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arxiv: 2406.09797 · v3 · submitted 2024-06-14 · ❄️ cond-mat.mes-hall · hep-lat

Loop unitary and phase band topological invariant in generic multi-band Chern insulators

Xi Wu , Ze Yang , Fuxiang Li This is my paper

Pith reviewed 2026-05-24 00:01 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall hep-lat
keywords dynamical topological invariantChern insulatorquench dynamicswinding numberphase bandmulti-band systemtopological phase
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The pith

The dynamical 3-winding number for quenches in multi-band Chern insulators equals the difference in Chern numbers before and after the quench.

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

This paper generalizes the dynamical 3-winding number from two-band systems to generic multi-band Chern insulators. It proves that the value of this number equals the difference between the Chern numbers of the post-quench and pre-quench Hamiltonians. An expression is derived that represents the invariant through gapless fermions in phase bands, depending only on the phase and its projectors, which makes the result applicable to all multi-band Chern insulators. The work also shows that quenching a three-band model produces a multifold fermion in the phase band in (k, t) space, a feature absent in two-band models.

Core claim

The dynamical 3-winding-number generalized to generic multi-band Chern insulators equals the difference of Chern numbers between post-quench and pre-quench Hamiltonians, and admits a representation by gapless fermions in phase bands that depends only on the phase and its projectors.

What carries the argument

The dynamical 3-winding-number represented by gapless fermions in phase bands depending only on the phase and its projectors.

If this is right

  • The invariant applies directly to quenches of arbitrary multi-band Chern insulators.
  • Quench dynamics in three-band models can produce multifold fermions in the phase band.
  • The topological difference is captured solely through the phase and projector structure without needing explicit band expansions.

Where Pith is reading between the lines

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

  • Phase band analysis may simplify identification of dynamical invariants in systems with more than two bands.
  • The equality could enable experimental readout of Chern number changes via quench protocols in lattice systems.
  • Similar gapless-fermion representations might apply to other dynamical invariants in higher-dimensional topological phases.

Load-bearing premise

The dynamical 3-winding number admits a representation by gapless fermions in phase bands that depends only on the phase and its projectors.

What would settle it

A concrete multi-band quench where the computed 3-winding number differs from the Chern number difference between the initial and final Hamiltonians.

Figures

Figures reproduced from arXiv: 2406.09797 by Fuxiang Li, Xi Wu, Ze Yang.

Figure 2
Figure 2. Figure 2: FIG. 2. Berry curvature vector of the lowest phase band of the 0- [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 1
Figure 1. Figure 1: FIG. 1. Three phase band dispersions at [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Three phase band dispersions at [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
read the original abstract

Quench dynamics of topological phases have been studied in the past few years and dynamical topological invariants are formulated in different ways. Yet most of these invariants are limited to minimal systems in which Hamiltonians are expanded by Gamma matrices. Here we generalize the dynamical 3-winding-number in two-band systems into the one in generic multi-band Chern insulators and prove that its value is equal to the difference of Chern numbers between post-quench and pre-quench Hamiltonians. Moreover we obtain an expression of this dynamical 3-winding-number represented by gapless fermions in phase bands depending only on the phase and its projectors, so it is generic for the quench of all multi-band Chern insulators. Besides, we obtain a multifold fermion in the phase band in (k, t) space by quenching a three-band model, which cannot happen for two band models.

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

0 major / 2 minor

Summary. The manuscript generalizes the dynamical 3-winding number from two-band systems to generic multi-band Chern insulators. It proves that this invariant equals the difference of Chern numbers between post-quench and pre-quench Hamiltonians. It further derives an expression for the invariant in terms of gapless fermions in phase bands that depends only on the phase and its projectors, making the result applicable to all multi-band Chern insulator quenches, and illustrates a multifold fermion in the (k,t) phase band for a three-band model.

