- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Research

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Wiener-Hopf factorization and non-Hermitian topology for Amoeba formulation in one-dimensional multiband systems

Shin Kaneshiro* and Robert Peters

  • *Contact author: kaneshiro.shin.88a@st.kyoto-u.ac.jp
PDF

Phys. Rev. Research 8, 013292 – Published 17 March, 2026

DOI: https://doi.org/10.1103/s43l-h6z6

Abstract

The non-Hermitian skin effect, characterized by the extensive localization of bulk modes at the boundaries, has attracted significant attention as a hallmark feature of non-Hermitian topology. This localization invalidates the conventional Bloch band theory, necessitating an analysis under open boundary conditions even in the thermodynamic limit. The Amoeba formulation addresses this challenge by computing the spectral potential rather than the spectrum itself. Based on the (strong) Szegö limit theorem and its topological generalization, this approach reduces the evaluation of the potential to an optimization problem involving the Ronkin function. However, while the generalized Szegö limit theorem is formally applicable in arbitrary dimensions, its implementation is limited to single-band systems, and its applicability to multiband systems remains unclear even in one-dimensional systems. In this paper, we establish the Wiener-Hopf factorization (WHF) of the non-Bloch Hamiltonian as a powerful framework, providing a unified and rigorous foundation for Amoeba analysis in one-dimensional multiband systems. By combining the WHF with Hermitian doubling, we first elucidate the applicability criteria for the generalized Szegö limit theorem in multiband systems. We then show that the WHF provides the natural mathematical origin for the symmetry-decomposed Ronkin function in symmetry class AII†, leading to a rigorous proof of the generalized Szegö limit theorem for these systems and opening a path toward systematic generalizations to other symmetry classes.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (100)

