Dynamical crossover in the transient quench dynamics of short-range transverse-field Ising models

In this work, we follow up on a recent numerical demonstration where out-of-time-order correlators (OTOC) of a single-site are shown to exhibit type-I dynamical phase transition (DPT-I) for (non-)integrable short-range transverse-field Ising model (TFIM). Given the requirement of sophisticated probe techniques to measure OTOC, we ask whether simpler single-site probes could be constructed to detect quantum phase transitions (QPT), e.g. magnetization per site. Because magnetization for short-range TFIM cannot exhibit DPT-I when quenched from polarized states, we question whether the transient regime could encode information about the underlying QPT. Decay rates of single-site observables exhibit a cusplike feature which separates two dynamical phases, ordered and disordered, both of which have distinct nonequilibrium responses. We construct a dynamical order parameterlike quantity that exhibits a scaling law at the vicinity of the cusp. These signatures suggest a crossover that is originated from the underlying QPT and encoded into the transient regime of one-point observables in short-range TFIM. When integrability is strongly broken, the crossover boundary turns into a region that separates two other dynamical regions that act like dynamically-ordered and -disordered phases.