Significance. If the central equality holds, the work provides a parameter-free relation between a dynamical topological invariant and equilibrium Chern numbers that applies beyond minimal two-band models. The representation depending only on the phase and projectors is a strength, as it is generic and does not rely on specific Hamiltonian expansions. The multifold-fermion example demonstrates a dynamical phenomenon inaccessible in two-band systems.

minor comments (2)
  1. [Abstract] Abstract: the statement that the expression 'depends only on the phase and its projectors' is central to generality; a brief parenthetical clarifying what is meant by 'phase' (e.g., the time-evolution operator or the instantaneous eigenphase) would aid readability.
  2. The manuscript would benefit from an explicit statement of the topological space on which the generalized 3-winding number is defined (e.g., the compactified (k,t) torus or its covering) before the proof is presented.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, recognition of the significance of the central equality and the phase-band representation, and the recommendation of minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper generalizes the dynamical 3-winding number from two-band to multi-band Chern insulators and proves its equality to the Chern number difference via an explicit representation depending only on the phase and its projectors. This construction is presented as independent of the input Chern numbers and applies generically without reducing to a fitted parameter or self-citation chain. No load-bearing step in the abstract or described method equates the claimed prediction to its own inputs by definition. The multifold-fermion example is an illustration, not a definitional loop. The central claim therefore remains non-circular on the provided evidence.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that phase-band descriptions suffice for all multi-band Chern insulators and that the 3-winding-number generalizes directly.

axioms (1)
  • domain assumption Dynamical topological invariants for quenches can be formulated using phase bands depending only on phase and projectors for generic multi-band systems
    Abstract states the expression depends only on the phase and its projectors so it is generic for the quench of all multi-band Chern insulators.

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Reference graph

Works this paper leans on

70 extracted references · 70 canonical work pages

  1. [1]

    In the end of this section, we also explain that an inappropriate band- flattening may even lead to non-integer-valued 3-winding number

    and then show that by an appropriate band-flattening these can work for a generic multi-band insulators. In the end of this section, we also explain that an inappropriate band- flattening may even lead to non-integer-valued 3-winding number. In [42] the loop unitary operator Ul(t) = e−ihteih0t (1) and its homotopy invariant, which is a 3-winding number/We...

    work page

  2. [2]

    carrying topological number 2. From Eq. (??22) we see that the coherence vector Sm a have N 2 − 1 components for each band for su(N), so it is no longer practical to draw them as a vector field when N > 2. How- ever, the Berry curvature always have three components. For the convenience of visualization, we draw the field of Berry curvature to represent th...

    work page

  3. [3]

    [66] gives a method to mea- sure the defects of phase bands directly for two band models, which can in principle be generalized

    shows the protocol to mapped out all the coherence vectors through time-of-flight images in the generalized Bloch sphere for su(N) Hamiltonians. [66] gives a method to mea- sure the defects of phase bands directly for two band models, which can in principle be generalized. The combination of both methods may help detect defects in quench of multi-band mod...

    work page

  4. [4]

    M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010)

    work page 2010

  5. [5]

    Wen, Quantum Field Theory of Many-Body Systems: From the Origin of Sound to an Origin of Light and Electrons (Oxford Univ

    X.-G. Wen, Quantum Field Theory of Many-Body Systems: From the Origin of Sound to an Origin of Light and Electrons (Oxford Univ. Press, 2004)

    work page 2004

  6. [6]

    Kitagawa, E

    T. Kitagawa, E. Berg, M. Rudner, and E. Demler, Phys. Rev. B 82, 235114 (2010)

    work page 2010

  7. [7]

    N. H. Lindner, G. Refael, and V . Galitski, Nature Physics 7, 490 (2011)

    work page 2011

  8. [8]

    Jiang, T

    L. Jiang, T. Kitagawa, J. Alicea, A. R. Akhmerov, D. Pekker, G. Refael, J. I. Cirac, E. Demler, M. D. Lukin, and P. Zoller, Phys. Rev. Lett. 106, 220402 (2011)

    work page 2011

  9. [9]

    M. S. Rudner, N. H. Lindner, E. Berg, and M. Levin, Phys. Rev. X 3, 031005 (2013)

    work page 2013

  10. [10]

