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  • Letter
  • Open Access

Spontaneous Chern-Euler duality transitions

Kang Yang1,*,†, Zhi Li2, Peng Xue3, Emil J. Bergholtz4, and Piet W. Brouwer1

  • *Present address: Westlake University.
  • †Contact author: yangkang@westlake.edu.cn
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Phys. Rev. B 114, L171409 – Published 30 September, 2026

DOI: https://doi.org/10.1103/g3pz-tnq7

Abstract

Topological phase transitions are typically characterized by abrupt changes in a quantized invariant. Here we report a contrasting paradigm in spontaneous parity-time symmetry-breaking transitions induced by non-Hermiticity. The topological invariant remains conserved, but its nature transitions between the Chern number, characteristic of chiral transport in complex bands, and the Euler number, which characterizes the number of nodal points in pairs of real bands. This conserved morphing of topological properties applies to all pairs of bands whose spectral gaps to other bands remain open during the transition. It features qualitative changes in the non-Abelian geometric phases, which quantize in different forms across the transition. Our findings establish a generic topological duality principle governing transitions across symmetry classes and reveal unique connections among topology, symmetry, and non-Abelian gauge structure in nonunitary processes.

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References (64)

  1. F. D. M. Haldane, Nobel lecture: Topological quantum matter, Rev. Mod. Phys. 89, 040502 (2017).
  2. N. P. Armitage, E. J. Mele, and A. Vishwanath, Weyl and Dirac semimetals in three-dimensional solids, Rev. Mod. Phys. 90, 015001 (2018).
  3. M.-A. Miri and A. Alù, Exceptional points in optics and photonics, Science 363, eaar7709 (2019).
  4. L. Feng, R. El-Ganainy, and L. Ge, Non-Hermitian photonics based on parity–time symmetry, Nat. Photon. 11, 752 (2017).
  5. R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S. Rotter, and D. N. Christodoulides, Non-Hermitian physics and PT symmetry, Nat. Phys. 14, 11 (2018).
  6. Ş. K. Özdemir, S. Rotter, F. Nori, and L. Yang, Parity–time symmetry and exceptional points in photonics, Nat. Mater. 18, 783 (2019).
  7. E. J. Bergholtz, J. C. Budich, and F. K. Kunst, Exceptional topology of non-Hermitian systems, Rev. Mod. Phys. 93, 015005 (2021).
  8. C. M. Bender and D. W. Hook, PT-symmetric quantum mechanics, Rev. Mod. Phys. 96, 045002 (2024).
  9. C. M. Bender, Making sense of non-Hermitian Hamiltonians, Rep. Prog. Phys. 70, 947 (2007).
  10. A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, Observation of PT-symmetry breaking in complex optical potentials, Phys. Rev. Lett. 103, 093902 (2009).
  11. L. Jin and Z. Song, Solutions of PT-symmetric tight-binding chain and its equivalent Hermitian counterpart, Phys. Rev. A 80, 052107 (2009).
  12. C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, Observation of parity–time symmetry in optics, Nat. Phys. 6, 192 (2010).
  13. Y. D. Chong, L. Ge, and A. D. Stone, PT-symmetry breaking and laser-absorber modes in optical scattering systems, Phys. Rev. Lett. 106, 093902 (2011).
  14. H. Hodaei, A. U. Hassan, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D. N. Christodoulides, and M. Khajavikhan, Enhanced sensitivity at higher-order exceptional points, Nature (London) 548, 187 (2017).
  15. Z. Xiao, H. Li, T. Kottos, and A. Alù, Enhanced sensing and nondegraded thermal noise performance based on PT-symmetric electronic circuits with a sixth-order exceptional point, Phys. Rev. Lett. 123, 213901 (2019).
  16. W. Tang, X. Jiang, K. Ding, Y.-X. Xiao, Z.-Q. Zhang, C. T. Chan, and G. Ma, Exceptional nexus with a hybrid topological invariant, Science 370, 1077 (2020).
