- Physical Review Journals
All JournalsPhysics Magazine

Physical Review B

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

Topological zero modes from the interplay of PT symmetry and anti-PT symmetry

H. C. Wu1 and L. Jin2,*

  • *Contact author: jinliang@nankai.edu.cn
PDF

Phys. Rev. B 112, 125146 – Published 19 September, 2025

DOI: https://doi.org/10.1103/pv42-w9r9

Abstract

Parity-time symmetry and anti-parity-time symmetry are complementary but generally incompatible. In this work, we investigate the coexistence of parity-time symmetry and anti-parity-time symmetry. These symmetries impose intriguing constraints on the spectrum, and the Bloch bands appear in opposite and conjugate tetrads. The bulk states, affected by the non-Hermiticity, are entirely complex; however, the topological edge states, protected by sublattice symmetry, are exactly zero. We demonstrate a two-leg ladder consisting of two Su-Schrieffer-Heeger chains connected by radiative and dissipative cross-couplings. The non-Hermiticity facilitates the formation of a 2Z point gap topology, and the topological zero modes are robust against disorder.

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. R. Lin, T. Tai, L. Li, and C. H. Lee, Topological non-Hermitian skin effect, Front. Phys. 18, 53605 (2023).
  5. N. Okuma and M. Sato, Non-Hermitian topological phenomena: A review, Annu. Rev. Condens. Matter Phys. 14, 83 (2023).
  6. X. Zhang, T. Zhang, M.-H. Lu, and Y.-F. Chen, A review on non-Hermitian skin effect, Adv. Phys. X 7, 2109431 (2022).
  7. T. E. Lee, Anomalous edge state in a non-Hermitian lattice, Phys. Rev. Lett. 116, 133903 (2016).
  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. S. Yao and Z. Wang, Edge states and topological invariants of non-Hermitian systems, Phys. Rev. Lett. 121, 086803 (2018).
  10. L. Jin and Z. Song, Bulk-boundary correspondence in a non-Hermitian system in one dimension with chiral inversion symmetry, Phys. Rev. B 99, 081103(R) (2019).
  11. C. H. Lee and R. Thomale, Anatomy of skin modes and topology in non-Hermitian systems, Phys. Rev. B 99, 201103(R) (2019).
  12. K. Yokomizo and S. Murakami, Non-Bloch band theory of non-Hermitian systems, Phys. Rev. Lett. 123, 066404 (2019).
  13. F. Song, S. Yao, and Z. Wang, Non-Hermitian skin effect and chiral damping in open quantum systems, Phys. Rev. Lett. 123, 170401 (2019).
  14. F. Song, S. Yao, and Z. Wang, Non-Hermitian topological invariants in real space, Phys. Rev. Lett. 123, 246801 (2019).
  15. N. Okuma, K. Kawabata, K. Shiozaki, and M. Sato, Topological origin of non-Hermitian skin effects, Phys. Rev. Lett. 124, 086801 (2020).
  16. K. Zhang, Z. Yang, and C. Fang, Correspondence between winding numbers and skin modes in non-Hermitian systems, Phys. Rev. Lett. 125, 126402 (2020).
  17. 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).
  18. 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)
  19. K. Wang, A. Dutt, C. C. Wojcik, and S. Fan, Topological complex-energy braiding of non-Hermitian bands, Nature (London) 598, 59 (2021).
  20. C.-X. Guo, S. Chen, K. Ding, and H. Hu, Exceptional non-Abelian topology in multiband non-Hermitian systems, Phys. Rev. Lett. 130, 157201 (2023).
  21. Q. Zhang, Y. Li, H. Sun, X. Liu, L. Zhao, X. Feng, X. Fan, and C. Qiu, Observation of acoustic non-Hermitian Bloch braids and associated topological phase transitions, Phys. Rev. Lett. 130, 017201 (2023).