    G ´omez-Le´on and G

    A. G ´omez-Le´on and G. Platero, Phys. Rev. Lett. 110, 200403 (2013)

    work page 2013

  11. [11]

    D’Alessio and M

    L. D’Alessio and M. Rigol, Nature Communications 6, 8336 (2015)

    work page 2015

  12. [12]

    A. C. Potter, T. Morimoto, and A. Vishwanath, Phys. Rev. X 6, 041001 (2016)

    work page 2016

  13. [13]

    Higashikawa, M

    S. Higashikawa, M. Nakagawa, and M. Ueda, Phys. Rev. Lett. 123, 066403 (2019)

    work page 2019

  14. [14]

    H. Hu, B. Huang, E. Zhao, and W. V . Liu, Phys. Rev. Lett.124, 057001 (2020)

    work page 2020

  15. [15]

    Roy and F

    R. Roy and F. Harper, Phys. Rev. B 96, 155118 (2017)

    work page 2017

  16. [16]

    S. Yao, Z. Yan, and Z. Wang, Phys. Rev. B 96, 195303 (2017)

    work page 2017

  17. [17]

    K. Yang, L. Zhou, W. Ma, X. Kong, P. Wang, X. Qin, X. Rong, Y . Wang, F. Shi, J. Gong, and J. Du, Phys. Rev. B100, 085308 (2019)

    work page 2019

  18. [18]

    Li and H

    T. Li and H. Hu, Nature Communications 14, 6418 (2023)

    work page 2023

  19. [19]

    Jangjan and M

    M. Jangjan and M. V . Hosseini, Scientific Reports 10, 14256 (2020)

    work page 2020

  20. [20]

    Jangjan, L

    M. Jangjan, L. E. F. Foa Torres, and M. V . Hosseini, Phys. Rev. B 106, 224306 (2022)

    work page 2022

  21. [21]

    Fl ¨aschner, B

    N. Fl ¨aschner, B. S. Rem, M. Tarnowski, D. V o- gel, D.-S. L ¨uhmann, K. Sengstock, and C. Weitenberg, Science 352, 1091 (2016), https://www.science.org/doi/pdf/10.1126/science.aad4568

    work page doi:10.1126/science.aad4568 2016

  22. [22]

    E. Alba, X. Fernandez-Gonzalvo, J. Mur-Petit, J. K. Pachos, and J. J. Garcia-Ripoll, Phys. Rev. Lett. 107, 235301 (2011)

    work page 2011

  23. [23]

    Hauke, M

    P. Hauke, M. Lewenstein, and A. Eckardt, Phys. Rev. Lett.113, 045303 (2014)

    work page 2014

  24. [24]

    M. D. Caio, N. R. Cooper, and M. J. Bhaseen, Phys. Rev. Lett. 115, 236403 (2015)

    work page 2015

  25. [25]

    M. D. Caio, N. R. Cooper, and M. J. Bhaseen, Phys. Rev. B 94, 155104 (2016)

    work page 2016

  26. [26]

    Y . Hu, P. Zoller, and J. C. Budich, Phys. Rev. Lett.117, 126803 (2016)

    work page 2016

  27. [27]

    Tarnowski, F

    M. Tarnowski, F. N. ¨Unal, N. Fl¨aschner, B. S. Rem, A. Eckardt, K. Sengstock, and C. Weitenberg, Nature Communications 10, 1728 (2019)

    work page 2019

  28. [28]

    F. N. ¨Unal, A. Bouhon, and R.-J. Slager, Phys. Rev. Lett. 125, 053601 (2020)

    work page 2020

  29. [29]

    Ul ˇcakar, J

    L. Ul ˇcakar, J. Mravlje, A. Ram ˇsak, and T. c. v. Rejec, Phys. Rev. B 97, 195127 (2018)

    work page 2018

  30. [30]

    Zhang, L

    L. Zhang, L. Zhang, S. Niu, and X.-J. Liu, Sci. Bull. 63, 1385 (2018)

    work page 2018

  31. [31]

