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arxiv: 2510.07214 · v1 · pith:7X6KFD2Tnew · submitted 2025-10-08 · ❄️ cond-mat.mes-hall · math-ph· math.MP

Topology of the generalized Brillouin zone of one-dimensional models

Heming Wang , Janet Zhong , Shanhui Fan This is my paper

Pith reviewed 2026-05-22 13:06 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall math-phmath.MP
keywords non-Hermitian systemsgeneralized Brillouin zonepoint gaptopologyone-dimensional modelssublattice symmetryopen boundary conditions
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The pith

The generalized Brillouin zone in non-Hermitian one-dimensional models can disconnect into more connected components than the number of bands due to point-gap features.

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

This paper studies the topology of generalized Brillouin zones in generic non-Hermitian one-dimensional models. It proves sufficient conditions under which the GBZ remains connected. It also demonstrates that point-gap features can cause the GBZ to split into disconnected parts whose number exceeds the number of bands. This topology permits a line gap to close in an open-boundary spectrum with sublattice symmetry while the point-gap topology stays unchanged. The results question standard pictures of bands and gaps in non-Hermitian systems.

Core claim

In generic non-Hermitian one-dimensional models the generalized Brillouin zone can become disconnected and have more connected components than the number of bands. This results from the point-gap features of the band structure. The novel GBZ topology is applied to show that the line gap of an open-boundary spectrum with sublattice symmetry may be closed without changing its point-gap topology.

What carries the argument

the connected components of the generalized Brillouin zone, whose number and connectivity are governed by point-gap features of the non-Hermitian band structure

If this is right

  • Sufficient conditions exist that guarantee connectivity of the GBZ for certain models.
  • Disconnected GBZs allow the line gap of open-boundary spectra to close under sublattice symmetry without altering point-gap topology.
  • Topological invariants and open-boundary braiding tied to the GBZ must be re-examined.
  • Conventional understanding of bands and gaps in non-Hermitian systems is challenged by these GBZ features.

Where Pith is reading between the lines

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

  • Redefinition of topological invariants for open non-Hermitian systems may become necessary once GBZ disconnections are accounted for.
  • Analogous multiple-component GBZs could appear in models with other symmetries or in two or more dimensions.
  • Photonic or mechanical lattice experiments could directly image the disconnected components of the GBZ.
  • Device design relying on protected gaps in non-Hermitian systems may need to incorporate GBZ connectivity as an additional design variable.

Load-bearing premise

Point-gap features of the band structure directly set the number of connected components of the generalized Brillouin zone independently of the number of bands.

What would settle it

A non-Hermitian one-dimensional model that exhibits point gaps yet produces a connected GBZ with no more components than bands, or in which closing the line gap alters the point-gap topology.

Figures

Figures reproduced from arXiv: 2510.07214 by Heming Wang, Janet Zhong, Shanhui Fan.

Figure 1
Figure 1. Figure 1: FIG. 1. Connectivity of the GBZ. (a) Euler diagram show [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Example of a disconnected GBZ. The model is Eq. [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Transitions between connected and disconnected [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Line-gap transitions in sublattice-symmetric models. (a) The OBC spectra of Eq. ( [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. (a) The OBC spectrum of the model [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
read the original abstract

The generalized Brillouin zones (GBZs) are integral in the analysis of non-Hermitian band structures. Conventional wisdom suggests that the GBZ should be connected, where each point can be indexed by the real part of the wavevector, similar to the Brillouin zone. Here we demonstrate rich topological features of the GBZs in generic non-Hermitian one-dimensional models. We prove and discuss a set of sufficient conditions for the model to ensure the connectivity of its GBZ. In addition, we show that the GBZ can become disconnected and have more connected components than the number of bands, which results from the point-gap features of the band structure. This novel GBZ topology is applied to further demonstrate a counterintuitive effect, where the line gap of an open-boundary spectrum with sublattice symmetry may be closed without changing its point-gap topology. Our results challenge the current understanding of bands and gaps in non-Hermitian systems and highlight the need to further investigate the topological effects associated with the GBZ including topological invariants and open-boundary braiding.

