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/ts (ts is the timescale of the protocol), across their first-order quantum transitions (FOQTs) at h = 0. The KZ protocol starts at time ti < 0 from the negatively magnetized ground state for hi = ti/ts < 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 ts 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 ts, as h* ∼ 1/ ln ts. Moreover, in the large-ts limit, the time dependence of the magnetization is described by a universal function of Ω = t/τs, with τs = ts/ ln ts.