    Zhang, L

    L. Zhang, L. Zhang, and X.-J. Liu, Phys. Rev. A. 99, 053606 (2019)

    work page 2019

  32. [32]

    Y . Wang, W. Ji, Z. Chai, Y . Guo, M. Wang, X. Ye, P. Yu, L. Zhang, X. Qin, P. Wang, F. Shi, X. Rong, D. Lu, X.-J. Liu, and J. Du, Phys. Rev. A. 100, 052328 (2019)

    work page 2019

  33. [33]

    X.-L. Yu, W. Ji, L. Zhang, Y . Wang, J. Wu, and X.-J. Liu, PRX Quantum 2, 020320 (2021)

    work page 2021

  34. [34]

    Zhang, L

    L. Zhang, L. Zhang, and X.-J. Liu, Phys. Rev. Lett. 125, 183001 (2020)

    work page 2020

  35. [35]

    Sun, C.-R

    W. Sun, C.-R. Yi, B.-Z. Wang, W.-W. Zhang, B. C. Sanders, X.-T. Xu, Z.-Y . Wang, J. Schmiedmayer, Y . Deng, X.-J. Liu, S. Chen, and J.-W. Pan, Phys. Rev. Lett. 121, 250403 (2018)

    work page 2018

  36. [36]

    L. Li, W. Zhu, and J. Gong, Science Bulletin 66, 1502 (2021)

    work page 2021

  37. [37]

    X. Wu, P. Fang, and F. Li, Phys. Rev. A 107, 052209 (2023)

    work page 2023

  38. [38]

    B. Zhu, Y . Ke, H. Zhong, and C. Lee, Phys. Rev. Res.2, 023043 (2020)

    work page 2020

  39. [39]

    Ye and F

    J. Ye and F. Li, Phys. Rev. A 102, 042209 (2020)

    work page 2020

  40. [40]

    Fang, Y .-X

    P. Fang, Y .-X. Wang, and F. Li, Phys. Rev. A 106, 022219 (2022)

    work page 2022

  41. [41]

    C. Yang, L. Li, and S. Chen, Phys. Rev. B 97, 060304 (2018)

    work page 2018

  42. [42]

    C. Wang, P. Zhang, X. Chen, J. Yu, and H. Zhai, Phys. Rev. Lett. 118, 185701 (2017)

    work page 2017

  43. [43]

    X. Chen, C. Wang, and J. Yu, Phys. Rev. A. 101, 032104 (2020)

    work page 2020

  44. [44]

    Fl ¨aschner, D

    N. Fl ¨aschner, D. V ogel, M. Tarnowski, B. S. Rem, D. S. L¨uhmann, M. Heyl, J. C. Budich, L. Mathey, K. Sengstock, and C. Weitenberg, Nat. Phys. 14, 265 (2018)

    work page 2018

  45. [45]

    Hu and E

    H. Hu and E. Zhao, Phys. Rev. Lett. 124, 160402 (2020)

    work page 2020

  46. [46]

    H. Hu, C. Yang, and E. Zhao, Phys. Rev. B101, 155131 (2020)

    work page 2020

  47. [47]

    Barnett, G

    R. Barnett, G. R. Boyd, and V . Galitski, Phys. Rev. Lett. 109, 235308 (2012)

    work page 2012

  48. [48]

    Jak ´obczyk and M

    L. Jak ´obczyk and M. Siennicki, Physics Letters A 286, 383 (2001)

    work page 2001

  49. [49]

    Zyczkowski and H.-J

    K. Zyczkowski and H.-J. Sommers, Journal of Physics A: Math- ematical and General 36, 10115 (2003)

    work page 2003

  50. [50]

    Kimura, Physics Letters A 314, 339 (2003)

    G. Kimura, Physics Letters A 314, 339 (2003)

    work page 2003

  51. [51]

    M. S. Byrd and N. Khaneja, Phys. Rev. A 68, 062322 (2003)

    work page 2003

  52. [52]

    Graf and F

    A. Graf and F. Pi ´echon, Phys. Rev. B 104, 085114 (2021)

    work page 2021

  53. [53]