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 studies the topology of the generalized Brillouin zone (GBZ) in one-dimensional non-Hermitian models. It proves a set of sufficient conditions ensuring GBZ connectivity and constructs an explicit multi-band example in which the GBZ is disconnected, possessing more connected components than the number of bands; this disconnection is directly linked to point-gap features of the band structure while preserving the algebraic degree of the secular polynomial. The same topology is used to exhibit a counterintuitive closing of the line gap in an open-boundary spectrum possessing sublattice symmetry without altering the point-gap topology.

Significance. If the derivations hold, the work provides a concrete advance by supplying explicit sufficient conditions for GBZ connectivity (Section 3) together with a reproducible multi-band construction that ties extra GBZ components to point gaps. These results challenge the conventional expectation that GBZs remain connected and indexed by the real part of the wavevector, and they motivate further study of GBZ-related topological invariants and open-boundary braiding phenomena.

minor comments (2)
  1. The numerical verification of the open-boundary spectrum in the multi-band example would benefit from an explicit statement of the system size and the method used to extract the GBZ components (e.g., root-finding precision or contour integration parameters).
  2. Notation for the characteristic equation and the secular polynomial should be unified across Section 3 and the example; currently the same quantity appears under two different symbols.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive and accurate summary of our manuscript, as well as for recommending minor revision. The referee correctly identifies the key contributions: sufficient conditions for GBZ connectivity, an explicit multi-band example with disconnected GBZ tied to point-gap features, and the counterintuitive line-gap closing under sublattice symmetry without altering point-gap topology. We address the report below.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper derives sufficient conditions for GBZ connectivity via direct algebraic analysis of the characteristic equation in Section 3 and constructs explicit multi-band examples where disconnection arises from point-gap topology while preserving polynomial degree. These steps rely on standard definitions of the GBZ, point gaps, and open-boundary spectra without any parameter fitting, self-referential definitions, or load-bearing self-citations that reduce the central claims to their own inputs. The results are self-contained mathematical statements supported by verifiable constructions and numerical checks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The analysis relies on established definitions from non-Hermitian band theory without introducing new free parameters or invented entities; proofs of connectivity conditions are the main addition.

axioms (1)
  • domain assumption Standard definitions and topological properties of generalized Brillouin zones and point gaps in non-Hermitian one-dimensional systems
    The paper invokes these as background to prove sufficient conditions for connectivity and to link disconnection to point gaps.

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    Relation between the paper passage and the cited Recognition theorem.

    We prove and discuss a set of sufficient conditions for the model to ensure the connectivity of its GBZ. In addition, we show that the GBZ can become disconnected and have more connected components than the number of bands, which results from the point-gap features of the band structure.

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Works this paper leans on

64 extracted references · 64 canonical work pages

  1. [1]

    B. A. Bernevig and T. L. Hughes,Topological Insulators and Topological Superconductors(Princeton University Press, Princeton, 2013)

    work page 2013

  2. [2]

    Bansil, H

    A. Bansil, H. Lin, and T. Das, Colloquium: Topological band theory, Rev. Mod. Phys.88, 021004 (2016)

    work page 2016

  3. [3]

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

    work page 2010

  4. [4]

    Cayssol and J

    J. Cayssol and J. N. Fuchs, Topological and geometrical aspects of band theory, Journal of Physics: Materials4, 034007 (2021)

    work page 2021

  5. [5]

    Ashida, Z

    Y. Ashida, Z. Gong, and M. Ueda, Non-Hermitian physics, Advances in Physics69, 249 (2020)

    work page 2020

  6. [6]

    Wang and Y

    Q. Wang and Y. D. Chong, Non-Hermitian photonic lat- tices: tutorial, Journal of the Optical Society of America B40, 1443 (2023)

    work page 2023

  7. [7]

    H. Wang, X. Zhang, J. Hua, D. Lei, M. Lu, and Y. Chen, Topological physics of non-Hermitian optics and photon- ics: a review, Journal of Optics23, 123001 (2021)

    work page 2021

  8. [8]

    E. J. Bergholtz, J. C. Budich, and F. K. Kunst, Ex- ceptional topology of non-Hermitian systems, Reviews of Modern Physics93, 015005 (2021)

    work page 2021

  9. [9]

    K. Ding, C. Fang, and G. Ma, Non-Hermitian topol- ogy and exceptional-point geometries, Nature Reviews Physics4, 745 (2022)

    work page 2022

  10. [10]

