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  • Open Access

Theory of irreversibility in quantum many-body systems

Takato Yoshimura1,2,* and Lucas Sá3,†

  • 1All Souls College, Oxford OX1 4AL, United Kingdom
  • 2Rudolf Peierls Centre for Theoretical Physics, University of Oxford, 1 Keble Road, Oxford OX1 3NP, United Kingdom
  • 3TCM Group, Cavendish Laboratory, University of Cambridge, JJ Thomson Avenue, Cambridge CB3 0HE, United Kingdom

  • *Contact author: takato.yoshimura@physics.ox.ac.uk
  • †Contact author: ld710@cam.ac.uk
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Phys. Rev. E 111, 064135 – Published 27 June, 2025

DOI: https://doi.org/10.1103/82f6-qdyd

Abstract

We address the longstanding challenge in quantum many-body theory of reconciling unitary dynamics with irreversible relaxation. In classical chaos, the unitary evolution operator develops Ruelle-Pollicott (RP) resonances inside the unit circle in the continuum limit, leading to mixing. In the semiclassical limit, chaotic single-particle quantum systems relax with the same RP resonances. In contrast, the theory of quantum many-body RP resonances and their link to irreversibility remain underdeveloped. Here, we relate the spectral form factor to the sum of autocorrelation functions and, in generic many-body lattice systems without conservation laws, we argue that all quantum many-body RP resonances converge inside the unit disk, highlighting the role of nonunitary and the thermodynamic limit. While we conjecture this picture to be general, we analytically prove the emergence of irreversibility in the random phase model (RPM), a paradigmatic Floquet quantum circuit model, in the limit of large local Hilbert space dimension. To this end, we couple it to local environments and compute the exact time evolution of autocorrelation functions, the dissipative form factor, and out-of-time-order correlation functions (OTOCs). Although valid for any dissipation strength, we then focus on weak dissipation to clarify the origin of irreversibility in unitary systems. When the dissipationless limit is taken after the thermodynamic limit, the unitary quantum map develops an infinite tower of decaying RP resonances—chaotic systems display so-called anomalous relaxation. We identify the exact RP resonances in the RPM and prove that the same RP resonances are obtained from operator truncation. We also show that the OTOC in the RPM can undergo a two-stage relaxation, and that during the second stage, the approach to the stationary value is again controlled by the leading RP resonance. Finally, we demonstrate how conservation laws, many-body localization, and nonlocal interactions merge the leading RP resonance into the unit circle, thereby suppressing anomalous relaxation. Our findings clarify the microscopic origin of macroscopic relaxation in quantum systems and emphasize the role of RP resonances as a largely unexplored signature of many-body quantum chaos.

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Phys. Rev. E 111, 064135– Published 27 June, 2025
  • Received 28 January 2025
  • Accepted 5 June 2025
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