
+learning procedure with flash block allocation in the SSD. As flash
+block allocation decides the distribution of mapped PPAs, LeaFTL
+will allocate consecutive PPAs to contiguous LPAs at its best effort,
+for increasing the possibility of learning a space-efficient index seg-
+ment. Similar to existing page-level mapping [20, 23], LeaFTL stores
+the learned index segments in flash blocks for recovery. Overall,
+we make the following contributions:
+• We present a learning-based FTL, it can learn various data access
+patterns and turn them into index segments for reducing the
+storage cost of the mapping table.
+• We develop an error-tolerant address translation mechanism to
+handle address mispredictions caused by the learned indexes,
+with minimal extra flash accesses.
+• We preserve the core FTL functions, and enable the coordination
+between the learning procedure of the address mapping table
+with the flash block allocation and GC to maximize the efficiency
+of the learned FTL.
+• We manage the learned segments in an optimized log-structured
+manner, and enable compaction to further improve the space
+efficiency for the address mapping.
+We implement LeaFTL with a validated SSD simulator Wisc-
+Sim [27] and evaluate its efficiency with a variety of popular storage
+workloads. We also develop a system prototype with a real 1TB
+open-channel SSD to verify the functions of LeaFTL and validate
+its efficiency with real data-intensive applications, such as the key-
+value store and transactional database. Our evaluation with the
+real SSD shows similar benefits as that of the SSD simulator imple-
+mentation. We demonstrate that LeaFTL reduces the storage cost
+of the address mapping in the FTL by 2.9× on average. The saved
+memory space benefits the utilization of the precious SSD DRAM,
+and further improves the storage performance by 1.4× on average.
+We also show that LeaFTL does not affect the SSD lifetime, and its
+
+LeaFTL: A Learning-based Flash-Translation Layer for Solid-State Drives
+flash
+flash
+flash
+flash
+Flash
+Flash
+Flash
+Flash
+DRAM
+Flash
+Controller
+SSD Controller/Firmware
+PCIe Interface
+Embedded 
+Processor
+Internal Bus
+DRAM 
+Controller
+Block I/O
+Figure 2: The internal system architecture of SSDs.
+learning procedure introduces negligible performance overhead
+to the storage processor in the SSD controllers. The codebase of
+LeaFTL is available at https://github.com/platformxlab/LeaFTL.
+2
+BACKGROUND AND MOTIVATION
+Flash-Based Solid-State Drive. An SSD has three major parts
+(see Figure 2): a set of flash memory packages, an SSD controller
+with embedded processors, and a set of flash controllers. With the
+nature of NAND Flash, when a free page is written, the page cannot
+be written again until that page is erased. However, erase operation
+is performed only at a block granularity. As the erase operation is
+expensive, writes are issued to free flash pages erased in advance
+(i.e., out-of-place write). GC will be performed to clean the stale
+data. As each flash block has limited endurance, it is important for
+them to age uniformly (i.e., wear leveling). SSDs have a logical-
+to-physical address mapping table to index flash pages. All these
+functions are managed by the FTL in the SSD firmware.
+Modern SSD controllers have general-purpose embedded pro-
+cessors (e.g., ARM processors). The processors help with issuing
+I/O requests, translating LPAs to PPAs, and handling GC and wear-
+leveling. SSDs also have limited DRAM capacities to cache the
+mapping table and the application data.
+Address Mapping Table in the FTL. The address mapping table
+in FTL generally has three types: page-level mapping, block-level
+mapping, and hybrid mapping. The page-level mapping enables di-
+rect LPA-PPA mapping for fast lookup. However, each entry usually
+takes 8 bytes (4 bytes for LPA, 4 bytes for PPA), and the entire map-
+ping table requires large storage space. The block-level mapping
+significantly reduces the mapping table size. However, it introduces
+additional overhead for the page lookup in the flash block. The hy-
+brid mapping takes advantages of both page-level and block-level
+mapping. It uses log blocks to store new writes, and index them
+with the page-level mapping. The log blocks will be moved into
+data blocks that are indexed with block-level mapping. This incurs
+significant GC overhead. Therefore, modern SSDs commonly use
+the page-level mapping scheme.
+Metadata Structures for Flash Management. The FTL usually
+employs four metadata structures (see Figure 3): (1) the address
+mapping cache ( 1 AMC) for caching the address mapping table
+in the SSD DRAM; (2) the global mapping directory ( 2 GMD) for
+tracking the locations of the address mapping table pages in the
+Address Mapping  
+Cache (AMC)
+1
+Global Mapping 
+Directory (GMD)
+2
+Block Validity 
+Counter (BVC)
+3
+Page Validity 
+Table (PVT)
+4
+LPA
+PPA
+...
+...
+LX
+PY
+...
+...
+LPA
+PPA
+...
+...
+VX
+PZ
+...
+...
+PBA
+Counter
+...
+...
+...
+...
+...
+...
+PBA
+Bitmap
+...
+...
+PB
+...
+...
+...
+Data Structures in the FTL of Modern SSDs
+Flash Memory
+Data Blocks
+Address Mapping Blocks
+Validity Blocks
+Figure 3: The common data structures in the FTL of SSDs.
+SSD; (3) the block validity counter ( 3 BVC) for tracking the number
+of valid pages for each flash block for assisting the GC in the SSD;
+and (4) the page validity table ( 4 PVT), which uses bitmaps to
+track the valid pages in each flash block. During the GC, the FTL
+will check the 3 BVC to select candidate flash blocks, and migrate
+their valid pages to free flash blocks. After that, it will erase these
+selected flash blocks, and mark them as free blocks.
+Limited DRAM Capacity in SSD Controllers. It is hard to provi-
+sion large DRAM inside SSD controllers, due to their hardware con-
+straints and limited budgets for power and hardware cost [12, 41, 60].
+Thus, SSD controllers often use on-demand caching to maintain
+the recently accessed metadata and data in the SSD DRAM.
+Among all the metadata structures, the address mapping table
+has the largest memory footprint. As discussed, 1 AMC caches the
+recently accessed mapping table entries. If a mapping entry is not
+cached, the FTL will locate the corresponding address mapping ta-
+ble pages stored in the flash blocks, and place the mapping entry in
+the 1 AMC. As we scale the SSD capacity, the DRAM challenge will
+become even worse. To overcome this challenge, various optimiza-
+tions on the mapping table have been proposed [9, 25, 29, 31, 38, 39]
+to improve the utilization of the SSD DRAM. However, most of
+them cannot automatically capture diverse data access patterns at
+runtime, leaving a large room for improvement.
+3
+DESIGN AND IMPLEMENTATION
+To develop LeaFTL in the SSD controller, we have to overcome the
+following research challenges.
+• LeaFTL should be able to automatically capture diverse data
+access patterns, and generate memory-efficient address mapping
+(§3.1, §3.2, §3.3, and §3.4).
+• LeaFTL may incur address mispredictions, which could incur
+additional flash accesses. LeaFTL should be tolerant of errors and
+have low misprediction penalty (§3.5).
+• LeaFTL should work coordinately with other core FTL functions
+that include GC and wear leveling (§3.6).
+• LeaFTL should be lightweight and not incur much extra overhead
+to storage operations (§3.7, §3.8 and §3.9).
+
+Jinghan Sun, Shaobo Li, Yunxin Sun, Chao Sun, Dejan Vucinic, and Jian Huang
+(a) Precise Linear Approximation 
+(b) Inaccurate Linear Approximation 
+Figure 4: Visualization of learned index segments.
+1
+2
+4
+8
+16
+32
+64
+128
+256
+512 1024 2048
+Length of Learned Segments
+0
+20
+40
+60
+80
+100
+Percentage of 
+Segments (%)
+=0, #Segments=5540
+=4, #Segments=4267
+=8, #Segments=3718
+Figure 5: Aggregated distribution of learned segments.
+3.1
+Key Ideas of LeaFTL
+Instead of using the space-consuming one-to-one mapping in the
+page-level mapping, the key idea of LeaFTL is to exploit learning
+techniques to identify various LPA-PPA mapping patterns and build
+efficient learned address mapping entries. Modern SSD controllers
+usually have a data buffer for grouping writes and write the large
+data chunk at once for exploiting the internal flash parallelisms.
+LeaFTL utilizes this data buffer to collect LPA-to-PPA mappings for
+learning index segments for free, and does not introduce extra data
+collection overhead (see the details in §3.3).
+As shown in Figure 4 (a), the PPA of an LPA can be obtained
+with the expression: 𝑃𝑃𝐴 = 𝑓 (𝐿𝑃𝐴) = ⌈𝐾 ∗ 𝐿𝑃𝐴 + 𝐼⌉, 𝐿𝑃𝐴 ∈
+[𝑆𝐿𝑃𝐴,𝑆𝐿𝑃𝐴 + 𝐿], where [𝑆𝐿𝑃𝐴,𝑆𝐿𝑃𝐴 + 𝐿] denotes the interval (𝐿)
+of LPAs, 𝐾 is the slope, and 𝐼 is the intercept. As discussed in §1,
+each learned index segment can be represented in 8 bytes: 1 byte for
+𝑆𝐿𝑃𝐴 and 𝐿, respectively; 2 bytes for 𝐾, and 4 bytes for 𝐼. The size
+of 𝑆𝐿𝑃𝐴 is reduced from 4 bytes to 1 byte with our optimizations
+on the segment management (see §3.4).
+We can relax the linear regression to capture more flash access
+patterns, which further reduces the learned address mapping table
+size. As shown in Figure 4 (b), the linear regression can learn a
+pattern with guaranteed error bound [−𝛾,𝛾]. As we increase 𝛾, we
+can cover more flash access patterns. We applied the relaxed linear
+regression with different 𝛾 values to a variety of storage workloads
+(see §4.1), our experimental results demonstrate that the number
+of learned index segments is gradually decreased, as we increase 𝛾.
+Figure 5 shows that 98.2–99.2% of the learned index segments cover
+Segment
+SLPA
+L
+K
+I
+1B
+1B
+2B
+4B
+Type
+LPAs
+PPAs
+Index Segment
+Accurate
+[0, 1, 2, 3]
+[32, 33, 34, 35]
+Approximate
+[0, 1, 4, 5]
+[64, 65, 66, 67]
+0
+3
+1.00
+32
+0
+5
+0.56
+64
+Figure 6: Types of learned segments in LeaFTL.
+up to 128 LPA-PPA mapping entries, demonstrating the potential
+advantages of the learning-based approach.
+As for random access patterns, LeaFTL will transfer the learned
+segments into single-point segments. And these linear segments
+do not require more storage space than the page-level mapping.
+3.2
+Learned Index Segment
+Types of Learned Index Segment. The mapping table of LeaFTL
+is built with learned index segments. It has two types of segments:
+accurate and approximate segments, as shown in Figure 6. Both of
+them are learned with piecewise linear regression technique [64].
+As for the accurate index segments, given an LPA, we can pre-
+cisely get the corresponding PPA with 𝑓 (𝐿𝑃𝐴) = ⌈𝐾 ∗ 𝐿𝑃𝐴 + 𝐼⌉.
+For example, when the LPA is 2 in Figure 6, we can directly get the
+PPA value of 34 with ⌈1.00 ∗ 2 + 32⌉. In this example, the learned
+segment has 𝐿 = 3 and it indexes 4 LPA-PPA mappings. If 𝐿 = 0,
+the learned segment will become a single-point segment, the slope
+𝐾 = 0, and we will get its PPA with 𝑃𝑃𝐴 = 𝐼.
+As for approximate index segments, we use the same formula
+𝑓 (𝐿𝑃𝐴) = ⌈𝐾 ∗𝐿𝑃𝐴+𝐼⌉ to calculate the PPA. However, the returned
+PPA may not be the exact corresponding PPA. It has an error bound
+[−𝛾,𝛾] guaranteed by the linear regression, and 𝛾 is configurable.
+For example, given 𝐿𝑃𝐴 = 4 in Figure 6, the value of the PPA is
+67, according to the calculation ⌈4 ∗ 0.56 + 64⌉. However, the real
+PPA should be 66. We define this as address misprediction. We will
+discuss how we handle the address misprediction with reduced
+miss penalty in §3.5.
+Size of Learned Index Segment. As discussed in §3.1, each seg-
+ment can be expressed in (𝑆𝐿𝑃𝐴, 𝐿, 𝐾, 𝐼). The starting LPA will take
+4 bytes. We can further reduce this size by partitioning a range of
+LPAs into small groups, and each LPA group represents a certain
+number of contiguous LPAs. Therefore, we can index an LPA with
+its offset in a corresponding group. In LeaFTL, each group repre-
+sents 256 contiguous LPAs. Thus, 𝑆𝐿𝑃𝐴 can be indexed by the offset
+(28 = 256) in the group, which takes only 1 byte. We use 256 as the
+group size, because the length of the learned segments is usually
+less than 256 (see Figure 5).
+Given an LPA, we can get its offset in the group with (𝐿𝑃𝐴 𝑚𝑜𝑑
+256). In LeaFTL, we set the 𝐿 as 1 byte. Thus, each segment can
+index 256 LPA-PPA mappings. We use a 16-bit floating point to
+store the value of the slope 𝐾. And the intercept 𝐼 of a segment
+can be represented in 4 bytes. Therefore, in combination with 𝑆𝐿𝑃𝐴,
+both accurate and approximate segments can be encoded with 8
+bytes (see Figure 6), which are memory aligned.
+
+LeaFTL: A Learning-based Flash-Translation Layer for Solid-State Drives
+(a) Unoptimized learned segments
+(b) Optimized learned segments with sorting
+Learned Segments
+78
+32  33
+76
+Flush
+Data Buffer
+115
+34  38
+Flash Block
+78
+32
+33
+76
+115
+34
+38
+...
+LPA 78
+32
+33
+76 115 34
+38
+Learned Segments
+Flush
+Data Buffer
+Flash Block
+32
+33
+34
+38
+76
+78
+115
+...
+LPA 78
+32
+33
+76 115 34
+38
+115
+32  33  34  38 76  78
+Figure 7: An example of reducing the number of learned seg-
+ments via exploiting the flash block allocation.
+LeaFTL uses the least significant bit of the 𝐾 to indicate segment
+types (0 for accurate segments, 1 for approximate segments). This
+has negligible impact on the address translation accuracy, because
+𝐾 ∈ [0, 1], which will only affect the tenth digit after decimal point.
+3.3
+Improve the Learning Efficiency
+To further reduce the number of learned segments, LeaFTL performs
+optimizations to improve its learning efficiency of address mappings
+by exploiting the flash block allocation in SSD controllers, as shown
+in Figure 7. Flash pages are usually buffered in the SSD controller
+and written to flash chips at a flash block granularity, for utilizing
+the internal bandwidth and avoiding the open-block problem [6,
+22, 37, 48]. This allows LeaFTL to learn more space-efficient index
+segments (i.e., index segments can cover more LPA-PPA mappings)
+by reordering the flash pages with their LPAs in the data buffer.
+As shown in Figure 7 (a), LeaFTL learns 5 index segments (78), (32,
+33), (76), (115), and (34, 38) with 𝛾 = 4. After sorting the pages in
+the data buffer shown in Figure 7 (b), LeaFTL generates 3 index
+segments (32, 33, 34, 38), (76, 78), and (115).
+To develop the optimized learned segments, LeaFTL sorts the
+flash pages in ascending order of their LPAs in the data buffer (8MB
+by default). When pages in the data buffer is flushed to the flash
+chips, their PPAs are in ascending order. This ensures a mono-
+tonic address mapping between LPAs and PPAs, which reduces the
+number of index segments.
+3.4
+Manage Learned Index Segments
+Upon new data updates or GC in the SSD, the learned index seg-
+ments need to be updated, due to the intrinsic property (i.e., out-of-
+place update) of SSDs. Unfortunately, the direct updates to learned
+index segments are expensive, since we have to relearn the in-
+dex segments with new PPAs. This relearning procedure not only
+consumes extra compute cycles, but also involves additional flash
+accesses, since we have to access the corresponding flash pages to
+obtain accurate PPAs for some of the LPAs in the index segment
+being updated. For instance, for in-place update to an approximate
+Level 0
+Level 1
+0     63
+100     200 230    255
+16   127
+206    240
+non-overlapping 
+at each level
+segments can overlap 
+across levels
+Figure 8: The learned index segments are managed in a log-
+structured manner in LeaFTL.
+segment, it can incur 21 flash accesses on average when relearn-
+ing. In-place update also breaks the existing LPA-to-PPA mapping
+patterns, which results in 1.2× additional segments and memory
+footprint, according to our experiments with various workloads.
+To address this challenge, we manage the learned index segments
+in a log-structured manner, as shown in Figure 8. Therefore, the
+newly learned index segments will be appended to the log structure
+(level 0 in Figure 8) and used to index the updated LPA-PPA map-
+pings, while the existing learned segments (level 1 and lower levels
+in Figure 8) can still serve address translations for LPAs whose map-
+pings have not been updated. Such a structure supports concurrent
+lookups as enabled in the traditional log-structured merge tree. As
+we insert the newly learned index segments at the top level of the
+log-structured tree, this minimizes the impact on other segments.
+Log-Structured Mapping Table. The log-structured mapping ta-
+ble has multiple levels to maintain the temporal order of index seg-
+ments. As discussed, the topmost level has the most recent learned
+index segments, and the lower level stores the older segments. For
+the segments on the same level, LeaFTL ensures that they are sorted
+and do not have overlapped LPAs. This is for fast location of the
+corresponding learned index segments in each level. For the seg-
+ments across the levels, they may have overlapped LPAs, due to the
+nature of the log-structured organization. And the segments with
+overlapped LPA-PPA mappings will be compacted periodically for
+space reclamation (see its detailed procedure in §3.7).
+Manage Two Types of Index Segments. LeaFTL manages the ac-
+curate and approximate index segments in the same log-structured
+mapping table, as they can be encoded in the same format. For each
+accurate segment, we can directly infer its indexed LPAs with the
+𝑆𝐿𝑃𝐴, 𝐾, and 𝐿, since it has a regular pattern. However, for approx-
+imate index segments, we only have the knowledge of the starting
+LPA and the end LPA with 𝑆𝐿𝑃𝐴 + 𝐿. Its encoded LPAs cannot be
+directly inferred from their metadata (𝑆𝐿𝑃𝐴, 𝐿, 𝐾, 𝐼), since they are
+learned from irregular access patterns and may have mispredictions.
+If two approximate segments have overlapping LPA ranges, we
+could obtain inaccurate PPAs from the learned index segments.
+As shown in Figure 9 (a), given an LPA with the value 105, we
+will check the segment at Level 0 and may get an inaccurate PPA.
+This will also affect the efficiency of the segment compaction, with
+which we eliminate duplicated entries between segments.
+To address this challenge, LeaFTL uses a Conflict Resolution
+Buffer (CRB) for each LPA group to store the LPAs indexed by each
+approximate segment. The main purpose of CRB is to help LeaFTL
+check whether a given LPA belongs to one approximate segment.
+The CRB is a nearly-sorted list [10] by the starting LPAs of its ap-
+proximate segments. To be specific, the CRB ensures the following
+
+Jinghan Sun, Shaobo Li, Yunxin Sun, Chao Sun, Dejan Vucinic, and Jian Huang
+100
+6
+K1
+I1
+ [100, 101, 103, 104, 106]
+102
+6
+K2
+I2
+[102, 105, 107, 108]
+L0
+L1
+LPAs
+Lookup (LPA = 105)
+(a) Approximate index segments that index overlapped LPAs.
+Conflict Resolution Buffer
+100
+101
+103
+104
+106
+null
+102
+105
+107 108
+null
+...
+Lookup (LPA = 105)
+102
+6
+K2
+I2
+(b) Resolve the conflict between approximate segments with CRB
+Figure 9: A case study of conflict resolution buffer for ap-
+proximate learned index segments.
+properties: (1) the LPAs belong to the same approximate segment
+are stored contiguously; (2) different approximate segments are
+sorted by their starting LPA, and CRB uses a 𝑛𝑢𝑙𝑙 byte to separate
+these segments; (3) it does not have redundant LPAs, which means
+an LPA will appear at most once in the CRB. This is achieved by
+removing existing same LPAs when we insert new approximate
+segments into the CRB.
+However, if the 𝑆𝐿𝑃𝐴 of a new approximate segment is the same
+as any starting LPAs that have been stored in the CRB, LeaFTL will
+update the 𝑆𝐿𝑃𝐴 of the old segment with the adjacent LPA. Take
+Figure 9 (b) as an example, upon a new approximate segment with
+𝑆𝐿𝑃𝐴 = 100, we will update the 𝑆𝐿𝑃𝐴 of the existing segment to 101,
+and then insert the new segment into the CRB. In this case, LeaFTL
+will ensure each approximate segment will have its unique 𝑆𝐿𝑃𝐴.
+This will facilitate the approximate LPA-PPA address translation
+with high accuracy confidence.
+Since CRB is nearly sorted, its insertion, deletion, and lookup
+operations are fast. The CRB is also space efficient, as each LPA
+(the offset in its corresponding LPA group) will take only one byte,
+and it guarantees that there are no redundant LPAs. Therefore, the
+CRB will maximally store 256 LPAs. Our experiments with a variety
+of storage workloads show that the CRB will take 13.9 bytes on
+average, as shown in Figure 10.
+Given an LPA, in order to identify which approximate index
+segment it belongs to, LeaFTL will check the CRB with binary
+search. Once the LPA is found, LeaFTL will search to its left until
+identifying the 𝑆𝐿𝑃𝐴, and this 𝑆𝐿𝑃𝐴 will be the starting LPA of
+the corresponding approximate segment, as shown in Figure 9 (b).
+Therefore, CRB can assist LeaFTL to resolve the LPA lookups.