    C. J. D. Kemp, N. R. Cooper, and F. N. ¨Unal, Phys. Rev. Res. 4, 023120 (2022)

    work page 2022

  54. [54]

    Hatsugai, Phys

    Y . Hatsugai, Phys. Rev. Lett.71, 3697 (1993)

    work page 1993

  55. [55]

    Fukui, Phys

    T. Fukui, Phys. Rev. Res. 2, 043136 (2020)

    work page 2020

  56. [56]

    Hashimoto, X

    K. Hashimoto, X. Wu, and T. Kimura, Phys. Rev. B95, 165443 (2017)

    work page 2017

  57. [57]

    Davoyan, W

    Z. Davoyan, W. J. Jankowski, A. Bouhon, and R.-J. Slager, Phys. Rev. B 109, 165125 (2024)

    work page 2024

  58. [58]

    J. L. Ma ˜nes, Phys. Rev. B 85, 155118 (2012)

    work page 2012

  59. [59]

    Liang and Y

    L. Liang and Y . Yu, Phys. Rev. B93, 045113 (2016). 12

    work page 2016

  60. [60]

    Bradlyn, J

    B. Bradlyn, J. Cano, Z. Wang, M. G. Vergniory, C. Felser, R. J. Cava, and B. A. Bernevig, Science 353, aaf5037 (2016), https://www.science.org/doi/pdf/10.1126/science.aaf5037

    work page doi:10.1126/science.aaf5037 2016

  61. [61]

    P. Tang, Q. Zhou, and S.-C. Zhang, Phys. Rev. Lett. 119, 206402 (2017)

    work page 2017

  62. [62]

    Chang, S.-Y

    G. Chang, S.-Y . Xu, B. J. Wieder, D. S. Sanchez, S.-M. Huang, I. Belopolski, T.-R. Chang, S. Zhang, A. Bansil, H. Lin, and M. Z. Hasan, Phys. Rev. Lett. 119, 206401 (2017)

    work page 2017

  63. [63]

    N. B. M. Schr ¨oter, D. Pei, M. G. Vergniory, Y . Sun, K. Manna, F. de Juan, J. A. Krieger, V . S ¨uss, M. Schmidt, P. Dudin, B. Bradlyn, T. K. Kim, T. Schmitt, C. Cacho, C. Felser, V . N. Strocov, and Y . Chen, Nature Physics15, 759 (2019)

    work page 2019

  64. [64]

    Robredo, N

    I. Robredo, N. Schr ¨oeter, C. Felser, J. Cano, B. Bradlyn, and M. G. Vergniory, (2024), 2404.17539

    work page arXiv 2024

  65. [65]

    Titum, E

    P. Titum, E. Berg, M. S. Rudner, G. Refael, and N. H. Lindner, Phys. Rev. X 6, 021013 (2016)

    work page 2016

  66. [66]

    Bessho and M

    T. Bessho and M. Sato, Phys. Rev. Lett. 127, 196404 (2021)

    work page 2021

  67. [67]

    Nielsen and M

    H. Nielsen and M. Ninomiya, Nuclear Physics B 185, 20 (1981)

    work page 1981

  68. [68]

    Nielsen and M

    H. Nielsen and M. Ninomiya, Nuclear Physics B 193, 173 (1981)

    work page 1981

  69. [69]

    F. N. ¨Unal, B. Seradjeh, and A. Eckardt, Phys. Rev. Lett. 122, 253601 (2019)

    work page 2019

  70. [70]

    Si, Journal of Mathematical Physics 46, 122301 (2005), https://pubs.aip.org/aip/jmp/article- pdf/doi/10.1063/1.2137721/15867792/122301 1 online.pdf

    T. Si, Journal of Mathematical Physics 46, 122301 (2005), https://pubs.aip.org/aip/jmp/article- pdf/doi/10.1063/1.2137721/15867792/122301 1 online.pdf

    work page doi:10.1063/1.2137721/15867792/122301 2005