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arxiv: 2604.23209 · v1 · submitted 2026-04-25 · ⚛️ physics.optics · cond-mat.mes-hall· cond-mat.mtrl-sci

Non-Hermitian corner skin effect in a two-dimensional photonic crystal

Huyen Thanh Phan , Katsunori Wakabayashi This is my paper

Pith reviewed 2026-05-08 07:32 UTC · model grok-4.3

classification ⚛️ physics.optics cond-mat.mes-hallcond-mat.mtrl-sci
keywords non-Hermitian skin effectphotonic crystalpoint gapcorner localizationmagneto-optical materialstopological edge statescomplex eigenfrequenciesnon-Hermitian topology
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The pith

In a two-dimensional photonic crystal with loss, the non-Hermitian skin effect localizes waves at both edges and corners due to point gaps in complex eigenfrequencies.

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

This paper numerically examines electromagnetic waves in a two-dimensional photonic crystal made from lossy magneto-optical materials. It shows that the non-Hermitian skin effect, which localizes wavefunctions at boundaries, appears not only at edges but also at corners due to nontrivial point gaps in the complex eigenfrequency spectrum. Such corner localization has no equivalent in Hermitian systems. The work also finds non-Hermitian topological edge states stemming from the topology of the bulk bands. By using a continuous photonic system rather than discrete models, it offers a practical path toward observing these effects in experiments.

Core claim

In a two-dimensional non-Hermitian photonic crystal composed of lossy magneto-optical materials, the complex eigenfrequencies exhibit point gaps that protect the non-Hermitian skin effect, causing electromagnetic wavefunctions to localize at both the edges and corners of finite structures. Additionally, the nontrivial topology of the bulk bands gives rise to non-Hermitian topological edge states. This skin effect at corners is a feature unique to non-Hermitian systems and is demonstrated through numerical simulations in a continuous medium.

What carries the argument

Point gaps in the complex eigenfrequency spectrum that protect the localization of modes at boundaries including corners in the non-Hermitian photonic crystal.

Load-bearing premise

The numerical discretization and material loss model faithfully reproduce the point-gap topology and skin localization without spurious numerical artifacts or unmodeled dissipation channels.

What would settle it

Direct observation in a fabricated sample of whether wave intensity localizes at corners when point gaps are present in the complex frequency spectrum, versus delocalization if the gaps close.

Figures

Figures reproduced from arXiv: 2604.23209 by Huyen Thanh Phan, Katsunori Wakabayashi.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) Schematic of a 1D reciprocal complex band structure, view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. (a)Schematic of one unit cell of investigated 2D non view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Photonic band structure for the infinite structure (black) and ribbon structure (red) at view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. (a) Complex geometrical phase in view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. (a) Schematic of investigated finite structure, system size is 20 view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. (a) Schematic of eigenfrequencies of the infinite structure view at source ↗
read the original abstract

We numerically study topological effects of electromagnetic (EM) waves in a two-dimensional (2D) non-Hermitian photonic crystal (PhC) composed of lossy magneto-optical materials. In this system, not only the EM wavefunctions but also the complex eigenfrequencies exhibit nontrivial topological properties. We demonstrate that the non-Hermitian skin effect, protected by point gaps in the complex eigenfrequency spectrum, emerges at both the edges and corners of truncated structures. This phenomenon has no counterpart in Hermitian systems. In addition, we identify non-Hermitian topological edge states originating from the nontrivial topology of the bulk bands. While most previous studies of non-Hermitian topology have focused on tight-binding models, our work addresses a continuous photonic system, providing a more realistic platform and offering a concrete route toward experimental realization of non-Hermitian effects.

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 numerically investigates topological properties of electromagnetic waves in a 2D photonic crystal composed of lossy magneto-optical materials. It claims to demonstrate a non-Hermitian skin effect, protected by point gaps in the complex eigenfrequency spectrum, that appears at both edges and corners of truncated structures (with no Hermitian counterpart), along with non-Hermitian topological edge states arising from the nontrivial topology of the bulk bands. The work emphasizes the continuous nature of the photonic system as a realistic platform for experimental realization.

