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

TL;DR

The paper investigates non-equilibrium Kibble-Zurek dynamics when a one-dimensional quantum Ising chain is driven across a first-order quantum transition by a time-dependent longitudinal field $h(t)=t/t_s$. Through exact-diagonalization across multiple boundary conditions, it demonstrates a thermodynamic-limit scaling regime in which the magnetization dynamics collapse onto a universal function of the scaling variable $\\Omega=t/\\tau_s$ with $\\tau_s=t_s/\\ln t_s$, and where the magnetization switching occurs at $h_*(t_s)\sim 1/\\ln t_s$, independently of boundary conditions. The study distinguishes this TL behavior—spinodal-like and controlled by multi-kink states—from the finite-size OFSS regime, which depends on the gap scaling $\\Delta(L)$ and boundary conditions. It also shows that while OFSS can be well described by two-level or kink-based models in particular BCs, the TL dynamics require a broader many-body perspective, especially for OFBC and ABC where kink towers dominate. Overall, the results reveal a BC-independent, universal nonequilibrium scaling regime at FOQTs in the TL with potential experimental relevance for quantum simulators and qubit arrays.

Abstract

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 time scale 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 multi-kink 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 $h*\sim 1/\ln t_s$. Moreover, in the large-$t_s$ limit, the time-dependence of the magnetization is described by a universal function of $Ω= t/τ_s$, with $τ_s = t_s/\ln t_s$.

Figures (13)

• Figure 1: Rescaled central longitudinal magnetization $M_c$ defined in Eq. \ref{['MMcdef']}, for chains with OBC, plotted vs $h(t)$, for $t_s = 25$ (top) and $200$ (bottom). We report curves for different lattice sizes up to $L=22$ (see legend). With increasing $L$, the data appear to converge to an asymptotic curve, which we consider as the infinite-size limit at fixed $h(t)$. Unless otherwise specified, all numerical data shown here and in the following figures have been obtained fixing $g=0.5$.
• Figure 2: Rescaled central magnetization $M_c$ for chains with OBC and $L=22$, plotted vs $h(t)$ (top) and vs the rescaled variable $\Omega(t) = h(t) \, \ln t_s$ (bottom). The curves correspond to different time scales, up to $t_s = 200$ (see legend). For these values of $t_s$ the data for chains with 22 sites provide, with good approximation, the time behavior of $M_c$ in the infinite-size limit.
• Figure 3: Rescaled central magnetization for $t_s = 25$ (top) and $200$ (bottom) for EFBC. Results should be compared with those for OBC, reported in Fig. \ref{['OBCinfL']}. Note that convergence is faster in the EFBC case. Indeed, while EFBC data for $L=22$ and $t_s=200$ are clearly asymptotic, in the OBC case significant finite-size effects are still present for $L=22$ when $t_s = 200$ (see the bottom panel of Fig. \ref{['OBCinfL']}).
• Figure 4: Infinite-size rescaled central magnetization for different values of $t_s$, vs $h(t)$ (top) and $\Omega(t)$ (bottom). The data have been obtained for EFBC. Results should be compared with those for OBC, reported in Fig. \ref{['OBCtssca']}. Also for EFBC, we observe a reasonable scaling when the data are plotted as a function of the rescaled variable $\Omega(t)$ (bottom panel).
• Figure 5: Rescaled central magnetization for a chain with OFBC, as a function of the rescaled variable $\widehat{\Phi}$, defined in Eq. \ref{['katdef']}. We report results for two values of $\Upsilon$, defined in Eq. (\ref{['upsilondef']}): $\Upsilon = 0.5$ (top) and $1$ (bottom). The colored dashed-dotted curves have been obtained by numerically solving the KZ dynamics in the full Hilbert space, for different system sizes (see legend). The continuous black line (1-kink) is the result of a computation performed in the restricted single-kink model discussed in App. \ref{['AppOFSS']}, with an appropriate rescaling of $\widehat{\Phi}$ and $\Upsilon$ (see text).