    Okuma and M

    N. Okuma and M. Sato, Non-Hermitian topological phe- nomena: A review, Annual Review of Condensed Matter Physics14, 83 (2023)

    work page 2023

  11. [11]

    Zhang, T

    X. Zhang, T. Zhang, M.-H. Lu, and Y.-F. Chen, A review on non-Hermitian skin effect, Advances in Physics: X7, 2109431 (2022)

    work page 2022

  12. [12]

    H. Wang, J. Zhong, and S. Fan, Non-Hermitian photonic band winding and skin effects: a tutorial, Advances in Optics and Photonics16, 659 (2024)

    work page 2024

  13. [13]

    R. Lin, T. Tai, L. Li, and C. H. Lee, Topological non- Hermitian skin effect, Frontiers of Physics18, 53605 (2023)

    work page 2023

  14. [14]

    Z. Gong, Y. Ashida, K. Kawabata, K. Takasan, S. Hi- gashikawa, and M. Ueda, Topological phases of non- Hermitian systems, Physical Review X8, 031079 (2018)

    work page 2018

  15. [15]

    J. T. Gohsrich, A. Banerjee, and F. K. Kunst, The non- Hermitian skin effect: A perspective, Europhysics Letters 150, 60001 (2025)

    work page 2025

  16. [16]

    Zhang and Z

    Y. Zhang and Z. Wei, Non-Hermitian skin effect in non- Hermitian optical systems, Laser & Photonics Reviews 19, 2400099 (2025)

    work page 2025

  17. [17]

    Yao and Z

    S. Yao and Z. Wang, Edge states and topological invari- ants of non-Hermitian systems, Physical Review Letters 121, 086803 (2018)

    work page 2018

  18. [18]

    Yokomizo and S

    K. Yokomizo and S. Murakami, Non-Bloch band theory of non-Hermitian systems, Physical Review Letters123, 066404 (2019)

    work page 2019

  19. [19]

    Z. Yang, K. Zhang, C. Fang, and J. Hu, Non-Hermitian bulk-boundary correspondence and auxiliary general- ized brillouin zone theory, Physical Review Letters125, 226402 (2020)

    work page 2020

  20. [20]

    Zhang, Z

    K. Zhang, Z. Yang, and C. Fang, Correspondence be- tween winding numbers and skin modes in non-Hermitian systems, Physical Review Letters125, 126402 (2020)

    work page 2020

  21. [21]

    Chen and Y

    H. Chen and Y. Zhang, Real spectra in one-dimensional single-band non-hermitian hamiltonians, Phys. Rev. B 108, 115114 (2023)

    work page 2023

  22. [22]

    L. Zhou, R. Jing, and S. Wu, Topological characteriza- tion of phase transitions and critical edge states in one- dimensional non-hermitian systems with sublattice sym- metry (2025), arXiv:2509.01174 [cond-mat.mes-hall]

    work page arXiv 2025

  23. [23]

    Guo, C.-H

    C.-X. Guo, C.-H. Liu, X.-M. Zhao, Y. Liu, and S. Chen, Exact solution of non-Hermitian systems with general- ized boundary conditions: Size-dependent boundary ef- fect and fragility of the skin effect, Physical Review Let- ters127, 116801 (2021)

    work page 2021

  24. [24]

    J. L. Ullman, A problem of schmidt and spitzer, Bulletin of the American Mathematical Society73, 883 (1967)

    work page 1967

  25. [25]

    Tai and C

    T. Tai and C. H. Lee, Zoology of non-Hermitian spec- tra and their graph topology, Physical Review B107, L220301 (2023)

    work page 2023

  26. [26]

    Xiong and H

    Y. Xiong and H. Hu, Graph morphology of non-hermitian bands, Phys. Rev. B109, L100301 (2024)

    work page 2024

  27. [27]

    Zhong, H

    J. Zhong, H. Wang, A. N. Poddubny, and S. Fan, Topo- logical nature of edge states for one-dimensional sys- tems without symmetry protection, Phys. Rev. Lett.135, 016601 (2025)

    work page 2025

  28. [28]

    Verma and M

    S. Verma and M. J. Park, Topological phase transitions of generalized brillouin zone, Communications Physics7, 21 (2024)

    work page 2024

  29. [29]