Significance. If the numerical observations are robust, the results would provide a concrete continuous-system example of point-gap-protected skin localization at corners in non-Hermitian photonics, extending beyond discrete tight-binding models and offering a potential route to experimental tests in magneto-optical structures.

major comments (2)
  1. [Numerical results section (eigenfrequency spectra and localization plots)] The manuscript provides no convergence tests, mesh-refinement studies, or validation against known Hermitian limits for the computed complex eigenfrequencies. This is load-bearing for the central claim because the observed point gaps and corner localization could arise from discretization artifacts in the continuous EM system rather than genuine topological protection.
  2. [Results on truncated structures (edge and corner modes)] No error bars, sensitivity analysis to material-loss parameters, or checks against unmodeled radiative losses are reported for the skin-effect localization lengths. Without these, it is impossible to confirm that the claimed edge- and corner-skin accumulation is topologically protected rather than numerically induced.
minor comments (2)
  1. [Abstract and Introduction] The abstract and introduction would benefit from a brief statement of the specific numerical method (e.g., plane-wave expansion, finite-element, or FDTD) and the form of the loss tensor.
  2. [Figures showing band structures] Figure captions for the complex-frequency spectra should explicitly note the Brillouin-zone path and any symmetry reductions due to the magneto-optical bias.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful review and constructive suggestions, which highlight important aspects of numerical validation in our study of the non-Hermitian skin effect in a continuous photonic crystal. We will strengthen the manuscript by adding the requested convergence and sensitivity analyses to better support the robustness of our numerical observations.

read point-by-point responses

  1. Referee: [Numerical results section (eigenfrequency spectra and localization plots)] The manuscript provides no convergence tests, mesh-refinement studies, or validation against known Hermitian limits for the computed complex eigenfrequencies. This is load-bearing for the central claim because the observed point gaps and corner localization could arise from discretization artifacts in the continuous EM system rather than genuine topological protection.

    Authors: We agree that explicit numerical convergence tests are essential to substantiate the claims. In the revised manuscript, we will add a new subsection on numerical methods that includes mesh-refinement studies demonstrating convergence of the complex eigenfrequencies and point gaps with increasing resolution. We will also present validation in the Hermitian limit (vanishing material loss), where the skin effect disappears and standard Hermitian topological features are recovered. These checks confirm that the reported point gaps and corner localizations are not discretization artifacts. revision: yes

  2. Referee: [Results on truncated structures (edge and corner modes)] No error bars, sensitivity analysis to material-loss parameters, or checks against unmodeled radiative losses are reported for the skin-effect localization lengths. Without these, it is impossible to confirm that the claimed edge- and corner-skin accumulation is topologically protected rather than numerically induced.

    Authors: We acknowledge the need for additional robustness checks on the localization lengths. The revised manuscript will include sensitivity plots showing the dependence of edge and corner skin localization on the material loss parameter, with the accumulation remaining stable across a physically relevant range of loss values. Numerical error estimates for the localization lengths will be added. For unmodeled radiative losses, our 2D model is designed to isolate the effects of material-induced non-Hermiticity under appropriate boundary conditions; we will add a clarifying discussion of this modeling choice and note that incorporating radiative losses would require a separate 3D treatment, which lies beyond the present scope. revision: partial

Circularity Check

0 steps flagged

No significant circularity in numerical demonstration of non-Hermitian skin effect

full rationale

The paper reports a numerical study of eigenfrequencies and wavefunction localization in a continuous 2D photonic crystal with lossy magneto-optical materials. The claimed point-gap-protected skin effect at edges and corners is obtained directly from discretization and truncation of the structure; no equations, fitted parameters, or self-citations are shown that would make the output equivalent to the input by construction. The work is self-contained as an observation in a realistic EM platform, with no load-bearing analytical steps that reduce to tautology.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review yields no explicit free parameters, axioms, or invented entities; the study implicitly relies on Maxwell's equations with complex permittivity tensors for lossy magneto-optical media.

