Translation symmetry restoration under random unitary dynamics

TL;DR

This paper investigates how space-time symmetries, specifically translation symmetry, are restored in many-body quantum systems under random unitary dynamics, revealing exact timescales and phenomena like the quantum Mpemba effect.

Contribution

It extends the symmetry restoration framework from internal to space-time symmetries and provides exact results for the timescales of translation symmetry restoration in random unitary circuits.

Findings

Restoration of translation symmetry occurs on timescales proportional to subsystem volume.

For large subsystems, symmetry restoration time is independent of initial state and matches thermalization time.

Identifies and characterizes the quantum Mpemba effect in non-integrable systems.

Abstract

The finite parts of a large, locally interacting many-body system prepared out-of-equilibrium eventually equilibrate. Characterising the underlying mechanisms of this process and its timescales, however, is particularly hard as it requires to decouple universal features from observable-specific ones. Recently, new insight came by studying how certain symmetries of the dynamics that are broken by the initial state are restored at the level of the reduced state of a given subsystem. This provides a high level, observable-independent probe. Until now this idea has been applied to the restoration of internal symmetries, e.g. U(1) symmetries related to charge conservation. Here we show that that the same logic can be applied to the restoration of space-time symmetries, and hence can be used to characterise the relaxation of fully generic systems. We illustrate this idea by considering the paradigmatic example of "generic" many-body dynamics, i.e. a local random unitary circuit, where our method leads to exact results. We show that the restoration of translation symmetry in these systems only happens on time-scales proportional to the subsystem's volume. In fact, for large enough subsystems the time of symmetry restoration becomes initial-state independent (as long as the latter breaks the symmetry at time zero) and coincides with the thermalisation time. For intermediate subsystems, however, one can observe the so-called "quantum Mpemba effect", where the state of the system restores a symmetry faster if it is initially more asymmetric. We provide the first exact characterisation of this effect in a non-integrable system.