pith. sign in

arxiv: 2604.08043 · v1 · submitted 2026-04-09 · ❄️ cond-mat.mes-hall · cond-mat.mtrl-sci· cond-mat.str-el

Optical Hall absorption sum rule and spectral compensation in time-reversal-breaking moir\'e and Hofstadter systems

Yixin Zhang , H. Huang This is my paper

Pith reviewed 2026-05-10 18:14 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall cond-mat.mtrl-scicond-mat.str-el
keywords optical Hall conductivitysum rulemoiré materialsHofstadter modeltopological bandscircular dichroismspectral compensationanomalous Hall absorption
0
0 comments X p. Extension

The pith

The first-frequency moment of the antisymmetric optical conductivity vanishes exactly in zero-field moiré models with topological bands and equals a value fixed by flux density in Hofstadter models.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper derives a sum rule constraining the first frequency moment of the antisymmetric optical conductivity, whose imaginary part sets the strength of chirality-dependent absorption. In zero-field moiré continuum models that host topological bands this moment is exactly zero, so any anomalous Hall absorption at low frequencies must be balanced by spectral weight of the opposite sign at higher frequencies. In Hofstadter models placed in a uniform magnetic field the same moment instead takes a universal value set solely by the flux density per plaquette, independent of other microscopic details. The resulting link between low- and high-frequency parts of the spectrum supplies a concrete way to test models of circular dichroism and to check for Landau-level mixing in these time-reversal-breaking systems.

Core claim

We formulate the corresponding first-frequency-moment constraint for the antisymmetric optical conductivity, whose imaginary part governs chirality-dependent absorption. For a zero-field moiré continuum model hosting topological bands, the moment vanishes exactly, implying that any low-frequency anomalous Hall absorption must be compensated by higher-frequency spectral weight of the opposite sign. For a Hofstadter model under a uniform magnetic field, the same moment takes a universal value fixed by the magnetic flux density, independent of microscopic model details. By linking low- and high-frequency spectral contributions, this optical Hall absorption sum rule provides a rigorous framework

What carries the argument

The first-frequency-moment sum rule for the antisymmetric optical conductivity, derived via linear response, that forces exact cancellation in zero-field moiré models or a flux-determined value in Hofstadter models.

If this is right

  • Any low-frequency anomalous Hall absorption in zero-field moiré systems must be exactly compensated by higher-frequency absorption of the opposite sign.
  • In Hofstadter systems the moment supplies a direct, model-independent measure of the magnetic flux density.
  • The sum rule connects low- and high-frequency parts of the Hall absorption spectrum.
  • Deviations from the predicted moment can be used to diagnose Landau-level mixing.
  • The constraint quantifies limits on circular dichroism in these topological systems.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Spectroscopy experiments on moiré heterostructures could integrate the Hall absorption spectrum to check whether the moment vanishes as predicted.
  • The compensation requirement may appear in other zero-field topological band systems that share similar continuum descriptions.
  • Analogous moment constraints could be worked out for additional response functions in time-reversal-broken materials.
  • The rule offers a benchmark for interpreting data from circularly polarized optical probes in related quantum materials.

Load-bearing premise

The sum rule assumes the antisymmetric optical conductivity is well-defined by standard linear response and that the chosen moiré continuum and Hofstadter models contain all relevant physics without extra interactions that would change the moment.

What would settle it

A measurement of the frequency-integrated first moment of the antisymmetric conductivity in a zero-field moiré sample that yields a nonzero value, or a value in a Hofstadter sample that deviates from the flux-density prediction, would falsify the sum rule.

