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  • Letter
  • Open Access

Quantum geometric tensor and wavepacket dynamics in two-dimensional non-Hermitian systems

Y.-M. Robin Hu, Elena A. Ostrovskaya, and Eliezer Estrecho

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Phys. Rev. Research 7, L012067 – Published 12 March, 2025

DOI: https://doi.org/10.1103/PhysRevResearch.7.L012067

Abstract

The quantum geometric tensor (QGT) characterizes the local geometry of quantum states, and its components directly account for the dynamical effects observed, e.g., in condensed-matter systems. In this work, we address the problem of extending the QGT formalism to non-Hermitian systems with gain and loss. In particular, we investigate a wavepacket dynamics in two-band non-Hermitian systems to elucidate how non-Hermiticity affects the definition of QGT. We employ first-order perturbation theory to account for nonadiabatic corrections due to interband mixing. Our results suggest that two different generalizations of the QGT, one defined using only the right eigenstates and the other one using both the left and right eigenstates, both play a significant role in wavepacket dynamics. We then determine the accuracy of the perturbative approach by simulating a wavepacket dynamics in a well-studied physical non-Hermitian system—exciton polaritons in a semiconductor microcavity. Our work aids deeper understanding of quantum geometry and dynamical behavior in non-Hermitian systems.

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References (93)

