Out-of-equilibrium dynamics across the first-order quantum transitions of one-dimensional quantum Ising models

We study the out-of-equilibrium dynamics of one-dimensional quantum Ising models in a transverse field g. driven by a time-dependent longitudinal field h across their magnetic first-order quantum transition at h = 0 for sufficiently small values of [g]. We consider nearest-neighbor Ising chains of size L with periodic boundary conditions. We focus on the out-of-equilibrium behavior arising from Kibble-Zurek protocol, in which h is varied linearly in time with a timescale 1,, ie, h(t)=t/t(s),. The system starts from the ground state at h(i) equivalent to h(t(i)) < 0 , where the longitudinal magnetization M is negative. Then it evolves unitarily up to positive values of h(t), where M(t) becomes eventually positive. We identify several scaling regimes characterized by a nontrivial interplay be-tween the size 1. and the timescale 1,, which can be observed when the system is close to one of the many avoided level crossings that occur for h >= 0 In the L -> infinity limit, all these crossings approach h = 0, making the study of the thermodynamic limit, defined as the limit L -> infinity keeping 1 and 1, constant, problematic. We study this limit numerically, by first determining the large-1. quantum evolution at fixed t,, and then analyzing its behavior with increasing 1,. Our analysis shows that the system switches from the initial state with to a positively magnetized state at h = h(*)(t(s)) > 0, where h(*)(t(s)) decreases with increasing t(s), apparently as h(star)similar to 1/ln t(s). This suggests the existence of a scaling behavior in terms of the rescaled time Omega = t ln t(s)/t(s). The numerical results also show that the system converges to a nontrivial stationary state in the large-t limit, characterized by an energy significantly larger than that of the corresponding homogeneously magnetized ground state.