  17. P. Delplace, T. Yoshida, and Y. Hatsugai, Symmetry-protected multifold exceptional points and their topological characterization, Phys. Rev. Lett. 127, 186602 (2021).
  18. I. Mandal and E. J. Bergholtz, Symmetry and higher-order exceptional points, Phys. Rev. Lett. 127, 186601 (2021).
  19. Y. X. Zhao and Y. Lu, PT-symmetric real Dirac fermions and semimetals, Phys. Rev. Lett. 118, 056401 (2017).
  20. J. Ahn, S. Park, and B.-J. Yang, Failure of Nielsen-Ninomiya theorem and fragile topology in two-dimensional systems with space-time inversion symmetry: Application to twisted bilayer graphene at magic angle, Phys. Rev. X 9, 021013 (2019).
  21. A. Bouhon, Q. Wu, R.-J. Slager, H. Weng, O. V. Yazyev, and T. Bzdušek, Non-Abelian reciprocal braiding of Weyl points and its manifestation in ZrTe, Nat. Phys. 16, 1137 (2020).
  22. Y. X. Zhao, Equivariant PT-symmetric real Chern insulators, Front. Phys. 15, 13603 (2020).
  23. F. N. Ünal, A. Bouhon, and R.-J. Slager, Topological Euler class as a dynamical observable in optical lattices, Phys. Rev. Lett. 125, 053601 (2020).
  24. K. Wang, J.-X. Dai, L. B. Shao, S. A. Yang, and Y. X. Zhao, Boundary criticality of PT-invariant topology and second-order nodal-line semimetals, Phys. Rev. Lett. 125, 126403 (2020).
  25. M. Ezawa, Topological Euler insulators and their electric circuit realization, Phys. Rev. B 103, 205303 (2021).
  26. W. Zhao, Y.-B. Yang, Y. Jiang, Z. Mao, W. Guo, L. Qiu, G. Wang, L. Yao, L. He, Z. Zhou, et al., Quantum simulation for topological Euler insulators, Commun. Phys. 5, 223 (2022).
  27. K. Yang, Z. Li, J. L. K. König, L. Rødland, M. Stålhammar, and E. J. Bergholtz, Homotopy, symmetry, and non-Hermitian band topology, Rep. Prog. Phys. 87, 078002 (2024).
  28. H. Shen, B. Zhen, and L. Fu, Topological band theory for non-Hermitian Hamiltonians, Phys. Rev. Lett. 120, 146402 (2018).
  29. C. C. Wojcik, X.-Q. Sun, T. Bzdušek, and S. Fan, Homotopy characterization of non-Hermitian Hamiltonians, Phys. Rev. B 101, 205417 (2020).
  30. Z. Li and R. S. K. Mong, Homotopical characterization of non-Hermitian band structures, Phys. Rev. B 103, 155129 (2021).
  31. K. Wang, A. Dutt, C. C. Wojcik, and S. Fan, Topological complex-energy braiding of non-Hermitian bands, Nature (London) 598, 59 (2021).
  32. Y. S. Patil, J. Höller, P. A. Henry, C. Guria, Y. Zhang, L. Jiang, N. Kralj, N. Read, and J. G. Harris, Measuring the knot of non-Hermitian degeneracies and non-commuting braids, Nature (London) 607, 271 (2022).
  33. K. Ding, C. Fang, and G. Ma, Non-Hermitian topology and exceptional-point geometries, Nat. Rev. Phys. 4, 745 (2022).
  34. 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).
  35. A. Cerjan, S. Huang, M. Wang, K. P. Chen, Y. Chong, and M. C. Rechtsman, Experimental realization of a Weyl exceptional ring, Nat. Photon. 13, 623 (2019).
  36. G. Tarnopolsky, A. J. Kruchkov, and A. Vishwanath, Origin of magic angles in twisted bilayer graphene, Phys. Rev. Lett. 122, 106405 (2019).
  37. Y.-H. Zhang, D. Mao, and T. Senthil, Twisted bilayer graphene aligned with hexagonal boron nitride: Anomalous Hall effect and a lattice model, Phys. Rev. Res. 1, 033126 (2019).
  38. A. Bouhon and R.-J. Slager, Multi-gap topological conversion of Euler class via band-node braiding: Minimal models, PT-linked nodal rings, and chiral heirs, arXiv:2203.16741.
  39. A. Bouhon, T. Bzdušek, and R.-J. Slager, Geometric approach to fragile topology beyond symmetry indicators, Phys. Rev. B 102, 115135 (2020).
  40. P. W. Brouwer and V. Dwivedi, Homotopic classification of band structures: Stable, fragile, delicate, and stable representation-protected topology, Phys. Rev. B 108, 155137 (2023).
  41. J. X. Dai, C. Zhang, and Y. X. Zhao, Asymmetric real topology of conduction and valence bands, Phys. Rev. B 112, 035134 (2025).