Figures

Figures reproduced from arXiv: 2604.08043 by H. Huang, Yixin Zhang.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) Continuum model description: electrons move in [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. (a) Non-interacting band structure of twisted MoTe [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. (a) Hofstadter energy spectrum versus magnetic flux [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
read the original abstract

Optical spectroscopy provides a powerful, contact-free probe of topological quantum states, yet exact constraints on antisymmetric Hall absorption remain much less well developed than their longitudinal counterparts. Motivated by earlier Hall-conductivity sum rules, we formulate the corresponding first-frequency-moment constraint for the antisymmetric optical conductivity, whose imaginary part governs chirality-dependent absorption. We then demonstrate this sum rule in two classes of time-reversal-breaking topological systems. For a zero-field moir\'e continuum model hosting topological bands, the moment vanishes exactly, implying that any low-frequency anomalous Hall absorption must be compensated by higher-frequency spectral weight of the opposite sign. For a Hofstadter model under a uniform magnetic field, the same moment takes a universal value fixed by the magnetic flux density, independent of microscopic model details. By linking low- and high-frequency spectral contributions, this optical Hall absorption sum rule provides a rigorous framework for quantifying circular dichroism constraints and diagnosing Landau-level mixing. Our results show how a known Hall spectral constraint acquires new and experimentally relevant content in modern interacting topological materials.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper formulates a first-frequency-moment sum rule for the antisymmetric optical conductivity (whose imaginary part controls circular dichroism) in time-reversal-breaking systems. It proves that the moment vanishes exactly in zero-field moiré continuum models with topological bands, implying exact compensation of any low-frequency anomalous Hall absorption by opposite-sign weight at higher frequencies. In Hofstadter models the same moment equals a universal value set solely by the magnetic flux density, independent of microscopic details. The results are presented as exact identities within the chosen models and are motivated by earlier Hall sum rules.

Significance. If the derivations hold, the sum rule supplies a parameter-free, model-independent constraint that directly links low- and high-frequency spectral weight in two experimentally relevant classes of topological systems. The exact vanishing in the moiré case and the flux-only dependence in the Hofstadter case are strengths that could be used to benchmark circular-dichroism data and to diagnose Landau-level mixing without additional fitting parameters.

major comments (2)
  1. [§3] §3 (Moiré continuum model derivation): The exact vanishing of the first moment is obtained by integrating the antisymmetric conductivity over the entire frequency axis inside the continuum Hilbert space. Because the model is a low-energy effective theory whose high-energy states are unphysical continuum approximations, the mathematical identity does not automatically guarantee the physical spectral compensation asserted in the abstract and conclusion; the paper must explicitly discuss the cutoff dependence or embedding into a lattice to support the experimental interpretation.
  2. [§4] §4 (Hofstadter model): The claim that the moment equals a universal value fixed only by flux density is load-bearing for the 'independent of microscopic details' statement. The derivation should be checked against an explicit lattice regularization or against a second, independent Hofstadter Hamiltonian to confirm that no hidden model-specific terms survive the frequency integration.
minor comments (2)
  1. [Abstract] The abstract and introduction should cite the specific earlier Hall sum rules that motivate the present work (e.g., the longitudinal or symmetric-conductivity moments) so that the novelty of the antisymmetric extension is immediately clear.
  2. [§2] Notation for the antisymmetric conductivity (real/imaginary parts, frequency conventions) is introduced without a dedicated equation; adding an explicit definition early in §2 would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major point below and will revise the manuscript to incorporate the suggested clarifications, which will improve the discussion of the models' physical applicability.

read point-by-point responses

  1. Referee: §3 (Moiré continuum model derivation): The exact vanishing of the first moment is obtained by integrating the antisymmetric conductivity over the entire frequency axis inside the continuum Hilbert space. Because the model is a low-energy effective theory whose high-energy states are unphysical continuum approximations, the mathematical identity does not automatically guarantee the physical spectral compensation asserted in the abstract and conclusion; the paper must explicitly discuss the cutoff dependence or embedding into a lattice to support the experimental interpretation.

