- Physical Review Journals
All JournalsPhysics Magazine

Physical Review X

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

Algebraic Non-Hermitian Skin Effect and Generalized Fermi Surface Formula in Arbitrary Dimensions

Kai Zhang, Chang Shu, and Kai Sun*

  • *Contact author: sunkai@umich.edu
PDF

Phys. Rev. X 15, 031039 – Published 11 August, 2025

DOI: https://doi.org/10.1103/cwwd-bclc

Abstract

The non-Hermitian skin effect characterized by a proliferation of exponentially localized edge modes in open-boundary systems has led to the discovery of numerous novel physical phenomena that challenge the limits of conventional band theory. In sharp contrast to this familiar exponential localization, we report a distinct phenomenon—the algebraic non-Hermitian skin effect—which arises generically in non-Hermitian systems with two or more spatial dimensions. In such cases, the amplitude of skin modes typically decays from the boundary following a power law, rather than an exponential form—a behavior not captured by existing theoretical frameworks. To bridge this gap and describe the transition in localization from one to higher dimensions, we develop a generalized Fermi surface framework applicable to open-boundary systems in arbitrary dimensions. This framework not only reproduces known results for the exponential skin effect in 1D, but also predicts a new class of skin effects with algebraic decay in 2D and above. We demonstrate this framework in both tight-binding and continuum models in two and three dimensions. This investigation not only unveils a novel category of the non-Hermitian skin effect but also offers a comprehensive theoretical structure that describes skin effects in any non-Hermitian system, irrespective of its spatial dimensionality.

View figure in article

Physics Subject Headings (PhySH)

Popular Summary

In quantum mechanics, time evolution is typically described by mathematical operators known as Hermitian Hamiltonians, ensuring real energy values and stable dynamics. However, real systems often interact with messy environments that introduce loss, gain, or decoherence. These effects are more accurately captured using non-Hermitian Hamiltonians, which allow complex energy spectra and lead to surprising behaviors. One such phenomenon is the non-Hermitian skin effect, where many eigenstates—specific states with a definite value for a given observable—cluster at the system’s boundary. While this effect is well understood in 1D systems, its behavior in higher dimensions has remained largely uncharted. Here, we go beyond past analyses and identify a new behavior in higher-dimensional systems.

Specifically, we discover a new form of boundary localization, which we call the algebraic non-Hermitian skin effect. Unlike the familiar skin effect where eigenstates decay exponentially from the edge, the states we study decay algebraically, following a power law. This means they are more spread out yet remain localized at the boundaries. These extended modes lead to longer-range correlations within the system, offering new possibilities for controlling quantum entanglement and wave transport in open systems subject to loss and gain. To understand this new effect, we develop a detailed theoretical framework that lets us construct these algebraically decaying boundary states in any number of dimensions.

Our analysis reveals deep connections between the way these modes decay and the system’s geometry and topology. This discovery broadens our understanding of non- Hermitian physics and opens the door to engineering novel wave-localization effects in fields such as photonics, acoustics, and quantum materials.

Article Text

References (104)