  1. Y. Ashida, Z. Gong, and M. Ueda, Non-Hermitian physics, Adv. Phys. 69, 249 (2020).
  2. E. J. Bergholtz, J. C. Budich, and F. K. Kunst, Exceptional topology of non-Hermitian systems, Rev. Mod. Phys. 93, 015005 (2021).
  3. K. Ding, C. Fang, and G. Ma, Non-Hermitian topology and exceptional-point geometries, Nat. Rev. Phys. 4, 745 (2022).
  4. Z. Gong, Y. Ashida, K. Kawabata, K. Takasan, S. Higashikawa, and M. Ueda, Topological phases of non-Hermitian systems, Phys. Rev. X 8, 031079 (2018).
  5. K. Kawabata, K. Shiozaki, M. Ueda, and M. Sato, Symmetry and topology in non-Hermitian physics, Phys. Rev. X 9, 041015 (2019).
  6. V. M. M. Alvarez, J. E. B. Vargas, and L. E. F. F. Torres, Non-Hermitian robust edge states in one dimension: Anomalous localization and eigenspace condensation at exceptional points, Phys. Rev. B 97, 121401(R) (2018).
  7. S. Yao and Z. Wang, Edge states and topological invariants of non-Hermitian systems, Phys. Rev. Lett. 121, 086803 (2018).
  8. F. K. Kunst, E. Edvardsson, J. C. Budich, and E. J. Bergholtz, Biorthogonal bulk-boundary correspondence in non-Hermitian systems, Phys. Rev. Lett. 121, 026808 (2018).
  9. K. Yokomizo and S. Murakami, Non-Bloch band theory of non-Hermitian systems, Phys. Rev. Lett. 123, 066404 (2019).
  10. C. H. Lee and R. Thomale, Anatomy of skin modes and topology in non-Hermitian systems, Phys. Rev. B 99, 201103(R) (2019).
  11. F. K. Kunst and V. Dwivedi, Non-Hermitian systems and topology: A transfer-matrix perspective, Phys. Rev. B 99, 245116 (2019).
  12. F. Song, S. Yao, and Z. Wang, Non-Hermitian topological invariants in real space, Phys. Rev. Lett. 123, 246801 (2019).
  13. M. Brandenbourger, X. Locsin, E. Lerner, and C. Coulais, Non-reciprocal robotic metamaterials, Nat. Commun. 10, 4608 (2019).
  14. L. E. F. F. Torres, Perspective on topological states of non-Hermitian lattices, J. Phys.: Mater. 3, 014002 (2020).
  15. K. Zhang, Z. Yang, and C. Fang, Correspondence between winding numbers and skin modes in non-Hermitian systems, Phys. Rev. Lett. 125, 126402 (2020).
  16. Z. Yang, K. Zhang, C. Fang, and J. Hu, Non-Hermitian bulk-boundary correspondence and auxiliary generalized Brillouin zone theory, Phys. Rev. Lett. 125, 226402 (2020).
  17. N. Okuma, K. Kawabata, K. Shiozaki, and M. Sato, Topological origin of non-Hermitian skin effects, Phys. Rev. Lett. 124, 086801 (2020).
  18. L. Li, C. H. Lee, S. Mu, and J. Gong, Critical non-Hermitian skin effect, Nat. Commun. 11, 5491 (2020).
  19. K. Kawabata, M. Sato, and K. Shiozaki, Higher-order non-Hermitian skin effect, Phys. Rev. B 102, 205118 (2020).
  20. S. Longhi, Non-Bloch-band collapse and chiral Zener tunneling, Phys. Rev. Lett. 124, 066602 (2020).
  21. T. Yoshida, T. Mizoguchi, and Y. Hatsugai, Mirror skin effect and its electric circuit simulation, Phys. Rev. Res. 2, 022062(R) (2020).
  22. N. Okuma and M. Sato, Hermitian zero modes protected by non-normality: Application of pseudospectra, Phys. Rev. B 102, 014203 (2020).
  23. A. Ghatak, M. Brandenbourger, J. van Wezel, and C. Coulais, Observation of non-Hermitian topology and its bulk-edge correspondence in an active mechanical metamaterial, Proc. Natl. Acad. Sci. USA 117, 29561 (2020).
  24. S. Weidemann, M. Kremer, T. Helbig, T. Hofmann, A. Stegmaier, M. Greiter, R. Thomale, and A. Szameit, Topological funneling of light, Science 368, 311 (2020).
  25. T. Helbig, T. Hofmann, S. Imhof, M. Abdelghany, T. Kiessling, L. W. Molenkamp, C. H. Lee, A. Szameit, M. Greiter, and R. Thomale, Generalized bulk-boundary correspondence in non-Hermitian topolectrical circuits, Nat. Phys. 16, 747 (2020).
