- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Research

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
  • Open Access

Path integral approach to quantum thermalization

Alexander Altland

Kun Woo Kim

Tobias Micklitz

  • Institut für Theoretische Physik, Universität zu Köln, Zülpicher Straße 77, 50937 Cologne, Germany

PDF

Phys. Rev. Research 8, 023076 – Published 27 April, 2026

DOI: https://doi.org/10.1103/94k5-q37p

Abstract

We introduce a quasiclassical Green's function approach describing the unitary yet irreversible dynamics of quantum systems effectively acting as their own environment. Combining a variety of concepts of quantum many-body theory, notably the nonlinear σ model of disordered systems, the GΣ formalism for strong correlations, and real-time path integration, the theory is capable of describing a wide range of system classes and disorder models. It extends previous work beyond perturbation theory (in inverse Hilbert-space dimensions), enabling a description of thermalization dynamics from short scattering times, through the onset of ergodicity at an effective “Thouless time,” up to the many-body Heisenberg time. We illustrate the approach with two case studies: (1) a brickwork model of unitarily coupled quantum circuits with and without conserved symmetries and (2) an array of capacitively coupled quantum dots. Using the spectral form factor as a test observable, we find good agreement with numerical simulations. We present our formalism in a self-contained and pedagogical manner, aiming to provide a transferable toolbox for the first-principles description of many-body chaotic quantum systems in regimes of strong entanglement.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (27)

  1. T. Mori, T. N. Ikeda, E. Kaminishi, and M. Ueda, Thermalization and prethermalization in isolated quantum systems: A theoretical overview, J. Phys. B: At. Mol. Opt. Phys. 51, 112001 (2018).
  2. V. França, Making quantum dynamics exact, Physics 14, 60 (2021).
  3. R. Nandkishore and D. A. Huse, Many-body localization and thermalization in quantum statistical mechanics, Annu. Rev. Condens. Matter Phys. 6, 15 (2015).
  4. A. Chan, A. De Luca, and J. T. Chalker, Solution of a minimal model for many-body quantum chaos, Phys. Rev. X 8, 041019 (2018).
  5. A. Chan, A. De Luca, and J. Chalker, Spectral statistics in spatially extended chaotic quantum many-body systems, Phys. Rev. Lett. 121, 060601 (2018).
  6. A. J. Friedman, A. Chan, A. De Luca, and J. Chalker, Spectral statistics and many-body quantum chaos with conserved charge, Phys. Rev. Lett. 123, 210603 (2019).
  7. F. Fritzsch and T. Prosen, Eigenstate thermalization in dual-unitary quantum circuits: Asymptotics of spectral functions, Phys. Rev. E 103, 062133 (2021).
  8. A. Altland and B. Simons, Condensed Matter Field Theory (Cambridge University Press, Cambridge, 2023).
  9. A. I. Larkin and Y. N. Ovchinnikov, Quasiclassical method in the theory of superconductivity, Sov. Phys. JETP 28, 1200 (1969).
  10. K. B. Efetov, Sypersymmetry in Disorder and Chaos (Cambridge University Press, Cambridge, 1997).
  11. V. Rosenhaus, An introduction to the SYK model, J. Phys. A: Math. Theor. 52, 323001 (2019).
  12. J. M. Luttinger and J. C. Ward, Ground-state energy of a many-fermion system. II, Phys. Rev. 118, 1417 (1960).
  13. Within our later path integral formulation, ψμ,ta are represented by Grassmann integration variables, hence the denotation “fields.” Alternatively, one may think of them as fermionic or bosonic creation and annihilation operators.
  14. Depending on the context, 〈⋯〉 will denote path integration, ensemble averaging, or both. In cases of ambiguity we occasionally denote the latter by 〈⋯〉H.
  15. F. Haake, Quantum Signatures of Chaos, 4th ed. (Springer, Berlin, 2018).
  16. A. V. Andreev and B. L. Altshuler, Spectral statistics beyond random matrix theory, Phys. Rev. Lett. 75, 902 (1995).
  17. The disorder-averaged saddle point Green's function is block diagonal in retarded/advanced space, G=diag(GR,GA) with GR≠GA. While the microscopic theory is invariant under unitary rotations mixing the two causal sectors, these stationary configurations spontaneously break the symmetry down to transformations within the retarded/advanced structure. The corresponding Goldstone modes rotating in R/A are the dominant fluctuations of the ergodic phase.
  18. A. M. Perelomov, Coherent states for arbitrary Lie group, Commun. Math. Phys. 26, 222 (1972).
  19. A. Kamenev, Field Theory of Non-Equilibrium Systems (Cambridge University Press, Cambridge, 2011).
  20. N. Tyagi and K. Wharton, Spacetime path integrals for entangled states, Found. Phys. 52, 9 (2022).
  21. A. G. Green, C. A. Hooley, J. Keeling, and S. H. Simon, Feynman path integrals over entangled states, arXiv:1607.01778.
  22. M. R. Zirnbauer, Supersymmetry for systems with unitary disorder: Circular ensembles, J. Phys. A: Math. Gen. 29, 7113 (1996).
  23. A. Altland, S. Gnutzmann, F. Haake, and T. Micklitz, A review of sigma models for quantum chaotic dynamics, Rep. Prog. Phys. 78, 086001 (2015).
  24. A. Altland and A. Kamenev, Wigner-Dyson statistics from the Keldysh σ-model, Phys. Rev. Lett. 85, 5615 (2000).
  25. Here, the “disconnected” contraction between the adjacent matrices BBt1†=BtuBu1† is an example of a contribution producing a free running summation over the dummy u index. Such contributions are cancelled by the measure of the B integral, as discussed above.
  26. A. Altland, K. W. Kim, and T. Micklitz, Matlab script, datasets, figures for “Path integral approach for quantum thermalization” [Data set], Zenodo (2025), https://doi.org/10.5281/zenodo.15782178.
  27. S. Müller, S. Heusler, A. Altland, P. Braun, and F. Haake, Periodic-orbit theory of universal level correlations in quantum chaos, New J. Phys. 11, 103025 (2009).

Outline

Information

Phys. Rev. Research 8, 023076– Published 27 April, 2026
  • Received 11 August 2025
  • Revised 27 August 2025
  • Accepted 25 February 2026
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