  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. B. Zhen, C. W. Hsu, Y. Igarashi, L. Lu, I. Kaminer, A. Pick, S.-L. Chua, J. D. Joannopoulos, and M. Soljačić, Spawning rings of exceptional points out of Dirac cones, Nature (London) 525, 354 (2015).
  5. H. Zhou, C. Peng, Y. Yoon, C. W. Hsu, K. A. Nelson, L. Fu, J. D. Joannopoulos, M. Soljačić, and B. Zhen, Observation of bulk Fermi arc and polarization half charge from paired exceptional points, Science 359, 1009 (2018).
  6. A. Cerjan, S. Huang, M. Wang, K. P. Chen, Y. Chong, and M. C. Rechtsman, Experimental realization of a Weyl exceptional ring, Nat. Photonics 13, 623 (2019).
  7. L. Huang, S. Huang, C. Shen, S. Yves, A. S. Pilipchuk, X. Ni, S. Kim, Y. K. Chiang, D. A. Powell, J. Zhu, Y. Cheng, Y. Li, A. F. Sadreev, A. Alù, and A. E. Miroshnichenko, Acoustic resonances in non-Hermitian open systems, Nat. Rev. Phys. 6, 11 (2024).
  8. L. Zhang, Y. Yang, Y. Ge, Y.-J. Guan, Q. Chen, Q. Yan, F. Chen, R. Xi, Y. Li, D. Jia, S.-Q. Yuan, H.-X. Sun, H. Chen, and B. Zhang, Acoustic non-Hermitian skin effect from twisted winding topology, Nat. Commun. 12, 6297 (2021).
  9. C. Scheibner, A. Souslov, D. Banerjee, P. Surówka, W. T. M. Irvine, and V. Vitelli, Odd elasticity, Nat. Phys. 16, 475 (2020).
  10. C. Scheibner, W. T. M. Irvine, and V. Vitelli, Non-Hermitian band topology and skin modes in active elastic media, Phys. Rev. Lett. 125, 118001 (2020).
  11. D. Zhou and J. Zhang, Non-Hermitian topological metamaterials with odd elasticity, Phys. Rev. Res. 2, 023173 (2020).
  12. D. Banerjee, V. Vitelli, F. Jülicher, and P. Surówka, Active viscoelasticity of odd materials, Phys. Rev. Lett. 126, 138001 (2021).
  13. S. Shankar, A. Souslov, M. J. Bowick, M. C. Marchetti, and V. Vitelli, Topological active matter, Nat. Rev. Phys. 4, 380 (2022).
  14. F. Song, S. Yao, and Z. Wang, Non-Hermitian skin effect and chiral damping in open quantum systems, Phys. Rev. Lett. 123, 170401 (2019).
  15. C.-H. Liu, K. Zhang, Z. Yang, and S. Chen, Helical damping and dynamical critical skin effect in open quantum systems, Phys. Rev. Res. 2, 043167 (2020).
  16. T. Haga, M. Nakagawa, R. Hamazaki, and M. Ueda, Liouvillian skin effect: Slowing down of relaxation processes without gap closing, Phys. Rev. Lett. 127, 070402 (2021).
  17. W.-T. Xue, Y.-M. Hu, F. Song, and Z. Wang, Non-Hermitian edge burst, Phys. Rev. Lett. 128, 120401 (2022).
  18. K. Kawabata, T. Numasawa, and S. Ryu, Entanglement phase transition induced by the non-Hermitian skin effect, Phys. Rev. X 13, 021007 (2023).
  19. D. C. Brody, Biorthogonal quantum mechanics, J. Phys. A 47, 035305 (2013).
  20. N. Hatano and D. R. Nelson, Localization transitions in non-Hermitian quantum mechanics, Phys. Rev. Lett. 77, 570 (1996).
  21. K. Esaki, M. Sato, K. Hasebe, and M. Kohmoto, Edge states and topological phases in non-Hermitian systems, Phys. Rev. B 84, 205128 (2011).
  22. H. Shen, B. Zhen, and L. Fu, Topological Band Theory for Non-Hermitian Hamiltonians, Phys. Rev. Lett. 120, 146402 (2018).
  23. 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).
  24. Y. Nagai, Y. Qi, H. Isobe, V. Kozii, and L. Fu, DMFT reveals the non-Hermitian topology and Fermi arcs in heavy-fermion systems, Phys. Rev. Lett. 125, 227204 (2020).
  25. S. Yao and Z. Wang, Edge states and topological invariants of non-Hermitian systems, Phys. Rev. Lett. 121, 086803 (2018).
  26. 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).