+3.5
+Handle Address Misprediction
+As discussed in §3.2, the mapping table entries encoded with ap-
+proximate segments may occasionally incur mispredictions and
+return an approximated PPA. These approximate segments have a
+guaranteed error bound [−𝛾,𝛾], where 𝛾 is a constant value that
+can be specified in the linear regression algorithm. To verify the
+correctness of the address translation, a simple method is to access
+MSR-hm
+MSR-src2
+MSR-prxy
+MSR-prn
+MSR-usr
+FIU-home
+FIU-mail
+0
+100
+200
+300
+CRB Size (in Bytes)
+Average
+99 Percentile
+Figure 10: The distribution of CRB sizes for different storage
+workloads, when we set 𝛾 = 4 in LeaFTL.
+PPA1
+PPA2
+PPA3
+PPA4
+PPA5
+Data Blocks
+Data
+OOB
+Flash Page
+LPA2
+LPA4
+LPA
+Reverse Mapping
+Figure 11: The out-of-band (OOB) metadata organization. It
+stores the reverse mapping for its neighbor PPAs.
+the flash page with the predicted PPA, and use the reverse mapping
+(its corresponding LPA) stored in the OOB metadata of the flash
+page to check whether the LPA matches or not. In this case, upon
+a PPA misprediction, we need log(𝛾) flash accesses on average to
+identify the correct PPA.
+To avoid extra flash accesses for address mispredictions, LeaFTL
+leverages the OOB of the flash page to store the reverse mappings
+of its neighbor PPAs. This is developed based on the insight that:
+with a 𝑃𝑃𝐴𝑙𝑒𝑎𝑟𝑛𝑒𝑑 obtained from an approximate segment, its er-
+ror bound [−𝛾,𝛾] guarantees that the correct PPA is in the range
+of [𝑃𝑃𝐴𝑙𝑒𝑎𝑟𝑛𝑒𝑑 − 𝛾, 𝑃𝑃𝐴𝑙𝑒𝑎𝑟𝑛𝑒𝑑 + 𝛾], as discussed in Figure 4 (b).
+Thus, upon a misprediction, LeaFTL will read the flash page with
+𝑃𝑃𝐴𝑙𝑒𝑎𝑟𝑛𝑒𝑑, and use its OOB to find the correct PPA. In this case,
+LeaFTL ensures that it will incur only one extra flash access for
+address mispredictions.
+This is a feasible approach, as the OOB size is usually 128–256
+bytes in modern SSDs. As each LPA takes 4 bytes, we can store
+32–64 reverse mapping entries in the OOB. We show the OOB
+organization of LeaFTL in Figure 11. For the flash page 𝑃𝑃𝐴𝑋 , the
+first 2𝛾 + 1 entries in its OOB correspond to the LPAs for the flash
+pages [𝑃𝑃𝐴𝑋 − 𝛾, 𝑃𝑃𝐴𝑋 + 𝛾]. For the flash pages at the beginning
+and end of a flash block, we may not be able to obtain the reverse
+mapping of their neighbor PPAs. We place the 𝑛𝑢𝑙𝑙 bytes in the
+corresponding entry of the OOB.
+3.6
+Preserve Other Core FTL Functions
+LeaFTL preserves the core functions such as GC and wear leveling
+in an FTL. It follows the same GC and wear leveling policies in
+modern SSDs. When the number of free blocks in an SSD is below
+a threshold (usually 15-40% of the total flash blocks), the SSD con-
+troller will trigger the GC execution. LeaFTL employs the greedy
+algorithm [5] to select the candidate blocks which have the minimal
+
+LeaFTL: A Learning-based Flash-Translation Layer for Solid-State Drives
+ALGORITHM 1: LeaFTL operations
+Input: 𝑔𝑟𝑜𝑢𝑝𝑠 ← 𝐿𝑒𝑎𝐹𝑇𝐿 𝑔𝑟𝑜𝑢𝑝 𝑝𝑎𝑟𝑡𝑖𝑡𝑖𝑜𝑛𝑠
+// Insert/Update Segment in the LeaFTL
+1 Function 𝑠𝑒𝑔_𝑢𝑝𝑑𝑎𝑡𝑒(𝑠𝑒𝑔𝑚𝑒𝑛𝑡,𝑙𝑒𝑣𝑒𝑙):
+2
+𝑠𝑒𝑔_𝑝𝑜𝑠 = 𝑏𝑖𝑛𝑎𝑟𝑦_𝑠𝑒𝑎𝑟𝑐ℎ(𝑙𝑒𝑣𝑒𝑙,𝑠𝑒𝑔𝑚𝑒𝑛𝑡.𝑆𝐿𝑃𝐴)
+3
+𝑙𝑒𝑣𝑒𝑙.𝑖𝑛𝑠𝑒𝑟𝑡 (𝑠𝑒𝑔𝑚𝑒𝑛𝑡,𝑠𝑒𝑔_𝑝𝑜𝑠)
+4
+if 𝑛𝑜𝑡 𝑠𝑒𝑔𝑚𝑒𝑛𝑡.𝑎𝑐𝑐𝑢𝑟𝑎𝑡𝑒 then
+5
+Insert LPAs into CRB and remove redundant LPAs
+6
+if 𝑠𝑒𝑔𝑚𝑒𝑛𝑡.𝑆𝐿𝑃𝐴 exists in CRB then
+7
+Update the 𝑆𝐿𝑃𝐴 of the old segment
+8
+𝑣𝑖𝑐𝑡𝑖𝑚_𝑠𝑒𝑔𝑚𝑒𝑛𝑡𝑠 ← All segments that overlap the 𝑠𝑒𝑔𝑚𝑒𝑛𝑡
+starting with 𝑠𝑒𝑔_𝑝𝑜𝑠
+9
+foreach 𝑣𝑖𝑐𝑡𝑖𝑚 ∈ 𝑣𝑖𝑐𝑡𝑖𝑚_𝑠𝑒𝑔𝑚𝑒𝑛𝑡𝑠 do
+10
+𝑠𝑒𝑔_𝑚𝑒𝑟𝑔𝑒 (𝑠𝑒𝑔𝑚𝑒𝑛𝑡, 𝑣𝑖𝑐𝑡𝑖𝑚)
+// if marked as removable by seg_merge()
+11
+if 𝑣𝑖𝑐𝑡𝑖𝑚.𝐿 = −1 then
+12
+𝑙𝑒𝑣𝑒𝑙.𝑟𝑒𝑚𝑜𝑣𝑒 (𝑣𝑖𝑐𝑡𝑖𝑚)
+13
+if 𝑠𝑒𝑔𝑚𝑒𝑛𝑡.𝑜𝑣𝑒𝑟𝑙𝑎𝑝𝑠 (𝑣𝑖𝑐𝑡𝑖𝑚) then
+14
+Pop 𝑣𝑖𝑐𝑡𝑖𝑚 to the next level
+15
+if 𝑣𝑖𝑐𝑡𝑖𝑚 has overlaps in the next level then
+16
+Create level for 𝑣𝑖𝑐𝑡𝑖𝑚 to avoid recursion
+// Lookup LPA in the LeaFTL
+17 Function 𝑙𝑜𝑜𝑘𝑢𝑝(𝑙𝑝𝑎):
+18
+foreach 𝑙𝑒𝑣𝑒𝑙 ∈ 𝑔𝑟𝑜𝑢𝑝𝑠 [𝑙𝑝𝑎 𝑚𝑜𝑑 256] do
+19
+𝑠𝑒𝑔_𝑝𝑜𝑠 = 𝑏𝑖𝑛𝑎𝑟𝑦_𝑠𝑒𝑎𝑟𝑐ℎ(𝑙𝑒𝑣𝑒𝑙,𝑙𝑝𝑎)
+20
+𝑠𝑒𝑔𝑚𝑒𝑛𝑡 = 𝑙𝑒𝑣𝑒𝑙.𝑔𝑒𝑡_𝑠𝑒𝑔𝑚𝑒𝑛𝑡 (𝑠𝑒𝑔_𝑝𝑜𝑠)
+21
+if ℎ𝑎𝑠_𝑙𝑝𝑎(𝑠𝑒𝑔𝑚𝑒𝑛𝑡, 𝑙𝑝𝑎) then
+22
+return 𝑠𝑒𝑔𝑚𝑒𝑛𝑡.𝑡𝑟𝑎𝑛𝑠𝑙𝑎𝑡𝑒𝑃𝑃𝐴(𝑙𝑝𝑎)
+// LeaFTL Compaction
+23 Function 𝑠𝑒𝑔_𝑐𝑜𝑚𝑝𝑎𝑐𝑡():
+24
+foreach 𝑔𝑟𝑜𝑢𝑝 ∈ 𝑔𝑟𝑜𝑢𝑝𝑠 do
+25
+foreach 𝑢𝑝𝑝𝑒𝑟_𝑙𝑒𝑣𝑒𝑙,𝑙𝑜𝑤𝑒𝑟_𝑙𝑒𝑣𝑒𝑙 ∈ 𝑔𝑟𝑜𝑢𝑝 do
+26
+foreach 𝑠𝑒𝑔𝑚𝑒𝑛𝑡 ∈ 𝑢𝑝𝑝𝑒𝑟_𝑙𝑒𝑣𝑒𝑙 do
+27
+𝑠𝑒𝑔_𝑢𝑝𝑑𝑎𝑡𝑒 (𝑠𝑒𝑔𝑚𝑒𝑛𝑡,𝑙𝑜𝑤𝑒𝑟_𝑙𝑒𝑣𝑒𝑙)
+28
+if 𝑢𝑝𝑝𝑒𝑟_𝑙𝑒𝑣𝑒𝑙 is empty then
+29
+𝑔𝑟𝑜𝑢𝑝.𝑟𝑒𝑚𝑜𝑣𝑒 (𝑢𝑝𝑝𝑒𝑟_𝑙𝑒𝑣𝑒𝑙)
+number of valid pages, for reducing the data movement overhead
+at GC. As the GC move the valid pages from the candidate blocks
+to the free blocks, LeaFTL places these valid pages into the DRAM
+buffer, sort them by their LPAs, and learn a new index segment.
+The learning procedure is the same as we build index segments for
+new flash writes/updates. Thus, the address mapping of the valid
+pages is updated after the GC.
+LeaFTL also ensures all the flash blocks age at the same rate
+(i.e., wear leveling). It uses the throttling and swapping mechanism
+developed in existing GC, in which the cold data blocks (i.e., blocks
+not frequently accessed) will be migrated to hot blocks (i.e., blocks
+that experience more wear). LeaFTL will learn new indexes for
+these swapped blocks and insert them into the mapping table to
+update their address mappings.
+3.7
+LeaFTL Operations
+Now we describe the LeaFTL operations, including segment cre-
+ation, insert/update, LPA lookup, and compaction. We discuss their
+procedures, and use examples to illustrate each of them, respec-
+tively. We present their detailed procedures in Algorithm 1 and 2.
+ALGORITHM 2: Segment Merge
+// Check if Segment Contains LPA
+1 Function ℎ𝑎𝑠_𝑙𝑝𝑎(𝑠𝑒𝑔, 𝑙𝑝𝑎):
+2
+𝑎𝑐𝑐 ← 𝑠𝑒𝑔.𝑎𝑐𝑐𝑢𝑟𝑎𝑡𝑒
+3
+if 𝑙𝑝𝑎 ∉ [𝑠𝑒𝑔.𝑆𝐿𝑃𝐴,𝑠𝑒𝑔.𝑆𝐿𝑃𝐴 + 𝑠𝑒𝑔.𝐿] 𝑜𝑟
+(𝑛𝑜𝑡 𝑎𝑐𝑐 & 𝑐ℎ𝑒𝑐𝑘 (𝐶𝑅𝐵) 𝑓 𝑎𝑖𝑙𝑒𝑑) 𝑜𝑟
+(𝑎𝑐𝑐 & (𝑙𝑝𝑎 − 𝑠𝑒𝑔.𝑆𝐿𝑃𝐴) 𝑚𝑜𝑑 ⌈
+1
+𝑠𝑒𝑔.𝐾 ⌉ ≠ 0) then
+4
+𝑟𝑒𝑡𝑢𝑟𝑛 𝐹𝑎𝑙𝑠𝑒
+5
+𝑟𝑒𝑡𝑢𝑟𝑛 𝑇𝑟𝑢𝑒
+// Convert Segment into a Temporary Bitmap
+6 Function 𝑔𝑒𝑡_𝑏𝑖𝑡𝑚𝑎𝑝(𝑠𝑒𝑔, 𝑠𝑡𝑎𝑟𝑡, 𝑒𝑛𝑑):
+7
+𝑏𝑚 ← 𝑏𝑖𝑡𝑚𝑎𝑝 𝑜𝑓 𝑙𝑒𝑛𝑔𝑡ℎ (𝑒𝑛𝑑 − 𝑠𝑡𝑎𝑟𝑡 + 1)
+8
+foreach 𝑙𝑝𝑎 ∈ [𝑠𝑡𝑎𝑟𝑡,𝑒𝑛𝑑] do
+9
+if ℎ𝑎𝑠_𝑙𝑝𝑎(𝑠𝑒𝑔, 𝑙𝑝𝑎) then
+10
+𝑏𝑚[𝑙𝑝𝑎 − 𝑠𝑡𝑎𝑟𝑡 ] = 1
+11
+else
+12
+𝑏𝑚[𝑙𝑝𝑎 − 𝑠𝑡𝑎𝑟𝑡 ] = 0
+13
+return 𝑏𝑚
+// Merge a New Segment with an Old Segment
+14 Function 𝑠𝑒𝑔_𝑚𝑒𝑟𝑔𝑒(𝑛𝑒𝑤, 𝑜𝑙𝑑):
+15
+𝑠𝑡𝑎𝑟𝑡 ← 𝑚𝑖𝑛(𝑛𝑒𝑤.𝑆𝐿𝑃𝐴, 𝑜𝑙𝑑.𝑆𝐿𝑃𝐴)
+16
+𝑒𝑛𝑑 ← 𝑚𝑎𝑥 (𝑛𝑒𝑤.𝑆𝐿𝑃𝐴 + 𝑛𝑒𝑤.𝐿, 𝑜𝑙𝑑.𝑆𝐿𝑃𝐴 + 𝑜𝑙𝑑.𝐿)
+17
+𝑏𝑚𝑛𝑒𝑤 ← 𝑔𝑒𝑡_𝑏𝑖𝑡𝑚𝑎𝑝 (𝑛𝑒𝑤, 𝑠𝑡𝑎𝑟𝑡, 𝑒𝑛𝑑)
+18
+𝑏𝑚𝑜𝑙𝑑 ← 𝑔𝑒𝑡_𝑏𝑖𝑡𝑚𝑎𝑝 (𝑜𝑙𝑑, 𝑠𝑡𝑎𝑟𝑡, 𝑒𝑛𝑑)
+19
+𝑏𝑚𝑜𝑙𝑑 ← 𝑏𝑚𝑜𝑙𝑑 & ¬𝑏𝑚𝑛𝑒𝑤
+20
+𝑓 𝑖𝑟𝑠𝑡, 𝑙𝑎𝑠𝑡 ← the first and last valid bit of 𝑏𝑚𝑜𝑙𝑑
+21
+𝑜𝑙𝑑.𝑆𝐿𝑃𝐴, 𝑜𝑙𝑑.𝐿 ← 𝑓 𝑖𝑟𝑠𝑡 + 𝑠𝑡𝑎𝑟𝑡, 𝑙𝑎𝑠𝑡 − 𝑓 𝑖𝑟𝑠𝑡
+22
+if no valid bits in 𝑜𝑙𝑑 then
+23
+𝑜𝑙𝑑.𝐿 ← −1
+// mark it as removable
+24
+if 𝑛𝑜𝑡 𝑜𝑙𝑑.𝑎𝑐𝑐𝑢𝑟𝑎𝑡𝑒 then
+25
+Remove outdated LPAs in CRB
+Creation of Learned Segments. Once the data buffer of the SSD
+controller is filled, LeaFTL takes the LPAs and PPAs of the flash
+pages in the buffer as the input. It sorts the LPA-PPA mappings
+by reordering the flash pages with their LPAs (see §3.3), and uses
+greedy piecewise linear regression [64] to learn the index segment.
+Insert/Update of Learned Segments. When we insert or update
+a new learned index segment, we will place it in the topmost level
+of the log-structured mapping table. Since each level of the map-
+ping table is sorted, we can quickly identify its insert location via
+a binary search (line 2 in Algorithm 1). If the new segment is ap-
+proximate, LeaFTL will update the CRB for future lookups (line
+4-7 in Algorithm 1). After that, LeaFTL will check whether the
+new segment overlaps with existing segments. If yes, LeaFTL will
+identify the overlapped LPAs. The overlap detection is performed
+by the comparison between the LPA range of the new segment and
+[𝑆𝐿𝑃𝐴,𝑆𝐿𝑃𝐴 +𝐿] of the adjacent segments. We group these overlap-
+ping segments as a list of victim segments (line 8 in Algorithm 1).
+LeaFTL will merge segments to remove outdated LPAs (line 10 in
+Algorithm 1 and line 14-25 in Algorithm 2).
+To fulfill the segment merge, LeaFTL will use the 𝑆𝐿𝑃𝐴, 𝐿, and 𝐾
+to reconstruct the list of the encoded LPAs in the victim segment.
+And it will create a bitmap to index these encoded LPAs (line 6-13
+in Algorithm 2). Given an accurate segment with 𝑆𝐿𝑃𝐴 = 100, 𝐾 =
+0.5, 𝐿 = 6, we can infer that its encoded LPAs are [100, 102, 104, 106].
+We can transfer the LPA list to the bitmap [1010101]. If the victim
+
+Jinghan Sun, Shaobo Li, Yunxin Sun, Chao Sun, Dejan Vucinic, and Jian Huang
+MSR-hm
+MSR-src2
+MSR-prxy
+MSR-prn
+MSR-usr
+FIU-home
+FIU-mail
+0
+5
+10
+15
+20
+# of Levels 
+in Each Group
+Average
+99 Percentile
+Figure 12: A study of the number of levels in the log-
+structured mapping table for different storage workloads.
+L0
+0      63
+T0
+Initial Snapshot
+T1
+Update LPAs 200 - 255
+L0
+0     63
+200  255
+T2
+Update LPAs 16 - 31
+L0
+16    31
+200  255
+L1
+0      63
+T4
+Update [72, 73, 80]
+L0
+16    31
+200  255
+L1
+0      63
+T6
+Lookup LPA 78
+L0
+L1
+T8
+Compaction
+Timeline
+Segments
+CRB
+T7
+Update LPAs 32 - 90
+75     82
+72     80
+16    31
+200  255
+0      63
+75     82
+72     80
+T5
+Lookup LPA 50
+L0
+L1
+16    31
+200  255
+0      63
+75     82
+72     80
+L0
+L1
+16    31
+200  255
+0      63
+75     82
+32     90
+L0
+16   31
+200  255
+0    15
+32   90
+Start      End
+Accurate Segment
+Start      End
+Approximate Segment
+72 73 80
+/ 75 78 82
+72 73 80
+/
+75 78 82
+72 73 80
+/
+75 78 82
+75 78 82
+T3
+Update [75, 78, 82]
+L0
+16    31
+200  255
+L1
+0      63
+75     82
+75 78 82
+Figure 13: Examples that involve update/insert, lookup, and
+compaction operations in LeaFTL.
+segment is an approximate segment, LeaFTL will leverage the 𝑆𝐿𝑃𝐴,
+𝐿, and the LPAs stored in the CRB to reconstruct the encoded LPAs.
+Afterwards, LeaFTL will conduct a comparison between the bitmaps
+to identify the overlapped LPAs (line 15-19 in Algorithm 2).
+During the segment merge, LeaFTL will update the 𝑆𝐿𝑃𝐴 and 𝐿
+of the old segments accordingly, remove the outdated LPAs from
+CRB for approximate segments. Note that we do not update the 𝐾
+and 𝐼 for the victim segments during the merge.
+After the merge, (1) if the victim segment does not contain any
+valid LPA (𝐿 is negative), it will be removed from the mapping
+table (line 11-12 in Algorithm 1). (2) If the victim segment has
+valid LPAs but their range still overlaps with the new segment,
+the victim segment will be moved to the next level in the log-
+structured mapping table (line 13-16 in Algorithm 1). To avoid
+recursive updates across the levels, we create a new level for the
+victim segment if it also overlaps with segments in the next level.
+According to our study of diverse workloads, this will not create
+many levels in the mapping table (see Figure 12). (3) If the victim
+segment has valid LPAs and they do not overlap with the new
+segment, we do not need to perform further operations. This is
+because the victim segment is updated with new 𝑆𝐿𝑃𝐴 and 𝐿 during
+segment merge (line 20-25 in Algorithm 2), and the new segment
+insertion keeps each level sorted (line 3 in Algorithm 1).
+To facilitate our discussion, we present a few examples in Fig-
+ure 13. At the initial stage, the mapping table has one segment that
+indexes the LPA range [0, 63]. At 𝑇1, the new segment [200, 255] is
+directly inserted into the topmost level, as it does not overlap with
+existing segments. At 𝑇2, we insert a new segment [16, 31] that has
+overlaps with the old segment [0, 63], LeaFTL conducts the segment
+merge procedure. After that, the old segment still has valid LPAs.
+Thus, it moves to level 1. At 𝑇3 and 𝑇4, we insert two approximate
+segments [75, 82] and [72, 80], LeaFTL will also insert their encoded
+LPAs into the CRB. The segment [75, 82] will be moved to the next
+level as it overlaps with the new segment [72, 80].