  42. F. Wilczek and A. Zee, Appearance of gauge structure in simple dynamical systems, Phys. Rev. Lett. 52, 2111 (1984).
  43. J. Milnor and J. Stasheff, Characteristic Classes, Annals of Mathematics Studies (Princeton University Press, Princeton, 1974).
  44. Q. Wu, A. A. Soluyanov, and T. Bzdušek, Non-Abelian band topology in noninteracting metals, Science 365, 1273 (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. T. Kato, Perturbation Theory for Linear Operators (Springer, Berlin, Heidelberg, 2013), Vol. 132.
  47. See Supplemental Material at http://link.aps.org/supplemental/10.1103/g3pz-tnq7 for review of homotopic classification, physical illustration of the proof, background knowledge on projection operators, continuity of vector spaces, realification of bundles, and connections on bundles, which includes Refs. [60, 61, 62, 63, 64].
  48. A. Hatcher, Vector bundles and K-theory, On the Internet at https://pi.math.cornell.edu/∼hatcher/VBKT/VBpage.html (2003).
  49. R. Bott and L. W. Tu, Differential Forms in Algebraic Topology (Springer, New York, NY, 1982), Vol. 82.
  50. Y. X. Zhao, A. P. Schnyder, and Z. D. Wang, Unified theory of PT and CP invariant topological metals and nodal superconductors, Phys. Rev. Lett. 116, 156402 (2016).
  51. K. Shiozaki, M. Sato, and K. Gomi, Topological crystalline materials: General formulation, module structure, and wallpaper groups, Phys. Rev. B 95, 235425 (2017).
  52. J. Ahn, S. Park, D. Kim, Y. Kim, and B.-J. Yang, Stiefel–Whitney classes and topological phases in band theory, Chin. Phys. B 28, 117101 (2019).
  53. S.-S. Chern, Characteristic classes of Hermitian manifolds, Ann. Math. 47, 85 (1946).
  54. D. Sticlet, F. Piéchon, J.-N. Fuchs, P. Kalugin, and P. Simon, Geometrical engineering of a two-band Chern insulator in two dimensions with arbitrary topological index, Phys. Rev. B 85, 165456 (2012).
  55. T. Fukui, Y. Hatsugai, and H. Suzuki, Chern numbers in discretized Brillouin zone: Efficient method of computing (spin) Hall conductances, J. Phys. Soc. Jpn. 74, 1674 (2005).
  56. Y. Wu, W. Liu, J. Geng, X. Song, X. Ye, C.-K. Duan, X. Rong, and J. Du, Observation of parity-time symmetry breaking in a single-spin system, Science 364, 878 (2019).
  57. K. Wang, J. L. K. König, K. Yang, L. Xiao, W. Yi, E. J. Bergholtz, and P. Xue, Observation of braid-protected unpaired exceptional points, Phys. Rev. Lett. 136, 056602 (2026).
  58. K. Yang, F. Song, and P. W. Brouwer, Stable boundary modes for fragile topology from spontaneous PT-symmetry breaking, arXiv:2601.02672.
  59. O. Zilberberg, S. Huang, J. Guglielmon, M. Wang, K. P. Chen, Y. E. Kraus, and M. C. Rechtsman, Photonic topological boundary pumping as a probe of 4D quantum Hall physics, Nature (London) 553, 59 (2018).
  60. E. M. Graefe, U. Günther, H. J. Korsch, and A. E. Niederle, A non-Hermitian PT symmetric Bose–Hubbard model: eigenvalue rings from unfolding higher-order exceptional points, J. Phys. A: Math. Theor. 41, 255206 (2008).
  61. H. Zhao, S. Longhi, and L. Feng, Robust light state by quantum phase transition in non-Hermitian optical materials, Sci. Rep. 5, 17022 (2015).
  62. S. Yao, F. Song, and Z. Wang, Non-Hermitian Chern bands, Phys. Rev. Lett. 121, 136802 (2018).
  63. M. Xie and A. H. MacDonald, Nature of the correlated insulator states in twisted bilayer graphene, Phys. Rev. Lett. 124, 097601 (2020).
  64. J. Hu, R.-Y. Zhang, Y. Wang, X. Ouyang, Y. Zhu, H. Jia, and C. T. Chan, Non-Hermitian swallowtail catastrophe revealing transitions among diverse topological singularities, Nat. Phys. 19, 1098 (2023).

Outline

Information

Phys. Rev. B 114, L171409– Published 30 September, 2026
  • Received 5 May 2025
  • Revised 27 August 2026
  • Accepted 1 September 2026
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