    Authors: We agree that the moiré continuum model is a low-energy effective theory and that the exact vanishing holds within its Hilbert space. The physical interpretation of spectral compensation therefore requires care regarding the model's range of validity. In the revised manuscript we will add an explicit discussion in §3 addressing cutoff dependence. We will note that the sum rule enforces compensation between low-frequency anomalous Hall absorption and higher-frequency weight up to the cutoff set by the continuum approximation (typically the moiré bandwidth). We will further explain that embedding the model into a full lattice regularization replaces the unphysical high-energy continuum states with lattice-scale physics, but the low-energy compensation mechanism for the topological bands remains a robust, testable prediction. This addition will support the experimental claims while clarifying the scope of the effective theory. revision: yes

  2. Referee: §4 (Hofstadter model): The claim that the moment equals a universal value fixed only by flux density is load-bearing for the 'independent of microscopic details' statement. The derivation should be checked against an explicit lattice regularization or against a second, independent Hofstadter Hamiltonian to confirm that no hidden model-specific terms survive the frequency integration.

    Authors: The derivation in §4 follows from the general structure of the Hofstadter Hamiltonian on a lattice with uniform flux, where the frequency-integrated antisymmetric conductivity reduces to a trace over the magnetic unit cell that depends only on the flux density. To explicitly verify the claimed independence from microscopic details, we will add a verification in the revised §4. We will consider an alternative Hofstadter Hamiltonian that includes next-nearest-neighbor hopping and demonstrate that the first moment remains unchanged, still fixed solely by the flux per plaquette. This explicit check will confirm that no model-specific terms survive the integration and will strengthen the universality statement. revision: yes

Circularity Check

0 steps flagged

No significant circularity; sum rule follows from linear response

full rationale

The derivation starts from the standard Kubo formula for the antisymmetric conductivity and integrates its first frequency moment over the model's Hilbert space. In the zero-field moiré case the integral vanishes by direct cancellation within the continuum Hamiltonian; in the Hofstadter case it equals the enclosed flux by lattice periodicity. Neither result is presupposed in the input definitions, nor obtained by fitting or by renaming a known empirical pattern. Earlier sum-rule literature is cited only for motivation; the central identities are recomputed explicitly inside the present models and do not reduce to those citations by construction. The continuum cutoff issue raised by the skeptic concerns physical applicability, not internal circularity of the mathematical steps.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard linear-response definitions of optical conductivity and the applicability of moment sum rules to the antisymmetric channel in the two model classes; no free parameters or new entities are introduced in the abstract.

axioms (1)
  • standard math The antisymmetric optical conductivity is well-defined and its frequency moments are constrained by general response-theory relations analogous to earlier Hall-conductivity sum rules.
    Explicitly motivated by earlier Hall-conductivity sum rules in the abstract.

pith-pipeline@v0.9.0 · 5497 in / 1223 out tokens · 57310 ms · 2026-05-10T18:14:41.970978+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

55 extracted references · 55 canonical work pages · 1 internal anchor

  1. [1]

    B. T. Thole, P. Carra, F. Sette, and G. van der Laan, X-ray circular dichroism as a probe of orbital magnetiza- tion, Physical Review Letters68, 1943 (1992)

    work page 1943

  2. [2]

    L. Wu, M. Salehi, N. Koirala, J. Moon, S. Oh, and N. P. Armitage, Quantized Faraday and Kerr rotation and ax- ion electrodynamics of a 3D topological insulator, Science 354, 1124 (2016)

    work page 2016

  3. [3]

    Dziom, A

    V. Dziom, A. Shuvaev, A. Pimenov, G. V. As- takhov, C. Ames, K. Bendias, J. B¨ ottcher, G. Tkachov, E. M. Hankiewicz, C. Br¨ une, H. Buhmann, and L. W. Molenkamp, Observation of the universal magnetoelec- tric effect in a 3D topological insulator, Nature Commu- nications8, 15197 (2017)

    work page 2017

  4. [4]

    Molignini, B

    P. Molignini, B. Lapierre, R. Chitra, and W. Chen, Prob- ing Chern number by opacity and topological phase tran- sition by a nonlocal Chern marker, SciPost Physics Core 6, 059 (2023)

    work page 2023

  5. [5]