  1. J. Provost and G. Vallee, Riemannian structure on manifolds of quantum states, Commun. Math. Phys. 76, 289 (1980).
  2. M. V. Berry, Quantal phase factors accompanying adiabatic changes, Proc. R. Soc. London A 392, 45 (1984).
  3. K. v. Klitzing, G. Dorda, and M. Pepper, New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance, Phys. Rev. Lett. 45, 494 (1980).
  4. R. B. Laughlin, Quantized Hall conductivity in two dimensions, Phys. Rev. B 23, 5632 (1981).
  5. D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, Quantized Hall conductance in a two-dimensional periodic potential, Phys. Rev. Lett. 49, 405 (1982).
  6. J. E. Avron, R. Seiler, and B. Simon, Homotopy and quantization in condensed matter physics, Phys. Rev. Lett. 51, 51 (1983).
  7. Q. Niu, D. J. Thouless, and Y.-S. Wu, Quantized Hall conductance as a topological invariant, Phys. Rev. B 31, 3372 (1985).
  8. D. Xiao, M.-C. Chang, and Q. Niu, Berry phase effects on electronic properties, Rev. Mod. Phys. 82, 1959 (2010).
  9. O. Bleu, D. D. Solnyshkov, and G. Malpuech, Measuring the quantum geometric tensor in two-dimensional photonic and exciton-polariton systems, Phys. Rev. B 97, 195422 (2018).
  10. G. Sundaram and Q. Niu, Wave-packet dynamics in slowly perturbed crystals: Gradient corrections and Berry-phase effects, Phys. Rev. B 59, 14915 (1999).
  11. D. Culcer, Y. Yao, and Q. Niu, Coherent wave-packet evolution in coupled bands, Phys. Rev. B 72, 085110 (2005).
  12. Y. Gao, S. A. Yang, and Q. Niu, Field induced positional shift of Bloch electrons and its dynamical implications, Phys. Rev. Lett. 112, 166601 (2014).
  13. Y. Gao, S. A. Yang, and Q. Niu, Geometrical effects in orbital magnetic susceptibility, Phys. Rev. B 91, 214405 (2015).
  14. O. Bleu, G. Malpuech, Y. Gao, and D. D. Solnyshkov, Effective theory of nonadiabatic quantum evolution based on the quantum geometric tensor, Phys. Rev. Lett. 121, 020401 (2018).
  15. A. Gianfrate, O. Bleu, L. Dominici, V. Ardizzone, M. De Giorgi, D. Ballarini, G. Lerario, K. W. West, L. N. Pfeiffer, D. D. Solnyshkov, D. Sanvitto, and G. Malpuech, Measurement of the quantum geometric tensor and of the anomalous Hall drift, Nature (London) 578, 381 (2020).
  16. M.-C. Chang and Q. Niu, Berry phase, hyperorbits, and the Hofstadter spectrum: Semiclassical dynamics in magnetic Bloch bands, Phys. Rev. B 53, 7010 (1996).
  17. M.-C. Chang and Q. Niu, Berry curvature, orbital moment, and effective quantum theory of electrons in electromagnetic fields, J. Phys.: Condens. Matter 20, 193202 (2008).
  18. S. Peotta and P. Törmä, Superfluidity in topologically nontrivial flat bands, Nat. Commun. 6, 8944 (2015).
  19. H. Tian, X. Gao, Y. Zhang, S. Che, T. Xu, P. Cheung, K. Watanabe, T. Taniguchi, M. Randeria, F. Zhang, C. N. Lau, and M. W. Bockrath, Evidence for Dirac flat band superconductivity enabled by quantum geometry, Nature (London) 614, 440 (2023).
  20. S. A. Chen and K. T. Law, Ginzburg-Landau theory of flat-band superconductors with quantum metric, Phys. Rev. Lett. 132, 026002 (2024).
  21. R. Jozsa, Fidelity for mixed quantum states, J. Mod. Opt. 41, 2315 (1994).
  22. S.-J. Gu, Fidelity approach to quantum phase transitions, Int. J. Mod. Phys. B 24, 4371 (2010).
  23. L. Wang, Y.-H. Liu, J. Imriška, P. N. Ma, and M. Troyer, Fidelity susceptibility made simple: A unified quantum Monte Carlo approach, Phys. Rev. X 5, 031007 (2015).
  24. B.-B. Wei and X.-C. Lv, Fidelity susceptibility in the quantum Rabi model, Phys. Rev. A 97, 013845 (2018).
  25. N. A. Khan, Fidelity susceptibility probes of dynamical quantum criticality, Chaos Solitons Fractals 183, 114975 (2024).
  26. N. Hatano and D. R. Nelson, Localization transitions in non-Hermitian quantum mechanics, Phys. Rev. Lett. 77, 570 (1996).
