Growth of the out-of-time-order commutator in many-body localized systems: a typicality and universality

Abstract

We investigate the typicality of the out-of-time-order commutator (OTOC) in the many-body localized (MBL) disordered systems by considering the estimate with a random pure state across individual disorder realizations. In the MBL phase of the disordered Heisenberg XXZ chain, we numerically observe a broken typicality of a general random pure state: one may need a particular type of random states to reproduce the probability distribution of the OTOC evaluated at a given time within the infinite-temperature thermal ensemble. This is in contrast to the observations in the ergodic phase where all examined random state preparations lead to the same distribution. On the other hand, despite the quantitative deviations in the MBL phase, the different state preparations exhibit a similar OTOC growth pattern which may characterize the MBL phase in realistic disordered systems. A power-law-type growth appears after a disorder-configuration dependent onset time, which for some disorder realizations is very close to the $t^2$-form of the OTOC growth derived in the effective Hamiltonian of a fully MBL system. We observe the same features also in the MBL phase of the disordered mixed-field Ising chain.