+LPA Lookup. LeaFTL conducts an LPA lookup from the top-
+most level of the mapping table with binary searches (line 19 in
+Algorithm 1). We will check whether the LPA is represented by the
+matched segment (line 21 in Algorithm 1, line 1-5 in Algorithm 2). If
+the 𝐿𝑃𝐴 ∈ [𝑆𝐿𝑃𝐴,𝑆𝐿𝑃𝐴 + 𝐿] of the segment, LeaFTL will check the
+least bit of its 𝐾. If the least bit of 𝐾 is 0, it is an accurate segment,
+and LeaFTL will use 𝑓 (𝐿𝑃𝐴) = ⌈𝐾 ∗ 𝐿𝑃𝐴 + 𝐼⌉ to get the accurate
+PPA (see §3.2). Otherwise, it is an approximate segment. LeaFTL
+will check the CRB to identify the 𝑆𝐿𝑃𝐴 of the segment, following
+the approach described in Figure 9 and §3.4. LeaFTL will use the
+same 𝑓 (𝐿𝑃𝐴) formula to obtain the PPA. If the LPA is not found in
+the top level of the mapping table, LeaFTL will search the lower
+levels until a segment is identified.
+We use Figure 13 to illustrate the lookup procedure. At 𝑇5, we
+conduct the address translation for 𝐿𝑃𝐴 = 50. However, none of
+the segments in the level 0 covers this LPA, LeaFTL will continue
+the search in the level 1 and find the accurate segment [0, 63]. At
+𝑇6, we do the address translation for 𝐿𝑃𝐴 = 78. LeaFTL finds that
+the LPA 78 is in the LPA range of the segment [72, 80]. Since this
+is an approximate segment, LeaFTL checks the CRB and finds this
+LPA is actually indexed by the segment [75, 82].
+With the PPA, LeaFTL will read the corresponding flash page and
+use the reversed mapping (its corresponding LPA) in its OOB to ver-
+ify the correctness of the address translation. Upon mispredictions,
+we will use the approach discussed in §3.5 to handle it.
+Segment Compaction. The purpose of the compaction is to
+merge segments with overlapped LPAs across different levels, which
+further saves memory space. LeaFTL will iteratively move the upper-
+level segments into the lower level, until the mapping table is fully
+compacted (line 27 in Algorithm 1). When an approximate segment
+is removed, its corresponding CRB entries will also be deleted. As
+shown in 𝑇7 of Figure 13, we insert a new segment [32, 90] which
+fully covers the LPA range of the segment [72, 80]. After merge,
+LeaFTL removes the old segment [72, 80]. However, some segments
+
+LeaFTL: A Learning-based Flash-Translation Layer for Solid-State Drives
+Conflict Resolution
+Buffer (CRB)
+Key Data Structures in LeaFTL 
+6
+Log-Structured
+Mapping Table
+5
+L0
+L1
+L2
+...
+Group
+0
+...
+CRB
+...
+...
+0   63
+...
+16  31
+...
+...
+64  95
+Figure 14: Key data structures used in LeaFTL.
+in the level 0 still overlap with the segments in the level 1. After 𝑇8,
+LeaFTL will remove outdated segments and LPAs.
+LeaFTL performs segment compaction after each 1 million writes
+by default. According to our experiments with various storage work-
+loads, the segment compaction of the entire mapping table will take
+4.1 milliseconds (the time of 20-40 flash writes) on average. Consider
+the low frequency (i.e., once per 1 million writes), the compaction
+incurs trivial performance overhead to storage operations.
+3.8
+Put It All Together
+LeaFTL is compatible with existing FTL implementations. As shown
+in Figure 14, it uses the log-structured mapping table ( 5 ) to replace
+the address mapping cache ( 1 in Figure 3), and employs CRB ( 6 )
+for assisting the address translation of approximate segments. The
+CRB requires trivial storage space in the SSD DRAM (see Figure 10).
+Read Operation. For a read request, LeaFTL will first check the
+data cache. For a cache hit, LeaFTL serves the read request with
+the cached flash page. Otherwise, LeaFTL will perform address
+translation with 5 (see §3.7). If there is a misprediction of PPA,
+LeaFTL checks the OOB of the mispredicted flash page, read the
+correct page (§3.5), and updates the data cache with the page.
+Write Operation. For a write request, LeaFTL buffers it in the
+data cache. Once the buffered writes reach the size of a flash block,
+LeaFTL will allocate a free block. It will sort the writes in the buffer
+based on their LPAs, and learn new index segments with the PPAs
+of the allocated flash block. This enables LeaFTL to group more LPA-
+PPA mappings in the same index segment. After that, LeaFTL will
+insert the new index segment in the mapping table, and flush the
+buffered data to the flash blocks. For those writes, LeaFTL will also
+check whether their LPAs exist in the mapping table. If yes, LeaFTL
+will update their corresponding entries in 3 BVC and 4 PVT to
+indicate that they become invalid and can be garbage collected in
+the future. Otherwise, the new learned segments will have their
+LPA-PPA mappings for future address translations.
+LeaFTL caches the mapping table in SSD DRAM for fast lookup.
+The table will also be stored in the flash blocks. LeaFTL utilizes the
+existing 2 GMD to index the translation pages. If a segment is not
+found in the cached mapping table, LeaFTL will fetch it from the
+translation blocks and place it in the cached mapping table.
+Crash Consistency and Recovery. Upon system crashes or power
+failures, LeaFTL guarantees the crash consistency of learned in-
+dexes. In order to ensure the data durability of DRAM buffer in
+SSD controllers, modern SSDs today have employed battery-backed
+DRAM and power loss protection mechanisms [1, 2]. With battery-
+backed DRAM, LeaFTL has sufficient time to persist the up-to-date
+mapping table to the flash blocks and record their PPAs in the GMD
+Table 1: SSD configurations in our simulator.
+Parameter
+Value
+Parameter
+Value
+Capacity
+2TB
+#Channels
+16
+Page size
+4KB
+OOB size
+128B
+DRAM size
+1GB
+Pages/block
+256
+Read latency
+20𝜇s
+Write latency
+200𝜇s
+Erase
+1.5 millisecs
+Overprovisioning ratio
+20%
+( 2 in Figure 3). During the data recovery, LeaFTL reads the GMD
+to locate its mapping table and place it into the DRAM.
+Without battery-backed DRAM, LeaFTL periodically flushes the
+learned mapping table and the Block Validity Counter ( 3 BVC in
+Figure 3) into the flash blocks. When GC is triggered, LeaFTL also
+flushes the updated mapping table and BVC into the flash blocks.
+Upon crashes, LeaFTL will scan all the flash blocks at the channel-
+level parallelism, and reconstruct an up-to-date BVC. LeaFTL is able
+to identify the flash blocks allocated since the last mapping table
+flush, by comparing the up-to-date BVC with the stored BVC in the
+SSD. Therefore, LeaFTL only needs to relearn the index segments
+for these recently allocated flash blocks and add them into the
+mapping table (see §3.4).
+3.9
+Implementation Details
+SSD Simulator. We implement LeaFTL based on a trace-driven
+simulator WiscSim [27], which has provided an event simulation
+environment for the end-to-end performance analysis of SSDs. We
+extend WiscSim by implementing an LRU-based read-write cache.
+LeaFTL also preserves the functions of existing FTL, such as GC and
+wear-leveling. To support the learned indexing, LeaFTL employs
+a simple linear regression algorithm [65], which incurs negligible
+computation overhead with modern storage processors (see §4.5).
+The error bound 𝛾 for learned segments is configurable, and we set
+it to 0 by default in LeaFTL.
+SSD Prototype. We also develop a real system prototype with
+an open-channel SSD to validate the functions and efficiency of
+LeaFTL. The SSD has 1TB storage capacity with 16 KB flash page
+size. It has 16 channels, each channel has 16K flash blocks, and each
+flash block has 256 pages. It enables developers to implement their
+own FTL in the host by providing basic I/O commands such as read,
+write, and erase. We implement LeaFTL with 4,016 lines of code
+using C programming language with the SDK library of the device.
+4
+EVALUATION
+Our evaluation shows that: (1) LeaFTL significantly reduces the
+address mapping table size, and the saved memory brings perfor-
+mance benefits (§4.2); (2) the benefits of LeaFTL are validated on a
+real SSD device (§4.3); (3) LeaFTL can achieve additional memory
+savings and performance benefits with larger error-tolerance, and
+it demonstrate generality for different SSD configurations (§4.4);
+(4) Its learning procedure does not introduce much extra overhead
+to the SSD controller (§4.5); (5) It has minimal negative impact on
+the SSD lifetime (§4.6).
+
+Jinghan Sun, Shaobo Li, Yunxin Sun, Chao Sun, Dejan Vucinic, and Jian Huang
+Table 2: Real workloads used in our real SSD evaluation.
+Workload
+Description
+OLTP [59]
+Transactional benchmark in the FileBench.
+CompFlow (CompF) [59]
+File accesses in a computation flow.
+TPCC [13]
+Online transaction queries in warehouses.
+AuctionMark (AMark) [13]
+Activity queries in an auction site.
+SEATS [13]
+Airline ticketing system queries.
+MSR-hm
+MSR-src2
+MSR-prxy
+MSR-prn
+MSR-usr
+FIU-home
+FIU-mail
+50x
+20x
+10x
+5x
+2x
+1x
+Memory Footprint 
+Reduction
+DFTL
+SFTL
+LeaFTL
+Figure 15: The reduction on the mapping table size of
+LeaFTL, in comparison with DFTL and SFTL.
+4.1
+Experiment Setup
+We examine the efficiency of LeaFTL with both the SSD simula-
+tor and real SSD prototype. As for the evaluation with the SSD
+simulator, we configure a 2TB SSD with 4KB flash pages and 1GB
+DRAM in the SSD controller. We list the core SSD parameters in
+Table 1. For other parameters, we use the default setting in the
+WiscSim. We use a variety of storage workloads that include the
+block I/O traces from enterprise servers from Microsoft Research
+Cambridge [45] and workload traces from computers at FIU [16].
+As for the evaluation with the real SSD prototype (see §3.9), we
+validate the benefits of LeaFTL using a set of real-world file system
+benchmarks and data intensive applications as shown in Table 2.
+Before we measure the performance, we run a set of workloads
+consisting of various real-world and synthetic storage workload
+traces to warm up the SSD and make sure the GC will be executed
+during the experiments.
+We compare LeaFTL with state-of-the-art page-level mapping
+schemes described as follows 1.
+• DFTL (Demand-based FTL) [20]: it uses a page-level mapping
+scheme, and caches the most recently used address translation
+entries in the SSD DRAM.
+• SFTL (Spatial-locality-aware FTL) [25]: it is a page-level map-
+ping that exploits the spatial locality and strictly sequential access
+patterns of workloads to condense mapping table entries.
+4.2
+Memory Saving and Performance
+We first evaluate the benefits of LeaFTL on the memory saving
+and storage performance with the SSD simulator. As shown in
+Figure 15, LeaFTL reduces the mapping table size by 7.5–37.7×,
+compared to the page-level mapping scheme DFTL. This is because
+LeaFTL can group a set of page-level mapping entries into an 8-
+byte segment. In comparison with SFTL, LeaFTL achieves up to
+5.3× (2.9× on average) reduction on the address mapping table for
+different storage workloads, when we set its 𝛾 = 0 (i.e., the learned
+1We do not compare LeaFTL with block-level and hybrid-level mappings, as they
+perform dramatically worse than the page-level mapping [20, 25].
+MSR-hm
+MSR-src2
+MSR-prxy
+MSR-prn
+MSR-usr
+FIU-home
+FIU-mail
+0.0
+0.5
+1.0
+Normalized Perf.
+DFTL
+SFTL
+LeaFTL
+(a) SSD performance when using its DRAM mainly for the address
+mapping table (lower is better).
+MSR-hm
+MSR-src2
+MSR-prxy
+MSR-prn
+MSR-usr
+FIU-home
+FIU-mail
+0.0
+0.5
+1.0
+Normalized Perf.
+DFTL
+SFTL
+LeaFTL
+(b) SSD performance when using its DRAM partially (up to 80%) for
+the address mapping table (lower is better).
+Figure 16: Performance improvement with LeaFTL.
+SEATS
+AMark
+TPCC
+OLTP
+CompF
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Normalized Perf.
+DFTL
+SFTL
+LeaFTL
+Figure 17: Performance on the real SSD prototype.
+99.9%
+99%
+90%
+60%
+30%
+0%
+Percentage of Storage Accesses
+100
+101
+102
+103
+Latency ( s)
+DFTL
+SFTL
+LeaFTL
+Figure 18: The latency distribution of storage accesses when
+running OLTP workload on the real SSD prototype.
+segments are 100% accurate). This is because LeaFTL captures more
+LPA-PPA mapping patterns.
+We now evaluate the performance benefit of LeaFTL from its
+saved memory space. We evaluate LeaFTL with two experimental
+settings: (1) the SSD DRAM is mainly used (as much as possible)
+for the mapping table; (2) the SSD DRAM is partially used for the
+mapping table, in which we ensure at least 20% of the DRAM will
+be used for the data caching.
+In the first setting, DRAM is almost used for mapping table in
+DFTL. As shown in Figure 16 (a), LeaFTL reduces the storage access
+latency by 1.6× on average (up to 2.7×), compared to SFTL. This
+is because LeaFTL saves more memory from the mapping table
+
+LeaFTL: A Learning-based Flash-Translation Layer for Solid-State Drives
+MSR-hm
+MSR-src2
+MSR-prxy
+MSR-prn
+MSR-usr
+FIU-home
+FIU-mail
+SEATS
+AMark
+TPCC
+OLTP
+CompF
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Memory Footprint 
+Reduction
+=0
+=1
+=4
+=16
+SSD Simulator
+Real SSD
+Figure 19: The reduction of the mapping table size of LeaFTL
+with different 𝛾 (lower is better).
+=0
+=1
+=4
+=16
+0%
+20%
+40%
+60%
+80%
+100%
+Percentage of
+Segments
+Accurate
+Approximate
+Figure 20: The distribution of learned segments.
+than SFTL. SFTL slightly outperforms DFTL, because it reduces the
+mapping table size by compressing mapping entries with grouping
+strictly sequential data accesses. In the second setting, as shown in
+Figure 16 (b), LeaFTL obtains 1.4× (up to 3.4×) and 1.6× (up to 4.9×)
+performance speedup, compared to SFTL and DFTL, respectively.
+4.3
+Benefits on the Real SSD Prototype
+We validate the benefits of LeaFTL on the real SSD prototype with
+real workloads (see Table 2). They include filesystem benchmark
+suite FileBench [59], and transactional database workloads from
+BenchBase [13, 61]. All these workloads run on the ext4 file system.
+With FileBench, we run OLTP and CompFlow (CompF) workloads
+to read/write 10GB files. With BenchBase, we run TPCC, Auction-
+Mark (AMark), and SEATS workloads on MySQL, and their data-
+base sizes are 10–30GB. These database workloads will generate
+37–230GB read traffic and 26–59GB write traffic to the SSD. We allo-
+cate 256MB DRAM to host the mapping table (for different DRAM
+sizes, see our sensitivity analysis in §4.4).
+We present the performance benefit of LeaFTL in Figure 17.
+Across all workloads, LeaFTL obtains 1.4× performance speedup
+on average (up to 1.5×), compared to SFTL and DFTL. Similar to
+our evaluation with the SSD simulator implementation, the per-
+formance benefit of LeaFTL comes from the memory saving from
+the address mapping table. And LeaFTL demonstrates comparable
+performance improvement on real SSD devices, in comparison with
+the SSD simulator in §4.2. We also show the latency distribution of
+storage accesses in Figure 18, when running the OLTP workload on
+the real SSD prototype. In comparison with existing FTL schemes,
+LeaFTL does not increase the tail latency of storage accesses. And
+the higher cache hit ratio of LeaFTL brings latency reduction for
+many storage accesses.
+4.4
+Sensitivity Analysis
+Vary the value of 𝛾. As we increase the value of 𝛾 from 0 to
+16, the size of the learned mapping table is reduced, as shown in
+MSR-hm
+MSR-src2
+MSR-prxy
+MSR-prn
+MSR-usr
+FIU-home
+FIU-mail
+SEATS
+AMark
+TPCC
+OLTP
+CompF
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Normalized Perf.
+=0
+=1
+=4
+=16
+SSD Simulator
+Real SSD
+Figure 21: Performance with various 𝛾 (lower is better).
+256MB
+512MB
+1024MB
+(a) Various DRAM size
+0.0
+0.5
+1.0
+Normalized Perf.
+4KB
+8KB
+16KB
+(b) Various flash page size
+0.0
+0.5
+1.0
+Normalized Perf.
+DFTL
+SFTL
+LeaFTL
+Figure 22: SSD performance with different DRAM capacity
+and flash page size (lower is better).
+Figure 19. LeaFTL achieves 1.3× reduction on average (1.2× on
+the real SSD) with 𝛾 = 16, compared to that of 𝛾 = 0. The saved
+memory with a larger 𝛾 is achieved by learning a wider range
+of LPAs into approximate segments. To further understand this,
+we profile the distribution of segments learned by LeaFTL with
+different values of 𝛾, as shown in Figure 20. When 𝛾 = 0, all the
+segments are accurate. When 𝛾 = 16, 26.5% of the learned segments
+are approximate on average, and LeaFTL delivers 1.3× improvement
+on storage performance (1.2× with workloads on the real SSD), in
+comparison with the case of 𝛾 = 0 (see Figure 21).
+Vary the SSD DRAM capacity. We now conduct the sensitivity
+analysis of SSD DRAM by varying its capacity from 256MB to 1GB
+on the real SSD prototype. As shown in Figure 22 (a), LeaFTL always
+outperforms DFTL and SFTL as we vary the SSD DRAM capacity.
+As we increase the DRAM capacity, the storage workloads are still
+bottlenecked by the available memory space for the data caching.
+LeaFTL can learn various data access patterns and significantly
+reduce the address mapping table size, the saved memory further
+benefits data caching.
+Vary the flash page size. In this experiment, we fix the number
+of flash pages, and vary the flash page size from 4KB to 16KB in the
+SSD simulator, as SSD vendors usually use larger flash pages for
+increased SSD capacity. We use the simulator for this study, since
+the flash page size of the real SSD is fixed. As shown in Figure 22
+(b), LeaFTL always performs the best in comparison with DFTL and
+SFTL. As we increase the flash page size to 16KB, we can cache less
+number of flash pages with limited DRAM capacity. Thus, LeaFTL
+experiences a slight performance drop. As we fix the total SSD
+
+Jinghan Sun, Shaobo Li, Yunxin Sun, Chao Sun, Dejan Vucinic, and Jian Huang
+1
+5
+10
+15
+20
+25
+30
+35
+(a) Number of Levels
+99.99%
+99.9%
+99%
+90%
+0%
+Percentage of
+Lookups
+MSR-prn
+MSR-usr
+MSR-src2
+MSR-hm
+MSR-prxy
+FIU-home
+FIU-mail
+0.0
+0.5
+1.0
+1.5
+(b) LPA Lookup Overhead (%)
+99.99%
+99.9%
+99%
+90%
+0%
+Percentage of
+Lookups
+SEATS      
+CompF
+OLTP
+TPCC
+AMark
+Figure 23: Performance overhead of the LPA lookup.
+MSR-hm
+MSR-src2
+MSR-prxy
+MSR-prn
+MSR-usr
+FIU-home
+FIU-mail
+SEATS
+AMark
+TPCC
+OLTP
+CompF
+0
+5
+10
+15
+20
+Misprediction (%)
+=0
+=1
+=4
+=16
+SSD Simulator
+Real SSD
+Figure 24: Misprediction ratio of flash pages access.
+capacity and vary the page size, LeaFTL outperforms SFTL by 1.2×
+and 1.1× for the page size of 8KB and 16KB, respectively.
+4.5
+Overhead Source in LeaFTL
+We evaluate the overhead sources in LeaFTL in three aspects: (1)
+the performance overhead of the learning procedure in LeaFTL;
+(2) the LPA lookup overhead in the learned segments; and (3) the
+overhead caused by the address misprediction in LeaFTL.
+We evaluate the performance of segment learning and address
+lookup on an ARM Cortex-A72 core. This core is similar to the
+storage processor used in modern SSDs. The learning time for a
+batch of 256 mapping entries is 9.8–10.8 𝜇s (see Table 3). As we
+learn one batch of index segments for every 256 flash writes, the
+learning overhead is only 0.02% of their flash write latency.
+In LeaFTL, the LPA lookup is 40.2–67.5 ns, as the binary search of
+segments is fast and some segments can be cached in the processor
+cache. The lookup time is slightly higher as we increase𝛾, due to the
+additional CRB accesses. We also profile the cumulative distribution
+function (CDF) of the number of levels to lookup for each LPA
+lookup, and present the results in Figure 23 (a). For most of the
+tested workloads, 90% of the mapping table lookup can be fulfilled
+at the topmost level, and 99% of the lookups are within 10 levels.
+Although MSR-prn workload requires more lookups than other
+workloads, it only checks 1.4 levels on average. We also evaluate
+the performance overhead of the LPA lookup on the real SSD, and
+show the results in Figure 23 (b). The extra lookup overhead for each
+flash read is 0.21% on average. And for 99.99% of all the lookups,
+the additional overhead is less than 1% of the flash access latency.
+Table 3: Overhead source of LeaFTL with an ARM core.
+𝛾
+0
+1
+4
+Learning (256 LPAs)
+9.8 𝜇s
+10.8 𝜇s
+10.8 𝜇s
+Lookup (per LPA)
+40.2 ns
+60.5 ns
+67.5 ns
+LeaFTL also has low misprediction ratios with approximate seg-
+ments. This is because LeaFTL can still learn accurate segments
+even if 𝛾 > 0, and not all entries in the approximate segments
+will result in misprediction. As shown in Figure 24, most of the
+workloads achieve less than 10% misprediction ratio when 𝛾 = 16.
+We obtain similar misprediction ratio on the real SSD prototype.
+Note that each misprediction only incurs one flash read access with
+the help of our proposed OOB verification.
+4.6
+Impact on SSD Lifetime
+The flash blocks of an SSD can only undergo a certain amount of
+writes. In this experiment, we use the write amplification factor
+(WAF, the ratio between the actual and requested flash writes) to
+evaluate the SSD lifetime. The SSD will age faster if the WAF is
+larger. As shown Figure 25, the WAF of LeaFTL is comparable to
+DFTL and SFTL. DFTL has larger WAF in most workloads. SFTL
+and LeaFTL occasionally flush translation pages to the flash blocks,
+but the cost is negligible.