  27. C. M. Bender and S. Boettcher, Real spectra in non-Hermitian Hamiltonians having PT symmetry, Phys. Rev. Lett. 80, 5243 (1998).
  28. C. M. Bender, S. Boettcher, and P. N. Meisinger, PT-symmetric quantum mechanics, J. Math. Phys. 40, 2201 (1999).
  29. D. C. Brody, Biorthogonal quantum mechanics, J. Phys. A: Math. Theor. 47, 035305 (2014).
  30. 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).
  31. A. Ghatak and T. Das, New topological invariants in non-Hermitian systems, J. Phys.: Condens. Matter 31, 263001 (2019).
  32. E. J. Bergholtz, J. C. Budich, and F. K. Kunst, Exceptional topology of non-Hermitian systems, Rev. Mod. Phys. 93, 015005 (2021).
  33. Ş. K. Özdemir, S. Rotter, F. Nori, and L. Yang, Parity–time symmetry and exceptional points in photonics, Nat. Mater. 18, 783 (2019).
  34. 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).
  35. K. Kawabata, K. Shiozaki, M. Ueda, and M. Sato, Symmetry and topology in non-Hermitian physics, Phys. Rev. X 9, 041015 (2019).
  36. 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).
  37. 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).
  38. D. S. Borgnia, A. J. Kruchkov, and R.-J. Slager, Non-Hermitian boundary modes and topology, Phys. Rev. Lett. 124, 056802 (2020).
  39. K. Zhang, Z. Yang, and C. Fang, Correspondence between winding numbers and skin modes in non-Hermitian systems, Phys. Rev. Lett. 125, 126402 (2020).
  40. K. Zhang, Z. Yang, and C. Fang, Universal non-Hermitian skin effect in two and higher dimensions, Nat. Commun. 13, 2496 (2022).
  41. C. H. Lee and R. Thomale, Anatomy of skin modes and topology in non-Hermitian systems, Phys. Rev. B 99, 201103 (2019).
  42. N. Pernet, P. St-Jean, D. D. Solnyshkov, G. Malpuech, N. Carlon Zambon, Q. Fontaine, B. Real, O. Jamadi, A. Lemaître, M. Morassi, L. Le Gratiet, T. Baptiste, A. Harouri, I. Sagnes, A. Amo, S. Ravets, and J. Bloch, Gap solitons in a one-dimensional driven-dissipative topological lattice, Nat. Phys. 18, 678 (2022).
  43. P. Kokhanchik, D. Solnyshkov, and G. Malpuech, Non-Hermitian skin effect induced by Rashba-Dresselhaus spin-orbit coupling, Phys. Rev. B 108, L041403 (2023).
  44. S. Manna and B. Roy, Inner skin effects on non-Hermitian topological fractals, Commun. Phys. 6, 10 (2023).
  45. E. M. Graefe, H. J. Korsch, and A. E. Niederle, Mean-field dynamics of a non-Hermitian Bose-Hubbard dimer, Phys. Rev. Lett. 101, 150408 (2008).
  46. 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).
  47. 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).
  48. S. Longhi, Non-Hermitian skin effect and self-acceleration, Phys. Rev. B 105, 245143 (2022).
  49. C. Guria, Q. Zhong, S. K. Ozdemir, Y. S. S. Patil, R. El-Ganainy, and J. G. E. Harris, Resolving the topology of encircling multiple exceptional points, Nat. Commun. 15, 1369 (2024).
  50. 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. B 104, 125416 (2021).
  51. J. Zhong, C. C. Wojcik, D. Cheng, and S. Fan, Numerical and theoretical study of eigenenergy braids in two-dimensional photonic crystals, Phys. Rev. B 108, 195413 (2023).
  52. A. Ghatak, M. Brandenbourger, J. van Wezel, and C. Coulais, Observation of non-Hermitian topology and its bulkedge correspondence in an active mechanical metamaterial, Proc. Natl. Acad. Sci. USA 117, 29561 (2020).
  53. T. Hofmann, T. Helbig, F. Schindler, N. Salgo, M. Brzezińska, M. Greiter, T. Kiessling, D. Wolf, A. Vollhardt, A. Kabaši, C. H. Lee, A. Bilušić, R. Thomale, and T. Neupert, Reciprocal skin effect and its realization in a topolectrical circuit, Phys. Rev. Res. 2, 023265 (2020).
  54. P. A. McClarty and J. G. Rau, Non-Hermitian topology of spontaneous magnon decay, Phys. Rev. B 100, 100405 (2019).
  55. T. Yu, J. Zou, B. Zeng, J. Rao, and K. Xia, Non-Hermitian topological magnonics, Phys. Rep. 1062, 1 (2024).