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Reference graph

Works this paper leans on

58 extracted references · 58 canonical work pages

  1. [1]

    L. Fu, C. L. Kane, and E. J. Mele, Topological insulators in three dimensions, Phys. Rev. Lett.98, 106803 (2007)

    work page 2007

  2. [2]

    Y . L. Chen, J. G. Analytis, J.-H. Chu, Z. K. Liu, S.-K. Mo, X. L. Qi, H. J. Zhang, D. H. Lu, X. Dai, Z. Fang, S. C. Zhang, I. R. Fisher, Z. Hussain, and Z.-X. Shen, Experimental realization of a three-dimensional topological insulator, bi2te3, Science325, 178 (2009)

    work page 2009

  3. [3]

    D. Kong, J. C. Randel, H. Peng, J. J. Cha, S. Meister, K. Lai, Y . Chen, Z.-X. Shen, H. C. Manoharan, and Y . Cui, Topolog- ical insulator nanowires and nanoribbons, Nano Lett.10, 329 (2010), pMID: 20030392

    work page 2010

  4. [4]

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

    work page 2010

  5. [5]

    Qi and S.-C

    X.-L. Qi and S.-C. Zhang, Topological insulators and supercon- ductors, Rev. Mod. Phys.83, 1057 (2011)

    work page 2011

  6. [6]

    Ando, Topological insulator materials, J

    Y . Ando, Topological insulator materials, J. Phys. Soc. Jpn.82, 102001 (2013)

    work page 2013

  7. [7]

    Liu, H.-Y

    F. Liu, H.-Y . Deng, and K. Wakabayashi, Topological photonic crystals with zero berry curvature, Phys. Rev. B97, 035442 (2018)

    work page 2018

  8. [8]

    Ozawa, H

    T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. C. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, and I. Carusotto, Topological photonics, Rev. Mod. Phys.91, 015006 (2019)

    work page 2019

  9. [9]

    H. T. Phan, F. Liu, and K. Wakabayashi, Valley-dependent cor- ner states in honeycomb photonic crystals without inversion symmetry, Opt. Express29, 18277 (2021)

    work page 2021

  10. [10]

    Tang, X.-T

    G.-J. Tang, X.-T. He, F.-L. Shi, J.-W. Liu, X.-D. Chen, and J.- W. Dong, Topological photonic crystals: Physics, designs, and applications, Laser Photonics Rev.16, 2100300 (2022)

    work page 2022

  11. [11]

    Z. Lan, M. L. Chen, F. Gao, S. Zhang, and W. E. Sha, A brief review of topological photonics in one, two, and three dimen- sions, Rev. in Phys.9, 100076 (2022)

    work page 2022

  12. [12]

    H. T. Phan, K. Koizumi, F. Liu, and K. Wakabayashi, Topologi- cal edge and corner states in biphenylene photonic crystal, Opt. Express32, 2223 (2024)

    work page 2024

  13. [13]

    H. Xue, Y . Yang, and B. Zhang, Topological acoustics, Nat. Rev. Mater.7, 974 (2022)

    work page 2022

  14. [14]

    Z. Yang, F. Gao, X. Shi, X. Lin, Z. Gao, Y . Chong, and B. Zhang, Topological acoustics, Phys. Rev. Lett.114, 114301 (2015)

    work page 2015

  15. [15]

    Chen, Z.-G

    Y .-F. Chen, Z.-G. Chen, H. Ge, C. He, X. Li, M.-H. Lu, X.- C. Sun, S.-Y . Yu, and X. Zhang, Various topological phases and their abnormal effects of topological acoustic metamaterials, In- terdisciplinary Materials2, 179 (2023)

    work page 2023

  16. [16]

    Gao, Y .-G

    F. Gao, Y .-G. Peng, X. Xiang, X. Ni, C. Zheng, S. Yves, X.- F. Zhu, and A. Alù, Acoustic higher-order topological insula- tors induced by orbital-interactions, Adv. Mater.36, 2312421 (2024)

    work page 2024

  17. [17]

    Bansil, H

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

    work page 2016

  18. [18]

    Delplace, D

    P. Delplace, D. Ullmo, and G. Montambaux, Zak phase and the existence of edge states in graphene, Phys. Rev. B84, 195452 (2011)

    work page 2011

  19. [19]

    Obana, F

    D. Obana, F. Liu, and K. Wakabayashi, Topological edge states in the su-schrieffer-heeger model, Phys. Rev. B100, 075437 (2019)

    work page 2019

  20. [20]

    Koizumi, H

    K. Koizumi, H. T. Phan, K. Nishigomi, and K. Wakabayashi, Topological edge and corner states in the biphenylene network, Phys. Rev. B109, 035431 (2024)

    work page 2024

  21. [21]

    Yoshida, Y

    A. Yoshida, Y . Otaki, R. Otaki, and T. Fukui, Edge states, cor- ner states, and flat bands in a two-dimensionalPT-symmetric system, Phys. Rev. B100, 125125 (2019)

    work page 2019

  22. [22]