+5
+DISCUSSION
+Why Linear Regression. Unlike deep neural networks, the lin-
+ear regression used in LeaFTL is simple and lightweight, which
+takes only a few microseconds to learn an index segment with
+embedded ARM processors available in modern SSD controllers.
+In addition, the linear regression algorithm has been well studied,
+and offers guaranteed error bounds for its learned results. LeaFTL
+is the first work that uses learning techniques to solve a critical
+system problem (i.e., address mapping) in SSDs.
+Adaptivity of LeaFTL. LeaFTL focuses on the page-level address
+translation, its design and implementation will not be affected by
+the low-level flash memory organization (i.e., TLC/QLC). As we
+use TLC/QLC technique to further increase the SSD capacity, the
+address mapping issue will become more critical, since the SSD
+DRAM capacity does not scale well and becomes the bottleneck for
+caching address mappings and user data.
+Recovery of Learned Index Segments. As discussed in §3.8, us-
+ing a battery or large capacitor to preserve and persist the cached
+segments upon failures or crashes will simplify the recovery pro-
+cedure significantly. In our real SSD prototype, we do not assume
+the battery-backed DRAM is available. Thus, we follow the conven-
+tional recovery approach in modern SSDs [20, 23], and scan flash
+blocks in parallel by utilizing the channel-level parallelism.
+When we run real workloads like TPCC on the SSD prototype,
+we intentionally reboot the system after running the workload for
+a period of time (0.5-3 hours). We find that the system can recover
+in 15.8 minutes on average whenever the reboot happens. This
+is similar to the time of recovering the conventional page-level
+mapping table in DFTL [20]. This is mostly caused by scanning the
+blocks in a channel (70MB/s per channel in our SSD prototype),
+and the time for reconstructing recently learned segments is rela-
+tively low (101.3 milliseconds on average). We believe the recovery
+
+LeaFTL: A Learning-based Flash-Translation Layer for Solid-State Drives
+MSR-hm
+MSR-src2
+MSR-prxy
+MSR-prn
+MSR-usr
+FIU-home
+FIU-mail
+SEATS
+AMark
+TPCC
+OLTP
+CompF
+0.0
+0.5
+1.0
+1.5
+Write 
+Amplification
+DFTL
+SFTL
+LeaFTL
+SSD Simulator
+Real SSD
+Figure 25: Write amplification factor of LeaFTL.
+time is not much of a concern as the recovery does not happen
+frequently in reality. And the recovery can be accelerated as we
+increase the channel-level bandwidth. In addition, if an SSD can
+tolerate more data losses, we can still ensure the crash consistency
+by only loading the stored index segments from flash chips, which
+requires minimum recovery time.
+6
+RELATED WORK
+Address Translation for SSDs. A variety of FTL optimizations
+have been proposed [8, 12, 20, 25, 28, 34, 49, 50]. These works ex-
+ploited the data locality of flash accesses to improve the cache
+efficiency of the mapping table. However, most of them were devel-
+oped with human-driven heuristics. An alternative approach is to
+integrate application semantics into the FTL, such as content-aware
+FTL [7]. However, they were application specific and required signif-
+icant changes to the FTL. LeaFTL is a generic solution and does not
+require application semantics in its learning. Researchers proposed
+to integrate the FTL mapping table into the host [18, 23, 26, 66]. Typi-
+cal examples include DFS [26], Nameless writes [66], FlashMap [23],
+and FlatFlash [4]. LeaFTL is orthogonal to them and can be applied
+to further reduce their memory footprint.
+Machine Learning for Storage. Recent studies have been using
+learning techniques to build indexes such as B-trees, log-structured
+merge tree, hashmaps, and bloom filters [11, 14, 15, 32, 33, 42]
+for in-memory datasets, identify optimal cache replacement and
+prefetching policies [40, 53, 56, 57], facilitate efficient storage har-
+vesting [52], and drive the development of software-defined stor-
+age [24]. LeaFTL applies learning techniques to optimize the address
+mapping. However, unlike existing optimizations [43, 63] such as
+learned page table for virtual memory that used deep neural net-
+works to learn the patterns, LeaFTL provides a lightweight solution.
+SSD Hardware Development. For the recent SSD innovations [3,
+17, 19, 47] like Z-SSD [55], KVSSD [35], and ZNS SSD [21], DRAM
+capacity and storage processor are still the main constraints in SSD
+controllers. As we scale the storage capacity, the challenge with
+the address translation becomes only worse. Researchers recently
+deployed hardware accelerators inside SSD controllers for near-
+data computing [36, 41, 54, 58]. We wish to extend LeaFTL with
+in-storage accelerators to deploy more powerful learning models
+as the future work.
+7
+CONCLUSION
+We present a learning-based flash translation layer, named LeaFTL
+for SSDs. LeaFTL can automatically learn different flash access
+patterns and build space-efficient indexes, which reduces the ad-
+dress mapping size and improves the caching efficiency in the SSD
+controller. Our evaluation shows that LeaFTL improves the SSD
+performance by 1.4× on average for a variety of storage workloads.
+ACKNOWLEDGMENTS
+We thank the anonymous reviewers for their helpful comments
+and feedback. This work is partially supported by the NSF CAREER
+Award 2144796, CCF-1919044, and CNS-1850317.
+REFERENCES
+[1] 2019. A Closer Look At SSD Power Loss Protection. https://www.kingston.com/
+en/blog/servers-and-data-centers/ssd-power-loss-protection.
+[2] 2020. Harnessing Microcontrollers to Deliver Intelligent SSD Power Management
+and PLP Capabilities. https://www.atpinc.com/de/about/stories/microcontroller-
+SSD-power-loss-protection.
+[3] 3D NAND – An Overview. 2022.
+https://www.simms.co.uk/tech-talk/3d-nand-overview/.
+[4] Ahmed Abulila, Vikram Sharma Mailthoday, Zaid Qureshi, Jian Huang, Nam Sung
+Kim, Jin jun Xiong, and Wen mei Hwu. 2019. FlatFlash: Exploiting the Byte-
+Accessibility of SSDs within A Unified Memory-Storage Hierarchy. In Proceedings
+of the 24th ACM International Conference on Architectural Support for Programming
+Languages and Operating Systems (ASPLOS’19). Providence, RI.
+[5] Nitin Agrawal, Vijayan Prabhakaran, Ted Wobber, John D. Davis, Mark Manasse,
+and Rina Panigrahy. 2008. Design Tradeoffs for SSD Performance. In Proceedings
+of the USENIX 2008 Annual Technical Conference (ATC’08). Boston, Massachusetts.
+[6] Yu Cai, Saugata Ghose, Erich F Haratsch, Yixin Luo, and Onur Mutlu. 2017. Error
+characterization, mitigation, and recovery in flash-memory-based solid-state
+drives. Proc. IEEE 105, 9 (2017), 1666–1704.
+[7] Feng Chen, Tian Luo, and Xiaodong Zhang. 2011. CAFTL: A Content-Aware
+Flash Translation Layer Enhancing the Lifespan of Flash Memory based Solid
+State Drives. In Proceedings of the 9th USENIX Conference on File and Storage
+Technologies (FAST’11). San Jose, CA.
+[8] Renhai Chen, Zhiwei Qin, Yi Wang, Duo Liu, Zili Shao, and Yong Guan. 2014. On-
+demand block-level address mapping in large-scale NAND flash storage systems.
+IEEE Trans. Comput. 64, 6 (2014), 1729–1741.
+[9] Tae-Sun Chung, Dong-Joo Park, and Jongik Kim. 2011. LSTAFF*: An Efficient
+Flash Translation Layer for Large Block Flash Memory. In Proceedings of the 2011
+ACM Symposium on Applied Computing (SAC’11). TaiChung Taiwan.
+[10] Curtis R Cook and Do Jin Kim. 1980. Best sorting algorithm for nearly sorted
+lists. Commun. ACM 23, 11 (1980), 620–624.
+[11] Yifan Dai, Yien Xu, Aishwarya Ganesan, Ramnatthan Alagappan, Brian Kroth,
+Andrea Arpaci-Dusseau, and Remzi Arpaci-Dusseau. 2020. From WiscKey to
+Bourbon: A Learned Index for Log-Structured Merge Trees. In Proceedings of
+the 14th USENIX Symposium on Operating Systems Design and Implementation
+(OSDI’20). Virtual Event.
+[12] Niv Dayan, Philippe Bonnet, and Stratos Idreos. 2016. GeckoFTL: Scalable Flash
+Translation Techniques For Very Large Flash Devices. In Proceedings of the Inter-
+national Conference on Management of Data (SIGMOD’16). San Francisco, CA.
+[13] Djellel Eddine Difallah, Andrew Pavlo, Carlo Curino, and Philippe Cudré-
+Mauroux. 2013. OLTP-Bench: An Extensible Testbed for Benchmarking Relational
+Databases. PVLDB 7, 4 (2013).
+[14] Paolo Ferragina, Fabrizio Lillo, and Giorgio Vinciguerra. 2020. Why Are Learned
+Indexes So Effective?. In Proceedings of the 37th International Conference on
+Machine Learning (ICML’20). Virtual Event.
+[15] Paolo Ferragina and Giorgio Vinciguerra. 2020. The PGM-Index: A Fully-Dynamic
+Compressed Learned Index with Provable Worst-Case Bounds. Proceedings of
+the VLDB Endowment 13, 8 (April 2020).
+[16] FIU. 2009. FIU Server Traces.
+[17] Flash Memory. 2022. https://en.wikipedia.org/wiki/Flash_memory.
+[18] Fusion-io Directcache: Transparent Storage Accelerator. 2011.
+http://www.fusionio.com/systems/directcache/.
+[19] Gartner. 2017. Forecast Overview: NAND Flash, Worldwide, 2017.
+https:
+//www.gartner.com/doc/3745121/forecast-overview-nand-flash-worldwide
+[20] Aayush Gupta, Youngjae Kim, and Bhuvan Urgaonkar. 2009. DFTL: A Flash
+Translation Layer Employing Demand-based Selective Caching of Page-level
+Address Mappings. In Proceedings of the 14th International Conference on Archi-
+tectural Support for Programming Languages and Operating Systems (ASPLOS’09).
+Washington, DC.
+[21] Kyuhwa Han, Hyunho Gwak, Dongkun Shin, and Joo-Young Hwang. 2021. ZNS+:
+Advanced Zoned Namespace Interface for Supporting In-Storage Zone Com-
+paction. In 15th {USENIX} Symposium on Operating Systems Design and Imple-
+mentation (OSDI’21). 147–162.
+[22] Jian Huang, Anirudh Badam, Laura Caulfield, Suman Nath, Sudipta Sengupta,
+Bikash Sharma, and Moinuddin K. Qureshi. 2017. FlashBlox: Achieving Both
+Performance Isolation and Uniform Lifetime for Virtualized SSDs. In Proceedings
+
+Jinghan Sun, Shaobo Li, Yunxin Sun, Chao Sun, Dejan Vucinic, and Jian Huang
+of the 15th Usenix Conference on File and Storage Technologies (FAST’17). Santa
+clara, CA.
+[23] Jian Huang, Anirudh Badam, Moinuddin K. Qureshi, and Karsten Schwan. 2015.
+Unified Address Translation for Memory-mapped SSDs with FlashMap. In Pro-
+ceedings of the 42nd Annual International Symposium on Computer Architecture
+(ISCA’15). Portland, OR.
+[24] Jian Huang, Daixuan Li, and Jinghan Sun. 2022. Learning to Drive Software-
+Defined Storage. Workshop on Machine Learning for Systems at NIPS’22 (2022).
+[25] Song Jiang, Lei Zhang, XinHao Yuan, Hao Hu, and Yu Chen. 2011. S-FTL: An
+Efficient Address Translation for Flash Memory by Exploiting Spatial Locality.
+In Proceedings of the 2011 IEEE 27th Symposium on Mass Storage Systems and
+Technologies (MSST’11). IEEE Computer Society.
+[26] William K. Josephson, Lars A. Bongo, Kai Li, and David Flynn. 2010. DFS: A
+File System for Virtualized Flash Storage. ACM Trans. on Storage 6, 3 (2010),
+14:1–14:25.
+[27] Jun He, Sudarsun Kannan, Andrea C. Arpaci-Dusseau, Remzi H. Arpaci-Dusseau.
+2017. The Unwritten Contract of Solid State Drives. In Proceedings of the Twelfth
+European Conference on Computer Systems (EuroSys’17). Belgrade, Serbia.
+[28] Dawoon Jung, Jeong-UK Kang, Heeseung Jo, Jin-Soo Kim, and Joonwon Lee.
+2010. Superblock FTL: A superblock-based flash translation layer with a hybrid
+address translation scheme. ACM Transactions on Embedded Computing Systems
+(TECS) 9, 4 (2010), 1–41.
+[29] Jeong-Uk Kang, Heeseung Jo, Jinsoo Kim, and Joonwon Lee. 2006. A Superblock-
+Based Flash Translation Layer for NAND Flash Memory. In Proceedings of the
+6th International Conference on Embedded Software (EMSOFT’06). Seoul, South
+Korea.
+[30] Luyi Kang, Yuqi Xie, Weiwei Jia, Xiaohao Wang, Jongryool Kim, Changhwan
+Youn, Myeong Joon Kang, Jin Lim, Bruce Jacob, and Jian Huang. 2021. IceClave: A
+Trusted Execution Environment for In-Storage Computing. In Proceedings of the
+54th Annual IEEE/ACM International Symposium on Microarchitecture (MICRO’21).
+Virtual Event.
+[31] Jesung Kim, Jong Min Kim, S.H. Noh, Sang Lyul Min, and Yookun Cho. 2002. A
+space-efficient flash translation layer for CompactFlash systems. IEEE Transac-
+tions on Consumer Electronics 48, 2 (2002).
+[32] Andreas Kipf, Ryan Marcus, Alexander van Renen, Mihail Stoian, Alfons Kemper,
+Tim Kraska, and Thomas Neumann. 2020. RadixSpline: A Single-Pass Learned
+Index. In Proceedings of the Third International Workshop on Exploiting Artificial
+Intelligence Techniques for Data Management (aiDM ’20). Portland, Oregon.
+[33] Tim Kraska, Alex Beutel, Ed H. Chi, Jeffrey Dean, and Neoklis Polyzotis. 2018.
+The Case for Learned Index Structures. In Proceedings of the 2018 International
+Conference on Management of Data (SIGMOD’18). Houston, TX, USA.
+[34] Hunki Kwon, Eunsam Kim, Jongmoo Choi, Donghee Lee, and Sam H Noh. 2010.
+Janus-FTL: Finding the optimal point on the spectrum between page and block
+mapping schemes. In Proceedings of the tenth ACM international conference on
+Embedded software. 169–178.
+[35] Samsung Memory Solutions Lab. 2017. Samsung Key Value SSD enables High Per-
+formance Scaling. https://www.samsung.com/semiconductor/global.semi.static/
+Samsung_Key_Value_SSD_enables_High_Performance_Scaling-0.pdf (2017).
+[36] Joo Hwan Lee, Hui Zhang, Veronica Lagrange, Praveen Krishnamoorthy, Xi-
+aodong Zhao, and Yang Seok Ki. 2020. SmartSSD: FPGA accelerated near-storage
+data analytics on SSD. IEEE Computer architecture letters 19, 2 (2020), 110–113.
+[37] Sungjin Lee, Ming Liu, Sangwoo Jun, Shuotao Xu, Jihong Kim, and Arvind. 2016.
+Application-managed flash. In Proceedings of the 14th USENIX Conference on File
+and Storage Technologies (FAST’16). 339–353.
+[38] Sungjin Lee, Dongkun Shin, Young-Jin Kim, and Jihong Kim. 2008. LAST: Locality-
+Aware Sector Translation for NAND Flash Memory-Based Storage Systems. In
+Proceedings of the SIGOPS Operating Systems Review (2008).
+[39] Sang-Won Lee, Dong-Joo Park, Tae-Sun Chung, Dong-Ho Lee, Sangwon Park,
+and Ha-Joo Song. 2007. A Log Buffer-Based Flash Translation Layer Using
+Fully-Associative Sector Translation. ACM Transactions on Embedded Computing
+Systems 6, 3 (2007), 18:1–18:27.
+[40] Evan Liu, Milad Hashemi, Kevin Swersky, Parthasarathy Ranganathan, and Jun-
+whan Ahn. 2020. An imitation learning approach for cache replacement. In
+International Conference on Machine Learning. PMLR, 6237–6247.
+[41] Vikram Sharma Mailthoday, Zaid Qureshi, Weixin Liang, Ziyan Feng, Simon Gar-
+cia de Gonzalo, Youjie Li, Hubertus Franke, Jinjun Xiong, Jian Huang, and Wen
+mei Hwu. 2019. DeepStore: In-Storage Acceleration for Intelligent Queries. In
+Proceedings of the 52nd IEEE/ACM International Symposium on Microarchitecture
+(MICRO’19). Columbus, OH.
+[42] Ryan Marcus, Emily Zhang, and Tim Kraska. 2020. CDFShop: Exploring and
+Optimizing Learned Index Structures. In Proceedings of the 2020 ACM SIGMOD
+International Conference on Management of Data (SIGMOD’20). Portland, OR, USA.
+https://doi.org/10.1145/3318464.3384706
+[43] Artemiy Margaritov, Dmitri Ustiugov, Edouard Bugnion, and Boris Grot. 2018.
+Virtual Address Translation via Learned Page Table Indexes. In Proceedings of
+the Workshop on ML for Systems at NeurIPS. Montreal, Canada.
+[44] Kiran Kumar Matam, Gunjae Koo, Haipeng Zha, Hung-Wei Tseng, and Murali
+Annavaram. 2019. GraphSSD: Graph Semantics Aware SSD. In Proceedings of
+the 46th International Symposium on Computer Architecture (ISCA’19). Phoenix,
+Arizona.
+[45] Microsoft. 2007. MSR Cambridge Traces.
+[46] Jian Ouyang, Shiding Lin, Song Jiang, Yong Wang, Wei Qi, Jason Cong, and
+Yuanzheng Wang. 2014. SDF: Software-Defined Flash for Web-Scale Internet
+Storage Systems. In Proceedings of 19th International Conference on Architectural
+Support for Programming Language and Operating Systems (ASPLOS’14). Salt Lake
+City, UT.
+[47] Over 50 years of development history of Flash Memory Technology. 2019.
+https://www.elinfor.com/knowledge/over-50-years-of-development-history-
+of-flash-memory-technology-p-11271.
+[48] Nikolaos Papandreou, Haralampos Pozidis, Nikolas Ioannou, Thomas Parnell,
+Roman Pletka, Milos Stanisavljevic, Radu Stoica, Sasa Tomic, Patrick Breen, Gary
+Tressler, et al. 2020. Open block characterization and read voltage calibration of
+3D QLC NAND flash. In 2020 IEEE International Reliability Physics Symposium
+(IRPS). IEEE, 1–6.
+[49] Chanik Park, Wonmoon Cheon, Jeonguk Kang, Kangho Roh, Wonhee Cho, and
+Jin-Soo Kim. 2008. A reconfigurable FTL (flash translation layer) architecture
+for NAND flash-based applications. ACM Transactions on Embedded Computing
+Systems (TECS) 7, 4 (2008), 1–23.
+[50] Zhiwei Qin, Yi Wang, Duo Liu, and Zili Shao. 2010. Demand-based block-level
+address mapping in large-scale NAND flash storage systems. In Proceedings of
+the eighth IEEE/ACM/IFIP international conference on Hardware/software codesign
+and system synthesis.
+[51] Benjamin Reidys, Peng Liu, and Jian Huang. 2022. RSSD: Defend against Ran-
+somware with Hardware-Isolated Network-Storage Codesign and Post-Attack
+Analysis. In Proceedings of the 27th ACM International Conference on Architec-
+tural Support for Programming Languages and Operating Systems (ASPLOS’22).
+Lausanne, Switzerland.
+[52] Benjamin Reidys, Jinghan Sun, Anirudh Badam, Shadi Noghabi, and Jian Huang.
+2022. BlockFlex: Enabling Storage Harvesting with Software-Defined Flash
+in Modern Cloud Platforms. In Proceedings of the 16th USENIX Symposium on
+Operating Systems Design and Implementation (OSDI’22). Carlsbad, CA.
+[53] Liana V Rodriguez, Farzana Yusuf, Steven Lyons, Eysler Paz, Raju Rangaswami,
+Jason Liu, Ming Zhao, and Giri Narasimhan. 2021. Learning Cache Replacement
+with CACHEUS. In 19th USENIX Conference on File and Storage Technologies
+(FAST’21). 341–354.
+[54] Zhenyuan Ruan, Tong He, and Jason Cong. 2019. INSIDER: Designing In-Storage
+Computing System for Emerging High-Performance Drive. In Proceedings of the
+2019 USENIX Annual Technical Conference (USENIX ATC’19). Renton, WA.
+[55] Samsung Z-NAND. 2019. https://www.samsung.com/semiconductor/ssd/z-ssd/.
+[56] Subhash Sethumurugan, Jieming Yin, and John Sartori. 2021. Designing a Cost-
+Effective Cache Replacement Policy using Machine Learning. In 2021 IEEE Inter-
+national Symposium on High-Performance Computer Architecture (HPCA). IEEE,
+291–303.
+[57] Zhan Shi, Xiangru Huang, Akanksha Jain, and Calvin Lin. 2019. Applying deep
+learning to the cache replacement problem. In Proceedings of the 52nd Annual
+IEEE/ACM International Symposium on Microarchitecture. 413–425.
+[58] smartssd 2018. SmartSSD Computational Storage Drive. https://www.xilinx.com/
+applications/data-center/computational-storage/smartssd.html.