  27. K. Yokomizo and S. Murakami, Non-Bloch band theory of non-Hermitian systems, Phys. Rev. Lett. 123, 066404 (2019).
  28. C. H. Lee and R. Thomale, Anatomy of skin modes and topology in non-Hermitian systems, Phys. Rev. B 99, 201103(R) (2019).
  29. D. S. Borgnia, A. J. Kruchkov, and R.-J. Slager, Non-Hermitian boundary modes and topology, Phys. Rev. Lett. 124, 056802 (2020).
  30. K. Zhang, Z. Yang, and C. Fang, Correspondence between winding numbers and skin modes in non-Hermitian systems, Phys. Rev. Lett. 125, 126402 (2020).
  31. N. Okuma, K. Kawabata, K. Shiozaki, and M. Sato, Topological origin of non-Hermitian skin effects, Phys. Rev. Lett. 124, 086801 (2020).
  32. 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).
  33. 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).
  34. S. Longhi, Probing non-Hermitian skin effect and non-Bloch phase transitions, Phys. Rev. Res. 1, 023013 (2019).
  35. T.-S. Deng and W. Yi, Non-Bloch topological invariants in a non-Hermitian domain wall system, Phys. Rev. B 100, 035102 (2019).
  36. 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).
  37. 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).
  38. 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. U.S.A. 117, 29561 (2020).
  39. L. Li, C. H. Lee, S. Mu, and J. Gong, Critical non-Hermitian skin effect, Nat. Commun. 11, 5491 (2020).
  40. L. Li, C. H. Lee, and J. Gong, Impurity induced scale-free localization, Commun. Phys. 4, 42 (2021).
  41. J. Claes and T. L. Hughes, Skin effect and winding number in disordered non-Hermitian systems, Phys. Rev. B 103, L140201 (2021).
  42. M. Lu, X.-X. Zhang, and M. Franz, Magnetic suppression of non-Hermitian skin effects, Phys. Rev. Lett. 127, 256402 (2021).
  43. L. Li and C. H. Lee, Non-Hermitian pseudo-gaps, Sci. Bull. 67, 685 (2022).
  44. R. Yang, J. W. Tan, T. Tai, J. M. Koh, L. Li, S. Longhi, and C. H. Lee, Designing non-Hermitian real spectra through electrostatics, Sci. Bull. 67, 1865 (2022).
  45. 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).
  46. Z. Gu, H. Gao, H. Xue, J. Li, Z. Su, and J. Zhu, Transient non-Hermitian skin effect, Nat. Commun. 13, 7668 (2022).
  47. N. Cheng, C. Shu, K. Zhang, X. Mao, and K. Sun, Universal spectral moment theorem and its applications in non-Hermitian systems, Phys. Rev. Lett. 133, 216401 (2024).
  48. K. Ochkan, R. Chaturvedi, V. Könye, L. Veyrat, R. Giraud, D. Mailly, A. Cavanna, U. Gennser, E. M. Hankiewicz, B. Büchner, J. van den Brink, J. Dufouleur, and I. C. Fulga, Non-Hermitian topology in a multi-terminal quantum Hall device, Nat. Phys. 20, 395 (2024).
  49. N. Okuma and M. Sato, Non-Hermitian topological phenomena: A review, Annu. Rev. Condens. Matter Phys. 14, 83 (2023).
  50. R. Lin, T. Tai, L. Li, and C. H. Lee, Topological non-Hermitian skin effect, Front. Phys. 18, 53605 (2023).
  51. X. Zhang, T. Zhang, M.-H. Lu, and Y.-F. Chen, A review on non-Hermitian skin effect, Adv. Phys. 7, 2109431 (2022).
  52. T. E. Lee, Anomalous edge state in a non-Hermitian lattice, Phys. Rev. Lett. 116, 133903 (2016).
  53. 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, Phys. Rev. B 97, 121401(R) (2018).
  54. Y. Xiong, Why does bulk boundary correspondence fail in some non-Hermitian topological models, J. Phys. Comm. 2, 035043 (2018).
  55. C.-K. Chiu, J. C. Y. Teo, A. P. Schnyder, and S. Ryu, Classification of topological quantum matter with symmetries, Rev. Mod. Phys. 88, 035005 (2016).
