Thermalisation Dynamics and Spectral Statistics of Extended Systems with Thermalising Boundaries

TL;DR

This paper investigates how extended quantum systems thermalize or fail to do so when coupled to a thermal bath at the boundaries, analyzing spectral properties and identifying conditions for non-ergodic behavior and rapid thermalization.

Contribution

It introduces a framework using quantum channels to analyze boundary-driven thermalization, classifies strongly non-ergodic circuits, and characterizes near dual-unitary circuits with rapid thermalization.

Findings

Strongly non-ergodic circuits do not thermalize and deviate from random matrix predictions.

Almost dual-unitary circuits thermalize exponentially fast, with observable and spectral properties approaching equilibrium.

Provides a complete classification of strongly non-ergodic circuits for qubits.

Abstract

We study thermalisation and spectral properties of extended systems connected, through their boundaries, to a thermalising Markovian bath. Specifically, we consider periodically driven systems modelled by brickwork quantum circuits where a finite section (block) of the circuit is constituted by arbitrary local unitary gates while its complement, which plays the role of the bath, is dual-unitary. We show that the evolution of local observables and the spectral form factor are determined by the same quantum channel, which we use to characterise the system's dynamics and spectral properties. In particular, we identify a family of fine-tuned quantum circuits -- which we call strongly non-ergodic -- that fails to thermalise even in this controlled setting, and, accordingly, their spectral form factor does not follow the random matrix theory prediction. We provide a set of necessary conditions on the local quantum gates that lead to strong non-ergodicity, and in the case of qubits, we provide a complete classification of strongly non-ergodic circuits. We also study the opposite extreme case of circuits that are almost dual-unitary, i.e., where thermalisation occurs with the fastest possible rate. We show that, in these systems, local observables and spectral form factor approach respectively thermal values and random matrix theory prediction exponentially fast. We provide a perturbative characterisation of the dynamics and, in particular, of the time-scale for thermalisation.