+[59] Vasily Tarasov, Erez Zadok, and Spencer Shepler. 2016. Filebench: A flexible
+framework for file system benchmarking. The USENIX Magazine 41, 1 (2016).
+[60] Usman Saleem, Advanced SSD Buying Guide - NAND Types, DRAM Cache, HMB
+Explained. 2022. https://appuals.com/ssd-buying-guide/.
+[61] Dana Van Aken, Djellel E. Difallah, Andrew Pavlo, Carlo Curino, and Philippe
+Cudré-Mauroux. 2015. BenchPress: Dynamic Workload Control in the OLTP-
+Bench Testbed. In Proceedings of the 2015 ACM SIGMOD International Conference
+on Management of Data (SIGMOD’15).
+[62] Xiaohao Wang, Yifan Yuan, You Zhou, Chance C. Coats, and Jian Huang. 2019.
+Project Almanac: A Time-Traveling Solid-State Drive. In Proceedings of the 14th
+European Conference on Computer Systems (EuroSys’19). Dresden, Germany.
+[63] Nan Wu and Yuan Xie. 2021. A Survey of Machine Learning for Computer
+Architecture and Systems. CoRR abs/2102.07952 (2021). https://arxiv.org/abs/
+2102.07952
+[64] Qing Xie, Chaoyi Pang, Xiaofang Zhou, Xiangliang Zhang, and Ke Deng. 2014.
+Maximum Error-Bounded Piecewise Linear Representation for Online Stream
+Approximation. Proceedings of the VLDB Journal 23, 6 (Dec. 2014).
+[65] Qing Xie, Chaoyi Pang, Xiaofang Zhou, Xiangliang Zhang, and Ke Deng. 2014.
+Maximum error-bounded piecewise linear representation for online stream ap-
+proximation. The VLDB journal 23, 6 (2014), 915–937.
+[66] Yiying Zhang, Leo Prasath Arulraj, Andrea C. Arpaci-Dusseau, and Remzi H.
+Arpaci-Dusseau. 2012. De-indirection for Flash-based SSDs with Nameless Writes.
+In Proceedings of the 10th USENIX Conference on File and Storage Technologies
+(FAST’12). San Jose, CA.
+

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+arXiv:2301.03778v1  [quant-ph]  10 Jan 2023
+Letter
+Optics Letters
+1
+Efficient and robust chiral discrimination by
+invariant-based inverse engineering
+HANG XU1, XUE-KE SONG1,2, LIU YE1, AND DONG WANG1,3
+1School of Physics and Optoelectronics Engineering, Anhui University, Hefei 230601, China
+2Corresponding author: [email protected]
+3Corresponding author: [email protected]
+Compiled January 11, 2023
+We propose an accurate and convenient method to
+achieve 100% discrimination of chiral molecules with
+Lewis-Riesenfeld invariant. By reversely designing the
+pulse scheme of handed resolution, we obtain the pa-
+rameters of the three-level Hamiltonians to achieve this
+goal. For the same initial state, we can completely trans-
+fer its population to one energy level for left-handed
+molecules, while transfer it to another energy level for
+right-handed molecules.
+Moreover, this method can
+be further optimized when errors exist, and it shows
+that the optimal method are more robust against these
+errors than the counterdiabatic and original invariant-
+based shortcut schemes. This provides an effective, ac-
+curate, and robust method to distinguish the handed-
+ness of molecules.
+© 2023 Optica Publishing Group
+http://dx.doi.org/10.1364/ao.XX.XXXXXX
+Chirality, which was first proposed by Pasteur in 1848 [1]
+originating from symmetry breaking [2], is a very important
+concept or attribute in natural science. It has attracted exten-
+sive attentions in specific fields of physics, materials science
+[3], chemistry [4], biology [5], and medicine [6]. In principle,
+when the atomic distribution and chemical bond structure of
+two molecules are symmetrical in the mirror image but cannot
+coincide, these molecules possess chirality with left (L) hand-
+edness or right (R) handedness. Generally, molecules with dif-
+ferent chirality show the same physical and chemical proper-
+ties. However, in some specific cases, they show dramatically
+opposite properties, especially biological activity [7]. The drug
+molecules must match the geometric structure of the receptor
+(reactive substance) molecules in order to have the proper effi-
+cacy.
+In recent years, there are many studies [8–18] to use quan-
+tum coherent manipulation techniques to realize the effective
+discrimination of chiral molecules, including adiabatic pas-
+sages [19], counter-diabatic driving [20–23], composite pulses
+[24–26], etc. In 2019, Vitanov et al. [9] proposed an efficient
+chiral resolution using delayed pulses based on the principle of
+counter-diabatic quantum driving. In 2019, Ye et al. [10] showed
+two dynamic methods to achieve inner-state enantioseparation
+in the case that the handedness system is reduced to a effec-
+tive two-level system. In 2020, Torosov et al. [11] introduced
+a method for the chiral molecule detection using sequences of
+three pulses, and the composite pulses are used to realize the
+robustness to the area error.
+In this paper, we propose an efficient and robust chiral res-
+olution method based on optimal Lewis-Riesenfeld invariant
+(LRI) shortcut.
+For the three-level Hamiltonians of the left-
+handed and right-handed molecules, we can design the invari-
+ants of the corresponding L and R systems [27–32], respectively.
+The systems are evolved along eigenstates of their respective
+invariants from the same initial energy level, while they will
+reach to different final energy levels with regard to different
+chiral molecules. This means that a 100% chiral resolution is
+achieved. The advantage of LRI is that it has a large parameter
+selections to be further optimized with respect to various con-
+trol errors. Taking systematic and detuning errors into account,
+we find that the optimal invariant shortcut scheme are more ro-
+bust against these errors compared to the counter-diabatic and
+the original invariant shortcuts.
+Let us consider a typical cyclic three-level system [33], as
+shown in Fig. 1. The Hamiltonian, in the bases {|1⟩ , |2⟩ , |3⟩},
+reads
+HL,R
+0
+= ¯h
+
+
+
+
+
+0
+Ωp
+∓Ωqeiγ
+Ωp
+0
+Ωs
+∓Ωqe−iγ
+Ωs
+0
+
+
+
+
+ ,
+(1)
+where the superscripts L and R denote the left-handedness and
+right-handedness. Ωp, Ωs , and Ωq represent the Rabi frequen-
+cies of the three energy level transitions, respectively. The sign
+− or + of Ωq represents L or R handedness. γ is the phase of
+Ωq, in this paper, we set γ = π/2 and Ωp = Ωs = Ω. Therefore,
+the simplified Hamiltonian is
+HL,R = ¯h
+
+
+
+
+
+0
+Ω
+∓iΩq
+Ω
+0
+Ω
+±iΩq
+Ω
+0
+
+
+
+
+ ,
+(2)
+In order to achieve accurate chiral resolution, the goal is that
+after applying the same specific pulse to the two chiral systems,
+the final state of the left-handedness system is completely at one
+energy level, and the final state of the right-handedness system
+
+Letter
+Optics Letters
+2
+ 
+! 
+!!
+"!"
+!"
+!!
+! 
+ 
+#
+#
+$
+$
+%&'
+%('
+Fig. 1. Schematic diagram of chiral molecules with L (a) and
+R (b) handedness in three different energy levels. Their dipole
+transitions are mirror symmetric, with the same Ωp and Ωs
+but the Ωq with opposite sign.
+is completely at another energy level, so that we can determine
+its chirality by measuring the energy spectrum of the system.
+First, we consider the L chiral system. The invariant is
+IL =
+
+
+
+
+
+0
+sin ϕ sin θ
+−i cos ϕ
+sin ϕ sin θ
+0
+sin ϕ cos θ
+i cos ϕ
+sin ϕ cos θ
+0
+
+
+
+
+ .
+(3)
+The eigenstates of the invariant are
+��φL
+0
+� =
+
+
+
+
+
+− sin ϕ cos θ
+i cos ϕ
+sin ϕsinθ
+
+
+
+
+ ,
+��φL±
+� =
+1
+√
+2
+
+
+
+
+
+cos ϕ cos θ ± i sin θ
+i sin ϕ
+− cos ϕ sin θ ± i cos θ
+
+
+
+
+ ,
+(4)
+with corresponding eigenvalues µ0 = 0 and µ± = ±1. By solv-
+ing the dynamical equation [31], the following constraint condi-
+tions are obtained:
+Ω = ˙ϕ/(sin θ − cos θ),
+Ωq = ˙ϕ cot ϕ(sin θ + cos θ)/(sin θ − cos θ) − ˙θ,
+(5)
+where the dot represents the derivative with respect to time.
+When the above conditions are satisfied, we can write the gen-
+eral solution
+��ψL(t)
+�
+of Schrödinger [27] as
+���ψL(t)
+�
+= ∑
+j=0,±
+Bjeiηj(t) ���φL
+j (t)
+�
+,
+(6)
+where Bj are time-independent constants, and ηj(t) are the so-
+called LR phases which satisfy
+˙ηj(t) = 1
+¯h
+�
+φL
+j (t)
+��� i¯h ∂
+∂t − HL ���φL
+j (t)
+�
+.
+(7)
+Thereby, we can get
+η0(t) = 0,
+η±(t) = ± � t
+0 dt′ [ ˙ϕ csc ϕ(sin θ + cos θ)/(cos θ − sin θ)].
+(8)
+0
+0.25
+0.5
+0.75
+1
+t (T)
+0
+0.5
+1
+Populations of L
+0
+0.25
+0.5
+0.75
+1
+t (T)
+0
+0.5
+1
+Populations of R
+(b)
+(a)
+Fig. 2. Schematic diagram of energy level populations of the
+L(R) systems using SPS. (a) Populations vs the time t of L sys-
+tem; (b) Populations vs the time t of R system. Red dashed,
+green solid, blue dotted lines stand for the populations of |1⟩,
+|2⟩, and |3⟩, respectively.
+It can be seen from the Eq. (6) that if the L-handed system is
+initially in an eigenstate
+���φL
+j (t)
+�
+, it will also be in this eigenstate
+at any time after time evolution. As for the eigenstate
+��φL
+0 (t)
+�
+,
+if the boundary conditions of the parameters are chosen as
+ϕ(0) = 0,
+ϕ(T) = π/2,
+θ(T) = π/2,
+(9)
+where T is final time moment, the L system will completely
+transfer to the level |3⟩ if initially in the level |2⟩. Second, let
+us consider the R system. We set its invariant as
+IR =
+
+
+
+
+
+0
+sin ϕ cos θ
+i cos ϕ
+sin ϕ cos θ
+0
+sin ϕ sin θ
+−i cos ϕ
+sin ϕ sin θ
+0
+
+
+
+
+ .
+(10)
+Similarly, we can obtain the eigenstates of this invariant IR:
+��φR
+0
+� =
+
+
+
+
+
+sin ϕsinθ
+i cos ϕ
+− sin ϕ cos θ
+
+
+
+
+ ,
+��φR±
+� =
+1
+√
+2
+
+
+
+
+
+− cos ϕ sin θ ± i cos θ
+i sin ϕ
+cos ϕ cos θ ± i sin θ
+
+
+
+
+ ,
+(11)
+with corresponding eigenvalues µ0 = 0 and µ± = ±1. For the
+R system, we find that the parameter constraints and LR phases
+of R system are exactly the same as those of L system, as shown
+in Eq. (5) and (8). This means that if we drive the L or R sys-
+tem to evolve along the eigenstate
+��φL
+0
+�
+or
+��φR
+0
+�
+, we can apply
+the same pulse scheme by inversely solving the constraint con-
+ditions in Eq. (5).
+Now, we pay attention to the eigenstate
+��φR
+0 (t)
+�
+. If we have
+the same boundary condition in Eq. (9), the R system will com-
+pletely transfer from |2⟩ to |1⟩ for t ∈ [0, T], which is completely
+different from the target energy level of the L system. That is to
+say, we can apply the same pulse to a pair of L and R systems
+when they are initially at the level |2⟩ by choosing the invariant
+parameters to fulfill the boundary condition in Eq. (9). This can
+drive the L system to fully evolve to the level |3⟩, while drive
+the R system to fully evolve to the level |1⟩. Finally, their hand-
+edness can be determined by measuring their energy spectrum.
+As a result, the 100% chiral discrimination is reached.
+
+Letter
+Optics Letters
+3
+-0.2
+-0.1
+0
+0.1
+0.2
+error amplitude α
+0.9
+0.95
+1
+Fidelity
+Fig. 3. The systematic error amplitude α vs fidelity of different
+schemes: SPS (blue, dotted line), OSE (red, dashed line), and
+CD (green, solid line).
+Here, we consider a simple parameter scheme (SPS) to show
+how to achieve an efficient chiral discrimination by invariant-
+based inverse engineering. When we choose
+ϕ(t) = πt
+2T ,
+θ(t) = π
+2 ,
+(12)
+to satisfy the boundary conditions in Eq. (9). Inversely, we can
+get the parameters of Hamiltonian, from the Eq. (5), as
+Ω = π
+2T ,
+Ωq = π
+2 cot πt
+2T ,
+(13)
+where T is pulse duration and t ∈ [0, T]. In Fig. 2, we plot
+the evolution curve of the level population of the L and R sys-
+tems. It can be seen that the two systems are initially at the
+same level |2⟩. At t = T, the population of the L system com-
+pletely transfers to level |3⟩, while the population of the R sys-
+tem completely transfers to level |1⟩. Therefore, through mea-
+suring their energy spectrum or population of the system, we
+can determine its chirality: if the population of the state |3⟩ is
+1, this is a left-handed system, and if the population of the state
+|1⟩ is 1, it is a right-handed system.
+On the other hand, when we consider the influence of con-
+trol errors that may occur in the experiment on the fidelity (or
+discrimination) of the resolution scheme, it is necessary to op-
+timize the LRI scheme with respect to these errors. We use
+a new Hamiltonian H′ to indicate the existence of errors, i.e.,
+H → H′ = H + He, where He is error Hamiltonian. The fidelity
+is generally defined as
+F =
+���
+ψ(T)
+�� ψ′(T)
+���2,
+(14)
+where |ψ(T)⟩ is target state and |ψ′(T)⟩ is actual state of system
+at the final moment T. Using perturbation theory [30], we have
+FL,R ≈ 1 − 1
+¯h2 ∑
+±
+����
+� T
+0 dt
+�
+φL,R
+0
+(t)
+��� He
+���φL,R
+j
+(t)
+�
+eiηj(t)
+����
+2
+.
+(15)
+We first consider the influence of systematic error. In this case,
+the error Hamiltonian is described as
+HL,R
+e
+= αHL,R,
+(16)
+-1
+-0.5
+0
+0.5
+1
+error amplitude δ (1/T)
+0.9
+0.95
+1
+Fidelity
+Fig. 4. The detuning error amplitude δ vs fidelity of different
+schemes: SPS (blue, dotted line), OSD (red, dashed line), and
+CD (green, solid line).
+where, α is a dimensionless parameter, representing the ampli-
+tude of systematic error. Combining Eqs. (15) and (16), we can
+get
+FL = FR = 1 − α2
+����
+� T
+0 ( ˙θ sin ϕ + i ˙ϕ)eiη+(t)dt
+����
+2
+.
+(17)
+Obviously, the fidelity of the target level for the L and R sys-
+tems is affected by the systematic error in the same way. There-
+fore, we only need to analyze the influence of error on the fi-
+delity of the L or R system. The systematic error sensitivity is
+defined as
+qα = − ∂2FL,R
+2∂α2 |α=0 = − ∂FL,R
+∂(α2) |α=0.
+(18)
+The smaller the sensitivity, the smaller the impact of error on
+fidelity. Then we have
+qα =
+����
+� T
+0 ( ˙θ sin ϕ + i ˙ϕ)eiη+(t)dt
+����
+2
+.
+(19)
+To meet the boundary conditions, we still choose
+ϕ(t) = πt
+2T .
+(20)
+We do not set the form of θ(t) at first, but try the Fourier series
+type of Ansatz with regard to the LR phase η+
+η+(t) = −[n sin(3ϕ) − ϕ],
+(21)
+where n is a real number that can be chosen freely. From the
+Eq. (8), the parameter θ(t) takes the form
+θ(t) = arccot3n cos(3ϕ) sin ϕ − sin ϕ + 1
+3n cos(3ϕ) sin ϕ − sin ϕ − 1,
+(22)
+which satisfies the boundary condition θ(T) = π/2.
+Based
+on the above equations, we can calculate the systematic error
+sensitivity qα numerically. When n = 1.07, the systematic er-
+ror sensitivity reaches the minimum value of 0.52, which is
+defined as the optimal scheme for systematic error sensitiv-
+ity (OSS). In Fig. 3, we compare the influence of systematic
+error on the fidelity or discrimination with several coherent
+
+Letter
+Optics Letters
+4
+Fig. 5. Fidelity FL,R vs the systematic error amplitude α and
+detuning error amplitude δ by LRI scheme of n = 1.10. The
+yellow area in the middle corresponds to FL,R ≥ 0.99.
+control schemes, including OSS, SPS, and the counter-dabatic
+(CD) shortcut method in Ref. [9]. We can observe that all these
+schemes can achieve 100% discrimination in the absence of the
+error, and the OSE scheme is the most robust against systematic
+error, followed by SPS, and finally CD.
+Another important error in experiment is the detuning error.
+In this case, the error Hamiltonian is
+HL,R
+e
+= δ¯h(|3⟩ ⟨3| − |1⟩ ⟨1|),
+(23)
+where δ represents the detuning amplitude, and its unit is 1/T.
+In the same way, we can obtain the fidelity as
+FL = FR
+= 1 − δ2
+4
+���
+� T
+0 [cos(2θ) sin(2ϕ) + 2i sin(2θ) sin ϕ]eiη+(t)dt
+���
+2
+.
+(24)
+And we have
+qδ =
+����
+� T
+0 [cos(2θ) sin(2ϕ) + 2i sin(2θ) sin ϕ]eiη+(t)dt
+����
+2
+.
+(25)
+Here, the parameters ϕ and η+ are chosen as the same forms
+of Eqs. (20) and (21). We can find that, the detuning error sen-
+sitivity reaches the minimum value 0 when n = 1.12. We call
+the corresponding parameter scheme as the optimal scheme for
+detuning error (OSD). In Fig. 4, we compare the influence of
+detuning error on the fidelity or discrimination with OSD, SPS,
+and CD control schemes. Again, the OSD scheme is the most
+robust against the detuning error. Furthermore, we plot how
+the fidelity is affected by the systematic error and detuning er-
+ror in Fig. 5. It can be seen that the optimal scheme shows high
+robustness against these two errors with a broad range of high
+efficiencies over 99% .
+In conclusion, we propose a highly efficient and robust chiral
+discrimination method for the cyclic three-level systems of chi-
+ral molecules based on the invariant-based inverse engineering.
+Through applying to the same pulse on the three-level system,
+molecules with different chirality will transit to different energy
+levels. The L system stay in |3⟩ and R system stay in |1⟩ at the
+final time from the same initial state. We can realize the 100%
+chiral discrimination of molecules by measuring population or
+energy spectrum. Moreover, we can design the corresponding
+optimization schemes with respect to different experimental er-
+rors. By comparison, the optimization schemes are superior to
+the SPS and the CD schemes.
+Funding.
+This study was supported by the National Natural Sci-
+ence Foundation of China (Grant No. 12004006, No. 12075001, and
+No. 12175001), Anhui Provincial Key Research and Development Plan
+(Grant No. 2022b13020004), and the Anhui Provincial Natural Science
+Foundation (Grant No. 2008085QA43).
+Disclosures.
+The authors declare no conflicts of interest.
+Data Availability Statement.
+Data underlying the results pre-
+sented in this Letter are not publicly available at this time but may be
+obtained from the authors upon reasonable request.
+REFERENCES
+1.
+L. Pasteur, Ann. Chim. Phys. 24, 442 (1848).
+2.
+R. F. Dashen, Phys. Rev. D 3, 1879 (1971).
+3.
+M. Fu, F. Liu, and L. Hu, Compos. Sci. Technol. 160, 111 (2018).
+4.
+Z.-G. Gu, C. Zhan, J. Zhang, and X. Bu, Chem. Soc. Rev. 45, 3122
+(2016).
+5.
+T. J. Leitereg, D. G. Guadagni, J. Harris, T. R. Mon, and R. Teranishi,
+J. Agric. Food Chem. 19, 785 (1971).
+6.
+A. J. Hutt and S. C. Tan, Drugs 52, 1 (1996).
+7.
+J. Gal, Chirality 12, 959 (2012).
+8.
+M. Shapiro, E. Frishman, and P. Brumer, Phys. Rev. Lett. 84, 1669
+(2000).
+9.
+N. V. Vitanov and M. Drewsen, Phys. Rev. Lett. 122, 173202 (2019).
+10.
+C. Ye, Q. Zhang, Y.-Y. Chen, and Y. Li, Phys. Rev. A 100, 043403
+(2019).
+11.
+B. T. Torosov, M. Drewsen, and N. V. Vitanov, Phys. Rev. A 101,
+063401 (2020).
+12.
+B. T. Torosov, M. Drewsen, and N. V. Vitanov, Phys. Rev. Res. 2,
+043235 (2020).
+13.
+J.-L. Wu, Y. Wang, J. Song, Y. Xia, S.-L. Su, and Y.-Y. Jiang, Phys.
+Rev. A 100, 043413 (2019).
+14.
+J.-L. Wu, S.-L. Su, Y. Xia, Y.-Y. Jiang, and J. Song, Opt. Express 28,
+33475 (2020).
+15.
+J.-L. Wu, Y. Wang, J.-X. Han, C. Wang, S.-L. Su, Y. Xia, Y.-Y. Jiang,
+and J. Song, Phys. Rev. Appl. 13, 044021 (2020).
+16.
+Y.-H. Kang, Z.-C. Shi, J. Song, and Y. Xia, Opt. Lett. 45, 4952 (2020).
+17.
+C. Ye, Q.-S. Zhang, Y.-Y. Chen, and Y. Li, Phys. Rev. Res. 2, 033064
+(2020).