  22. Y. Long, H. Xue, and B. Zhang, Unsupervised learning of topological non-Abelian braiding in non-Hermitian bands, Nat. Mach. Intell. 6, 904 (2024).
  23. J. Chen, Z. Wang, Y.-T. Tan, C. Wang, and J. Ren, Machine learning of knot topology in non-Hermitian band braids, Commun. Phys. 7, 209 (2024).
  24. 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).
  25. 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).
  26. D. S. Borgnia, A. J. Kruchkov, and R.-J. Slager, Non-Hermitian boundary modes and topology, Phys. Rev. Lett. 124, 056802 (2020).
  27. X. Zhang, G. Li, Y. Liu, T. Tai, R. Thomale, and C. H. Lee, Tidal surface states as fingerprints of non-Hermitian nodal knot metals, Commun. Phys. 4, 47 (2021).
  28. Q. Lin, T. Li, L. Xiao, K. Wang, W. Yi, and P. Xue, Topological phase transitions and mobility edges in non-Hermitian quasicrystals, Phys. Rev. Lett. 129, 113601 (2022).
  29. F. Schindler, K. Gu, B. Lian, and K. Kawabata, Hermitian bulk-non-Hermitian boundary correspondence, PRX Quantum 4, 030315 (2023).
  30. S. Hamanaka, T. Yoshida, and K. Kawabata, Non-Hermitian topology in Hermitian topological matter, Phys. Rev. Lett. 133, 266604 (2024).
  31. D. C. Ohnmacht, V. Wilhelm, H. Weisbrich, and W. Belzig, Non-Hermitian topology in multiterminal superconducting junctions, Phys. Rev. Lett. 134, 156601 (2025).
  32. J. Zhong, H. Wang, A. N. Poddubny, and S. Fan, Topological nature of edge states for one-dimensional systems without symmetry protection, Phys. Rev. Lett. 135, 016601 (2025).
  33. X. Liu, J. Wang, R. Mao, H. Hu, S.-Y. Zhu, X. Xu, H. Cai, and D.-W. Wang, Zak phase induced topological nonreciprocity, Phys. Rev. Lett. 134, 193602 (2025).
  34. H. Jia, J. Hu, R.-Y. Zhang, Y. Xiao, D. Wang, M. Wang, S. Ma, X. Ouyang, Y. Zhu, and C. T. Chan, Unconventional topological edge states in one-dimensional non-Hermitian gapless systems stemming from nonisolated hypersurface singularities, Phys. Rev. Lett. 134, 206603 (2025).
  35. 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).
  36. H. Shen, B. Zhen, and L. Fu, Topological band theory for non-Hermitian Hamiltonians, Phys. Rev. Lett. 120, 146402 (2018).
  37. W. Zhu, X. Fang, D. Li, Y. Sun, Y. Li, Y. Jing, and H. Chen, Simultaneous observation of a topological edge state and exceptional point in an open and non-Hermitian acoustic system, Phys. Rev. Lett. 121, 124501 (2018).
  38. Z. Dong, Z. Li, F. Yang, C.-W. Qiu, and J. S. Ho, Sensitive readout of implantable microsensors using a wireless system locked to an exceptional point, Nat. Electron. 2, 335 (2019).
  39. K. Yokomizo and S. Murakami, Topological semimetal phase with exceptional points in one-dimensional non-Hermitian systems, Phys. Rev. Res. 2, 043045 (2020).
  40. M. Parto, Y. G. N. Liu, B. Bahari, M. Khajavikhan, and D. N. Christodoulides, Non-Hermitian and topological photonics: Optics at an exceptional point, Nanophotonics 10, 403 (2020).
  41. W. Liu, Y. Zhang, Z. Deng, J. Ye, K. Wang, B. Wang, D. Gao, and P. Lu, On-chip chiral mode switching by encircling an exceptional point in an anti-parity-time symmetric system, Laser Photonics Rev. 16, 2100675 (2022).
  42. H. Gao, K. Sun, D. Qu, K. Wang, L. Xiao, W. Yi, and P. Xue, Photonic chiral state transfer near the Liouvillian exceptional point, Phys. Rev. Lett. 134, 146602 (2025).
  43. K. Kawabata, K. Shiozaki, M. Ueda, and M. Sato, Symmetry and topology in non-Hermitian physics, Phys. Rev. X 9, 041015 (2019).
