Thermalization slowing down of weakly nonintegrable quantum spin dynamics

We study thermalization slowing down of a quantum many-body system upon approach to two distinct integrability limits. Motivated by previous studies of classical systems, we identify two thermalization time scales: one quantum Lyapunov time scale is extracted by quantifying operator growth in time in an appropriately defined basis, while another ergodization time scale is related to statistics of fluctuations of the time-evolved operator around its mean value based on the eigenstate thermalization hypothesis. Using a paradigmatic Quantum Ising chain we find that both timescales diverge upon approach to integrability. The relative strength of the divergence of the scales depends on the particular integrable limit. This allows us to define two different universality classes of quantum thermalization: short- and long-range networks.