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Liouvillian topology and nonreciprocal dynamics in open Floquet chains

Florian Koch1,2, Yu-Min Hu1, and Jan Carl Budich1,2,*

  • *Contact author: jcbudich@pks.mpg.de
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Phys. Rev. Research 8, 023270 – Published 11 June, 2026

DOI: https://doi.org/10.1103/284g-926d

Abstract

Open quantum systems far from thermal equilibrium can exhibit remarkable physical phenomena including topological properties without a direct equilibrium counterpart. Along these lines, in periodically driven-dissipative systems within the effective non-Hermitian (NH) Hamiltonian approximation spectral winding numbers have been linked to intriguing nonreciprocal transport properties. Here, going beyond an NH Hamiltonian description, we introduce and study a microscopic lattice model of a driven open quantum system described by a Markovian quantum master equation, which exhibits the mentioned spectral winding within an NH approximation. By encompassing quantum jump processes in the topological analysis, we uncover a distinct jump-induced topological phase, which qualitatively corresponds to the richer nonreciprocal transport properties of the fully quantum model. In addition, we find that the NH skin effect, i.e., the accumulation of a macroscopic number of eigenstates at one end of the system, is already visible in the transient dynamics even for systems with periodic boundary conditions. Our results exemplify the subtle correspondence between NH topological properties and physical manifestations of Liouvillian topological properties in open quantum systems, thus providing a theoretical framework toward understanding unidirectional transport in quantum dissipative Floquet dynamics.

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References (75)