  44. H. Zhou and J. Y. Lee, Periodic table for topological bands with non-Hermitian symmetries, Phys. Rev. B 99, 235112 (2019).
  45. Y. C. Hu and T. L. Hughes, Absence of topological insulator phases in non-Hermitian PT-symmetric Hamiltonians, Phys. Rev. B 84, 153101 (2011).
  46. Y. V. Kartashov, V. V. Konotop, and L. Torner, Topological states in partially-PT-symmetric azimuthal potentials, Phys. Rev. Lett. 115, 193902 (2015).
  47. S. Malzard, C. Poli, and H. Schomerus, Topologically protected defect states in open photonic systems with non-Hermitian charge-conjugation and parity-time symmetry, Phys. Rev. Lett. 115, 200402 (2015).
  48. L. Xiao, X. Zhan, Z. H. Bian, K. K. Wang, X. Zhang, X. P. Wang, J. Li, K. Mochizuki, D. Kim, N. Kawakami, W. Yi, H. Obuse, B. C. Sanders, and P. Xue, Observation of topological edge states in parity-time-symmetric quantum walks, Nat. Phys. 13, 1117 (2017).
  49. H. Menke and M. M. Hirschmann, Topological quantum wires with balanced gain and loss, Phys. Rev. B 95, 174506 (2017).
  50. K. Takata and M. Notomi, Photonic topological insulating phase induced solely by gain and loss, Phys. Rev. Lett. 121, 213902 (2018).
  51. C.-X. Guo, X.-R. Wang, C. Wang, and S.-P. Kou, Non-Hermitian dynamic strings and anomalous topological degeneracy on a non-Hermitian toric-code model with parity-time symmetry, Phys. Rev. B 101, 144439 (2020).
  52. A. Stegmaier, S. Imhof, T. Helbig, T. Hofmann, C. H. Lee, M. Kremer, A. Fritzsche, T. Feichtner, S. Klembt, S. Höfling, I. Boettcher, I. C. Fulga, O. G. Schmidt, M. Greiter, T. Kiessling, A. Szameit, and R. Thomale, Topological defect engineering and PT symmetry in non-Hermitian electrical circuits, Phys. Rev. Lett. 126, 215302 (2021).
  53. X. Yang, J. Li, Y. Ding, M. Xu, X.-F. Zhu, and J. Zhu, Observation of transient parity-time symmetry in electronic systems, Phys. Rev. Lett. 128, 065701 (2022).
  54. A. Fritzsche, T. Biesenthal, L. J. Maczewsky, K. Becker, M. Ehrhardt, M. Heinrich, R. Thomale, Y. N. Joglekar, and A. Szameit, Parity-time-symmetric photonic topological insulator, Nat. Mater. 23, 377 (2024).
  55. Z. Lei, C. H. Lee, and L. Li, Activating non-Hermitian skin modes by parity-time symmetry breaking, Commun. Phys. 7, 100 (2024).
  56. W. Wang, X. Wang, and G. Ma, Anderson transition at complex energies in one-dimensional parity-time-symmetric disordered systems, Phys. Rev. Lett. 134, 066301 (2025).
  57. J. Li, W.-C. Jiang, Q.-X. Li, J.-J. Zhu, and K. Chang, Multiwrapped PT-symmetric topological insulator with complete spin polarization, Phys. Rev. B 111, 165135 (2025).
  58. V. V. Konotop and D. A. Zezyulin, Odd-time reversal PT symmetry induced by an anti-PT-symmetric medium, Phys. Rev. Lett. 120, 123902 (2018).
  59. Y. Choi, C. Hahn, J. W. Yoon, and S. H. Song, Observation of an anti-PT-symmetric exceptional point and energy-difference conserving dynamics in electrical circuit resonators, Nat. Commun. 9, 2182 (2018).
  60. Y. Li, Y.-G. Peng, L. Han, M.-A. Miri, W. Li, M. Xiao, X.-F. Zhu, J. Zhao, A. Alù, S. Fan, and C.-W. Qiu, Anti-parity-time symmetry in diffusive systems, Science 364, 170 (2019).
  61. A. Bergman, R. Duggan, K. Sharma, M. Tur, A. Zadok, and A. Alù, Observation of anti-parity-time-symmetry, phase transitions and exceptional points in an optical fibre, Nat. Commun. 12, 486 (2021).
  62. X.-W. Luo, C. Zhang, and S. Du, Quantum squeezing and sensing with pseudo-anti-parity-time symmetry, Phys. Rev. Lett. 128, 173602 (2022).