    J. Cai, E. Anderson, C. Wang, X. Zhang, X. Liu, W. Holtzmann, Y. Zhang, F. Fan, T. Taniguchi, K. Watanabe, Y. Ran, T. Cao, L. Fu, D. Xiao, W. Yao, and X. Xu, Signatures of fractional quantum anomalous Hall states in twisted MoTe2, Nature622, 63 (2023)

    work page 2023

  6. [6]

    Z. Tao, B. Shen, S. Jiang, T. Li, L. Li, L. Ma, W. Zhao, J. Hu, K. Pistunova, K. Watanabe, T. Taniguchi, T. F. Heinz, K. F. Mak, and J. Shan, Valley-coherent quantum anomalous hall state in ab-stacked mote 2/wse2 bilayers, Phys. Rev. X14, 011004 (2024)

    work page 2024

  7. [7]

    S.-K. Bac, F. Le Mardel´ e, J. Wang, M. Ozerov, K. Yoshimura, I. Mohelsk´ y, X. Sun, B. A. Piot, S. Wim- mer, A. Ney, T. Orlova, M. Zhukovskyi, G. Bauer, G. Springholz, X. Liu, M. Orlita, K. Park, Y.-T. Hsu, and B. A. Assaf, Probing Berry Curvature in Magnetic Topological Insulators through Resonant Infrared Mag- netic Circular Dichroism, Physical Review ...

    work page 2025

  8. [8]

    K. v. Klitzing, G. Dorda, and M. Pepper, New Method for High-Accuracy Determination of the Fine-Structure Constant Based on Quantized Hall Resistance, Physical Review Letters45, 494 (1980)

    work page 1980

  9. [9]

    Jungwirth, Q

    T. Jungwirth, Q. Niu, and A. H. MacDonald, Anomalous Hall Effect in Ferromagnetic Semiconductors, Physical Review Letters88, 207208 (2002)

    work page 2002

  10. [10]

    Onoda and N

    M. Onoda and N. Nagaosa, Topological Nature of Anomalous Hall Effect in Ferromagnets, Journal of the Physical Society of Japan71, 19 (2002)

    work page 2002

  11. [11]

    F. D. M. Haldane, Berry Curvature on the Fermi Surface: Anomalous Hall Effect as a Topological Fermi-Liquid Property, Physical Review Letters93, 206602 (2004)

    work page 2004

  12. [12]

    Souza and D

    I. Souza and D. Vanderbilt, Dichroic$f$-sum rule and the orbital magnetization of crystals, Physical Review B 77, 054438 (2008)

    work page 2008

  13. [13]

    K. N. Okada, Y. Takahashi, M. Mogi, R. Yoshimi, A. Tsukazaki, K. S. Takahashi, N. Ogawa, M. Kawasaki, and Y. Tokura, Terahertz spectroscopy on Faraday and Kerr rotations in a quantum anomalous Hall state, Na- ture Communications7, 12245 (2016)

    work page 2016

  14. [14]

    Onishi and L

    Y. Onishi and L. Fu, Fundamental bound on topological gap (2023), arXiv:2306.00078 [cond-mat]. 6

    work page arXiv 2023

  15. [15]

    Ghosh, Y

    B. Ghosh, Y. Onishi, S.-Y. Xu, H. Lin, L. Fu, and A. Bansil, Probing quantum geometry through opti- cal conductivity and magnetic circular dichroism (2024), arXiv:2401.09689 [cond-mat]

    work page arXiv 2024

  16. [16]

    Verma and R

    N. Verma and R. Queiroz, Instantaneous Response and Quantum Geometry of Insulators (2024), arXiv:2403.07052

    work page arXiv 2024

  17. [17]

    J. Yu, J. Herzog-Arbeitman, and B. A. Bernevig, Uni- versal Wilson Loop Bound of Quantum Geometry, Physi- cal Review Letters135, 086401 (2025), arXiv:2501.00100 [cond-mat]

    work page arXiv 2025

  18. [18]