    Imura and Y

    K.-I. Imura and Y. Takane, Generalized bulk-edge corre- spondence for non-Hermitian topological systems, Phys- ical Review B100, 165430 (2019)

    work page 2019

  30. [30]

    C. Hou, L. Li, S. Chen, Y. Liu, L. Yuan, Y. Zhang, and Z. Ni, Deterministic bulk-boundary correspondences for skin and edge modes in a general two-band non-hermitian system, Phys. Rev. Res.4, 043222 (2022)

    work page 2022

  31. [31]

    Verma and M

    S. Verma and M. J. Park, Non-bloch band theory of subsymmetry-protected topological phases, Phys. Rev. B 110, 035424 (2024)

    work page 2024

  32. [32]

    Liu, L.-H

    C.-C. Liu, L.-H. Li, and J. An, Topological invariant for multiband non-hermitian systems with chiral symmetry, Phys. Rev. B107, 245107 (2023)

    work page 2023

  33. [33]

    Hu and Z

    Y.-M. Hu and Z. Wang, Green’s functions of multi- band non-Hermitian systems, Physical Review Research 5, 043073 (2023)

    work page 2023

  34. [34]

    P. Liu, X. Y. Zhang, X. Z. Hao, Y. H. Zhou, S. C. Hou, and X. X. Yi, Stable directional amplification in one- dimensional non-hermitian single-band systems, Phys. Rev. A107, 053515 (2023)

    work page 2023

  35. [35]

    Li and S

    H. Li and S. Wan, Exact formulas of the end-to-end green’s functions in non-hermitian systems, Phys. Rev. B105, 045122 (2022)

    work page 2022

  36. [36]

    Zirnstein and B

    H.-G. Zirnstein and B. Rosenow, Exponentially grow- ing bulk green functions as signature of nontrivial non- Hermitian winding number in one dimension, Physical Review B103, 195157 (2021)

    work page 2021

  37. [37]

    Chen and W

    C. Chen and W. Guo, Formal green’s function theory in non-hermitian lattice systems, Phys. Rev. B109, 205407 (2024)

    work page 2024

  38. [38]

    Huang, K

    J. Huang, K. Ding, J. Hu, and Z. Yang, Complex fre- quency fingerprint: Basic concept and theory (2025), arXiv:2411.12577 [cond-mat.mes-hall]

    work page arXiv 2025

  39. [39]

    W.-T. Xue, F. Song, Y.-M. Hu, and Z. Wang, Non- bloch edge dynamics of non-hermitian lattices (2025), arXiv:2503.13671 [quant-ph]

    work page arXiv 2025

  40. [40]

    Xue, M.-R

    W.-T. Xue, M.-R. Li, Y.-M. Hu, F. Song, and Z. Wang, Simple formulas of directional amplification from non- Bloch band theory, Physical Review B103, L241408 (2021)

    work page 2021

  41. [41]

    Fu and Y

    Y. Fu and Y. Zhang, Anatomy of open-boundary bulk 7 in multiband non-Hermitian systems, Physical Review B 107, 115412 (2023)

    work page 2023

  42. [42]

    L. Mao, T. Deng, and P. Zhang, Boundary condition independence of non-hermitian hamiltonian dynamics, Phys. Rev. B104, 125435 (2021)

    work page 2021

  43. [43]

    C. C. Wojcik, X.-Q. Sun, T. Bzduˇ sek, and S. Fan, Ho- motopy characterization of non-Hermitian Hamiltonians, Physical Review B101, 205417 (2020)

    work page 2020

  44. [44]

    Hu and E

    H. Hu and E. Zhao, Knots and non-Hermitian Bloch bands, Physical Review Letters126, 010401 (2021)

    work page 2021

  45. [45]

    Li and R

    Z. Li and R. S. K. Mong, Homotopical characterization of non-Hermitian band structures, Physical Review B103, 155129 (2021)

    work page 2021

  46. [46]

    Fu and Y

    Y. Fu and Y. Zhang, Braiding topology of non-hermitian open-boundary bands, Phys. Rev. B110, L121401 (2024)

    work page 2024

  47. [47]

    Wang and C

    A. Wang and C. Q. Chen, Observing non-bloch braids and phase transitions by precise manipulation of the non- hermitian boundary and size, Communications Physics8, 294 (2025)

    work page 2025

  48. [48]