  56. K. Kawabata, K. Shiozaki, M. Ueda, and M. Sato, Symmetry and topology in non-Hermitian physics, Phys. Rev. X 9, 041015 (2019).
  57. Y. Yi and Z. Yang, Non-Hermitian skin modes induced by on-site dissipations and chiral tunneling effect, Phys. Rev. Lett. 125, 186802 (2020).
  58. 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).
  59. L. Li, S. Mu, C. H. Lee, and J. Gong, Quantized classical response from spectral winding topology, Nat. Commun. 12, 5294 (2021).
  60. S. Longhi, D. Gatti, and G. D. Valle, Robust light transport in non-Hermitian photonic lattices, Sci. Rep. 5, 13376 (2015).
  61. A. McDonald, T. Pereg-Barnea, and A. A. Clerk, Phase-dependent chiral transport and effective non-Hermitian dynamics in a bosonic Kitaev-Majorana chain, Phys. Rev. X 8, 041031 (2018).
  62. J. Y. Lee, J. Ahn, H. Zhou, and A. Vishwanath, Topological correspondence between Hermitian and non-Hermitian systems: Anomalous dynamics, Phys. Rev. Lett. 123, 206404 (2019).
  63. T. Bessho and M. Sato, Nielsen-Ninomiya theorem with bulk topology: Duality in Floquet and non-Hermitian systems, Phys. Rev. Lett. 127, 196404 (2021).
  64. 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).
  65. K. Zhang, C. Fang, and Z. Yang, Dynamical degeneracy splitting and directional invisibility in non-Hermitian systems, Phys. Rev. Lett. 131, 036402 (2023).
  66. J. C. Budich and E. J. Bergholtz, Non-Hermitian topological sensors, Phys. Rev. Lett. 125, 180403 (2020).
  67. A. McDonald and A. A. Clerk, Exponentially-enhanced quantum sensing with non-Hermitian lattice dynamics, Nat. Commun. 11, 5382 (2020).
  68. L. Bao, B. Qi, and D. Dong, Exponentially enhanced quantum non-Hermitian sensing via optimized coherent drive, Phys. Rev. Appl. 17, 014034 (2022).
  69. S. Longhi, Self-healing of non-Hermitian topological skin modes, Phys. Rev. Lett. 128, 157601 (2022).
  70. Z. Gong, M. Bello, D. Malz, and F. K. Kunst, Anomalous behaviors of quantum emitters in non-Hermitian baths, Phys. Rev. Lett. 129, 223601 (2022).
  71. Z. Fang, C. Fang, and K. Zhang, Point-gap bound states in non-Hermitian systems, Phys. Rev. B 108, 165132 (2023).
  72. 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).
  73. W. Wang, X. Wang, and G. Ma, Non-Hermitian morphing of topological modes, Nature (London) 608, 50 (2022).
  74. K. Kawabata, M. Sato, and K. Shiozaki, Higher-order non-Hermitian skin effect, Phys. Rev. B 102, 205118 (2020).
  75. S. Yao, F. Song, and Z. Wang, Non-Hermitian Chern bands, Phys. Rev. Lett. 121, 136802 (2018).
  76. C. H. Lee, L. Li, and J. Gong, Hybrid higher-order skin-topological modes in nonreciprocal systems, Phys. Rev. Lett. 123, 016805 (2019).
  77. 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).
  78. L. Li, C. H. Lee, and J. Gong, Topological switch for non-Hermitian skin effect in cold-atom systems with loss, Phys. Rev. Lett. 124, 250402 (2020).
  79. D. Zou, T. Chen, W. He, J. Bao, C. H. Lee, H. Sun, and X. Zhang, Observation of hybrid higher-order skin-topological effect in non-Hermitian topolectrical circuits, Nat. Commun. 12, 7201 (2021).
  80. 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).
  81. K. Zhang, Z. Yang, and C. Fang, Universal non-Hermitian skin effect in two and higher dimensions, Nat. Commun. 13, 2496 (2022).
  82. Y.-C. Wang, J.-S. You, and H. H. Jen, A non-Hermitian optical atomic mirror, Nat. Commun. 13, 4598 (2022).