+18.
+Y. Guo, X. Gong, S. Ma, and C.-C. Shu, Phys. Rev. A 105, 013102
+(2022).
+19.
+N. V. Vitanov, L. P. Yatsenko, and K. Bergmann, Phys. Rev. A 68,
+043401 (2003).
+20.
+M. Demirplak and S. A. Rice, J. Phys. Chem. A 107, 9937 (2003).
+21.
+M. V. Berry, J. Phys. A 42, 365303 (2009).
+22.
+X. Chen, I. Lizuain, A. Ruschhaupt, D. Guéry-Odelin, and J. G. Muga,
+Phys. Rev. Lett. 105, 123003 (2010).
+23.
+X.-K. Song, Q. Ai, J. Qiu, and F.-G. Deng, Phys. Rev. A 93, 052324
+(2016).
+24.
+B. T. Torosov, S. Guérin, and N. V. Vitanov, Phys. Rev. Lett. 106,
+233001 (2011).
+25.
+G. T. Genov, D. Schraft, T. Halfmann, and N. V. Vitanov, Phys. Rev.
+Lett. 113, 043001 (2014).
+26.
+G. T. Genov, D. Schraft, N. V. Vitanov, and T. Halfmann, Phys. Rev.
+Lett. 118, 133202 (2017).
+27.
+H. R. Lewis and W. B. Riesenfeld, J. Math. Phys. 10, 1458 (1969).
+28.
+X. Chen, A. Ruschhaupt, S. Schmidt, A. del Campo, D. Guéry-Odelin,
+and J. G. Muga, Phys. Rev. Lett. 104, 063002 (2010).
+29.
+X. Chen and J. G. Muga, Phys. Rev. A 86, 033405 (2012).
+30.
+A. Ruschhaupt, X. Chen, D. Alonso, and J. G. Muga, New J. Phys. 14,
+093040 (2012).
+31.
+S.-F. Qi and J. Jing, Phys. Rev. A 105, 053710 (2022).
+32.
+Y.-H. Kang, Y.-H. Chen, X. Wang, J. Song, Y. Xia, A. Miranowicz, S.-B.
+Zheng, and F. Nori, Phys. Rev. Res. 4, 013233 (2022).
+33.
+R. Unanyan, L. Yatsenko, K. Bergmann, and B. Shore, Opt. Commun.
+139, 48 (1997).
+
++2
+ude0.950.9error
+0
+b00.85etunin0.8
+20
+litude0
+amp
+rrortematic-2
+-0.2
+sysLetter
+Optics Letters
+5
+FULL REFERENCES
+1.
+L. Pasteur, “On the Relations Crystalline Form, Chemical Composition
+and Direction of Polarization Rotatorie,” Ann. Chim. Phys. 24, 442–459
+(1848).
+2.
+R. F. Dashen, “Some features of chiral symmetry breaking,” Phys. Rev.
+D 3, 1879 (1971).
+3.
+M. Fu, F. Liu, and L. Hu, “A novel category of 3d chiral material with
+negative Poisson’s ratio,” Compos. Sci. Technol. 160, 111–118 (2018).
+4.
+Z.-G. Gu, C. Zhan, J. Zhang, and X. Bu, “Chiral chemistry of
+metal–camphorate frameworks,” Chem. Soc. Rev. 45, 3122–3144
+(2016).
+5.
+T. J. Leitereg, D. G. Guadagni, J. Harris, T. R. Mon, and R. Teranishi,
+“Chemical and sensory data supporting the difference between the
+odors of the enantiomeric carvones,” J. Agric. Food Chem. 19, 785–
+787 (1971).
+6.
+A. J. Hutt and S. C. Tan, “Drug chirality and its clinical significance,”
+Drugs 52, 1–12 (1996).
+7.
+J. Gal, “The discovery of stereoselectivity at biological receptors: Ar-
+naldo Piutti and the taste of the asparagine enantiomers—History and
+analysis on the 125th anniversary,” Chirality 12, 959–976 (2012).
+8.
+M. Shapiro, E. Frishman, and P. Brumer, “Coherently controlled asym-
+metric synthesis with achiral light,” Phys. Rev. Lett. 84, 1669 (2000).
+9.
+N. V. Vitanov and M. Drewsen, “Highly efficient detection and separa-
+tion of chiral molecules through shortcuts to adiabaticity,” Phys. Rev.
+Lett. 122, 173202 (2019).
+10.
+C. Ye, Q. Zhang, Y.-Y. Chen, and Y. Li, “Effective two-level models
+for highly efficient inner-state enantioseparation based on cyclic three-
+level systems of chiral molecules,” Phys. Rev. A 100, 043403 (2019).
+11.
+B. T. Torosov, M. Drewsen, and N. V. Vitanov, “Efficient and robust chi-
+ral resolution by composite pulses,” Phys. Rev. A 101, 063401 (2020).
+12.
+B. T. Torosov, M. Drewsen, and N. V. Vitanov, “Chiral resolution by
+composite Raman pulses,” Phys. Rev. Res. 2, 043235 (2020).
+13.
+J.-L. Wu, Y. Wang, J. Song, Y. Xia, S.-L. Su, and Y.-Y. Jiang, “Robust
+and highly efficient discrimination of chiral molecules through three-
+mode parallel paths,” Phys. Rev. A 100, 043413 (2019).
+14.
+J.-L. Wu, S.-L. Su, Y. Xia, Y.-Y. Jiang, and J. Song, “Discrimination of
+enantiomers through quantum interference and quantum Zeno effect,”
+Opt. Express 28, 33475–33489 (2020).
+15.
+J.-L. Wu, Y. Wang, J.-X. Han, C. Wang, S.-L. Su, Y. Xia, Y.-Y. Jiang,
+and J. Song, “Two-Path Interference for Enantiomer-Selective State
+Transfer of Chiral Molecules,” Phys. Rev. Appl. 13, 044021 (2020).
+16.
+Y.-H. Kang, Z.-C. Shi, J. Song, and Y. Xia, “Effective discrimination of
+chiral molecules in a cavity,” Opt. Lett. 45, 4952–4955 (2020).
+17.
+C. Ye, Q.-S. Zhang, Y.-Y. Chen, and Y. Li, “Fast enantioconversion of
+chiral mixtures based on a four-level double-∆ model,” Phys. Rev. Res.
+2, 033064 (2020).
+18.
+Y. Guo, X. Gong, S. Ma, and C.-C. Shu, “Cyclic three-level-pulse-area
+theorem for enantioselective state transfer of chiral molecules,” Phys.
+Rev. A 105, 013102 (2022).
+19.
+N. V. Vitanov, L. P. Yatsenko, and K. Bergmann, “Population transfer
+by an amplitude-modulated pulse,” Phys. Rev. A 68, 043401 (2003).
+20.
+M. Demirplak and S. A. Rice, “Adiabatic population transfer with con-
+trol fields,” J. Phys. Chem. A 107, 9937–9945 (2003).
+21.
+M. V. Berry, “Transitionless quantum driving,” J. Phys. A 42, 365303
+(2009).
+22.
+X. Chen, I. Lizuain, A. Ruschhaupt, D. Guéry-Odelin, and J. G. Muga,
+“Shortcut to Adiabatic Passage in Two- and Three-Level Atoms,” Phys.
+Rev. Lett. 105, 123003 (2010).
+23.
+X.-K. Song, Q. Ai, J. Qiu, and F.-G. Deng, “Physically feasible three-
+level transitionless quantum driving with multiple Schrödinger dynam-
+ics,” Phys. Rev. A 93, 052324 (2016).
+24.
+B. T. Torosov, S. Guérin, and N. V. Vitanov, “High-Fidelity Adiabatic
+Passage by Composite Sequences of Chirped Pulses,” Phys. Rev.
+Lett. 106, 233001 (2011).
+25.
+G. T. Genov, D. Schraft, T. Halfmann, and N. V. Vitanov, “Correction of
+Arbitrary Field Errors in Population Inversion of Quantum Systems by
+Universal Composite Pulses,” Phys. Rev. Lett. 113, 043001 (2014).
+26.
+G. T. Genov, D. Schraft, N. V. Vitanov, and T. Halfmann, “Arbitrarily
+Accurate Pulse Sequences for Robust Dynamical Decoupling,” Phys.
+Rev. Lett. 118, 133202 (2017).
+27.
+H. R. Lewis and W. B. Riesenfeld, “An exact quantum theory of the
+time-dependent harmonic oscillator and of a charged particle in a
+time-dependent electromagnetic field,” J. Math. Phys. 10, 1458–1473
+(1969).
+28.
+X. Chen, A. Ruschhaupt, S. Schmidt, A. del Campo, D. Guéry-Odelin,
+and J. G. Muga, “Fast Optimal Frictionless Atom Cooling in Harmonic
+Traps: Shortcut to Adiabaticity,” Phys. Rev. Lett. 104, 063002 (2010).
+29.
+X. Chen and J. G. Muga, “Engineering of fast population transfer in
+three-level systems,” Phys. Rev. A 86, 033405 (2012).
+30.
+A. Ruschhaupt, X. Chen, D. Alonso, and J. G. Muga, “Optimally robust
+shortcuts to population inversion in two-level quantum systems,” New
+J. Phys. 14, 093040 (2012).
+31.
+S.-F. Qi and J. Jing, “Accelerated adiabatic passage in cavity mag-
+nomechanics,” Phys. Rev. A 105, 053710 (2022).
+32.
+Y.-H. Kang, Y.-H. Chen, X. Wang, J. Song, Y. Xia, A. Miranowicz, S.-
+B. Zheng, and F. Nori, “Nonadiabatic geometric quantum computation
+with cat-state qubits via invariant-based reverse engineering,” Phys.
+Rev. Res. 4, 013233 (2022).
+33.
+R. Unanyan, L. Yatsenko, K. Bergmann, and B. Shore, “Laser-induced
+adiabatic atomic reorientation with control of diabatic losses,” Opt.
+Commun. 139, 48–54 (1997).
+

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+Adaptively Clustering Neighbor Elements for Image Captioning
+Zihua Wang1,2
+Xu Yang1
+Haiyang Xu2*
+Hanwang Zhang3
+Chenliang Li2
+Songfang Huang2
+Fei Huang2
+Yu Zhang1*
+1 School of Computer Science & Engineering, Key Lab of Computer Network
+& Information Integration (Ministry of Education), Southeast Univ., Nanjing, China
+2Alibaba Group
+3 School of Computer Science & Engineering, Nanyang Technological Univ., Singapore.
+{zihua, 101013120, zhang yu}@seu.edu.cn,{shuofeng.xhy, lcl193798,
+songfang.hsf, f.huang}@alibaba-inc.com, [email protected]
+Abstract
+We design a novel global-local Transformer named Ada-
+ClustFormer (ACF) to generate captions. We use this name
+since each layer of ACF can adaptively cluster input el-
+ements to carry self-attention (Self-ATT) for learning lo-
+cal context. Compared with other global-local Transform-
+ers which carry Self-ATT in fixed-size windows, ACF can
+capture varying graininess, e.g., an object may cover dif-
+ferent numbers of grids or a phrase may contain diverse
+numbers of words. To build ACF, we insert a probabilis-
+tic matrix C into the Self-ATT layer.
+For an input se-
+quence {s1, ..., sN}, Ci,j softly determines whether the
+sub-sequence {si, ..., sj} should be clustered for carrying
+Self-ATT. For implementation, Ci,j is calculated from the
+contexts of {si, ..., sj}, thus ACF can exploit the input itself
+to decide which local contexts should be learned. By us-
+ing ACF to build the vision encoder and language decoder,
+the captioning model can automatically discover the hid-
+den structures in both vision and language, which encour-
+ages the model to learn a unified structural space for trans-
+ferring more structural commonalities. The experiment re-
+sults demonstrate the effectiveness of ACF that we achieve
+CIDEr of 137.8, which outperforms most SOTA captioning
+models and achieve comparable scores compared with some
+BERT-based models. The code will be available in the sup-
+plementary material.
+1. Introduction
+Image Captioning (IC) aims to learn a shared vision-
+language representation space for facilitating the transfer of
+multimodal knowledge to generate visually grounded sen-
+*Corresponding authors.
+tence [22]. Two prevailing deep learning techniques help
+the IC model learn such space.
+The first one is the vi-
+sion encoder-language decoder pipeline [41] which back-
+propagates the language semantic to the visual encoder
+and another one is the attention mechanism [46] which di-
+rectly bridges between vision and language domains for
+transferring multimodal knowledge.
+Transformers [39],
+which build the encoder and decoder based on dense at-
+tention operations, have both of the above-mentioned ad-
+vantages. Transformers have two types of attention opera-
+tions which are self-attention (Self-ATT) and cross-modal
+attention (Cross-ATT). From the perspective of structure
+learning, Self-ATT applies the fully connected (FC) graph
+prior to the data sequence.
+By using Self-ATT in both
+encoder and decoder, the graph structures of both vision
+and language data can be discovered and Cross-ATT helps
+transfer these structural commonalities for narrowing the
+modality gaps.
+Therefore, Transformer prevails in IC
+tasks [10,12,13,28].
+Interestingly, structure learning is one of the most sig-
+nificant research directions of IC since the paired vision-
+language data usually share a unified internal semantic
+structure although they have diverse external appearances.
+Thus, if this unified semantic structure is captured, more
+structural commonalities can be transferred for generating
+better captions. Motivated by this, various IC models are
+proposed to exploit scene graphs [5, 21, 49] or hierarchy
+trees [43, 51] to narrow the domain gap. However, such
+structures need additional well-trained parsers. Moreover,
+vision and language parsers usually have domain gaps that
+the parsed structures of the paired image-sentence may not
+match, which may even weaken the effectiveness of these
+IC models. We prefer an IC model that can adaptively dis-
+cover the unified semantic structures to remove the costs of
+the additional structure annotations and more importantly,
+arXiv:2301.01955v1  [cs.CV]  5 Jan 2023
+
+ (a) Fixed-Size Transformer
+s1 s2 s3 s4 s5 s6 s7
+s8
+Input
+1-st
+layer
+2-nd
+layer
+3-rd
+layer
+(b) ACF
+s1 s2 s3 s4 s5 s6 s7 s8
+Input
+1-st
+layer
+2-nd
+layer
+3-rd
+layer
+…
+…
+…
+…
+(c) ACF-based IC
+riding 
+a snow board
+on 
+snow
+A man 
+riding a snow board
+on snow 
+A man riding a snow board on snow.
+riding 
+a
+A
+man 
+snow
+board
+on 
+snow
+A man 
+Figure 1. (a) Transformer with fixed-size windows (size = 2); (b)
+ACF which adjusts the window size according to the input. (c)
+ACF-based IC. The left/right part shows how the vision/language
+ACFs cluster image grids/language words for transferring struc-
+tural commonalities.
+to learn a unified structure space for transferring structural
+commonalities.
+Transformer seems to be a good starting point since
+it can implicitly build graphs by Self-ATT. However, it
+exploits the FC graph prior, while the useful semantic
+structure is usually sparse and hierarchical like the scene
+graphs or trees.
+To discover more sparse structures, re-
+searchers design various global-local Transformers [20,29,
+33]. As sketched in Figure 1(a), these Transformers grad-
+ually merge the neighbor elements in fixed-size windows
+into bigger clusters and carry Self-ATT in each cluster. For
+example, the 1-st layer clusters 2 neighboring elements like
+{s1, s2} to carry Self-ATT for local contexts and the 2-
+nd layer merges {s1, s2} and {s3, s4} into a bigger one
+to learn more global context.
+Then a hierarchical struc-
+ture is built from lower to higher layers where local and
+global contexts are respectively captured. However, these
+Transformers still do not satisfy our requirement since vi-
+sion and language data have diverse graininess, e.g., objects
+may cover varying grids and phrases may compose different
+numbers of words, while fixed-size windows cannot effec-
+tively capture such varying graininess.
+To capture the varying graininess, we propose to
+Adaptively
+Cluster
+the
+neighbor
+elements
+to
+carry
+Self-ATT and named the novel Transformer as Ada-
+ClustFormer (ACF). As shown in Figure 1(b), in each
+layer, the window size is not fixed but can be adjusted
+to each specific input sequence, e.g., in the 1-st layer,
+{s1, s2, s3}, {s4}, {s5, s6}, {s7}, {s8} are respectively
+clustered. The higher layers merge small clusters into big-
+ger ones for global contexts, e.g., the 2-nd layer respectively
+merges {s1, s2, s3, s4, s5, s6}, {s7, s8} into two clusters to
+carry Self-ATT. To achieve this adaptive clustering, we in-
+sert a probabilistic clustering matrix C into the Self-ATT
+layer, where the probability Cij softly determines whether
+the sub-sequence {si, ..., sj} should be clustered or not. To
+calculate Cij, we consider whether the next element sj is
+similar to the mean-pooling of {si, ..., sj−1}. Thus ACF
+can adjust the window of Self-ATT based on each specific
+data sample.
+To construct an IC model based on ACF, besides build-
+ing 1-D ACF for the language decoder, we also extend it
+to the 2-D ACF as the vision encoder. In this way, both
+the visual encoder and language decoder can automatically
+discover the hidden structures of the image and language
+data. This means that the ACF model does not need any
+additional structure annotations as some previous IC mod-
+els [2, 5] but still exploits the sparse structures implied in
+both vision and language data. For example, as shown in
+Figure 1(c), a visual ACF can merge the smaller grids into
+bigger regions to capture both grid-level [15] and region-
+level [4] contexts. And the language one gradually clus-
+ters the single words into phrases to generate the captions
+in an imaginary phrase-by-phrase manner [38, 48]. More
+importantly, compared with certain global-local Transform-
+ers which are exclusively developed in vision and language
+domains [24, 47], the visual and language ACF exploit the
+same way to discover hidden structures. So, our ACF model
+is a homogeneous structure that helps transfer more struc-
+tural commonalities between vision and language domains,
+e.g., as shown in Figure 1(c), the patches of the object “snow
+board” is clustered in the image and correspondingly, the
+phrase “a snow board” is also clustered in the language do-
+main.
+In summary, our contributions can be listed as follows:
+• We propose ACF that can adaptively capture varying
+graininess.
+• We extend ACF to the 2-D case for building a homo-
+geneous IC model that learns unified structural space
+for transferring more structural commonalities.
+• The experimental results show that our ACF model
+outperforms the classic Transformers in IC.
+2. Related Work
+Image Captioning (IC). IC aims to generate descriptions
+according to the given images.
+Typically, an encoder-
+decoder paradigm is used to convert visual inputs to se-
+quence outputs. In the early stage, image features are ex-
+tracted by CNN-based encoders, as the input of the RNN-
+based decoders [4, 16, 35, 41]. For example, Up-Down [4]
+employs a Faster R-CNN [34] to extract image region fea-
+tures and LSTM networks to generate sentences.
+Nowadays, Transformer-based models have shown their
+
+might in Neural Language Process (NLP) and replace RNN-
+based decoders in IC [12, 14, 19]. Subsequently, more ad-
+vanced Transformer-based decoders are proposed, e.g., M2
+Transformer [8] proposes a meshed-memory Transformer
+to interact with the low-level and high-level features; X-
+Linear Transformer [31] selectively capitalizes the visual
+information from image regions by bilinear pooling.
+However, these models still use CNN-based feature ex-
+tractors.
+More recently, witnessing the boom of Vision
+Transformers (ViT) [9, 24], researchers use ViT-based vi-
+sual encoders for captioning. For instance, CPTR [23] in-
+troduces grid-based features that are extracted by ViT [9]
+instead of using the ROI-based features; DLCT [25] fuses
+the ROI-based features with the grid-based features to over-
+come the shortcoming of both features.
+Besides that,
+some models exploit the knowledge distilled from Vision-
+Language BERTs for better captions [18].
+VinVL [52]
+and GRIT [28] propose the object detection model in IC.
+ClipCAP [27] and LEMON [13] introduce large-scale pre-
+training on IC. Noteworthy, the methods above employ the
+ViT [9] or Swin Transformer [24] as their backbone. Thus,
+our ACF adopts the Swin Transformer as our encoder back-
+bone.
+Among the previous IC models, Auto-Parsing Network
+(APN) [48] has a similar motivation as ours, which also in-
+serts a clustering matrix into the Self-ATT layer. However,
+Ada-ClustFormer (ACF) calculates this matrix differently.
+APN only considers whether pairwise neighboring elements
+should be clustered or not, while we calculate this proba-
+bility from a more global scope. Specifically, we consider
+whether the next element is similar to the previous clustered
+elements. More importantly, we extend our ACF into the 2-
+D case, which can adaptively cluster the visual patches into
+regions, while APN only treats a sequence of ROI features
+as the visual input and still applies a 1-D clustering matrix
+to address it. More comparisons will be given in the supple-
+mentary material.
+Global-Local Transformer.
+To alleviate the fully con-
+nected graph prior in Transformer, researchers propose var-
+ious global-local Transformers to learn sparse structures of
+the language [6, 26]. For example, Global-local [26] intro-
+duces a fixed-size of the global and local attention model in
+neural machine translation. Longformer [6] proposes global
+and local window attentions, which can provide inductive
+bias and long sequence representation, respectively.
+Hi-
+Transformer [44] learns sentence-level and document-level
+semantics through the hierarchical structure.
+The global-local Transformer mechanism is also effec-
+tive in vision area [7, 25, 53]. Pairwise and patchwise self-
+attention are proposed in image recognition [53]. Further-
+more, GLiT [7] proposes to adaptively trade off the global
+and local information of the images. DLCT [25] explores
+the global and local information by the combination of grid-
+based features and ROI-based features.
+However, these models are exclusively developed in a
+single domain (either NLP or CV), while our ACF provides
+a general approach in both the vision and language domains.
+Thus, using ACF to build the IC model encourages learn-
+ing a unified structure space for transferring more structure
+commonalities.