  56. K. Yang, D. Varjas, E. J. Bergholtz, S. Morampudi, and F. Wilczek, Exceptional dynamics of interacting spin liquids, Phys. Rev. Res. 4, L042025 (2022).
  57. T. Gao, G. Li, E. Estrecho, T. C. H. Liew, D. Comber-Todd, A. Nalitov, M. Steger, K. West, L. Pfeiffer, D. W. Snoke, A. V. Kavokin, A. G. Truscott, and E. A. Ostrovskaya, Chiral modes at exceptional points in exciton-polariton quantum fluids, Phys. Rev. Lett. 120, 065301 (2018).
  58. R. Su, E. Estrecho, D. Biegańska, Y. Huang, M. Wurdack, M. Pieczarka, A. G. Truscott, T. C. H. Liew, E. A. Ostrovskaya, and Q. Xiong, Direct measurement of a non-Hermitian topological invariant in a hybrid light-matter system, Sci. Adv. 7, eabj8905 (2021).
  59. D. D. Solnyshkov, C. Leblanc, L. Bessonart, A. Nalitov, J. Ren, Q. Liao, F. Li, and G. Malpuech, Quantum metric and wave packets at exceptional points in non-Hermitian systems, Phys. Rev. B 103, 125302 (2021).
  60. Q. Liao, C. Leblanc, J. Ren, F. Li, Y. Li, D. Solnyshkov, G. Malpuech, J. Yao, and H. Fu, Experimental measurement of the divergent quantum metric of an exceptional point, Phys. Rev. Lett. 127, 107402 (2021).
  61. J. Cuerda, J. M. Taskinen, N. Källman, L. Grabitz, and P. Törmä, Pseudospin-orbit coupling and non-Hermitian effects in the quantum geometric tensor of a plasmonic lattice, Phys. Rev. B 109, 165439 (2024).
  62. J. Cuerda, J. M. Taskinen, N. Källman, L. Grabitz, and P. Törmä, Observation of quantum metric and non-Hermitian Berry curvature in a plasmonic lattice, Phys. Rev. Res. 6, L022020 (2024).
  63. N. Matsumoto, K. Kawabata, Y. Ashida, S. Furukawa, and M. Ueda, Continuous phase transition without gap closing in non-Hermitian quantum many-body systems, Phys. Rev. Lett. 125, 260601 (2020).
  64. B. Alon, R. Ilan, and M. Goldstein, Quantum metric dependent anomalous velocity in systems subject to complex electric fields, Phys. Rev. B 110, 245103 (2024).
  65. D. C. Brody and E.-M. Graefe, Information geometry of complex Hamiltonians and exceptional points, Entropy 15, 3361 (2013).
  66. D.-J. Zhang, Q.-h. Wang, and J. Gong, Quantum geometric tensor in PT-symmetric quantum mechanics, Phys. Rev. A 99, 042104 (2019).
  67. Y.-Q. Zhu, W. Zheng, S.-L. Zhu, and G. Palumbo, Band topology of pseudo-Hermitian phases through tensor Berry connections and quantum metric, Phys. Rev. B 104, 205103 (2021).
  68. Y.-C. Tzeng, C.-Y. Ju, G.-Y. Chen, and W.-M. Huang, Hunting for the non-Hermitian exceptional points with fidelity susceptibility, Phys. Rev. Res. 3, 013015 (2021).
  69. G. H. A. Fan and S. Liang, Complex Berry curvature pair and quantum Hall admittance in non-Hermitian systems, J. Phys. Commun. 4, 115006 (2020).
  70. C. Chen Ye, W. L. Vleeshouwers, S. Heatley, V. Gritsev, and C. Morais Smith, Quantum metric of non-Hermitian Su-Schrieffer-Heeger systems, Phys. Rev. Res. 6, 023202 (2024).
  71. Y. Xu, S.-T. Wang, and L.-M. Duan, Weyl exceptional rings in a three-dimensional dissipative cold atomic gas, Phys. Rev. Lett. 118, 045701 (2017).
  72. N. Silberstein, J. Behrends, M. Goldstein, and R. Ilan, Berry connection induced anomalous wave-packet dynamics in non-Hermitian systems, Phys. Rev. B 102, 245147 (2020).
  73. J.-H. Wang, Y.-L. Tao, and Y. Xu, Anomalous transport induced by non-Hermitian anomalous Berry connection in non-Hermitian systems, Chin. Phys. Lett. 39, 010301 (2022).
  74. I. Carusotto and C. Ciuti, Quantum fluids of light, Rev. Mod. Phys. 85, 299 (2013).
  75. Y.-M. R. Hu, E. A. Ostrovskaya, and E. Estrecho, Generalized quantum geometric tensor in a non-Hermitian exciton-polariton system [invited], Opt. Mater. Express 14, 664 (2024).
  76. F. Piéchon, A. Raoux, J.-N. Fuchs, and G. Montambaux, Geometric orbital susceptibility: Quantum metric without Berry curvature, Phys. Rev. B 94, 134423 (2016).