  26. T. Hofmann, T. Helbig, F. Schindler, N. Salgo, M. Brzezińska, M. Greiter, T. Kiessling, D. Wolf, A. Vollhardt, A. Kabaši, C. H. Lee, A. Bilušić, R. Thomale, and T. Neupert, Reciprocal skin effect and its realization in a topolectrical circuit, Phys. Rev. Res. 2, 023265 (2020).
  27. L. Xiao, T. Deng, K. Wang, G. Zhu, Z. Wang, W. Yi, and P. Xue, Non-Hermitian bulk-boundary correspondence in quantum dynamics, Nat. Phys. 16, 761 (2020).
  28. W. Gou, T. Chen, D. Xie, T. Xiao, T.-S. Deng, B. Gadway, W. Yi, and B. Yan, Tunable nonreciprocal quantum transport through a dissipative Aharonov-Bohm ring in ultracold atoms, Phys. Rev. Lett. 124, 070402 (2020).
  29. D. S. Borgnia, A. J. Kruchkov, and R.-J. Slager, Non-Hermitian boundary modes and topology, Phys. Rev. Lett. 124, 056802 (2020).
  30. N. Okuma and M. Sato, Quantum anomaly, non-Hermitian skin effects, and entanglement entropy in open systems, Phys. Rev. B 103, 085428 (2021).
  31. K. Kawabata, K. Shiozaki, and S. Ryu, Topological field theory of non-Hermitian systems, Phys. Rev. Lett. 126, 216405 (2021).
  32. R. Okugawa, R. Takahashi, and K. Yokomizo, Non-Hermitian band topology with generalized inversion symmetry, Phys. Rev. B 103, 205205 (2021).
  33. S. Liu, R. Shao, S. Ma, L. Zhang, O. You, H. Wu, Y. J. Xiang, T. J. Cui, and S. Zhang, Non-Hermitian skin effect in a non-Hermitian electrical circuit, Research 2021, 5608038 (2021).
  34. X. Zhang, Y. Tian, J.-H. Jiang, M.-H. Lu, and Y.-F. Chen, Observation of higher-order non-Hermitian skin effect, Nat. Commun. 12, 5377 (2021).
  35. L. S. Palacios, S. Tchoumakov, M. Guix, I. P. S. Sánchez, and A. G. Grushin, Guided accumulation of active particles by topological design of a second-order skin effect, Nat. Commun. 12, 4691 (2021).
  36. D. Wu, J.-M. Xie, Y.-Q. Zhou, and J.-H. An, Connections between the open-boundary spectrum and the generalized Brillouin zone in non-Hermitian systems, Phys. Rev. B 105, 045422 (2022).
  37. Q. Liang, D. Xie, Z. Dong, H. Li, H. Li, B. Gadway, W. Yi, and B. Yan, Dynamic signatures of non-Hermitian skin effect and topology in ultracold atoms, Phys. Rev. Lett. 129, 070401 (2022).
  38. F. Song, H.-Y. Wang, and Z. Wang, Non-Bloch PT symmetry: Universal threshold and dimensional surprise, in A Festschrift in Honor of the C N Yang Centenary, edited by F.-C. Chen, M.-L. Ge, B.-L. Gu, K. K. Phua, K. Young, and B.-F. Zhu (World Scientific, Singapore, 2022), pp. 299–311.
  39. S. Longhi, Non-Hermitian skin effect and self-acceleration, Phys. Rev. B 105, 245143 (2022).
  40. Z. Gu, H. Gao, H.-Y. Xue, J. Li, Z. Su, and J. Zhu, Transient non-Hermitian skin effect, Nat. Commun. 13, 7668 (2022).
  41. K. Kawabata, T. Numasawa, and S. Ryu, Entanglement phase transition induced by the non-Hermitian skin effect, Phys. Rev. X 13, 021007 (2023).
  42. C.-A. Li, B. Trauzettel, T. Neupert, and S.-B. Zhang, Enhancement of second-order non-Hermitian skin effect by magnetic fields, Phys. Rev. Lett. 131, 116601 (2023).
  43. S. Manna and B. Roy, Inner skin effects on non-Hermitian topological fractals, Commun. Phys. 6, 10 (2023).
  44. Y.-L. Zhang, L.-W. Wang, Y. Liu, Z.-X. Chen, and J.-H. Jiang, Hybrid skin-topological effect in non-Hermitian checkerboard lattices with large Chern numbers, arXiv:2411.07465.
  45. D. Nakamura, T. Bessho, and M. Sato, Bulk-boundary correspondence in point-gap topological phases, Phys. Rev. Lett. 132, 136401 (2024).
  46. Y. O. Nakai, N. Okuma, D. Nakamura, K. Shimomura, and M. Sato, Topological enhancement of nonnormality in non-Hermitian skin effects, Phys. Rev. B 109, 144203 (2024).
  47. S. Ishikawa and T. Yoshida, Non-Hermitian Z4 skin effect protected by glide symmetry, Phys. Rev. B 110, 115301 (2024).
  48. X. Zhang, B. Zhang, W. Zhao, and C. H. Lee, Observation of non-local impedance response in a passive electrical circuit, SciPost Phys. 16, 002 (2024).