+3. Ada-ClustFormer IC model
+Compared
+with
+the
+classic
+Transformer,
+Ada-
+ClustFormer
+(ACF)
+inserts
+an
+adaptively
+clustering
+matrix C into each self-attention (Self-ATT) layer to
+adaptively control the scope of Self-ATT. The calculation
+of C is detailed in Section 3.1 where we first show the 1-D
+language case and then extend it to the 2-D vision case. By
+stacking these revised Self-ATT layers, ACF can be built
+for constructing the vision encoder and language decoder
+for captioning (cf. Section 3.2).
+3.1. Ada-ClustFormer
+Multi-Head Attention (MHA). ACF is built based on
+Transformer, whose most elemental building block is the
+Multi-Head Attention (MHA). Given the query Q
+∈
+RNQ×d, key K ∈ RNK×d, and value V ∈ RNV ×d, MHA
+calculates the output Z = MHA(Q, K, V) as:
+Input:
+Q, K, V
+ATT:
+Al = Softmax(QWQ
+l (KWK
+l )T
+√
+d
+)
+Head :
+Hl = AlVWV
+l ,
+Multi-Head:
+H = [H1, H2, ..., Hh]WH,
+Output:
+Z = LN(H + Q),
+(1)
+where WQ
+l , WK
+l , WV
+l
+∈ Rd×dh, WH
+l
+∈ Rd×d are all learn-
+able parameters; h denotes the head number and dh = d/h;
+Al is the l-th attention matrix corresponding to the l-th head
+Hl; [·] is the concatenation operation; and LN denotes to the
+Layer Normalization.
+Given an input sequence S = {s1, ..., sN}, if Q =
+K = V = S, Eq. (1) is also called self-attention (Self-
+ATT). Self-ATT captures the global contexts between any
+two elements si and sj by calculating the pairwise atten-
+tion weight in the “ATT” operation. From the perspective
+of structure learning [5], single-head Self-ATT constructs
+a fully-connected (FC) graph where the nodes are the ele-
+ments of S and the pairwise edges are weighted by the pair-
+wise attention weight. Correspondingly, a h-head Self-ATT
+constructs h FC graphs with different edge weights.
+Adaptive Clustering Matrix C. To sparsify this FC-graph,
+researchers [9, 24] propose to carry Self-ATT in fixed-size
+windows, which is achieved by revising “Head” in Eq. (1):
+C-based Head :
+H = Softmax(A ⊗ C)VWV ,
+(2)
+
+where “⊗” denotes the element-wise production; C is a
+N × N binary clustering matrix that only the elements
+in the window can attend to each other, i.e., if the win-
+dow size is w, Ci,j = 1 if |i − j| ≤ w and Ci,j = 0
+if |i − j| > w. However, language or vision data usually
+have diverse graininess, e.g., a phrase may contain different
+numbers of words or an object may cover diverse spatial
+regions, while the fixed-size windows can not capture the
+varying graininess.
+To amend this, we revise the binary C to a probabilistic
+one where Ci,j softly determines whether to cluster the em-
+beddings from si to sj for carrying Self-ATT. Then if Ci,j
+is small, the pairwise attention in A between si and sj is
+weakened in Eq. (2), which means si and sj are less likely
+to stay in the same cluster. To adaptively decide the win-
+dow size according to each specific input for capturing the
+varying graininess, we use the input itself to calculate Ci,j:
+Ci,j = P(si, ..., sj) =
+j�
+k=i
+P(sk|si, ..., sk−1),
+(3)
+where the joint distribution is decomposed to the produc-
+tions of conditional distributions P(sk|si, ..., sk−1), which
+softly decides whether to merge a new element sk into
+the sub-sequence {si, ..., sk−1}.
+In the implementation,
+P(sk|si, ..., sk−1) is calculated as:
+P(sk|si, ..., sk−1) = Sigmoid(FC([sk, si:k−1])),
+(4)
+where si:k−1 is the mean pooling of the embeddings from
+si to sk−1. Intuitively, Eq. (4) exploits the context of the
+whole sub-sequence {si, ..., sk−1} to decide whether to
+merge a new element {sk} into this sub-sequence. Note
+that Eq. (3) and Eq. (4) only make sense when i < k. Since
+clustering the embeddings from si to sk equals to cluster-
+ing from sk to si, we set Ci,k = Ck,i if i > k and since a
+single element si is itself a cluster, we set Ci,i = 1.
+From Eq. (3), we can also find that:
+Ci,j =P(sj|si, ..., sj−1) × P(si, ..., sj−1)
+=P(sj|si, ..., sj−1) × Ci,j−1.
+(5)
+Since P(sj|si, ..., sj−1) ≤ 1, we have Ci,j ≤ Ci,j−1,
+which means that two elements in the shorter distance are
+more likely to be clustered for carrying Self-ATT. In this
+way, local contexts are encouraged to be captured, as is
+shown in Figure 2(a).
+Stacking Revised Self-ATT. To learn global contexts, we
+can stack these revised Self-ATT layers. When stacking,
+we hope that the higher layers will carry Self-ATT in bigger
+windows than the lower layers to capture the global con-
+texts [43, 48]. To achieve this, for the m-th layer, we re-
+calculate C(m) as ˜C(m):
+˜C(m) = (1 − C(m)) ˜C(m−1) + C(m).
+(6)
+s1 s2 s3 s4 s5 s6
+s1 s2 s3 s4 s5 s6
+C1,4 = C1,3 ×   P( s4 | s1, s2, s3)
+Sigmoid(FC([s4, s1:s3]))
+(a) Calculation of C1,4
+(b) C(2) ≥ C(1)
+1-st 
+layer
+2-nd 
+layer
+~
+~
+Figure 2. (a) shows how to calculate C1,4, where the shade denotes
+the probability value, the darker the color, the larger the probability
+value. (b) shows that the clustered elements in the lower layer will
+be further clustered in a higher layer, e.g., the color of {s1, s2, s3}
+in the 2-nd layer is darker than the 1-st layer.
+Horizontal
+Upsampling
+(a) Calculation of C1,4;1,3
+(b) Down-up Sampling Strategy
+Ph(v1;1, ..., v4;1)
+Pv(v1;1, ..., v1;3)
+s1 s2
+s4
+s3
+s1 s2
+C1,2
+C2,3
+∏
+∏
+C1,4;1,3
+Horizontal
+Upsampling
+(a) Calculation of C1,4;1,3
+(b) Down-up Sampling Strategy
+Ph(v1;1, ..., v4;1)
+Pv(v1;1, ..., v1;3)
+s1 s2
+s4
+s3
+s1 s2
+C1,2
+C2,3
+∏
+∏
+C1,4;1,3
+Figure 3. (a) The example of 2-D C, where C1,4;1,3 is used as
+the example, which is decomposed into vertical and horizontal di-
+rections probabilities. (b) Overview of the Down-Up Sampling
+Strategy.
+Then ˜C(m) is used in Eq. (2) when m > 1 and ˜C(1) =
+C(1). Since 0 ≤ C(m)
+i,j
+≤ 1, ˜C(m)
+i,j
+is a convex combination
+of ˜C(m−1)
+i,j
+and 1, which means that ˜C(m−1)
+i,j
+≤ ˜C(m)
+i,j
+≤ 1.
+If ˜C(m−1)
+i,j
+is large, i.e., the sub-sequence {si, ..., sj} should
+be clustered in the (m − 1)-th layer, then ˜C(m)
+i,j
+must be
+larger, i.e., {si, ..., sj} is also clustered in the m-th layer.
+For example, Figure 2(b) shows that the 2-nd layer will
+further cluster {s1, s2, s3} since ˜C(1)
+1,3 ≤ ˜C(2)
+1,3. Thus, the
+higher layers will carry Self-ATT in a bigger window than
+the lower layers to learn more global contexts.
+2-D Clustering Matrix. Eq. (3) shows how to calculate
+C when the input is a 1-D language sequence, next we
+extend it to the 2-D vision surface.
+Given a 2-D fea-
+ture map V
+= {v1,1, ..., vH,W }, we use Ci,j;x,y to de-
+note the probability that softly decides whether a sub-region
+{vi,x, ..., vj,y} should be clustered or not, which is:
+Ci,j;x,y = P(vi;x, ..., vj;y)
+=
+j
+�
+k=i
+y
+�
+u=x
+P(vk;u|vi;x, vi+1;x, ..., vk−1;u−1)
+(7)
+where i, j and x, y respectively denote the horizontal and
+vertical dimensions. To cover all the sub-regions in a H×W
+
+Image
+Self-ATT
+Add&LN
+1-D C
+Self-ATT
+Add&LN
+Words
+Cross-ATT
+Add&LN
+Captioning: Z
+me×
+Encoder
+Decoder
+md×
+Q,K,V
+Q,K,V
+K,V
+Q
+2-D C
+Figure 4. Overview of our ACF-based encoder-decoder IC model.
+The “Add&LN” is the Add and Layer Normalization. me/md rep-
+resent the number of the encoder/decoder layers, respectively.
+map, it requires applying O(H2 × W 2) times for Eq. (4) to
+get all the probabilities. To reduce the computation burden,
+we apply the independence assumption to decompose the
+2-D distribution into two independent ones, which respec-
+tively correspond to the horizontal and vertical dimensions:
+P(vi;x, ..., vj;y) = Ph(vi;x, ...vj;x)Pv(vi;x, ..., vi;y)
+=
+j
+�
+k=i
+Ph(vk;x|vi;x, ..., vk−1;x)
+y
+�
+u=x
+Pv(vi;x|vi;x, ..., vi;u−1),
+(8)
+In this way, we only need to apply O(H2 + W 2) times
+for Eq. (4) and once matrix production.
+Noteworthy, as
+sketched in Figure 2, for the 2-D region which spans the
+horizontal axis from i to j and the vertical axis from
+x to y, we use the left-most vertical and top-most hor-
+izontal to calculate two 1-D distributions and then mul-
+tiply them to get Ci,j;x,y.
+As Figure 3(a) shows, to
+calculate C1,4;1,3, for the vertical distribution Pv, the
+horizontal ordinate is fixed to 1 and the vertical or-
+dinate changes.
+Ph(vk;1|v1;1, ..., vk−1;1)|k=1,2,3,4 and
+Pv(v1;u|v1;1, ..., v1;u−1)|u=1,2,3 are calculated in the same
+way as Eq. (4). The above-mentioned symmetric character-
+istic is also applied.
+Down-Up Sampling Strategy.
+If the sequence (feature
+map) is too long (big), we can apply the Down-Up Sam-
+pling Strategy to reduce the computation cost. We use a 1-D
+language case as an example to show this strategy. For S =
+{s1, ..., sL}, we can downsample it to ¯S = {¯s1, ..., ¯sL/2}
+where ¯si is the mean pooling of s2∗i−1 and s2∗i. Then ¯S
+is used in Eq. (3) and Eq. (4) to get ¯
+C. To upsample ¯C to
+the original size, we set Ci,j = ¯
+C⌈i/2⌉,⌈j/2⌉. Figure 3(b)
+shows one simple case where L = 4.
+3.2. Encoder-Decoder Architecture
+As is shown in Figure 4, we apply the ACF to build the
+vision encoder and language decoder. Compared to the clas-
+sic Transformer, our ACF introduces clustering-restrained
+attention head. Specifically, in encoder, we calculate a 2-D
+clustering matrix C (cf. Eq. (7)) to softly cluster the ele-
+ments for carrying Self-ATT. Similarly, in decoder, the at-
+tention head is revised with the 1-D C (cf. Eq. (5)). The
+output of this encoder-decoder is used to calculate the word
+distributions Z.
+To train our IC model, we optimize the model by min-
+imizing the cross-entropy loss and maximizing the Rein-
+forcement learning (RL) [35] reward. First, we train the
+model by minimizing the cross-entropy loss:
+LCE = − log P(Z∗),
+(9)
+where Z∗ is the ground-truth captions. Then, we further
+train the model by minimizing the negative reward:
+Lrl = −EZs∼P (Z)(S(Z∗, Zs)),
+(10)
+where Zs is sampled from Z, E represents the mathemat-
+ical expectation, and S represents the evaluation metrics,
+e.g., CIDEr [40].
+4. Experiments
+4.1. Dataset, Metrics, and Settings
+MSCOCO. Following [8, 12, 14, 31, 48], we train and
+evaluate our model on MSCOCO [22], which contains
+123, 287 images, and each one is annotated with 5 cap-
+tions.
+In the experiments, we use the Karpathy split
+(113,287/5,000/5,000 train/val/test images) [16] for offline
+training and the official split (40775 test images) for online
+testing.
+Metrics.
+We adopt five widely-used metrics in caption-
+ing for evaluation, including BLEU [32], METOR [1],
+ROUGE-L [36], CIDEr [40], and SPICE [3].
+Settings. In the training process, we convert all the captions
+into lowercase and delete all the words that occur less than
+6 times. The remaining 9487 words are regarded as our
+vocabulary. We adopt Swin Transformer [24] as the visual
+encoder to extract the visual features. The size of the feature
+map is H × W = 12 × 12, and we apply the Down-Up
+Sampling Strategy (cf. Section 3.1). We train 20/25 epochs
+in the cross-entropy/RL stage. In the cross-entropy stage,
+the Adam optimizer is used with the learning rate of 5 ×
+10−5 and decays by 0.8 per 5 epochs. In the RL stage, the
+learning rate is initialized to 5 × 10−6 and we implement
+the same decay policy for 10 epochs. Then the “Reduce-
+On-Plateau” strategy is applied with a decay rate of 0.5 and
+patience of 3. The batch size is 40 at the whole training
+stage.
+
+Table 1. Comparison between with and without Ada-ClustFormer.
+Models
+me
+md
+B@4
+M
+R
+C
+S
+BASE
+6S
+6S
+40.0
+29.7
+59.6
+134.4
+23.4
+ACF 1
+6C
+6S
+40.3
+29.6
+59.6
+134.7
+23.5
+ACF 2
+6S
+6C
+40.2
+29.8
+59.9
+135.1
+23.7
+ACF
+6C
+6C
+41.1
+30.1
+60.2
+137.8
+24.1
+4.2. Ablation Studies
+We conduct extensive ablations for quantifying the dif-
+ference between classic self-attention (Self-ATT) layers and
+Ada-ClustFormer (ACF) layers (cf. Section 4.2.1), the im-
+pact of the depth of the ACF layers (cf. Section 4.2.2), and
+the impact of the orders of ACF and the Self-ATT layers (cf.
+Section 4.2.3).
+4.2.1
+Differences Between ACF and Self-ATT
+Comparing Methods.
+To evaluate the effectiveness of
+the ACF, we ablate our ACF with the following baselines:
+BASE: We employ 6 Self-ATT encoder layers and de-
+coder layers, which is shown in Table 1 as “6S”. ACF 1
+/ ACF 2: We replace the encoder/decoder with our ACF,
+which is represented as “6C”.
+Results. The results of the ablation are listed in Table 1.
+Compared with BASE, we can find that only using ACF
+encoder (ACF 1) or decoder (ACF 2) has marginal im-
+provements, which is 0.3 or 0.7 on CIDEr. However, when
+combining the ACF encoder and decoder to build a homo-
+geneous architecture ACF, the improvement is substantial,
+which is 3.4. This comparison suggests that a homogeneous
+model transfers more structural commonalities for better
+captions.
+4.2.2
+Impact of the Layer Depth
+Comparing Methods. ACF 3: We reduce the depth of the
+encoder and decoder layer to 3. ACF 4/ACF 5: The num-
+ber of the encoder/decoder layers is set to 3 and the number
+of the decoder/encoder layer remains 6.
+Results. From Table 2, we observe that stacking 6 layers
+generally outperforms the 3-layer case. Our method with
+6 ACF layers in the encoder and decoder achieves the best
+performance among them. We also further explore the in-
+fluence of me by fixing md = 6. We present the impact of
+the number of the encoder layers me in Figure 5. It sug-
+gests that CIDEr approximately linearly increases when me
+increases.
+4.2.3
+Impact of the Layer Order
+Comparing Methods. We discuss the combination of the
+ACF layers and the Self-ATT layers. We freeze the depth
+Table 2. The performances with different layer depth
+Models
+me
+md
+B@4
+M
+R
+C
+S
+ACF 3
+3C
+3C
+38.9
+28.4
+58.8
+132.3
+22.0
+ACF 4
+6C
+3C
+39.3
+28.9
+59.1
+135.9
+23.7
+ACF 5
+3C
+6C
+40.2
+29.8
+59.7
+136.0
+24.0
+ACF
+6C
+6C
+41.1
+30.1
+60.2
+137.8
+24.1
+Table 3. The impact of the layer orders.
+Models
+me
+md
+B@4
+M
+R
+C
+S
+ACF 5
+3C
+6C
+40.2
+29.8
+59.7
+136.0
+24.0
+ACF 6
+3C+ 3S
+6C
+40.7
+29.7
+59.9
+135.7
+23.8
+ACF 7
+3S+ 3C
+6C
+40.5
+29.9
+59.9
+136.1
+23.9
+ACF 2
+6S
+6C
+40.2
+29.8
+59.9
+135.1
+23.7
+ACF
+6C
+6C
+41.1
+30.1
+60.2
+137.8
+24.1
+of the decoder layer md = 6 and quantify the influence of
+the order of the encoders: ACF 5: It stacks 3 ACF lay-
+ers. ACF 6/ACF 7: Both of them have 3 ACF layers and
+3 Self-ATT layers.
+The difference between them is that
+ACF 7 encodes on 3 Self-ATT layers firstly.
+Results. The results are listed in Table 3, where we can see
+that the performances are not sensitive to the orders of ACF
+and Self-ATT layers, i.e., ACF 6 and ACF 7 differ only 0.4.
+We can also find that replacing all the Self-ATT layers with
+our ACF layers will achieve the best captioning quality.
+3
+4
+5
+6
+Number of encoder layers
+136.0
+136.5
+137.0
+137.5
+138.0
+CIDEr
+135.97
+136.6
+137.5
+137.83
+Figure 5. Impact of the number of encoder layers me.
+Qualitative Results. We visualize the hierarchical struc-
+tures of the image and the generated captions in Figure 6
+according to the 2-D and 1-D clustering matrix calculated
+from the 1-st, 3-rd, 5-th, and 6-th layers in encoder and de-
+coder. By inspecting the images and captions, we can find
+that the patches and the words are respectively clustered,
+e.g., in the left part of (b), the patches in the “motorcycles”
+region are clustered, and in the right part, the words “sit-
+ting on motorcycles” are clustered into a phrase. More im-
+portantly, when uniting the image and caption, we can find
+that structural commonalities are transferred, e.g., in (b),
+the “motorcycle” region helps generate the phrase “sitting
+on motorcycles”.
+
+A woman standing on the door of a train with a suitcase.
+a woman 
+standing on
+the door of a train
+with a suitcase
+standing on
+a woman
+the door of
+a
+a
+train with
+suitcase
+Ground-truth: A woman in white and 
+black dress with suitcase on train.
+BASE: A woman standing with a 
+suitcase.
+ACF: A woman standing on the door of 
+a train with a suitcase.
+Two people sitting on motorcycles next to a stop sign.
+sitting on
+Two people
+motorcycles
+a
+next to
+stop
+sign
+sitting on motorcycles
+next to a stop sign
+Ground-truth: Two people riding 
+motorcycles on a city street.
+BASE: Two people riding black 
+motorcycles.
+ACF: Two people sitting on 
+motorcycles next to a stop sign.
+Ground-truth: A man with a hat and 
+eye glasses holding a cell phone.
+BASE: A man with a cowboy hat 
+holding a cell phone.
+ACF: A man wearing a cowboy hat 
+taking a picture with a cell phone.
+A man wearing a cowboy hat taking a picture with a cell phone.
+wearing
+cowboy hat
+a
+taking
+cell
+picture with
+wearing a cowboy hat
+a man
+taking a picture
+a man
+a
+a
+phone
+with a cell phone
+Two
+people
+(b)
+(c)
+(a)
+taking a picture with a cell phone
+A man wearing a cowboy hat
+Two people sitting on motorcycles
+next to a stop sign
+a woman 
+standing on the door of a train
+with a suitcase
+Figure 6. Examples of the generated captions by BASE and ACF models. We visualize the 2-D C and 1-D C in the 1-st, 3-rd, 5-th, and
+6-th layers as the clustered patches.
+4.3. Comparisons with SOTA
+Comparing Methods. Nowadays, the SOTA of image cap-
+tioning has been updated quickly and these models can
+be categorized into 3 groups. The first one is the meth-
+ods which use ROI-based features, including Up-Down [4],
+ORT [12], AoANet [14], M2 Transformer [8], Tree-
+Transformer [43], APN [48], and X-Transformer [31].
+Among the above methods, Up-Down [4] deploys a famous
+architecture with a CNN-based encoder and an LSTM-
+based decoder.
+ORT [12] applies Transformer to lan-
+guage decoder.
+AoANet [14] and M2 Transformer [8]
+further improve the attention mechanism on the language
+decoder. Tree-Transformer [43] and APN [48] reveal the
+validity of the utilization of the sequence structure.
+To
+capture high-order interaction between sequence and re-
+gions, X-Transformer [31] introduces a bilinear pooling
+structure. The second group are the methods using grid-
+based features: CPTR [23], Dual-Global [45], DLCT [25],
+and PureT [42].
+Among them, Dual-Global [45] and
+DLCT [25] combine the grid-based features with the ROI-
+based features.
+PureT [42] end-to-end trains the whole
+model and PureT-standard/PureT-Swin respectively use
+Transformer [9]/Swin Transformer [24] as the vision en-
+coder to deal with the visual features, which is also ex-
+tracted from a Swin Transformer.
+The third group dis-
+tills the knowledge from large-scale pretraining models:
+RSTNet [54], and ViTCAP [10]. Accordingly, we seg-
+ment the performances into 3 parts in Table 4, where the
+top/middle/bottom parts are the ROI-based, grid-based, and
+BERT-based models. Note that for APN, besides reporting
+the results in their paper [48], which is got by using ROI-
+based features, we also report the performances using the
+same visual features as ours, which is denoted as “APN♯”.
