- 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
  • Letter
  • Open Access

Adiabatic transformations in dissipative and non-Hermitian phase transitions

Pavel Orlov1,2,3, Georgy V. Shlyapnikov2,4,5,6, and Denis V. Kurlov2,7

PDF

Phys. Rev. B 111, L081105 – Published 6 February, 2025

DOI: https://doi.org/10.1103/PhysRevB.111.L081105

Abstract

The quantum geometric tensor has established itself as a general framework for the analysis and detection of equilibrium phase transitions in isolated quantum systems. We propose a generalization of the quantum geometric tensor, which offers a universal approach to studying phase transitions in non-Hermitian quantum systems. Our generalization is based on the concept of the generator of adiabatic transformations and can be applied to systems described by either a Liouvillian superoperator or by an effective non-Hermitian Hamiltonian. We illustrate the proposed method by analyzing the non-Hermitian Su-Schrieffer-Heeger model and a generic quasifree dissipative fermionic system with a quadratic Liouvillian. Our findings reveal that this method effectively identifies phase transitions across all examined models, providing a universal tool for investigating general non-Hermitian systems.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (60)

  1. H. Stanley, Introduction to Phase Transitions and Critical Phenomena, International Series of Monographs on Physics (Oxford University Press, Oxford, UK, 1987).
  2. S. Sachdev, Quantum Phase Transitions, 2nd ed. (Cambridge University Press, Cambridge, UK, 2011).
  3. J. P. Provost and G. Vallee, Riemannian structure on manifolds of quantum states, Commun. Math. Phys. 76, 289 (1980).
  4. P. Zanardi and N. Paunković, Ground state overlap and quantum phase transitions, Phys. Rev. E 74, 031123 (2006).
  5. P. Zanardi, P. Giorda, and M. Cozzini, Information-theoretic differential geometry of quantum phase transitions, Phys. Rev. Lett. 99, 100603 (2007).
  6. L. C. Venuti and P. Zanardi, Quantum critical scaling of the geometric tensors, Phys. Rev. Lett. 99, 095701 (2007).
  7. D. J. Thouless, Quantization of particle transport, Phys. Rev. B 27, 6083 (1983).
  8. B. Bernevig and T. Hughes, Topological Insulators and Topological Superconductors (Princeton University Press, Princeton, NJ, 2013).
  9. A. C. M. Carollo and J. K. Pachos, Geometric phases and criticality in spin-chain systems, Phys. Rev. Lett. 95, 157203 (2005).
  10. A. Hamma, Berry phases and quantum phase transitions, arXiv:quant-ph/0602091.
  11. T. Prosen, Third quantization: A general method to solve master equations for quadratic open Fermi systems, New J. Phys. 10, 043026 (2008).
  12. S. Diehl, A. Micheli, A. Kantian, B. Kraus, H. P. Büchler, and P. Zoller, Quantum states and phases in driven open quantum systems with cold atoms, Nat. Phys. 4, 878 (2008).
  13. E. G. D. Torre, E. Demler, T. Giamarchi, and E. Altman, Quantum critical states and phase transitions in the presence of non-equilibrium noise, Nat. Phys. 6, 806 (2010).
  14. G. Dagvadorj, J. M. Fellows, S. Matyjaśkiewicz, F. M. Marchetti, I. Carusotto, and M. H. Szymańska, Nonequilibrium phase transition in a two-dimensional driven open quantum system, Phys. Rev. X 5, 041028 (2015).
  15. J. Jin, A. Biella, O. Viyuela, L. Mazza, J. Keeling, R. Fazio, and D. Rossini, Cluster mean-field approach to the steady-state phase diagram of dissipative spin systems, Phys. Rev. X 6, 031011 (2016).