    Uchida, T

    S. Uchida, T. Ido, H. Takagi, T. Arima, Y. Tokura, and S. Tajima, Optical spectra of la2−xsrxcuo4: Effect of car- rier doping on the electronic structure of the cuo 2 plane, Physical Review B43, 7942 (1991)

    work page 1991

  19. [19]

    Katsufuji, Y

    T. Katsufuji, Y. Okimoto, and Y. Tokura, Spectral Weight Transfer of the Optical Conductivity in Doped Mott Insulators, Physical Review Letters75, 3497 (1995)

    work page 1995

  20. [20]

    M. J. Rozenberg, G. Kotliar, and H. Kajueter, Transfer of spectral weight in spectroscopies of correlated electron systems, Physical Review B54, 8452 (1996)

    work page 1996

  21. [21]

    Reiche and W

    F. Reiche and W. Thomas, ¨Uber die Zahl der Dispersion- selektronen, die einem station¨ aren Zustand zugeordnet sind, Zeitschrift f¨ ur Physik34, 510 (1925)

    work page 1925

  22. [22]

    Giuliani and G

    G. Giuliani and G. Vignale,Quantum Theory of the Elec- tron Liquid(Cambridge University Press, Cambridge, 2005)

    work page 2005

  23. [23]

    Onishi and L

    Y. Onishi and L. Fu, Quantum weight (2024), arXiv:2401.13847 [cond-mat]

    work page arXiv 2024

  24. [24]

    Onishi and L

    Y. Onishi and L. Fu, Topological bound on structure fac- tor (2024), arXiv:2406.18654

    work page arXiv 2024

  25. [25]

    Onishi and L

    Y. Onishi and L. Fu, Universal relation between en- ergy gap and dielectric constant (2024), arXiv:2401.04180 [cond-mat]

    work page arXiv 2024

  26. [26]

    Qiu and F

    W.-X. Qiu and F. Wu, Quantum Geometry Probed by Chiral Excitonic Optical Response of Chern Insulators (2024), arXiv:2407.03317

    work page arXiv 2024

  27. [27]

    Souza, R

    I. Souza, R. M. Martin, and M. Stengel, Optical bounds on many-electron localization, SciPost Physics18, 127 (2025), arXiv:2407.17908 [cond-mat]

    work page arXiv 2025

  28. [28]

    Electronic bounds in magnetic crystals

    D. Passos and I. Souza, Electronic bounds in magnetic crystals (2025), arXiv:2509.16121 [cond-mat]

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  29. [29]

    H. D. Drew and P. Coleman, Sum Rule for the Optical Hall Angle, Physical Review Letters78, 1572 (1997)

    work page 1997

  30. [30]

    Lange, Memory-function approach to the Hall con- stant in strongly correlated electron systems, Physical Review B55, 3907 (1997)

    E. Lange, Memory-function approach to the Hall con- stant in strongly correlated electron systems, Physical Review B55, 3907 (1997)

    work page 1997

  31. [31]

    Lange and G

    E. Lange and G. Kotliar, Magneto-optical Sum Rules Close to the Mott Transition, Physical Review Letters 82, 1317 (1999)

    work page 1999

  32. [32]

    C. Xu, N. Mao, T. Zeng, and Y. Zhang, Multiple Chern bands in twisted MoTe$ 2$and possible non-Abelian states (2024), arXiv:2403.17003 [cond-mat]

    work page arXiv 2024

  33. [33]

    K. Kang, B. Shen, Y. Qiu, Y. Zeng, Z. Xia, K. Watan- abe, T. Taniguchi, J. Shan, and K. F. Mak, Evidence of the fractional quantum spin Hall effect in moir´ e MoTe2, Nature , 1 (2024)

    work page 2024

  34. [34]

    [14, 26, 35, 36, 50, 51]

    See Supplemental Material at http://link.aps.org/supplemental/xxx, for more de- tails about optical response calculations in moir´ e continuum model and Hofstadter model, the correlation between optical weight and band mixing, and the con- nection to spontaneous Chern insulator, which includes Refs. [14, 26, 35, 36, 50, 51]

    work page

  35. [35]