+Results.
+From Table 4, we can see that ACF is com-
+parable to most of state-of-the-art performance when
+compared with ROI and grid-based models.
+Moreover,
+
+STOPS
+OPSTOPSTOPTable 4. The performances of SOTA methods on MSCOCO Karpathy split.
+Models
+Cross-Entroy Loss
+CIDEr optimization
+B@4
+M
+R
+C
+S
+B@4
+M
+R
+C
+S
+ROI-based feature
+Up-Down [4]
+36.2
+27.0
+56.4
+113.5
+20.3
+36.3
+27.7
+56.9
+120.1
+21.4
+ORT [12]
+35.5
+28.0
+56.6
+115.4
+21.2
+38.6
+28.7
+58.4
+128.3
+22.6
+AoANet [14]
+37.2
+28.4
+57.5
+119.8
+21.4
+38.9
+29.2
+58.8
+129.8
+22.4
+M2 Transformer [8]
+-
+-
+-
+-
+-
+39.1
+29.2
+58.6
+131.2
+22.6
+CATT [50]
+37.3
+28.5
+57.4
+119.0
+21.5
+39.4
+29.3
+58.9
+131.7
+22.8
+APN [48]
+-
+-
+-
+-
+-
+39.6
+29.2
+59.1
+131.8
+23.0
+X-Transformer [31]
+38.2
+28.8
+58.0
+122.0
+21.9
+39.7
+29.5
+59.2
+132.8
+23.2
+Grid-based feature
+CPTR [23]
+-
+-
+-
+-
+-
+40.0
+29.1
+59.4
+129.4
+−
+APN♯ [48]
+-
+-
+-
+-
+-
+40.1
+29.4
+59.4
+133.2
+23.3
+Dual-Global [45]
+-
+-
+-
+-
+-
+40.3
+29.2
+59.4
+132.4
+23.3
+DLCT [25]
+-
+-
+-
+-
+-
+40.8
+29.9
+59.8
+137.5
+23.3
+End-to-End training
+PureT-standard [42]
+-
+-
+-
+-
+-
+40.3
+29.9
+59.9
+137.5
+23.8
+PureT-Swin [42]
+-
+-
+-
+-
+-
+40.9
+30.2
+60.1
+138.2
+24.2
+Visual-language BERT pretraining
+RSTNet [54]
+-
+-
+-
+-
+-
+40.1
+28.9
+59.5
+135.6
+23.3
+ViTCAP-small [10]
+35.7
+28.8
+57.6
+121.8
+22.1
+40.1
+29.4
+59.4
+133.1
+23.0
+ViTCAP-large [10]
+36.3
+29.3
+58.1
+125.2
+22.6
+41.2
+30.1
+60.1
+138.1
+24.1
+ACF
+38.1
+28.8
+58.4
+123.8
+21.8
+41.1
+30.1
+60.2
+137.8
+24.1
+Table 5. The scores on the MSCOCO online test server.
+Models
+B@4
+M
+R
+C
+c5
+c40
+c5
+c40
+c5
+c40
+c5
+c40
+Up-Down [4]
+36.9
+68.5
+27.6
+36.7
+57.1
+72.4
+117.9
+120.5
+SGAE [49]
+37.8
+68.7
+28.1
+37.0
+58.2
+73.1
+122.7
+125.5
+ETA [19]
+38.9
+70.2
+28.6
+38.0
+58.6
+73.9
+122.1
+124.4
+APN [48]
+38.9
+70.2
+28.8
+38.0
+58.7
+73.7
+126.3
+127.6
+NG-SAN [11]
+38.8
+70.2
+29.0
+38.4
+58.7
+74.0
+126.3
+128.6
+Dual-Global [45]
+39.1
+71.2
+28.9
+38.4
+58.9
+74.4
+126.3
+129.2
+AoANet [14]
+39.4
+71.2
+29.1
+38.5
+58.9
+74.5
+126.9
+129.6
+M2 Transformer [8]
+39.7
+72.8
+29.4
+39.0
+59.2
+74.8
+129.3
+132.1
+RSTNet [54]
+39.7
+72.5
+29.3
+38.7
+59.2
+74.2
+130.1
+132.4
+ACF
+39.0
+71.3
+29.2
+39.2
+59.2
+74.2
+130.2
+132.3
+ACF achieves comparable performances with ViTCAP-
+large [10] that distills knowledge from Google-CC [37],
+SBU Caption dataset [30], MSCOCO [22], and Visual
+Genome dataset [17], which uses 9.9M image-text pairs
+and 4.1M independent images to pretrain a detector-free IC
+model. However, we only use the captions from MSCOCO
+to train our ACF. Moreover, compared with APN♯ [48]
+which inserts an additional clustering matrix into the Self-
+ATT layers into the decoder, ACF achieves higher per-
+formance since it inserts the clustering matrix in both vi-
+sion encoder and language decoder to build a homogeneous
+model.
+Also, we submit the single-model results to the online
+server for testing, which is shown in Table 5. We can see
+that ACF achieves the best performance than the other mod-
+els, even we do not ensemble the results as AoANet [14],
+M2 Transformer [8], and RSTNet [54].
+Limitations and Potential Solutions. From Table 4, we
+can find that PureT-Swin [42] achieves higher CIDEr than
+ours. There are two major reasons cause this. Firstly, PureT-
+Swin extracts visual features from Swin Transformer [24]
+and then still uses Swin Transformer as the visual encoder
+to deal with the extracted features. For ACF, the used vision
+encoder is quite different from Swin Transformer that they
+apply shifted fixed-size windows, while we insert an adap-
+tive clustering matrix into the Transformer. In this way, the
+whole captioning model (including the vision extractor) is
+not a strictly homogeneous structure. Also, it can be seen
+that ACF outperforms PureT-standard which applies a stan-
+dard Transformer as the vision encoder, which means that
+once PureT is not homogeneous, their performance will be
+worse.
+Secondly, they end-to-end train the whole architecture by
+captioning data since Swin Transformer [24] provides well-
+trained parameters that PureT does not need to train their
+visual extractor from scratch. However, this requires heavy
+computation resources to end-to-end train the visual extrac-
+tor by image annotations while we now cannot afford such
+computation burdens. However, even with these two limi-
+tations, it can be found that ACF still achieves comparable
+performances compared with PureT.
+To solve these limitations, we prepare to extend the com-
+putation resources like the GPU servers to build a novel pure
+vision global-local Transformer where ACF prior is used
+to learn hierarchical structure. And then using this model
+to extract visual features for solving more vision-language
+
+tasks, e.g., by building a homogeneous ACF-based vision-
+language model.
+5. Conclusion
+We propose a novel global-local Transformer named as
+Ada-ClustFormer (ACF) that can adaptively cluster the in-
+put elements for carrying self-attention (Self-ATT) to learn
+global-local contexts. Specifically, this is achieved by in-
+serting a clustering matrix into the Self-ATT layer, where
+the probability terms are calculated from the input data and
+thus ACF can adaptively cluster the elements. Moreover,
+we use ACF to build an image captioning model to transfer
+more structural commonalities for better captions. The ex-
+periment results confirm the effectiveness of the proposed
+model.
+References
+[1] Abhaya Agarwal and Alon Lavie. Meteor: An automatic
+metric for mt evaluation with high levels of correlation with
+human judgments. Proceedings of WMT-08, 2007.
+[2] Mahtab Ahmed, Muhammad Rifayat Samee, and Robert E
+Mercer. You only need attention to traverse trees. In Pro-
+ceedings of the 57th Annual Meeting of the Association for
+Computational Linguistics, pages 316–322, 2019.
+[3] Peter Anderson, Basura Fernando, Mark Johnson, and
+Stephen Gould. Spice: Semantic propositional image cap-
+tion evaluation. In European conference on computer vision,
+pages 382–398. Springer, 2016.
+[4] Peter Anderson, Xiaodong He, Chris Buehler, Damien
+Teney, Mark Johnson, Stephen Gould, and Lei Zhang.
+Bottom-up and top-down attention for image captioning and
+visual question answering. In Proceedings of the IEEE con-
+ference on computer vision and pattern recognition, pages
+6077–6086, 2018.
+[5] Peter W Battaglia, Jessica B Hamrick, Victor Bapst, Al-
+varo Sanchez-Gonzalez, Vinicius Zambaldi, Mateusz Ma-
+linowski, Andrea Tacchetti, David Raposo, Adam Santoro,
+Ryan Faulkner, et al. Relational inductive biases, deep learn-
+ing, and graph networks. arXiv preprint arXiv:1806.01261,
+2018.
+[6] Iz Beltagy, Matthew E Peters, and Arman Cohan.
+Long-
+former: The long-document transformer.
+arXiv preprint
+arXiv:2004.05150, 2020.
+[7] Boyu Chen, Peixia Li, Chuming Li, Baopu Li, Lei Bai, Chen
+Lin, Ming Sun, Junjie Yan, and Wanli Ouyang. Glit: Neural
+architecture search for global and local image transformer.
+In Proceedings of the IEEE/CVF International Conference
+on Computer Vision, pages 12–21, 2021.
+[8] Marcella Cornia, Matteo Stefanini, Lorenzo Baraldi, and
+Rita Cucchiara. Meshed-memory transformer for image cap-
+tioning.
+In Proceedings of the IEEE/CVF Conference on
+Computer Vision and Pattern Recognition, pages 10578–
+10587, 2020.
+[9] Alexey Dosovitskiy, Lucas Beyer, Alexander Kolesnikov,
+Dirk Weissenborn, Xiaohua Zhai, Thomas Unterthiner,
+Mostafa Dehghani, Matthias Minderer, Georg Heigold, Syl-
+vain Gelly, Jakob Uszkoreit, and Neil Houlsby. An image is
+worth 16x16 words: Transformers for image recognition at
+scale. ICLR, 2021.
+[10] Zhiyuan Fang, Jianfeng Wang, Xiaowei Hu, Lin Liang, Zhe
+Gan, Lijuan Wang, Yezhou Yang, and Zicheng Liu. Injecting
+semantic concepts into end-to-end image captioning. In Pro-
+ceedings of the IEEE/CVF Conference on Computer Vision
+and Pattern Recognition, pages 18009–18019, 2022.
+[11] Longteng Guo, Jing Liu, Xinxin Zhu, Peng Yao, Shichen
+Lu, and Hanqing Lu. Normalized and geometry-aware self-
+attention network for image captioning. In Proceedings of
+the IEEE/CVF Conference on Computer Vision and Pattern
+Recognition, pages 10327–10336, 2020.
+[12] Simao Herdade, Armin Kappeler, Kofi Boakye, and Joao
+Soares. Image captioning: Transforming objects into words.
+In Advances in Neural Information Processing Systems,
+pages 11137–11147, 2019.
+[13] Xiaowei Hu, Zhe Gan, Jianfeng Wang, Zhengyuan Yang,
+Zicheng Liu, Yumao Lu, and Lijuan Wang.
+Scaling up
+vision-language pre-training for image captioning. In Pro-
+ceedings of the IEEE/CVF Conference on Computer Vision
+and Pattern Recognition, pages 17980–17989, 2022.
+[14] Lun Huang, Wenmin Wang, Jie Chen, and Xiao-Yong Wei.
+Attention on attention for image captioning. In Proceedings
+of the IEEE International Conference on Computer Vision,
+pages 4634–4643, 2019.
+[15] Huaizu Jiang, Ishan Misra, Marcus Rohrbach, Erik Learned-
+Miller, and Xinlei Chen. In defense of grid features for visual
+question answering. In Proceedings of the IEEE/CVF Con-
+ference on Computer Vision and Pattern Recognition, pages
+10267–10276, 2020.
+[16] Andrej Karpathy and Li Fei-Fei. Deep visual-semantic align-
+ments for generating image descriptions. In Proceedings of
+the IEEE conference on computer vision and pattern recog-
+nition, pages 3128–3137, 2015.
+[17] Ranjay Krishna, Yuke Zhu, Oliver Groth, Justin Johnson,
+Kenji Hata, Joshua Kravitz, Stephanie Chen, Yannis Kalan-
+tidis, Li-Jia Li, David A Shamma, et al.
+Visual genome:
+Connecting language and vision using crowdsourced dense
+image annotations. International Journal of Computer Vi-
+sion, 123(1):32–73, 2017.
+[18] Hwanhee Lee,
+Seunghyun Yoon,
+Franck Dernoncourt,
+Doo Soon Kim, Trung Bui, and Kyomin Jung. Vilbertscore:
+Evaluating image caption using vision-and-language bert. In
+Proceedings of the First Workshop on Evaluation and Com-
+parison of NLP Systems, pages 34–39, 2020.
+[19] Guang Li, Linchao Zhu, Ping Liu, and Yi Yang.
+Entan-
+gled transformer for image captioning. In Proceedings of
+the IEEE/CVF International Conference on Computer Vision
+(ICCV), October 2019.
+[20] Jinpeng Li, Yichao Yan, Shengcai Liao, Xiaokang Yang, and
+Ling Shao.
+Local-to-global self-attention in vision trans-
+formers. arXiv preprint arXiv:2107.04735, 2021.
+[21] Xiujun Li, Xi Yin, Chunyuan Li, Pengchuan Zhang, Xiaowei
+Hu, Lei Zhang, Lijuan Wang, Houdong Hu, Li Dong, Furu
+Wei, et al. Oscar: Object-semantics aligned pre-training for
+
+vision-language tasks. In European Conference on Computer
+Vision, pages 121–137. Springer, 2020.
+[22] Tsung-Yi Lin, Michael Maire, Serge Belongie, James Hays,
+Pietro Perona, Deva Ramanan, Piotr Doll´ar, and C Lawrence
+Zitnick. Microsoft coco: Common objects in context. In
+European conference on computer vision, pages 740–755.
+Springer, 2014.
+[23] Wei Liu, Sihan Chen, Longteng Guo, Xinxin Zhu, and Jing
+Liu. Cptr: Full transformer network for image captioning.
+arXiv preprint arXiv:2101.10804, 2021.
+[24] Ze Liu, Yutong Lin, Yue Cao, Han Hu, Yixuan Wei, Zheng
+Zhang, Stephen Lin, and Baining Guo. Swin transformer:
+Hierarchical vision transformer using shifted windows. In
+Proceedings of the IEEE/CVF International Conference on
+Computer Vision, pages 10012–10022, 2021.
+[25] Yunpeng Luo, Jiayi Ji, Xiaoshuai Sun, Liujuan Cao,
+Yongjian Wu, Feiyue Huang, Chia-Wen Lin, and Rongrong
+Ji. Dual-level collaborative transformer for image caption-
+ing.
+In Proceedings of the AAAI Conference on Artificial
+Intelligence, volume 35, pages 2286–2293, 2021.
+[26] Minh-Thang Luong, Hieu Pham, and Christopher D Man-
+ning. Effective approaches to attention-based neural machine
+translation. arXiv preprint arXiv:1508.04025, 2015.
+[27] Ron Mokady, Amir Hertz, and Amit H Bermano.
+Clip-
+cap:
+Clip prefix for image captioning.
+arXiv preprint
+arXiv:2111.09734, 2021.
+[28] Van-Quang Nguyen, Masanori Suganuma, and Takayuki
+Okatani.
+Grit:
+Faster and better image captioning
+transformer using dual visual features.
+arXiv preprint
+arXiv:2207.09666, 2022.
+[29] Xuan-Phi Nguyen, Shafiq Joty, Steven Hoi, and Richard
+Socher. Tree-structured attention with hierarchical accumu-
+lation. In International Conference on Learning Representa-
+tions, 2020.
+[30] Vicente Ordonez,
+Girish Kulkarni,
+and Tamara Berg.
+Im2text: Describing images using 1 million captioned pho-
+tographs. Advances in neural information processing sys-
+tems, 24, 2011.
+[31] Yingwei Pan, Ting Yao, Yehao Li, and Tao Mei. X-linear
+attention networks for image captioning. In CVPR, pages
+10971–10980, 2020.
+[32] Kishore Papineni, Salim Roukos, Todd Ward, and Wei-Jing
+Zhu. Bleu: a method for automatic evaluation of machine
+translation. In Proceedings of the 40th annual meeting of the
+Association for Computational Linguistics, pages 311–318,
+2002.
+[33] Samrudhdhi B Rangrej, Kevin J Liang, Tal Hassner, and
+James J Clark. Glitr: Glimpse transformers with spatiotem-
+poral consistency for online action prediction. arXiv preprint
+arXiv:2210.13605, 2022.
+[34] S Ren, K He, R Girshick, and J Sun. Towards real-time ob-
+ject detection with region proposal networks. Advances in
+neural information processing systems, 2015.
+[35] Steven J Rennie, Etienne Marcheret, Youssef Mroueh, Jerret
+Ross, and Vaibhava Goel. Self-critical sequence training for
+image captioning. In Proceedings of the IEEE conference on
+computer vision and pattern recognition, pages 7008–7024,
+2017.
+[36] Lin CY ROUGE.
+A package for automatic evaluation of
+summaries.
+In Proceedings of Workshop on Text Summa-
+rization of ACL, Spain, 2004.
+[37] Piyush Sharma, Nan Ding, Sebastian Goodman, and Radu
+Soricut. Conceptual captions: A cleaned, hypernymed, im-
+age alt-text dataset for automatic image captioning. In Pro-
+ceedings of the 56th Annual Meeting of the Association for
+Computational Linguistics (Volume 1: Long Papers), pages
+2556–2565, 2018.
+[38] Ying Hua Tan and Chee Seng Chan. Phrase-based image
+caption generator with hierarchical lstm network. Neurocom-
+puting, 333:86–100, 2019.
+[39] Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszko-
+reit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, and Illia
+Polosukhin. Attention is all you need. Advances in neural
+information processing systems, 30, 2017.
+[40] Ramakrishna Vedantam, C Lawrence Zitnick, and Devi
+Parikh. Cider: Consensus-based image description evalua-
+tion. In Proceedings of the IEEE conference on computer
+vision and pattern recognition, pages 4566–4575, 2015.
+[41] Oriol Vinyals, Alexander Toshev, Samy Bengio, and Du-
+mitru Erhan. Show and tell: A neural image caption gen-
+erator. In Proceedings of the IEEE conference on computer
+vision and pattern recognition, pages 3156–3164, 2015.
+[42] Yiyu Wang, Jungang Xu, and Yingfei Sun. End-to-end trans-
+former based model for image captioning. In Proceedings of
+the AAAI Conference on Artificial Intelligence, pages 2585–
+2594, Jun. 2022.
+[43] Yau-Shian Wang, Hung-Yi Lee, and Yun-Nung Chen. Tree
+transformer: Integrating tree structures into self-attention.
+arXiv preprint arXiv:1909.06639, 2019.
+[44] Chuhan Wu, Fangzhao Wu, Tao Qi, and Yongfeng Huang.
+Hi-transformer: hierarchical interactive transformer for effi-
+cient and effective long document modeling. arXiv preprint
+arXiv:2106.01040, 2021.
+[45] Tiantao Xian, Zhixin Li, Canlong Zhang, and Huifang Ma.
+Dual global enhanced transformer for image captioning.
+Neural Networks, 148:129–141, 2022.
+[46] Kelvin Xu, Jimmy Ba, Ryan Kiros, Kyunghyun Cho, Aaron
+Courville, Ruslan Salakhudinov, Rich Zemel, and Yoshua
+Bengio. Show, attend and tell: Neural image caption gen-
+eration with visual attention. In International conference on
+machine learning, pages 2048–2057. PMLR, 2015.
+[47] Jianwei Yang, Chunyuan Li, Pengchuan Zhang, Xiyang Dai,
+Bin Xiao, Lu Yuan, and Jianfeng Gao. Focal attention for
+long-range interactions in vision transformers. Advances in
+Neural Information Processing Systems, 34:30008–30022,
+2021.
+[48] Xu Yang, Chongyang Gao, Hanwang Zhang, and Jianfei Cai.
+Auto-parsing network for image captioning and visual ques-
+tion answering. In Proceedings of the IEEE/CVF Interna-
+tional Conference on Computer Vision, pages 2197–2207,
+2021.
+[49] Xu Yang, Kaihua Tang, Hanwang Zhang, and Jianfei Cai.
+Auto-encoding scene graphs for image captioning. In Pro-
+ceedings of the IEEE/CVF Conference on Computer Vision
+and Pattern Recognition, pages 10685–10694, 2019.
+
+[50] Xu Yang, Hanwang Zhang, Guojun Qi, and Jianfei Cai.
+Causal attention for vision-language tasks. In Proceedings
+of the IEEE/CVF Conference on Computer Vision and Pat-
+tern Recognition, pages 9847–9857, 2021.
+[51] Ting Yao, Yingwei Pan, Yehao Li, and Tao Mei.
+Hierar-
+chy parsing for image captioning.
+In Proceedings of the
+IEEE/CVF International Conference on Computer Vision,
+pages 2621–2629, 2019.
+[52] Pengchuan Zhang, Xiujun Li, Xiaowei Hu, Jianwei Yang,
+Lei Zhang, Lijuan Wang, Yejin Choi, and Jianfeng Gao.
+Vinvl: Revisiting visual representations in vision-language
+models.
+In Proceedings of the IEEE/CVF Conference on
+Computer Vision and Pattern Recognition, pages 5579–
+5588, 2021.
+[53] Hengshuang Zhao, Jiaya Jia, and Vladlen Koltun. Explor-
+ing self-attention for image recognition. In Proceedings of
+the IEEE/CVF Conference on Computer Vision and Pattern
+Recognition, pages 10076–10085, 2020.
+[54] Luowei Zhou, Hamid Palangi, Lei Zhang, Houdong Hu, Ja-
+son Corso, and Jianfeng Gao. Unified vision-language pre-
+training for image captioning and vqa. In Proceedings of
+the AAAI Conference on Artificial Intelligence, volume 34,
+pages 13041–13049, 2020.
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