  77. C. Leblanc, G. Malpuech, and D. D. Solnyshkov, Universal semiclassical equations based on the quantum metric for a two-band system, Phys. Rev. B 104, 134312 (2021).
  78. Y.-M. R. Hu, E. A. Ostrovskaya, and E. Estrecho, Wave-packet dynamics in a non-Hermitian exciton-polariton system, Phys. Rev. B 108, 115404 (2023).
  79. M. M. Sternheim and J. F. Walker, Non-Hermitian Hamiltonians, decaying states, and perturbation theory, Phys. Rev. C 6, 114 (1972).
  80. See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevResearch.7.L012067 for the detailed derivation of the equation of motion, explicit forms of the real and imaginary parts in the equation of motion, eigenvalues and eigenvectors of the Hamiltonian, and additional analysis of the simulation results.
  81. T. Gao, E. Estrecho, K. Y. Bliokh, T. C. H. Liew, M. D. Fraser, S. Brodbeck, M. Kamp, C. Schneider, S. Höfling, Y. Yamamoto, F. Nori, Y. S. Kivshar, A. G. Truscott, R. G. Dall, and E. A. Ostrovskaya, Observation of non-Hermitian degeneracies in a chaotic exciton-polariton billiard, Nature (London) 526, 554 (2015).
  82. J. A. Weideman and B. M. Herbst, Split-step methods for the solution of the nonlinear Schrödinger equation, SIAM J. Numer. Anal. 23, 485 (1986).
  83. T. R. Taha and X. Xu, Parallel split-step Fourier methods for the coupled nonlinear Schrödinger type equations, J. Supercomput. 32, 5 (2005).
  84. H. Terças, H. Flayac, D. D. Solnyshkov, and G. Malpuech, Non-Abelian gauge fields in photonic cavities and photonic superfluids, Phys. Rev. Lett. 112, 066402 (2014).
  85. Y.-M. R. Hu, E. Ostrovskaya, and E. Estrecho, Data of wavepacket cntre-of-mass motion in figure 2 (2024), 10.5281/zenodo.14523051.
  86. M. Steger, C. Gautham, D. W. Snoke, L. Pfeiffer, and K. West, Slow reflection and two-photon generation of microcavity exciton–polaritons, Optica 2, 1 (2015).
  87. E. S. Sedov, Y. G. Rubo, and A. V. Kavokin, Zitterbewegung of exciton-polaritons, Phys. Rev. B 97, 245312 (2018).
  88. S. Klembt, T. H. Harder, O. A. Egorov, K. Winkler, R. Ge, M. A. Bandres, M. Emmerling, L. Worschech, T. C. H. Liew, M. Segev, C. Schneider, and S. Höfling, Exciton-polariton topological insulator, Nature (London) 562, 552 (2018).
  89. L. Polimeno, G. Lerario, M. De Giorgi, L. De Marco, L. Dominici, F. Todisco, A. Coriolano, V. Ardizzone, M. Pugliese, C. T. Prontera, V. Maiorano, A. Moliterni, C. Giannini, V. Olieric, G. Gigli, D. Ballarini, Q. Xiong, A. Fieramosca, D. D. Solnyshkov, G. Malpuech et al., Tuning of the Berry curvature in 2D perovskite polaritons, Nat. Nanotechnol. 16, 1349 (2021).
  90. K. Łempicka-Mirek, M. Król, H. Sigurdsson, A. Wincukiewicz, P. Morawiak, R. Mazur, M. Muszyński, W. Piecek, P. Kula, T. Stefaniuk, M. Kamiska, L. D. Marco, P. G. Lagoudakis, D. Ballarini, D. Sanvitto, J. Szczytko, and B. Pietka, Electrically tunable Berry curvature and strong light-matter coupling in liquid crystal microcavities with 2D perovskite, Sci. Adv. 8, eabq7533 (2022).
  91. P. Bhalla, K. Das, D. Culcer, and A. Agarwal, Resonant second-harmonic generation as a probe of quantum geometry, Phys. Rev. Lett. 129, 227401 (2022).
  92. K. Das, S. Lahiri, R. B. Atencia, D. Culcer, and A. Agarwal, Intrinsic nonlinear conductivities induced by the quantum metric, Phys. Rev. B 108, L201405 (2023).
  93. N. Wang, D. Kaplan, Z. Zhang, T. Holder, N. Cao, A. Wang, X. Zhou, F. Zhou, Z. Jiang, C. Zhang, S. Ru, H. Cai, K. Watanabe, T. Taniguchi, B. Yan, and W. Gao, Quantum-metric-induced nonlinear transport in a topological antiferromagnet, Nature (London) 621, 487 (2023).

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Phys. Rev. Research 7, L012067– Published 12 March, 2025
  • Received 11 December 2024
  • Accepted 21 February 2025
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