    Y. Li, X. Ji, Y. Chen, X. Yan, and X. Yang, Topological energy braiding of non-bloch bands, Phys. Rev. B106, 195425 (2022)

    work page 2022

  49. [49]

    S. Shi, S. Chu, Y. Xie, and Y. Chen, The studies of topo- logical phases and energy braiding of non-hermitian mod- els using machine learning, Physica Scripta100, 015957 (2024)

    work page 2024

  50. [50]

    Kawabata, K

    K. Kawabata, K. Shiozaki, M. Ueda, and M. Sato, Sym- metry and topology in non-Hermitian physics, Physical Review X9, 041015 (2019)

    work page 2019

  51. [51]

    Okuma, K

    N. Okuma, K. Kawabata, K. Shiozaki, and M. Sato, Topological origin of non-Hermitian skin effects, Phys- ical Review Letters124, 086801 (2020)

    work page 2020

  52. [52]

    C. H. Lee and R. Thomale, Anatomy of skin modes and topology in non-Hermitian systems, Physical Review B 99, 201103 (2019)

    work page 2019

  53. [53]

    D. Wu, J. Xie, Y. Zhou, and J. An, Connections between the open-boundary spectrum and the generalized Bril- louin zone in non-Hermitian systems, Physical Review B 105, 045422 (2022)

    work page 2022

  54. [54]

    Lieu, Topological phases in the non-Hermitian Su- Schrieffer-Heeger model, Physical Review B97, 045106 (2018)

    S. Lieu, Topological phases in the non-Hermitian Su- Schrieffer-Heeger model, Physical Review B97, 045106 (2018)

    work page 2018

  55. [55]

    F. K. Kunst, E. Edvardsson, J. C. Budich, and E. J. Bergholtz, Biorthogonal bulk-boundary correspondence in non-Hermitian systems, Physical Review Letters121, 026808 (2018)

    work page 2018

  56. [56]

    V. M. Martinez Alvarez, J. E. Barrios Vargas, and L. E. F. Foa Torres, Non-Hermitian robust edge states in one dimension: Anomalous localization and eigenspace condensation at exceptional points, Physical Review B 97, 121401 (2018)

    work page 2018

  57. [57]

    L.-J. Lang, Y. Wang, H. Wang, and Y. D. Chong, Effects of non-Hermiticity on Su-Schrieffer-Heeger defect states, Physical Review B98, 094307 (2018)

    work page 2018

  58. [58]

    Longhi, Probing non-Hermitian skin effect and non- Bloch phase transitions, Physical Review Research1, 023013 (2019)

    S. Longhi, Probing non-Hermitian skin effect and non- Bloch phase transitions, Physical Review Research1, 023013 (2019)

    work page 2019

  59. [59]

    Zhong, H

    J. Zhong, H. Wang, and S. Fan, Pole and zero edge state invariant for one-dimensional non-Hermitian sublattice symmetry, Physical Review B110, 214113 (2024)

    work page 2024

  60. [60]

    Yokomizo and S

    K. Yokomizo and S. Murakami, Non-bloch band theory and bulk–edge correspondence in non-hermitian systems, Progress of Theoretical and Experimental Physics2020, 12A102 (2020)

    work page 2020

  61. [61]

    Guo, X.-X

    G.-F. Guo, X.-X. Bao, and L. Tan, Non-Hermitian bulk-boundary correspondence and singular behaviors of generalized brillouin zone, New Journal of Physics23, 123007 (2021)

    work page 2021

  62. [62]

    Yokomizo and S

    K. Yokomizo and S. Murakami, Topological semimetal phase with exceptional points in one-dimensional non- hermitian systems, Phys. Rev. Res.2, 043045 (2020)

    work page 2020

  63. [63]

    J. B. Conway,Functions of one complex variable (Springer-Verlag, New York, 1978)

    work page 1978

  64. [64]

    B¨ ottcher and B

    A. B¨ ottcher and B. Silbermann,Introduction to Large Truncated Toeplitz Matrices(Springer New York, New York, NY, 1999). 8 SUPPLEMENTARY MATERIALS Definitions Before stating and proving the theorems in the main text, we formalize some definitions that we will use below. Asingle-band model, or amodelfor short, is a Laurent polynomial inzgiven byE= P −p≤j≤...

    work page 1999