    Kameda, F

    T. Kameda, F. Liu, S. Dutta, and K. Wakabayashi, Topological edge states induced by the zak phase in a3B monolayers, Phys. Rev. B99, 075426 (2019)

    work page 2019

  23. [23]

    Z. Gong, Y . Ashida, K. Kawabata, K. Takasan, S. Higashikawa, and M. Ueda, Topological phases of non-hermitian systems, Phys. Rev. X8, 031079 (2018)

    work page 2018

  24. [24]

    H. Shen, B. Zhen, and L. Fu, Topological band theory for non- hermitian hamiltonians, Phys. Rev. Lett.120, 146402 (2018)

    work page 2018

  25. [25]

    Kawabata, K

    K. Kawabata, K. Shiozaki, M. Ueda, and M. Sato, Symmetry and topology in non-hermitian physics, Phys. Rev. X9, 041015 (2019)

    work page 2019

  26. [26]

    E. J. Bergholtz, J. C. Budich, and F. K. Kunst, Excep- tional topology of non-hermitian systems, Rev. Mod. Phys.93, 015005 (2021)

    work page 2021

  27. [27]

    H. T. Teo, H. Xue, and B. Zhang, Topological phase transition induced by gain and loss in a photonic chern insulator, Phys. Rev. A105, 053510 (2022)

    work page 2022

  28. [28]

    Wang and Y

    Q. Wang and Y . D. Chong, Non-hermitian photonic lattices: tu- torial, J. Opt. Soc. Am. B40, 1443 (2023)

    work page 2023

  29. [29]

    Nasari, G

    H. Nasari, G. G. Pyrialakos, D. N. Christodoulides, and M. Khajavikhan, Non-hermitian topological photonics, Opt. Mater. Express13, 870 (2023)

    work page 2023

  30. [30]

    Zhong, K

    J. Zhong, K. Wang, Y . Park, V . Asadchy, C. C. Wojcik, A. Dutt, and S. Fan, Nontrivial point-gap topology and non-hermitian skin effect in photonic crystals, Phys. Rev. B104, 125416 (2021)

    work page 2021

  31. [31]

    Zhang, Z

    K. Zhang, Z. Yang, and C. Fang, Universal non-hermitian skin effect in two and higher dimensions, Nat. Commun.13, 2496 (2022)

    work page 2022

  32. [32]

    Okuma, K

    N. Okuma, K. Kawabata, K. Shiozaki, and M. Sato, Topolog- ical origin of non-hermitian skin effects, Phys. Rev. Lett.124, 086801 (2020)

    work page 2020

  33. [33]

    Yao and Z

    S. Yao and Z. Wang, Edge states and topological invariants of non-hermitian systems, Phys. Rev. Lett.121, 086803 (2018)

    work page 2018

  34. [34]

    D. S. Borgnia, A. J. Kruchkov, and R.-J. Slager, Non-hermitian boundary modes and topology, Phys. Rev. Lett.124, 056802 (2020)

    work page 2020

  35. [35]

    Zhang, Z

    K. Zhang, Z. Yang, and C. Fang, Correspondence between winding numbers and skin modes in non-hermitian systems, Phys. Rev. Lett.125, 126402 (2020)

    work page 2020

  36. [36]

    Liang and G.-Y

    S.-D. Liang and G.-Y . Huang, Topological invariance and global berry phase in non-hermitian systems, Phys. Rev. A87, 012118 (2013)

    work page 2013

  37. [37]

    H. Zhao, S. Longhi, and L. Feng, Robust light state by quantum phase transition in non-hermitian optical materials, Sci. Rep.5, 17022 (2015)

    work page 2015

  38. [38]

    Leykam, K

    D. Leykam, K. Y . Bliokh, C. Huang, Y . D. Chong, and F. Nori, Edge modes, degeneracies, and topological numbers in non- hermitian systems, Phys. Rev. Lett.118, 040401 (2017)

    work page 2017

  39. [39]

    M. Pan, H. Zhao, P. Miao, S. Longhi, and L. Feng, Photonic zero mode in a non-hermitian photonic lattice, Nat. Commun. 9, 1308 (2018)

    work page 2018

  40. [40]