  49. Z. Wei, J.-Y. Fan, K. Cao, X.-R. Ma, C.-X. Guo, X.-P. Ren, and S.-P. Kou, A unified framework for the non-Hermitian localization: Boundary-insensitive modes and electric-magnetic analogy, arXiv:2505.10252.
  50. R. Shen, T. Chen, B. Yang, and C. H. Lee, Observation of the non-Hermitian skin effect and Fermi skin on a digital quantum computer, Nat. Commun. 16, 1340 (2025).
  51. K. Kawabata, N. Okuma, and M. Sato, Non-Bloch band theory of non-Hermitian Hamiltonians in the symplectic class, Phys. Rev. B 101, 195147 (2020).
  52. H. Hu and E. Zhao, Knots and non-Hermitian bloch bands, Phys. Rev. Lett. 126, 010401 (2021).
  53. K. Yokomizo and S. Murakami, Non-Bloch band theory in bosonic Bogoliubov-de Gennes systems, Phys. Rev. B 103, 165123 (2021).
  54. T. Li, J.-Z. Sun, Y.-S. Zhang, and W. Yi, Non-Bloch quench dynamics, Phys. Rev. Res. 3, 023022 (2021).
  55. G.-F. Guo, X.-X. Bao, and L. Tan, Non-Hermitian bulk-boundary correspondence and singular behaviors of generalized Brillouin zone, New J. Phys. 23, 123007 (2021).
  56. W.-T. Xue, M.-R. Li, Y.-M. Hu, F. Song, and Z. Wang, Simple formulas of directional amplification from non-Bloch band theory, Phys. Rev. B 103, L241408 (2021).
  57. K. Yokomizo and S. Murakami, Scaling rule for the critical non-Hermitian skin effect, Phys. Rev. B 104, 165117 (2021).
  58. Y. Li, X. Ji, Y. Chen, X. Yan, and X. Yang, Topological energy braiding of non-Bloch bands, Phys. Rev. B 106, 195425 (2022).
  59. H. Li and S. Wan, Exact formulas of the end-to-end Green's functions in non-Hermitian systems, Phys. Rev. B 105, 045122 (2022).
  60. H. Liu, M. Lu, Z.-Q. Zhang, and H. Jiang, Modified generalized Brillouin zone theory with on-site disorder, Phys. Rev. B 107, 144204 (2023).
  61. T. Tai and C. H. Lee, Zoology of non-Hermitian spectra and their graph topology, Phys. Rev. B 107, L220301 (2023).
  62. Y.-M. Hu and Z. Wang, Green's functions of multiband non-Hermitian systems, Phys. Rev. Res. 5, 043073 (2023).
  63. Y.-M. Hu, Y.-Q. Huang, W.-T. Xue, and Z. Wang, Non-Bloch band theory for non-Hermitian continuum systems, Phys. Rev. B 110, 205429 (2024).
  64. K. Matsushima and T. Yamada, Non-Bloch band theory for time-modulated discrete mechanical systems, J. Sound Vib. 595 118757 (2025).
  65. H.-R. Wang, Z. Wang, and Z. Wang, Non-Bloch self-energy of dissipative interacting fermions, arXiv:2411.13661.
  66. K. Roy, K. Gogoi, and S. Basu, Topological characterization of a non-Hermitian ladder via Floquet non-Bloch theory, Phys. Rev. B 111, 115424 (2025).
  67. Z. Yang, C. Lu, and X. Lu, Entanglement entropy on generalized Brillouin zone, Phys. Rev. B 110, 235127 (2024).
  68. Y. Fu and Y. Zhang, Braiding topology of non-Hermitian open-boundary bands, Phys. Rev. B 110, L121401 (2024).
  69. S.-X. Wang and Z. Yan, General theory for infernal points in non-Hermitian systems, Phys. Rev. B 110, L201104 (2024).
  70. S. Verma and M. J. Park, Non-Bloch band theory of sub-symmetry-protected topological phases, arXiv:2405.06240.
  71. Y.-M. Hu, H.-Y. Wang, Z. Wang, and F. Song, Geometric origin of non-Bloch PT symmetry breaking, Phys. Rev. Lett. 132, 050402 (2024).
  72. Q. Li, H. Jiang, and C. H. Lee, Phase-space generalized Brillouin zone for spatially inhomogeneous non-Hermitian systems, arXiv:2501.09785.
  73. J.-X. Zhong, J. Lin, K. Chen, J. Lu, K. Ding, and Y. Jing, Unveiling non-Hermitian band structures with non-Bloch supercells, arXiv:2510.20160.
  74. H. Meng, Y. S. Ang, and C. H. Lee, Generalized Brillouin zone fragmentation, arXiv:2508.13275.
  75. K. Roy, D. Halder, K. Gogoi, B. Tanatar, and S. Basu, Floquet non-Bloch formalism for a non-Hermitian ladder: From theoretical framework to topolectrical circuits, Phys. Rev. Res. 7, 043331 (2025).
