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

Dynamical phase transitions (DPTs) are generally defined under two different but related categories: DPT-I where the equilibrium value, or long-time average, of the order parameter as a function of the control parameter demonstrates a phase boundary; DPT-II where the Loschmidt return rate shows a cusp singularity in real-time dynamics. Here we follow up on a recent numerical demonstration [Phys. Rev. Lett. 121, 016801 (2018)] where out-of-time-order correlators (OTOC) of a single-site are shown to exhibit DPT-I when quenched from polarized states both for integrable and nonintegrable short-range interacting 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. Quenched from polarized states, longitudinal magnetization has a featureless steady state regime in time for short-range TFIM, which prevents it from exhibiting DPT-I. Thus, we further question whether the transient regimes of such non-equilibrium processes could encode information about the underlying QPT. We show that the decay rates of time-dependent and single-site observables exhibit a cusplike feature, a reminiscent of the singularity in the integrable TFIM. The cusp separates two dynamical regimes, 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 behavior 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 regimes.