Complex energy plane and topological invariant in non-Hermitian systems

Abstract

Non-Hermitian systems as theoretical models of open or dissipative systems exhibit rich novel physical properties and fundamental issues in condensed matter physics. We propose a generalized local–global correspondence between the pseudo-boundary states in the complex energy plane and topological invariants of quantum states. We find that the patterns of the pseudo-boundary states in the complex energy plane mapped to the Brillouin zone are topological invariants against the parameter deformation. We demonstrate this approach by the non-Hermitian Chern insulator model. We give the consistent topological phases obtained from the Chern number and vorticity. We also find some novel topological invariants embedded in the topological phases of the Chern insulator model, which enrich the phase diagram of the non-Hermitian Chern insulators model beyond that predicted by the Chern number and vorticity. We also propose a generalized vorticity and its flipping index to understand physics behind this novel local–global correspondence and discuss the relationships between the local–global correspondence and the Chern number as well as the transformation between the Brillouin zone and the complex energy plane. These novel approaches provide insights to how topological invariants may be obtained from local information as well as the global property of quantum states, which is expected to be applicable in more generic non-Hermitian systems.

Graphical abstract

Keywords

topological invariant / Chern number / non-Hermitian system

References

[1] R. El-Ganainy , K. G. Makris , M. Khajavikhan , Z. H. Musslimani , S. Rotter , and D. N. Christodoulides , NonHermitian physics and PT symmetry, Nat. Phys. 14 (1), 11 (2018)

[2] Z. Gong , Y. Ashida , K. Kawabata , K. Takasan , S. Higashikawa , and M. Ueda , Topological phases of nonHermitian systems, Phys. Rev. X 8 (3), 031079 (2018)

[3] K. Kawabata , K. Shiozaki , M. Ueda , and M. Sato , Symmetry and topology in non-Hermitian physics, Phys. Rev. X 9 (4), 041015 (2019)

[4] M. He , H. Sun , and L. H. Qing , Topological insulator: Spintronics and quantum computations, Front. Phys. 14 (4), 43401 (2019)

[5] V. Y. Chernyak , J. R. Klein , and N. A. Sinitsyn , Quantization and fractional quantization of currents in periodically driven stochastic systems (I): Average currents, J. Chem. Phys. 136 (15), 154107 (2012)

[6] J. Qi , H. Liu , H. Jiang , and X. C. Xie , Dephasing effects in topological insulators, Front. Phys. 14 (4), 43403 (2019)

[7] S. D. Liang and G. Y. Huang , Topological invariance and global Berry phase in non-Hermitian systems, Phys. Rev. A 87 (1), 012118 (2013)

[8] A. Mostafazadeh , Pseudo-Hermitian representation of quantum mechanics, Int. J. Geom. Methods Mod. Phys. 07 (07), 1191 (2020)

[9] A. Mostafazadeh , Energy observable for a quantum system with a dynamical Hilbert space and a global geometric extension of quantum theory, Phys. Rev. D 98 (4), 046022 (2018)

[10] Y. Chen and H. Zhai , Hall conductance of a non-Hermitian Chern insulator, Phys. Rev. B 98 (24), 245130 (2018)

[11] Y. X. Zhao , Equivariant PT-symmetric real Chern insulators, Front. Phys. 15 (1), 13603 (2020)

[12] A. Fan , G. Y. Huang , and S. D. Liang , Complex Berry curvature pair and quantum Hall admittance in nonHermitian systems, J. Phys. Commun. 4 (11), 115006 (2020)

[13] Q. Niu , Advances on topological materials, Front. Phys. 15 (4), 43601 (2020)

[14] Y. Xu , New physics in old material: Topological and superconducting properties of stanene, Front. Phys. 15 (5), 53202 (2020)

[15] M. Yang , X. L. Zhang , and W. M. Liu , Tunable topological quantum states in three- and two-dimensional materials, Front. Phys. 10 (2), 161 (2015)

[16] K. Kawabata , K. Shiozaki , and M. Ueda , Anomalous helical edge states in a non-Hermitian Chern insulator, Phys. Rev. B 98 (16), 165148 (2018)

[17] A. Ghatak and T. Das , New topological invariants in nonHermitian systems, J. Phys.: Condens. Matter 31, 263001 (2019)

[18] H. Shen , B. Zhen , and L. Fu , Topological band theory for non-Hermitian Hamiltonians, Phys. Rev. Lett. 120 (14), 146402 (2018)

[19] T. E. Lee , Anomalous edge state in a non-Hermitian lattice, Phys. Rev. Lett. 116 (13), 133903 (2016)

[20] V. M. M. Alvarez , J. E. B. Vargas , and L. E. F. F. Torres , Torres, Non-Hermitian robust edge states in one dimension: Anomalous localization and eigenspace condensation at exceptional points, Phys. Rev. B 97, 121401(R) (2018)

[21] V. M. M. Alvarez , J. E. B. Vargas , M. Berdakin , and L. E. F. F. Torres , Topological states of non-Hermitian systems, Eur. Phys. J. Spec. Top. 227 (12), 1295 (2018)

[22] K. Kawabata , K. Shiozaki , and M. Ueda , Anomalous helical edge states in a non-Hermitian Chern insulator, Phys. Rev. B 98 (16), 165148 (2018)

