Exploring dynamical phase transitions in the XY chain through a linear quench: Early and long-term perspectives

We investigate the nonequilibrium dynamics induced by a finite-time linear quench in the XY chain. Initially, we examine the dynamical quantum phase transition (DQPT-LE), characterized by the nonanalytic behavior of the Loschmidt amplitude. We find distinct behaviors of DQPT-LEs during and following the ramp. Following the ramp, the ramp crossing the critical point hc is the sufficient condition for the occurrence of DQPT-LE, but it is not during the ramp. Through adiabatic-impulse-adiabatic (AIA) approximation analysis, we establish that adequate distancing from the critical point is crucial for DQPT-LE manifestation during the ramp, elucidating the absence of DQPT-LE as the ramp gets faster. Additionally, we explore another type of dynamical phase transition (DQPT-OP), describing the long-term relaxation behavior of the order parameter. Our finding indicates that the asymptotic behavior of the time-dependent part induced by the linear quench is equivalent to that following a sudden quench, i.e., the time-dependent part exhibits power-law decays of -t-3/2 and similar to t-1/2 for the ramp to the commensurate and incommensurate phases, respectively. Moreover, we also delve into the steady part, which showcases nonanalytic singularities at the critical point.