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arXiv:2605.10346

Non-equilibrium scaling across first-order transitions with relativistic scalar fields

Leon J. Sieke, Jessica Fuchs, Lorenz von Smekal

Published May 11, 2026• 0 citations
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Abstract

We investigate the out-of-equilibrium dynamics of a relativistic $Z_2$-symmetric scalar field theory with Langevin dynamics in two and three spatial dimensions under linear driving across magnetic first-order phase transitions, close to and far below the critical temperature $T_c$. Using classical-statistical lattice simulations, we find that if the driving timescale is sufficiently fast, the system exhibits finite-time scaling behavior independent of temperature and dimensionality, identical to that observed in mean-field simulations. In slow quenches near $T_c$ this mean-field behavior crosses over to critical Kibble-Zurek scaling behavior, while for temperatures $T \ll T_c$ nucleation and growth dominate the transition dynamics, resulting in corrections to scaling. Near the transition point where the order parameter changes sign, the crossover between mean-field and critical out-of-equilibrium dynamics is found to be well described by the leading algebraic correction to Kibble-Zurek scaling. We find that universal non-equilibrium scaling behavior can be observed for $T \lesssim T_c$, provided the driving is fast enough to avoid nucleation but slow enough for correlations to form, and compute the associated universal scaling functions for the order parameter.

DOI
10.48550/arxiv.2605.10346

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We investigate the out-of-equilibrium dynamics of a relativistic Z2Z_2Z2​-symmetric scalar field theory with Langevin dynamics in two and three spatial dimensions under linear driving across magnetic first-order phase transitions, close to and far below the critical temperature TcT_cTc​. Using classical-statistical lattice simulations, we find that if the driving timescale is sufficiently fast, the system exhibits finite-time scaling behavior independent of temperature and dimensionality, identical to that observed in mean-field simulations. In slow quenches near TcT_cTc​ this mean-field behavior crosses over to critical Kibble-Zurek scaling behavior, while for temperatures T≪TcT \ll T_cT≪Tc​ nucleation and growth dominate the transition dynamics, resulting in corrections to scaling. Near the transition point where the order parameter changes sign, the crossover between mean-field and critical out-of-equilibrium dynamics is found to be well described by the leading algebraic correction to Kibble-Zurek scaling. We find that universal non-equilibrium scaling behavior can be observed for T≲TcT \lesssim T_cT≲Tc​, provided the driving is fast enough to avoid nucleation but slow enough for correlations to form, and compute the associated universal scaling functions for the order parameter.
The critical behaviour of a relativistic scalar field theory with Z2 symmetry is investigated near a second-order phase transition. Two sets of dynamic equations are employed, allowing to control conservation laws of both order parameter and energy density. We calculate spectral functions of the order parameter as well as unequal-time correlation functions of the energy momentum tensor at zero and non-vanishing spatial momenta from first-principles classical-statistical lattice simulations in real-time. For both, we investigate general properties and relevant degrees of freedom in distinct regions of the phase diagram. Close to the critical point, we find signatures of dynamic scaling behaviour and calculate the dynamic critical exponent z controlling the divergence of the critical time scale. For both the order parameter spectral function as well as the energy density autocorrelation, we extract universal dynamic scaling functions. Modifying the simulation framework to include dynamically changing temperature and external field allows us to study non-equilibrium phenomena. For the special case of instant quenches to the critical point, we identify universal scaling behaviour controlled by the initial magnetization, and calculate the related additional dynamic critical exponent. We extract the universal non-equilibrium scaling functions for the evolution of both the order parameter as well as the correlation length.
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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/tsh(t)=t/t_sh(t)=t/ts​ (tst_sts​ is the time scale of the protocol), across their first-order quantum transitions (FOQTs) at h=0h=0h=0. The KZ protocol starts at time ti<0t_i<0ti​<0 from the negatively magnetized ground state for hi=ti/ts<0h_i = t_i/t_s<0hi​=ti​/ts​<0. Then, the system evolves unitarily up to a time tf>0t_f > 0tf​>0, such that the magnetization of the state at time tft_ftf​ 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 ttt and tst_sts​ 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∗>0h(t)=h*>0h(t)=h∗>0, where h∗h*h∗ decreases with increasing tst_sts​, as h∗∼1/ln⁡tsh*\sim 1/\ln t_sh∗∼1/lnts​. Moreover, in the large-tst_sts​ limit, the time-dependence of the magnetization is described by a universal function of Ω=t/τsΩ= t/τ_sΩ=t/τs​, with τs=ts/ln⁡tsτ_s = t_s/\ln t_sτs​=ts​/lnts​.
We investigate the nonequilibrium dynamics of quantum spin chains during a round-trip protocol that slowly drives the system across a quantum first-order transition. Out-of-equilibrium scaling behaviors \`a la Kibble-Zurek for the single-passage protocol across the first-order transition have been previously determined. Here, we show that such scaling relations persist when the driving protocol is inverted and the transition is approached again by a far-from-equilibrium state. This results in a quasi-universality of the scaling functions, which keep some dependence on the details of the protocol at the inversion time. We explicitly determine such quasi-universal scaling functions by employing an effective two-level description of the many-body system near the transition. We discuss the validity of this approximation and how this relates to the observed scaling regime. Although our results apply to generic systems, we focus on the prototypical example of a 1D transverse field Ising model in the ferromagnetic regime, which we drive across the first-order transitions through a time-dependent longitudinal field.