    F. Wu, F. Qu, and A. H. MacDonald, Exciton band struc- ture of monolayer${\mathrm{MoS}} {2}$, Physical Re- view B91, 075310 (2015)

    work page 2015

  36. [36]

    E. M. Spanton, A. A. Zibrov, H. Zhou, T. Taniguchi, K. Watanabe, M. P. Zaletel, and A. F. Young, Obser- vation of fractional Chern insulators in a van der Waals heterostructure, Science360, 62 (2018)

    work page 2018

  37. [37]

    Kohn, Cyclotron Resonance and de Haas-van Alphen Oscillations of an Interacting Electron Gas, Physical Re- view123, 1242 (1961)

    W. Kohn, Cyclotron Resonance and de Haas-van Alphen Oscillations of an Interacting Electron Gas, Physical Re- view123, 1242 (1961)

    work page 1961

  38. [38]

    The same conclusion applies to lattice models whose ki- netic terms preserve unit-cell translation symmetry and do not contain density-dependent correlated hopping

    work page

  39. [39]

    F. Wu, T. Lovorn, E. Tutuc, I. Martin, and A. H. MacDonald, Topological Insulators in Twisted Transi- tion Metal Dichalcogenide Homobilayers, Physical Re- view Letters122, 086402 (2019)

    work page 2019

  40. [40]

    Martin and C

    I. Martin and C. D. Batista, Itinerant Electron-Driven Chiral Magnetic Ordering and Spontaneous Quantum Hall Effect in Triangular Lattice Models, Physical Re- view Letters101, 156402 (2008)

    work page 2008

  41. [41]

    H. D. Rosales, F. A. G. Albarrac´ ın, and P. Pujol, From frustrated magnetism to spontaneous Chern insulators, Physical Review B99, 035163 (2019)

    work page 2019

  42. [42]

    Raghu, X.-L

    S. Raghu, X.-L. Qi, C. Honerkamp, and S.-C. Zhang, Topological Mott Insulators, Physical Review Letters 100, 156401 (2008)

    work page 2008

  43. [43]

    Ren, H.-C

    Y. Ren, H.-C. Jiang, Z. Qiao, and D. N. Sheng, Or- bital Chern Insulator and Quantum Phase Diagram of a Kagome Electron System with Half-Filled Flat Bands, Physical Review Letters126, 117602 (2021)

    work page 2021

  44. [44]

    F. Wu, T. Lovorn, E. Tutuc, and A. H. MacDonald, Hub- bard Model Physics in Transition Metal Dichalcogenide Moir\’e Bands, Physical Review Letters121, 026402 (2018)

    work page 2018

  45. [45]

    Ciorciaro, T

    L. Ciorciaro, T. Smole´ nski, I. Morera, N. Kiper, S. Hies- tand, M. Kroner, Y. Zhang, K. Watanabe, T. Taniguchi, E. Demler, and A. ˙Imamo˘ glu, Kinetic magnetism in tri- angular moir´ e materials, Nature623, 509 (2023)

    work page 2023

  46. [46]

    C. R. Kometter, J. Yu, T. Devakul, A. P. Reddy, Y. Zhang, B. A. Foutty, K. Watanabe, T. Taniguchi, L. Fu, and B. E. Feldman, Hofstadter states and re- entrant charge order in a semiconductor moir´ e lattice, Nature Physics19, 1861 (2023)

    work page 2023

  47. [47]

    Singha, M

    A. Singha, M. Gibertini, B. Karmakar, S. Yuan, M. Polini, G. Vignale, M. I. Katsnelson, A. Pinczuk, L. N. Pfeiffer, K. W. West, and V. Pellegrini, Two-Dimensional Mott-Hubbard Electrons in an Artificial Honeycomb Lat- tice, Science332, 1176 (2011)

    work page 2011

  48. [48]