  1. T. Kitagawa, E. Berg, M. Rudner, and E. Demler, Topological characterization of periodically driven quantum systems, Phys. Rev. B 82, 235114 (2010).
  2. L. Jiang, T. Kitagawa, J. Alicea, A. R. Akhmerov, D. Pekker, G. Refael, J. I. Cirac, E. Demler, M. D. Lukin, and P. Zoller, Majorana fermions in equilibrium and in driven cold-atom quantum wires, Phys. Rev. Lett. 106, 220402 (2011).
  3. T. Kitagawa, M. A. Broome, A. Fedrizzi, M. S. Rudner, E. Berg, I. Kassal, A. Aspuru-Guzik, E. Demler, and A. G. White, Observation of topologically protected bound states in photonic quantum walks, Nat. Commun. 3, 882 (2012).
  4. M. S. Rudner, N. H. Lindner, E. Berg, and M. Levin, Anomalous edge states and the bulk-edge correspondence for periodically driven two-dimensional systems, Phys. Rev. X 3, 031005 (2013).
  5. S. Vajna and B. Dóra, Topological classification of dynamical phase transitions, Phys. Rev. B 91, 155127 (2015).
  6. J. C. Budich and M. Heyl, Dynamical topological order parameters far from equilibrium, Phys. Rev. B 93, 085416 (2016).
  7. I. Martin, G. Refael, and B. Halperin, Topological frequency conversion in strongly driven quantum systems, Phys. Rev. X 7, 041008 (2017).
  8. A. Quelle, C. Weitenberg, K. Sengstock, and C. M. Smith, Driving protocol for a Floquet topological phase without static counterpart, New J. Phys. 19, 113010 (2017).
  9. N. Fläschner, D. Vogel, M. Tarnowski, B. S. Rem, D.-S. Lühmann, M. Heyl, J. C. Budich, L. Mathey, K. Sengstock, and C. Weitenberg, Observation of dynamical vortices after quenches in a system with topology, Nat. Phys. 14, 265 (2018).
  10. V. Junk, P. Reck, C. Gorini, and K. Richter, Floquet oscillations in periodically driven Dirac systems, Phys. Rev. B 101, 134302 (2020).
  11. N. H. Lindner, G. Refael, and V. Galitski, Floquet topological insulator in semiconductor quantum wells, Nat. Phys. 7, 490 (2011).
  12. A. Gómez-León and G. Platero, Floquet-Bloch theory and topology in periodically driven lattices, Phys. Rev. Lett. 110, 200403 (2013).
  13. J. Cayssol, B. Dóra, F. Simon, and R. Moessner, Floquet topological insulators, Phys. Status Solidi RRL 7, 101 (2013).
  14. A. C. Potter, T. Morimoto, and A. Vishwanath, Classification of interacting topological Floquet phases in one dimension, Phys. Rev. X 6, 041001 (2016).
  15. A. Eckardt, Colloquium: Atomic quantum gases in periodically driven optical lattices, Rev. Mod. Phys. 89, 011004 (2017).
  16. S. Yao, Z. Yan, and Z. Wang, Topological invariants of Floquet systems: General formulation, special properties, and Floquet topological defects, Phys. Rev. B 96, 195303 (2017).
  17. T. Oka and S. Kitamura, Floquet engineering of quantum materials, Annu. Rev. Condens. Matter Phys. 10, 387 (2019).
  18. 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).
  19. H. Zhou and J. Y. Lee, Periodic table for topological bands with non-Hermitian symmetries, Phys. Rev. B 99, 235112 (2019).
  20. K. Kawabata, K. Shiozaki, M. Ueda, and M. Sato, Symmetry and topology in non-Hermitian physics, Phys. Rev. X 9, 041015 (2019).
  21. J. C. Budich and E. J. Bergholtz, Non-Hermitian topological sensors, Phys. Rev. Lett. 125, 180403 (2020).
  22. E. J. Bergholtz, J. C. Budich, and F. K. Kunst, Exceptional topology of non-Hermitian systems, Rev. Mod. Phys. 93, 015005 (2021).
  23. R. Lin, T. Tai, L. Li, and C. H. Lee, Topological non-Hermitian skin effect, Front. Phys. 18, 53605 (2023).
  24. N. Okuma and M. Sato, Non-Hermitian topological phenomena: A review, Annu. Rev. Condens. Matter Phys. 14, 83 (2023).
  25. J. Gong and Q.-H. Wang, Stabilizing non-Hermitian systems by periodic driving, Phys. Rev. A 91, 042135 (2015).
  26. S. Longhi, Floquet exceptional points and chirality in non-Hermitian Hamiltonians, J. Phys. Math. Theor. 50, 505201 (2017).
  27. B. Höckendorf, A. Alvermann, and H. Fehske, Topological origin of quantized transport in non-Hermitian Floquet chains, Phys. Rev. Res. 2, 023235 (2020).
  28. X. Zhang and J. Gong, Non-Hermitian Floquet topological phases: Exceptional points, coalescent edge modes, and the skin effect, Phys. Rev. B 101, 045415 (2020).
  29. H. Wu, B.-Q. Wang, and J.-H. An, Floquet second-order topological insulators in non-Hermitian systems, Phys. Rev. B 103, L041115 (2021).
  30. P. Sierant, G. Chiriacò, F. M. Surace, S. Sharma, X. Turkeshi, M. Dalmonte, R. Fazio, and G. Pagano, Dissipative Floquet dynamics: From steady state to measurement induced criticality in trapped-ion chains, Quantum 6, 638 (2022).
  31. L. Zhou and D.-J. Zhang, Non-Hermitian Floquet topological matter—A review, Entropy 25, 1401 (2023).
  32. F. Koch and J. C. Budich, Dissipative frequency converter: From Lindblad dynamics to non-Hermitian topology, Phys. Rev. Res. 6, 033124 (2024).
  33. G. K. Dash, S. Bid, and M. Thakurathi, Floquet exceptional topological insulator, Phys. Rev. B 109, 035418 (2024).
  34. C. C. Wanjura and A. Nunnenkamp, Unifying framework for non-Hermitian and Hermitian topology in driven-dissipative systems, arXiv:2509.19433.
  35. F. Song, S. Yao, and Z. Wang, Non-Hermitian skin effect and chiral damping in open quantum systems, Phys. Rev. Lett. 123, 170401 (2019).
  36. F. Minganti, A. Miranowicz, R. W. Chhajlany, I. I. Arkhipov, and F. Nori, Hybrid-Liouvillian formalism connecting exceptional points of non-Hermitian Hamiltonians and Liouvillians via postselection of quantum trajectories, Phys. Rev. A 101, 062112 (2020).
  37. T. Mori, Floquet states in open quantum systems, Annu. Rev. Condens. Matter Phys. 14, 35 (2023).
