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 $h\left(t\right)=t/t_s$ ($t_s$ 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 $h\left(t\right)=h_{★}>0$, where $h_{★}$ decreases with increasing $t_s$, as $h_{★}\sim 1/ln{t}_s$. Moreover, in the large-$t_s$ limit, the time dependence of the magnetization is described by a universal function of $\mathrm{\Omega }=t/{\tau }_s$, with ${\tau }_s=t_s/ln{t}_s$.

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• Quantum phase transitions
• Quantum quench
• Nonequilibrium systems
• Quantum many-body systems
• Quantum spin models