    Levin, B

    M. Levin, B. I. Halperin, and B. Rosenow, Particle-Hole Symmetry and the Pfaffian State, Physical Review Let- ters99, 236806 (2007)

    work page 2007

  49. [49]

    J. Yu, J. Herzog-Arbeitman, Y. H. Kwan, N. Regnault, and B. A. Bernevig, Moir\’e fractional Chern insulators. IV. Fluctuation-driven collapse in multiband exact diago- nalization calculations on rhombohedral graphene, Phys- ical Review B112, 075110 (2025)

    work page 2025

  50. [50]

    F. Xu, X. Chang, J. Xiao, Y. Zhang, F. Liu, Z. Sun, N. Mao, N. Peshcherenko, J. Li, K. Watanabe, T. Taniguchi, B. Tong, L. Lu, J. Jia, D. Qian, Z. Shi, Y. Zhang, X. Liu, S. Jiang, and T. Li, Interplay between topology and correlations in the second moir´ e band of 7 twisted bilayer MoTe2, Nature Physics21, 542 (2025)

    work page 2025

  51. [51]

    Wang and M

    T. Wang and M. P. Zaletel, Chiral superconductivity near a fractional Chern insulator (2025), arXiv:2507.07921 [cond-mat]. 8 SUPPLEMENTARY MATERIAL FOR “ OPTICAL HALL ABSORPTION SUM RULE AND SPECTRAL COMPENSATION IN TIME-REVERSAL-BREAKING MOIR ´E AND HOFSTADTER SYSTEMS” Appendix A: Optical response calculation in continuum models In the non-interacting li...

    work page arXiv 2025

  52. [52]

    Working in the Landau gaugeA= (0,−Bx,0), we span the unper- turbed Hilbert space using the Landau-level (LL) basis |nLL, ky⟩, wheren LL denotes the LL index

    and that the magnetic field points along the nega- tive ˆzdirection,B=−BˆzwithB >0. Working in the Landau gaugeA= (0,−Bx,0), we span the unper- turbed Hilbert space using the Landau-level (LL) basis |nLL, ky⟩, wheren LL denotes the LL index. To discretize the Hilbert space, we wrap theydirection into a cylin- der of circumferenceL y =N ya, whereN y is a l...

    work page

  53. [53]

    To prove this, we utilize the cyclic property of the trace to rewrite the term as Tr( ˆJ+P (d) ˜0 ˆJ−P (od) ˜1 )

    Exact V anishing of Mixed T erms:The mixed terms, such as Tr(P (d) ˜0 ˆJ−P (od) ˜1 ˆJ+), vanish identically at all orders of the periodic potential. To prove this, we utilize the cyclic property of the trace to rewrite the term as Tr( ˆJ+P (d) ˜0 ˆJ−P (od) ˜1 ). Because the diagonal pro- jectorP (d) ˜0 is defined purely by states|n LL⟩⟨nLL|, and the curre...

    work page

  54. [54]

    Exact Lowest-Order Diagonal Contribu- tion:The diagonal-diagonal trace captures the clas- sical transfer of spectral weight without quantum in- terference. Using ˆJ+|nLL⟩= √nLL + 1|nLL + 1⟩and ˆJ−|nLL⟩= √nLL|nLL −1⟩, it evaluates exactly to: W (dd) = ∞X nLL=0 νnLL [(n+ 1)w n+1 −nw n−1] (S3) To evaluate this to the lowest non-trivial order (O(V 2)), we not...

    work page

  55. [55]

    Because bothP (od) ˜0 andP (od) ˜1 are first- order in the periodic potential, this interference term is rigorouslyO(V 2)

    The Role of Quantum Interference:The re- maining contribution arises from the pure off-diagonal traceW (od−od). Because bothP (od) ˜0 andP (od) ˜1 are first- order in the periodic potential, this interference term is rigorouslyO(V 2). Crucially, this means the quantum interference acts at the exact same perturbative order as the classical change in purity...

    work page