  38. K. Monkman and M. Berciu, Limits of the non-Hermitian description of decay models, Phys. Rev. A 113, 032213 (2025).
  39. C. M. Dai, Z. C. Shi, and X. X. Yi, Floquet theorem with open systems and its applications, Phys. Rev. A 93, 032121 (2016).
  40. J. C. Budich, Y. Hu, and P. Zoller, Helical Floquet channels in 1D lattices, Phys. Rev. Lett. 118, 105302 (2017).
  41. F. Koch and J. C. Budich, Quantum non-Hermitian topological sensors, Phys. Rev. Res. 4, 013113 (2022).
  42. A. Chaduteau, D. K. K. Lee, and F. Schindler, Lindbladian versus postselected non-Hermitian topology, Phys. Rev. Lett. 136, 016603 (2026).
  43. W. Chen, M. Abbasi, S. Erdamar, J. Muldoon, Y. N. Joglekar, and K. W. Murch, Engineering nonequilibrium steady states through Floquet Liouvillians, Phys. Rev. Lett. 134, 090402 (2025).
  44. C. L. Kane and E. J. Mele, Quantum spin Hall effect in graphene, Phys. Rev. Lett. 95, 226801 (2005).
  45. C.-H. Liu, H. Hu, and S. Chen, Symmetry and topological classification of Floquet non-Hermitian systems, Phys. Rev. B 105, 214305 (2022).
  46. H. Geng, J. Y. Wei, M. H. Zou, L. Sheng, W. Chen, and D. Y. Xing, Nonreciprocal charge and spin transport induced by non-Hermitian skin effect in mesoscopic heterojunctions, Phys. Rev. B 107, 035306 (2023).
  47. T. E. Lee, Anomalous edge state in a non-Hermitian lattice, Phys. Rev. Lett. 116, 133903 (2016).
  48. S. Yao and Z. Wang, Edge states and topological invariants of non-Hermitian systems, Phys. Rev. Lett. 121, 086803 (2018).
  49. 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).
  50. K. Yokomizo and S. Murakami, Non-Bloch band theory of non-Hermitian systems, Phys. Rev. Lett. 123, 066404 (2019).
  51. C. H. Lee and R. Thomale, Anatomy of skin modes and topology in non-Hermitian systems, Phys. Rev. B 99, 201103 (2019).
  52. G. Lindblad, On the generators of quantum dynamical semigroups, Commun. Math. Phys. 48, 119 (1976).
  53. V. Gorini, A. Kossakowski, and E. C. G. Sudarshan, Completely positive dynamical semigroups of N-level systems, J. Math. Phys. 17, 821 (1976).
  54. W. Pauli, Zur quantenmechanik des magnetischen elektrons, Z. Phys. 43, 601 (1927).
  55. H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Clarendon Press, Oxford, 2009), 1st ed.
  56. C. W. Gardiner and P. Zoller, Quantum Noise: A Handbook of Markovian and Non-Markovian Quantum Stochastic Methods with Applications to Quantum Optics, Springer Complexity, 3rd ed. (Springer, Berlin, 2010).
  57. A. Schnell, A. Eckardt, and S. Denisov, Is there a Floquet Lindbladian? Phys. Rev. B 101, 100301 (2020).
  58. Y. Ashida, Z. Gong, and M. Ueda, Non-Hermitian physics, Adv. Phys. 69, 249 (2020).
  59. R. B. Griffiths, Consistent interpretation of quantum mechanics using quantum trajectories, Phys. Rev. Lett. 70, 2201 (1993).
  60. H. M. Wiseman, Quantum trajectories and quantum measurement theory, Quantum Semiclassical Opt. 8, 205 (1996).
  61. A. J. Daley, Quantum trajectories and open many-body quantum systems, Adv. Phys. 63, 77 (2014).
  62. A. Sergi, The density matrix in the non-Hermitian approach to open quantum system dynamics, Atti Accad. Peloritana Dei Pericolanti Cl. Sci. Fis. Mat. Nat. 97, 11 (2019).
  63. N. Okuma, K. Kawabata, K. Shiozaki, and M. Sato, Topological origin of non-Hermitian skin effects, Phys. Rev. Lett. 124, 086801 (2020).
  64. K. Zhang, Z. Yang, and C. Fang, Correspondence between winding numbers and skin modes in non-Hermitian systems, Phys. Rev. Lett. 125, 126402 (2020).
  65. R. A. Horn and C. R. Johnson, Topics in Matrix Analysis (Cambridge University Press, Cambridge, 1994).
  66. T. Kato, Perturbation Theory for Linear Operators, Classics in Mathematics (Springer , Berlin, 1995), Vol. 132.
  67. F. Benaych-Georges and R. R. Nadakuditi, The eigenvalues and eigenvectors of finite, low rank perturbations of large random matrices, Adv. Math. 227, 494 (2011).
  68. C. Mehl, V. Mehrmann, A. C. M. Ran, and L. Rodman, Eigenvalue perturbation theory of classes of structured matrices under generic structured rank one perturbations, Linear Algebra Appl. 435, 687 (2011).
  69. T. Tao, Outliers in the spectrum of iid matrices with bounded rank perturbations, Probab. Theory Relat. Fields 155, 231 (2013).
  70. C. Bordenave, F. Chapon, and M. Capitaine, Outliers of perturbations of banded Toeplitz matrices, arXiv:2410.16439.
  71. J. C. Budich, C. Laflamme, F. Tschirsich, S. Montangero, and P. Zoller, Synthetic helical liquids with ultracold atoms in optical lattices, Phys. Rev. B 92, 245121 (2015).
  72. M. Mancini, G. Pagano, G. Cappellini, L. Livi, M. Rider, J. Catani, C. Sias, P. Zoller, M. Inguscio, M. Dalmonte, and L. Fallani, Observation of chiral edge states with neutral fermions in synthetic Hall ribbons, Science 349, 1510 (2015).
  73. M. Nakagawa and M. Ueda, Topology of discrete quantum feedback control, Phys. Rev. X 15, 021016 (2025).
  74. A. I. Pavlov, Y. Gefen, and A. Shnirman, Topological transitions in quantum jump dynamics: Hidden exceptional points, Phys. Rev. B 111, 104301 (2025).
  75. F. Koch, J. C. Budich, and Y.-M. Hu, Data set for “Liouvillian topology and nonreciprocal dynamics in open Floquet chains” [Data set], Zenodo, 2026, doi:10.5281/zenodo.20325478.

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Phys. Rev. Research 8, 023270– Published 11 June, 2026
  • Received 1 December 2025
  • Accepted 9 May 2026
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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. Open access publication funded by Max Planck Society.

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