  16. L. R. Bakker, M. S. Bahovadinov, D. V. Kurlov, V. Gritsev, A. K. Fedorov, and D. O. Krimer, Driven-dissipative time crystalline phases in a two-mode bosonic system with Kerr nonlinearity, Phys. Rev. Lett. 129, 250401 (2022).
  17. A. S. Kazmina, I. V. Zalivako, A. S. Borisenko, N. A. Nemkov, A. S. Nikolaeva, I. A. Simakov, A. V. Kuznetsova, E. Y. Egorova, K. P. Galstyan, N. V. Semenin, A. E. Korolkov, I. N. Moskalenko, N. N. Abramov, I. S. Besedin, D. A. Kalacheva, V. B. Lubsanov, A. N. Bolgar, E. O. Kiktenko, K. Y. Khabarova, A. Galda et al., Demonstration of a parity-time-symmetry-breaking phase transition using superconducting and trapped-ion qutrits, Phys. Rev. A 109, 032619 (2024).
  18. M. Müller, S. Diehl, G. Pupillo, and P. Zoller, Engineered open systems and quantum simulations with atoms and ions, in Advances in Atomic, Molecular, and Optical Physics, edited by P. R. Berman, E. Arimondo, and C. C. Lin (Elsevier, 2012), Vol. 61, pp. 1–80.
  19. D. Leibfried, R. Blatt, C. Monroe, and D. Wineland, Quantum dynamics of single trapped ions, Rev. Mod. Phys. 75, 281 (2003).
  20. M.-A. Miri and A. Alù, Exceptional points in optics and photonics, Science 363, eaar7709 (2019).
  21. A. A. Clerk, K. W. Lehnert, P. Bertet, J. R. Petta, and Y. Nakamura, Hybrid quantum systems with circuit quantum electrodynamics, Nat. Phys. 16, 257 (2020).
  22. M. G. A. Paris, Quantum estimation for quantum technology, Int. J. Quantum Inf. 07, 125 (2009).
  23. P. M. Harrington, E. J. Mueller, and K. W. Murch, Engineered dissipation for quantum information science, Nat. Rev. Phys. 4, 660 (2022).
  24. D. Bures, An extension of Kakutani's theorem on infinite product measures to the tensor product of semifinite w*-algebras, Trans. Am. Math. Soc. 135, 199 (1969).
  25. A. Uhlmann, The “transition probability” in the state space of a *-algebra, Rep. Math. Phys. 9, 273 (1976).
  26. A. Uhlmann, Parallel transport and “quantum holonomy” along density operators, Rep. Math. Phys. 24, 229 (1986).
  27. E. Ercolessi and M. Schiavina, Geometry of mixed states for a q-bit and the quantum Fisher information tensor, J. Phys. A: Math. Theor. 45, 365303 (2012).
  28. A. Carollo, D. Valenti, and B. Spagnolo, Geometry of quantum phase transitions, Phys. Rep. 838, 1 (2020).
  29. P. Zanardi, L. C. Venuti, and P. Giorda, Bures metric over thermal state manifolds and quantum criticality, Phys. Rev. A 76, 062318 (2007).
  30. Z. Huang and D. P. Arovas, Topological indices for open and thermal systems via uhlmann's phase, Phys. Rev. Lett. 113, 076407 (2014).
  31. O. Viyuela, A. Rivas, and M. A. Martin-Delgado, Uhlmann phase as a topological measure for one-dimensional fermion systems, Phys. Rev. Lett. 112, 130401 (2014).
  32. O. Viyuela, A. Rivas, and M. A. Martin-Delgado, Two-dimensional density-matrix topological fermionic phases: Topological Uhlmann numbers, Phys. Rev. Lett. 113, 076408 (2014).
  33. B. Mera, C. Vlachou, N. Paunković, and V. R. Vieira, Uhlmann connection in fermionic systems undergoing phase transitions, Phys. Rev. Lett. 119, 015702 (2017).
  34. L. Banchi, P. Giorda, and P. Zanardi, Quantum information-geometry of dissipative quantum phase transitions, Phys. Rev. E 89, 022102 (2014).
  35. A. Carollo, B. Spagnolo, and D. Valenti, Uhlmann curvature in dissipative phase transitions, Sci. Rep. 8, 9852 (2018).
