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 $\left|g\right|$. 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 $t_s$, i.e., $h\left(t\right)=t/t_s$. The system starts from the ground state at $h_i\equiv h\left(t_i\right)<0$, where the longitudinal magnetization $M$ is negative. Then it evolves unitarily up to positive values of $h\left(t\right)$, where $M\left(t\right)$ becomes eventually positive. We identify several scaling regimes characterized by a nontrivial interplay between the size $L$ and the timescale $t_s$, which can be observed when the system is close to one of the many avoided level crossings that occur for $h\ge 0$. In the $L\to \infty$ limit, all these crossings approach $h=0^{+}$, making the study of the thermodynamic limit, defined as the limit $L\to \infty$ keeping $t$ and $t_s$ constant, problematic. We study this limit numerically, by first determining the large-$L$ quantum evolution at fixed $t_s$, and then analyzing its behavior with increasing $t_s$. Our analysis shows that the system switches from the initial state with $M<0$ to a positively magnetized state at $h=h_{★}\left(t_s\right)>0$, where $h_{★}\left(t_s\right)$ decreases with increasing $t_s$, apparently as $h_{★}\sim 1/ln{t}_s$. This suggests the existence of a scaling behavior in terms of the rescaled time $\mathrm{\Omega }=tln{t}_s/t_s$. 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.

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• First order phase transitions
• Nonequilibrium statistical mechanics
• Quantum phase transitions
• Finite-size scaling
• Quantum spin chains