Thermalization dynamics and spectral statistics of extended systems with thermalizing boundaries

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