  63. Y. Yang, X. Xie, Y. Li, Z. Zhang, Y. Peng, C. Wang, E. Li, Y. Li, H. Chen, and F. Gao, Radiative anti-parity-time plasmonics, Nat. Commun. 13, 7678 (2022).
  64. Y. Song, S. Ke, Y. Chen, and M. Wang, Mode-locking in anti-PT symmetric frequency lattices, Appl. Phys. Lett. 122, 151106 (2023).
  65. Z. Guo, F. Yang, H. Zhang, X. Wu, Q. Wu, K. Zhu, J. Jiang, H. Jiang, Y. Yang, Y. Li, and H. Chen, Level pinning of anti-PT-symmetric circuits for efficient wireless power transfer, Nat. Sci. Rev. 11, nwad172 (2023).
  66. M. A. Selim, M. Ehrhardt, Y. Ding, H. M. Dinani, Q. Zhong, A. Perez-Leija, S. K. Özdemir, M. Heinrich, A. Szameit, D. N. Christodoulides, and M. Khajavikhan, Selective filtering of photonic quantum entanglement via anti-parity-time symmetry, Science 387, 1424 (2025).
  67. B. Li, Y. Zhang, J. Wang, J. Liu, M. Li, W. Wang, J. Ma, P. Yuan, D. Zhang, H. Zhu, H. Zhang, and L. Qian, Non-Hermitian optical parametric systems with anti-parity-time symmetry, Phys. Rev. Res. 7, 023012 (2025).
  68. S. Lieu, Topological phases in the non-Hermitian Su-Schrieffer-Heeger model, Phys. Rev. B 97, 045106 (2018).
  69. L. Du, J.-H. Wu, M. Artoni, and G. C. L. Rocca, Phase-dependent topological interface state and spatial adiabatic passage in a generalized Su-Schrieffer-Heeger model, Phys. Rev. A 100, 012112 (2019).
  70. X. M. Yang, H. C. Wu, P. Wang, L. Jin, and Z. Song, Visualizing one-dimensional non-Hermitian topological phases, J. Phys. Commun. 4, 095005 (2020).
  71. Z. Xu, R. Zhang, S. Chen, L. Fu, and Y. Zhang, Fate of zero modes in a finite Su-Schrieffer-Heeger model with PT symmetry, Phys. Rev. A 101, 013635 (2020).
  72. H. C. Wu, L. Jin, and Z. Song, Topology of an anti-parity-time symmetric non-Hermitian Su-Schrieffer-Heeger model, Phys. Rev. B 103, 235110 (2021).
  73. X. Zhao, Y. Xing, L. Qi, S. Liu, S. Zhang, and H.-F. Wang, Real-potential-driven anti-PT-symmetry breaking in non-Hermitian Su-Schrieffer-Heeger model, New J. Phys. 23, 073043 (2021).
  74. Y. Li, Y. Ao, X. Hu, C. Lu, C. T. Chan, and Q. Gong, Unsupervised learning of non-Hermitian photonic bulk topology, Laser Photonics Rev. 17, 2300481 (2023).
  75. W. Yan, R. Chen, W. Liu, Y. Tan, Y. Kivshar, and F. Chen, Tunable edge states and topological solitons in non-Hermitian photonic lattices, Laser Photonics Rev. 19, 2402126 (2025).
  76. B. Hu, Z. Zhang, Z. Yue, D. Liao, Y. Liu, H. Zhang, Y. Cheng, X. Liu, and J. Christensen, Anti-parity-time symmetry in a Su-Schrieffer-Heeger sonic lattice, Phys. Rev. Lett. 131, 066601 (2023).
  77. J. Qian, J. Li, S.-Y. Zhu, J. Q. You, and Y.-P. Wang, Probing PT-symmetry breaking of non-Hermitian topological photonic states via strong photon-magnon coupling, Phys. Rev. Lett. 132, 156901 (2024).
  78. Y.-M. Hu, H.-Y. Wang, Z. Wang, and F. Song, Geometric origin of non-Bloch symmetry breaking, Phys. Rev. Lett. 132, 050402 (2024).
  79. M. Yang and C. H. Lee, Percolation-induced PT symmetry breaking, Phys. Rev. Lett. 133, 136602 (2024).
  80. C. Poli, M. Bellec, U. Kuhl, F. Mortessagne, and H. Schomerus, Selective enhancement of topologically induced interface states in a dielectric resonator chain, Nat. Commun. 6, 6710 (2015).
  81. P. St-Jean, V. Goblot, E. Galopin, A. Lemaitre, T. Ozawa, L. Le Gratiet, I. Sagnes, J. Bloch, and A. Amo, Lasing in topological edge states of a one-dimensional lattice, Nat. Photonics 11, 651 (2017).