  83. Z. Fang, M. Hu, L. Zhou, and K. Ding, Geometry-dependent skin effects in reciprocal photonic crystals, Nanophotonics 11, 3447 (2022).
  84. Q. Zhou, J. Wu, Z. Pu, J. Lu, X. Huang, W. Deng, M. Ke, and Z. Liu, Observation of geometry-dependent skin effect in non-Hermitian phononic crystals with exceptional points, Nat. Commun. 14, 4569 (2023).
  85. W. Wang, M. Hu, X. Wang, G. Ma, and K. Ding, Experimental realization of geometry-dependent skin effect in a reciprocal two-dimensional lattice, Phys. Rev. Lett. 131, 207201 (2023).
  86. T. Wan, K. Zhang, J. Li, Z. Yang, and Z. Yang, Observation of the geometry-dependent skin effect and dynamical degeneracy splitting, Sci. Bull. 68, 2330 (2023).
  87. Y. Qin, K. Zhang, and L. Li, Geometry-dependent skin effect and anisotropic Bloch oscillations in a non-Hermitian optical lattice, Phys. Rev. A 109, 023317 (2024).
  88. K. Zhang, Z. Yang, and K. Sun, Edge theory of non-Hermitian skin modes in higher dimensions, Phys. Rev. B 109, 165127 (2024).
  89. Z.-Y. Wang, J.-S. Hong, and X.-J. Liu, Symmetric non-Hermitian skin effect with emergent nonlocal correspondence, Phys. Rev. B 108, L060204 (2023).
  90. E. Zhao, Z. Wang, C. He, T. F. J. Poon, K. K. Pak, Y.-J. Liu, P. Ren, X.-J. Liu, and G.-B. Jo, Two-dimensional non-Hermitian skin effect in an ultracold Fermi gas, Nature (London) 637, 565 (2025).
  91. K. Yokomizo and S. Murakami, Non-Bloch bands in two-dimensional non-Hermitian systems, Phys. Rev. B 107, 195112 (2023).
  92. H. Jiang and C. H. Lee, Dimensional transmutation from non-Hermiticity, Phys. Rev. Lett. 131, 076401 (2023).
  93. H.-Y. Wang, F. Song, and Z. Wang, Amoeba formulation of non-Bloch band theory in arbitrary dimensions, Phys. Rev. X 14, 021011 (2024).
  94. H. Hu, Topological origin of non-Hermitian skin effect in higher dimensions and uniform spectra, Sci. Bull. 70, 51 (2025).
  95. Z. Xu, B. Pang, K. Zhang, and Z. Yang, Two-dimensional asymptotic generalized Brillouin zone theory, arXiv:2311.16868.
  96. K. Sun, A. Souslov, X. Mao, and T. C. Lubensky, Surface phonons, elastic response, and conformal invariance in twisted kagome lattices, Proc. Natl. Acad. Sci. U.S.A. 109, 12369 (2012).
  97. D. H. Lee and J. D. Joannopoulos, Simple scheme for surface-band calculations. I, Phys. Rev. B 23, 4988 (1981).
  98. Y. Hatsugai, Edge states in the integer quantum Hall effect and the Riemann surface of the Bloch function, Phys. Rev. B 48, 11851 (1993).
  99. V. Dwivedi and V. Chua, Of bulk and boundaries: Generalized transfer matrices for tight-binding models, Phys. Rev. B 93, 134304 (2016).
  100. F. K. Kunst and V. Dwivedi, Non-Hermitian systems and topology: A transfer-matrix perspective, Phys. Rev. B 99, 245116 (2019).
  101. A. Böttcher and B. Silbermann, Introduction to Large Truncated Toeplitz Matrices (Springer Science & Business Media, New York, 2012).
  102. 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).
  103. C. Shu, K. Zhang, and K. Sun, Ultra spectral sensitivity and non-local bi-impurity bound states from quasi-long-range non-Hermitian skin modes, arXiv:2409.13623.
  104. M. Li, J. Lin, and K. Ding, Algebraic skin effect in two-dimensional non-Hermitian metamaterials, arXiv:2501.13440.

Outline

Information

Phys. Rev. X 15, 031039– Published 11 August, 2025
  • Received 25 June 2024
  • Accepted 17 July 2025
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