Thermalization slowing down of weakly nonintegrable quantum spin dynamics

We study thermalization slowing down of a quantum many-body spin system upon approach to two distinct integrability limits. Motivated by previous studies of classical systems, we identify two thermalization timescales: one quantum Lyapunov timescale is extracted by quantifying operator growth in time on an appropriately defined basis, while another ergodization timescale is related to the 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. We investigate the relative strength of the divergence in the two limits and find that, despite significant qualitative differences in the mechanism of integrability breaking, the timescales diverge in a similar fashion. This allows us to establish a universality of integrability breaking in quantum spin dynamics.