  82. S. Weimann, M. Kremer, Y. Plotnik, Y. Lumer, S. Nolte, K. G. Makris, M. Segev, M. C. Rechtsman, and A. Szameit, Topologically protected bound states in photonic parity-time-symmetric crystals, Nat. Mater. 16, 433 (2017).
  83. H. Zhao, P. Miao, M. H. Teimourpour, S. Malzard, R. El-Ganainy, H. Schomerus, and L. Feng, Topological hybrid silicon microlasers, Nat. Commun. 9, 981 (2018).
  84. 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).
  85. M. Parto, S. Wittek, H. Hodaei, G. Harari, M. A. Bandres, J. Ren, M. C. Rechtsman, M. Segev, D. N. Christodoulides, and M. Khajavikhan, Edge-mode lasing in 1D topological active arrays, Phys. Rev. Lett. 120, 113901 (2018).
  86. G. Harari, M. A. Bandres, Y. Lumer, M. C. Rechtsman, Y. D. Chong, M. Khajavikhan, D. N. Christodoulides, and M. Segev, Topological insulator laser: Theory, Science 359, eaar4003 (2018); M. A. Bandres, S. Wittek, G. Harari, M. Parto, J. Ren, M. Segev, D. Christodoulides, and M. Khajavikhan, Topological insulator laser: Experiments, ibid. 359, eaar4005 (2018).
  87. Y. V. Kartashov and D. V. Skryabin, Two-dimensional topological polariton laser, Phys. Rev. Lett. 122, 083902 (2019).
  88. Y. Ota, K. Takata, T. Ozawa, A. Amo, Z. Jia, B. Kante, M. Notomi, Y. Arakawa, and S. Iwamoto, Active topological photonics, Nanophotonics 9, 547 (2020).
  89. X. Yin and C. Peng, Manipulating light radiation from a topological perspective, Photonics Res. 8, B25 (2020).
  90. H. Gao, H. Xue, Q. Wang, Z. Gu, T. Liu, J. Zhu, and B. Zhang, Observation of topological edge states induced solely by non-Hermiticity in an acoustic crystal, Phys. Rev. B 101, 180303(R) (2020).
  91. K. Xu, X. Zhang, K. Luo, R. Yu, D. Li, and H. Zhang, Coexistence of topological edge states and skin effects in the non-Hermitian Su-Schrieffer-Heeger model with long-range nonreciprocal hopping in topoelectric realizations, Phys. Rev. B 103, 125411 (2021).
  92. Z. Lin, J. Li, W. Song, S. Su, J. Sun, S. Wu, C. Huang, S. Zhu, and T. Li, Dissipative topological dynamics in optical waveguides: Sensitivity versus robustness, Phys. Rev. Lett. 134, 223802 (2025).
  93. Z. Lippo, E. L. Pereira, J. L. Lado, and G. Chen, Topological zero modes and correlation pumping in an engineered Kondo lattice, Phys. Rev. Lett. 134, 116605 (2025).
  94. S. Longhi, Erratic non-Hermitian skin localization, Phys. Rev. Lett. 134, 196302 (2025).
  95. S. Longhi, Non-Bloch-band collapse and chiral Zener tunneling, Phys. Rev. Lett. 124, 066602 (2020).
  96. K. Monkman and J. Sirker, Hidden zero modes and topology of multiband non-Hermitian systems, Phys. Rev. Lett. 134, 056601 (2025).
  97. C. H. Lee, S. Imhof, C. Berger, F. Bayer, J. Brehm, L. W. Molenkamp, T. Kiessling, and R. Thomale, Topolectrical circuits, Commun. Phys. 1, 39 (2018).
  98. D. Xie, W. Gou, T. Xiao, B. Gadway, and B. Yan, Topological characterizations of an extended Su–Schrieffer–Heeger model, npj Quantum Inf. 5, 55 (2019).
  99. F. Yang, Y.-C. Liu, and L. You, Anti-PT symmetry in dissipatively coupled optical systems, Phys. Rev. A 96, 053845 (2017).
  100. H. C. Wu, X. M. Yang, L. Jin, and Z. Song, Untying links through anti-parity-time-symmetric coupling, Phys. Rev. B 102, 161101(R) (2020).

Outline

Information

Phys. Rev. B 112, 125146– Published 19 September, 2025
  • Received 22 July 2025
  • Revised 4 September 2025
  • Accepted 5 September 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