[23] T. Liu , Y. R. Zhang , Q. Ai , Z. Gong , K. Kawabata , M. Ueda , and F. Nori , Second-order topological phases in nonHermitian systems, Phys. Rev. Lett. 122 (7), 076801 (2019)

[24] F. K. Kunst , E. Edvardsson , J. C. Budich , and E. J. Bergholtz , Biorthogonal bulk–boundary correspondence in non-Hermitian systems, Phys. Rev. Lett. 121 (2), 026808 (2018)

[25] S. Yao and Z. Wang , Edge states and topological invariants of non-Hermitian systems, Phys. Rev. Lett. 121 (8), 086803 (2018)

[26] S. Yao , F. Song , and Z. Wang , Non-Hermitian Chern bands, Phys. Rev. Lett. 121 (13), 136802 (2018)

[27] K. Esaki , M. Sato , K. Hasebe , and M. Kohmoto , Edge states and topological phases in non-Hermitian systems, Phys. Rev. B 84 (20), 205128 (2011)

[28] B. Zhu , R. Lu , and S. Chen, PT symmetry in the non-Hermitian Su–Schrieffer–Heeger model with complex boundary potentials, Phys. Rev. A 89 (6), 062102 (2014)

[29] H. Jiang , C. Yang , and S. Chen , Topological invariants and phase diagrams for one-dimensional two-band nonHermitian systems without chiral symmetry, Phys. Rev. A 98 (5), 052116 (2018)

[30] C. Yin , H. Jiang , L. Li , R. Lu , and S. Chen , Geometrical meaning of winding number and its characterization of topological phases in one-dimensional chiral nonHermitian systems, Phys. Rev. A 97 (5), 052115 (2018)

[31] F. Dangel , M. Wagner , H. Cartarius , J. Main , and G. Wunner , Topological invariants in dissipative extensions of the Su–Schrieffer–Heeger model, Phys. Rev. A 98 (1), 013628 (2018)

[32] S. Lieu , Topological phases in the non-Hermitian Su– Schrieffer–Heeger model, Phys. Rev. B 97 (4), 045106 (2018)

[33] R. Chen , C. Z. Chen , B. Zhou , and D. H. Xu , Finite-size effects in non-Hermitian topological systems, Phys. Rev. B 99 (15), 155431 (2019)

[34] 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 (4), 040401 (2017)

[35] J. Y. Lee , J. Ahn , H. Zhou , and A. Vishwanath , Topological correspondence between Hermitian and non-Hermitian systems: Anomalous dynamics, Phys. Rev. Lett. 123 (20), 206404 (2019)

[36] D. C. Brody , Biorthogonal quantum mechanics, J. Phys. A: Math. Theor. 47, 035305 (2014)

[37] D. C. Brody , Consistency of PT-symmetric quantum mechanics J. Phys. A: Math. Theor. 49, 10LT03 (2016)

[38] L. Zhang , L. Zhang , S. Niu , and X. J. Liu , Dynamical classification of topological quantum phases, Sci. Bull. (Beijing) 63 (21), 1385 (2018)

[39] E. Zeidler , Quantum Field Theory (I): Basics in Mathematics and Physics, Springer, (2006)

[40] A. Bohm , A. Mostafazadeh , H. Koizumi , Q. Niu , and J. Zwanziger , The Geometric Phase in Quantum Systems, Springer, New York, (2003)

[41] D. Xiao , M. C. Chang , and Q. Niu , Berry phase effects on electronic properties, Rev. Mod. Phys. 82 (3), 1959 (2010)

[42] G. von Gersdorff , S. Panahiyan , and W. Chen , Unification of topological invariants in Dirac models, Phys. Rev. B 103 (24), 245146 (2021)

[43] W. Chen , M. Legner , A. Ruegg , and M. Sigrist , Correlation length, universality classes, and scaling laws associated with topological phase transitions, Phys. Rev. B 95 (7), 075116 (2017)

[44] F. Bernardini , J. Mittleman , H. Rushmeier , C. Silva , and G. Taubin , The Ball–Pivoting algorithm for surface reconstruction, IEEE Trans. Vis. Comput. Graph. 5 (4), 349 (1999)

[45] W. Chen , M. Sigrist , and A. P. Schnyder , Scaling theory of Z₂ topological invariants, J. Phys.: Condens. Matter 28 (36), 365501 (2016)

[46] W. Chen and A. P. Schnyder , Universality classes of topological phase transitions with higher-order band crossing, New J. Phys. 21 (7), 073003 (2019)

[47] X. G. Wen , A theory of 2+1D bosonic topological orders, Natl. Sci. Rev. 3 (1), 68 (2016)

[48] X. G. Wen , Topological orders in rigid states, Int. J. Mod. Phys. B 04 (02), 239 (1990)

[49] S. Kou , Z. Weng , and X. Wen , Mutual Chern–Simons theory and its applications in condensed matter physics, Front. Phys. 2 (1), 31 (2007)

[50] A. Fan , S. D. Liang , submitted to Annalen der Physik

[51] A. Fan , Ph. D. dissertation, Sun Yat-Sen University, Guangzhou, China, 2021

RIGHTS & PERMISSIONS

2022 Higher Education Press