    Dangel, M

    F. Dangel, M. Wagner, H. Cartarius, J. Main, and G. Wun- ner, Topological invariants in dissipative extensions of the su- schrieffer-heeger model, Phys. Rev. A98, 013628 (2018)

    work page 2018

  41. [41]

    Longhi and L

    S. Longhi and L. Feng, Complex berry phase and imperfect non-hermitian phase transitions, Phys. Rev. B107, 085122 (2023)

    work page 2023

  42. [42]

    Wagner, F

    M. Wagner, F. Dangel, H. Cartarius, J. Main, and G. Wun- ner, Numerical calculation of the complex berry phase in non- hermitian systems, Acta Polytech.57, 470–476 (2017)

    work page 2017

  43. [43]

    K. Ding, Z. Q. Zhang, and C. T. Chan, Coalescence of excep- tional points and phase diagrams for one-dimensionalPT- symmetric photonic crystals, Phys. Rev. B92, 235310 (2015)

    work page 2015

  44. [44]

    Okugawa, R

    R. Okugawa, R. Takahashi, and K. Yokomizo, Second-order topological non-hermitian skin effects, Phys. Rev. B102, 241202 (2020)

    work page 2020

  45. [45]

    Kawabata, M

    K. Kawabata, M. Sato, and K. Shiozaki, Higher-order non- hermitian skin effect, Phys. Rev. B102, 205118 (2020)

    work page 2020

  46. [46]

    Zhu and J

    W. Zhu and J. Gong, Photonic corner skin modes in non- hermitian photonic crystals, Phys. Rev. B108, 035406 (2023)

    work page 2023

  47. [47]

    G.-G. Liu, S. Mandal, P. Zhou, X. Xi, R. Banerjee, Y .-H. Hu, M. Wei, M. Wang, Q. Wang, Z. Gao, H. Chen, Y . Yang, Y . Chong, and B. Zhang, Localization of chiral edge states by the non-hermitian skin effect, Phys. Rev. Lett.132, 113802 (2024)

    work page 2024

  48. [48]

    A. K. Zvezdin and V . A. Kotov,Modern Magnetooptics and Magnetooptical Materials, 1st ed. (CRC Press, 1997)

    work page 1997

  49. [49]

    O. V . Kotov and Y . E. Lozovik, Giant tunable nonreciprocity of light in weyl semimetals, Phys. Rev. B98, 195446 (2018)

    work page 2018

  50. [50]

    Y .-C. Hsue, A. J. Freeman, and B.-Y . Gu, Extended plane-wave expansion method in three-dimensional anisotropic photonic crystals, Phys. Rev. B72, 195118 (2005)

    work page 2005

  51. [51]

    Wang, G.-Y

    H.-X. Wang, G.-Y . Guo, and J.-H. Jiang, Band topology in clas- sical waves: Wilson-loop approach to topological numbers and fragile topology, New J. Phys.21, 093029 (2019)

    work page 2019

  52. [52]

    Fan, G.-Y

    A. Fan, G.-Y . Huang, and S.-D. Liang, Complex berry curvature pair and quantum hall admittance in non-hermitian systems, J. Phys. Commun.4, 115006 (2020)

    work page 2020

  53. [53]

    Hongming, Y

    W. Hongming, Y . Rui, H. Xiao, D. Xi, and F. Zhong, Quantum anomalous hall effect and related topological electronic states, Adv. Phys.64, 227 (2015)

    work page 2015

  54. [54]

    Bradlyn and M

    B. Bradlyn and M. Iraola, Lecture notes on Berry phases and topology, SciPost Phys. Lect. Notes , 51 (2022)

    work page 2022

  55. [55]

    Yokomizo and S

    K. Yokomizo and S. Murakami, Non-bloch band theory of non- hermitian systems, Phys. Rev. Lett.123, 066404 (2019)

    work page 2019

  56. [56]

    Yokomizo and S

    K. Yokomizo and S. Murakami, Non-bloch bands in two- dimensional non-hermitian systems, Phys. Rev. B107, 195112 (2023)

    work page 2023

  57. [57]

    Yokomizo, T

    K. Yokomizo, T. Yoda, and S. Murakami, Non-hermitian waves in a continuous periodic model and application to photonic crystals, Phys. Rev. Res.4, 023089 (2022)

    work page 2022

  58. [58]

    D. C. Brody, Biorthogonal quantum mechanics, J. Phys. A: Math. Theor.47, 035305 (2013)

    work page 2013