Thermalization rates and quantum Ruelle-Pollicott resonances: insights from operator hydrodynamics

TL;DR

This paper establishes a theoretical link between exponential decay rates of local expectation values in quantum systems and operator spreading, using a hydrodynamical approach and eigenvalues of a dynamical map, applicable to random circuits and Floquet systems.

Contribution

It introduces a hydrodynamical framework connecting decay rates to eigenvalues of a dissipative dynamical map in quantum many-body systems.

Findings

Decay rate $ar{g}$ is linked to the leading eigenvalue of a dynamical map.

Eigenvalue size is independent of weak dissipation details under certain conditions.

Results align with numerical studies of quantum Ruelle-Pollicott resonances.

Abstract

In thermalizing many-body quantum systems without conservation laws such as ergodic Floquet and random unitary circuits, local expectation values are predicted to decay to their equilibrium values exponentially quickly. In this work we derive a relationship between said exponential decay rate $\overline{g}$ and the operator spreading properties of a local unitary evolution. A hydrodynamical picture for operator spreading allows us to argue that, for random unitary circuits, $\overline{g}$ is encoded by the leading eigenvalue of a dynamical map obtained by enriching unitary dynamics with dissipation, in the limit of weak dissipation. We argue that the size of the eigenvalue does not depend on the details of this weak dissipation (given mild assumptions on properties of the ergodic dynamics), so long as it only suppresses large operators significantly. Our calculations are based on analytical results for random unitary circuits, but we argue that similar results hold for ergodic Floquet systems. These conjectures are in accordance with existing results which numerically obtain quantum many-body analogues of classical Ruelle-Pollicott resonances [T. Prosen J. Phys. A: Math. Gen. 35 L737 (2002), T. Mori, arXiv:2311.10304] by studying unitary evolutions subject to weak dissipation.