Thermalisation Dynamics and Spectral Statistics of Extended Systems with Thermalising Boundaries

Pavel Kos, Tomaz Prosen, Bruno Bertini

Published 2021-08-17Version 1

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) is generated by generic local unitary gates while the complement 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 localising -- 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 localisation, and in the case of qubits, we provide a complete classification of strongly localising circuits. We also study the opposite extreme case of circuits that are almost dual-unitary, i.e., where instead of being localised the information moves at the maximal speed allowed by the brick-work geometry. 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.