Kibble-Zurek dynamics across the first-order quantum transitions of quantum Ising chains in the thermodynamic limit

We study the out-of-equilibrium Kibble-Zurek (KZ) dynamics in quantum Ising chains in a transverse field, driven by a time-dependent longitudinal field $h(t)=t/{t}_{s}$ (${t}_{s}$ is the timescale of the protocol), across their first-order quantum transitions (FOQTs) at $h=0$. The KZ protocol starts at time ${t}_{i}<0$ from the negatively magnetized ground state for ${h}_{i}={t}_{i}/{t}_{s}<0$. Then, the system evolves unitarily up to a time ${t}_{f}>0$, such that the magnetization of the state at time ${t}_{f}$ is positive. In finite-size systems, the KZ dynamics develops out-of-equilibrium finite-size scaling (OFSS) behaviors. Their scaling variables depend either exponentially or with a power law on the size, depending on the boundary conditions (BC). The OFSS functions can be computed in effective models restricted to appropriate low-energy (magnetized and/or kink) states. The KZ scaling behavior drastically changes in the thermodynamic limit (TL), defined as the infinite-size limit keeping $t$ and ${t}_{s}$ fixed, which appears substantially unrelated with the OFSS regime, because it involves higher-energy multikink states, which are irrelevant in the OFSS limit. The numerical analyses of the KZ dynamics in the TL show the emergence of a quantum spinodal-like scaling behavior at the FOQTs for all considered BC, which is independent of the BC. The longitudinal magnetization changes sign at $h(t)={h}_{★}>0$, where ${h}_{★}$ decreases with increasing ${t}_{s}$, as ${h}_{★}\ensuremath{\sim}1/ln{t}_{s}$. Moreover, in the large-${t}_{s}$ limit, the time dependence of the magnetization is described by a universal function of $\mathrm{\ensuremath{\Omega}}=t/{\ensuremath{\tau}}_{s}$, with ${\ensuremath{\tau}}_{s}={t}_{s}/ln{t}_{s}$.