  36. 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).
  37. 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).
  38. D. Brody and E.-M. Graefe, Information geometry of complex Hamiltonians and exceptional points, Entropy 15, 3361 (2013).
  39. D. J. Zhang, Q. H. Wang, and J. Gong, Quantum geometric tensor in PT-symmetric quantum mechanics, Phys. Rev. A 99, 042104 (2019).
  40. 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).
  41. C. C. Ye, W. L. Vleeshouwers, S. Heatley, V. Gritsev, and C. M. Smith, Quantum metric of non-Hermitian Su-Schrieffer-Heeger systems, Phys. Rev. Res. 6, 023202 (2024).
  42. M. Kolodrubetz, D. Sels, P. Mehta, and A. Polkovnikov, Geometry and non-adiabatic response in quantum and classical systems, Phys. Rep. 697, 1 (2017).
  43. M. Pandey, P. W. Claeys, D. K. Campbell, A. Polkovnikov, and D. Sels, Adiabatic eigenstate deformations as a sensitive probe for quantum chaos, Phys. Rev. X 10, 041017 (2020).
  44. Y. Ashida, Z. Gong, and M. Ueda, Non-Hermitian physics, Adv. Phys. 69, 249 (2020).
  45. Note that, unlike in the usual case, there is no imaginary unit in the definition of the AGP.
  46. This is a consequence of the trace-preserving property of a Liouvillian. Indeed, Tr[L(ρ)]=0 for an arbitrary matrix ρ, so that L†(1̂)=0.
  47. See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevB.111.L081105 for details.
  48. In the case of a nondiagonalizable matrix K one needs to use the generalized eigenvectors of K and then define the AGP via the response of these eigenvectors to a parameter change.
  49. S. Lieu, Topological phases in the non-Hermitian Su-Schrieffer-Heeger model, Phys. Rev. B 97, 045106 (2018).
  50. C. Yin, H. Jiang, L. Li, R. Lü, and S. Chen, Geometrical meaning of winding number and its characterization of topological phases in one-dimensional chiral non-Hermitian systems, Phys. Rev. A 97, 052115 (2018).
  51. J.-P. Blaizot and G. Ripka, Quantum Theory of Finite Systems (MIT Press, Cambridge, MA, 1985).
  52. B. Horstmann, J. I. Cirac, and G. Giedke, Noise-driven dynamics and phase transitions in fermionic systems, Phys. Rev. A 87, 012108 (2013).
  53. L. Sá, Signatures of dissipative quantum chaos, arXiv:2311.01518.
  54. 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).
  55. Y.-M. R. Hu, E. A. Ostrovskaya, and E. Estrecho, Generalized quantum geometric tensor in a non-Hermitian exciton-polariton system, Opt. Mater. Express 14, 664 (2024).
  56. C. Mc Keever and M. Lubasch, Towards adiabatic quantum computing using compressed quantum circuits, PRX Quantum 5, 020362 (2024).
  57. V. M. Vyas and D. Roy, Topological aspects of periodically driven non-Hermitian Su-Schrieffer-Heeger model, Phys. Rev. B 103, 075441 (2021).
  58. R. Nehra and D. Roy, Topology of multipartite non-Hermitian one-dimensional systems, Phys. Rev. B 105, 195407 (2022).
  59. R. Nehra and D. Roy, Anomalous dynamical response of non-Hermitian topological phases, Phys. Rev. B 109, 094311 (2024).
  60. B. Hetényi and P. Lévay, Fluctuations, uncertainty relations, and the geometry of quantum state manifolds, Phys. Rev. A 108, 032218 (2023).

Outline

Information

Phys. Rev. B 111, L081105– Published 6 February, 2025
  • Received 18 June 2024
  • Revised 4 December 2024
  • Accepted 21 January 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