  76. D. Nakamura, Y. Tanaka, K. Shiozaki, and K. Kawabata, Nonsymmorphic topological phases of non-Hermitian systems, arXiv:2504.20743.
  77. C. H. Lee, L. Li, and J. Gong, Hybrid higher-order skin-topological modes in nonreciprocal systems, Phys. Rev. Lett. 123, 016805 (2019).
  78. W. Zhu and L. Li, A brief review of hybrid skin-topological effect, J. Phys.: Condens. Matter 36, 253003 (2024).
  79. T. Liu, Y.-R. Zhang, Q. Ai, Z. Gong, K. Kawabata, M. Ueda, and F. Nori, Second-order topological phases in non-Hermitian systems, Phys. Rev. Lett. 122, 076801 (2019).
  80. K. Yokomizo and S. Murakami, Non-Bloch bands in two-dimensional non-Hermitian systems, Phys. Rev. B 107, 195112 (2023).
  81. H. Jiang and C. H. Lee, Dimensional transmutation from non-Hermiticity, Phys. Rev. Lett. 131, 076401 (2023).
  82. Z. Xu, B. Pang, K. Zhang, and Z. Yang, Two-dimensional asymptotic generalized Brillouin zone theory, arXiv:2311.16868.
  83. K. Zhang, Z. Yang, and K. Sun, Edge theory of non-Hermitian skin modes in higher dimensions, Phys. Rev. B 109, 165127 (2024).
  84. K. Zhang, C. Shu, and K. Sun, Algebraic non-Hermitian skin effect and generalized Fermi surface formula in arbitrary dimensions, Phys. Rev. X 15, 031039 (2025).
  85. C. Wang, J. Pi, Q. Liu, Y. Li, and Y.-C. Liu, Universal theory for geometry-dependent non-Hermitian bands, arXiv:2506.22743.
  86. L. Li, Y. Wei, G. Wu, Y. Ruan, S. Chen, C. H. Lee, and Z. Ni, Exact solutions disentangle higher-order topology in two-dimensional non-Hermitian lattices, Phys. Rev. B 111, 075132 (2025).
  87. H.-Y. Wang, F. Song, and Z. Wang, Amoeba formulation of non-Bloch band theory in arbitrary dimensions, Phys. Rev. X 14, 021011 (2024).
  88. S.-X. Wang, Constraints of internal symmetry on the non-Hermitian skin effect and bidirectional skin effect under the action of the Hermitian conjugate of time-reversal symmetry, Phys. Rev. B 109, L081108 (2024).
  89. Y. Xiong and H. Hu, Graph morphology of non-Hermitian bands, Phys. Rev. B 109, L100301 (2024).
  90. Y. Xiong, Z.-Y. Xing, and H. Hu, Non-Hermitian skin effect in arbitrary dimensions: Non-Bloch band theory and classification, arXiv:2407.01296.
  91. H. Hu, Topological origin of non-Hermitian skin effect in higher dimensions and uniform spectra, Sci. Bull. 70, 51 (2025).
  92. A. Yang, Z. Fang, K. Zhang, and C. Fang, Tailoring bound state geometry in high-dimensional non-Hermitian systems, Commun. Phys. 8, 124 (2025).
  93. G. Szegö, Ein grenzwertsatz über die toeplitzschen determinanten einer reellen positiven funktion, Math. Ann. 76, 490 (1915).
  94. H. Widom, Asymptotic behavior of Block Toeplitz matrices and determinants, Adv. Math. 13, 284 (1974).
  95. H. Widom, Asymptotic behavior of Block Toeplitz matrices and determinants. II, Adv. Math. 21, 1 (1976).
  96. S. Kaneshiro and R. Peters, Symplectic-amoeba formulation of the non-Bloch band theory for one-dimensional two-band systems, Phys. Rev. B 112, 075408 (2025).
  97. A. Alase, E. Cobanera, G. Ortiz, and L. Viola, Wiener–Hopf factorization approach to a bulk-boundary correspondence and stability conditions for topological zero-energy modes, Ann. Phys. (NY) 458, 169457 (2023).
  98. E. Basor, J. Dubail, T. Emig, and R. Santachiara, Modified Szegő–Widom asymptotics for Block Toeplitz matrices with zero modes, J. Stat. Phys. 174, 28 (2019).
  99. 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).
  100. L. Li, C. Hou, G. Wu, Y. Ruan, S. Chen, L. Yuan, and Z. Ni, Dual bulk-boundary correspondence in a nonreciprocal spin-orbit coupled zigzag lattice, Phys. Rev. B 110, L041103 (2024).

Outline

Information

Phys. Rev. Research 8, 013292– Published 17 March, 2026
  • Received 14 November 2025
  • Accepted 13 February 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation