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arXiv:2305.12993 · Physical review. B./Physical review. B

Out-of-equilibrium scaling behavior arising during round-trip protocols across a quantum first-order transition

Francesco Tarantelli, Stefano Scopa

Published Sep 26, 2023• 8 citations
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Abstract

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.

DOI
10.1103/physrevb.108.104316

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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.
We study the out-of-equilibrium dynamics of one-dimensional quantum Ising models in a transverse field ggg, driven by a time-dependent longitudinal field hhh across their {\em magnetic} first-order quantum transition at h=0h=0h=0, for sufficiently small values of ∣g∣|g|∣g∣. We consider nearest-neighbor Ising chains of size LLL with periodic boundary conditions. We focus on the out-of-equilibrium behavior arising from Kibble-Zurek protocols, in which hhh is varied linearly in time with time scale tst_sts​, i.e., h(t)=t/tsh(t)=t/t_sh(t)=t/ts​. The system starts from the ground state at hi≡h(ti)<0h_i\equiv h(t_i)<0hi​≡h(ti​)<0, where the longitudinal magnetization MMM is negative. Then it evolves unitarily up to positive values of h(t)h(t)h(t), where M(t)M(t)M(t) becomes eventually positive. We identify several scaling regimes characterized by a nontrivial interplay between the size LLL and the time scale tst_sts​, which can be observed when the system is close to one of the many avoided level crossings that occur for h≥0h\ge 0h≥0. In the L→∞L\to\inftyL→∞ limit, all these crossings approach h=0+h=0^+h=0+, making the study of the thermodynamic limit, defined as the limit L→∞L\to\inftyL→∞ keeping ttt and tst_sts​ constant, problematic. We study such limit numerically, by first determining the large-LLL quantum evolution at fixed tst_sts​, and then analyzing its behavior with increasing tst_sts​. Our analysis shows that the system switches from the initial state with M<0M<0M<0 to a positively magnetized state at h=h∗(ts)>0h = h_*(t_s)>0h=h∗​(ts​)>0, where h∗(ts)h_*(t_s)h∗​(ts​) decreases with increasing tst_sts​, apparently as h∗∼1/ln⁡tsh_*\sim 1/\ln t_sh∗​∼1/lnts​. This suggests the existence of a scaling behavior in terms of the rescaled time Ω=tln⁡ts/tsΩ= t \ln t_s/t_sΩ=tlnts​/ts​. The numerical results also show that the system converges to a nontrivial stationary state in the large-ttt limit, characterized by an energy significantly larger than that of the corresponding homogeneously magnetized ground state.
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 address the out-of-equilibrium dynamics of many-body systems subject to time-dependent round-trip protocols across quantum and classical (thermal) phase transitions. They are realized by slowly changing one relevant parameter www across its critical point wc=0{w}_{c}=0wc​=0, linearly in time with a large timescale ts{t}_{s}ts​, from wi<0{w}_{i}<0wi​<0 to wf>0{w}_{f}>0wf​>0 and then back to wi<0{w}_{i}<0wi​<0, thus entailing multiple passages through the critical point. Analogously to the one-way Kibble-Zurek protocols across a critical point, round-trip protocols develop dynamic scaling behaviors at both classical and quantum transitions put forward within renormalization-group frameworks. The scaling scenario is analyzed within some paradigmatic models undergoing quantum and classical transitions belonging to the two-dimensional Ising universality class, such as one-dimensional quantum Ising models and fermionic wires, and two-dimensional classical Ising models (supplemented with a purely relaxational dynamics). While the dynamic scaling frameworks are similar for classical and quantum systems, substantial differences emerge due to the different nature of their dynamics, which is purely relaxational for classical systems (implying thermalization in the large-time limit at fixed model parameters), and unitary in the case of quantum systems. In particular, when the critical point separates two gapped (short-ranged) phases and the extreme value wf>0{w}_{f}>0wf​>0 is kept fixed in the large-ts{t}_{s}ts​ limit of the round-trip protocol, we observe hysteresislike scenarios in classical systems, while quantum systems do not apparently develop a sufficiently robust scaling limit along the return way, due to the presence of rapidly oscillating relative phases among the relevant quantum states.
We study the effects of symmetry-breaking perturbations in the out-of-equilibrium quantum dynamics of many-body systems slowly driven by a time-dependent symmetry-preserving parameter, across the quantum critical regime associated with a continuous quantum transition (CQT). For this purpose, we analyze the out-of-equilibrium dynamics arising from generalized Kibble-Zurek (KZ) protocols, within a dynamic renormalization-group framework allowing for finite-size systems. We show that the time dependence of generic observables develops an out-of-equilibrium finite-size scaling (FSS) behavior, arising from the interplay between the timescale ts{t}_{s}ts​ of the parameter variations in the KZ protocol, the size LLL of the system, and the strength hhh of the symmetry-breaking perturbation, in the limit of large ts{t}_{s}ts​ and LLL. Moreover, scaling arguments based on the first-order adiabatic approximation of slow variations in quantum systems allow us to characterize the approach to the adiabatic regimes for some limits of the model parameters (for example, when we take ${t}_{s}\ensuremath{\rightarrow}\ensuremath{\infty}$ before $L\ensuremath{\rightarrow}\ensuremath{\infty}$), predicting asymptotic power-law suppressions of the nonadiabatic behaviors in the adiabatic limits. This out-of-equilibrium FSS is supported by numerical analyses for the paradigmatic quantum Ising chain along generalized KZ protocols, with a time-dependent transverse field crossing its CQT, in the presence of a static longitudinal field breaking the Z2{\mathbb{Z}}_{2}Z2​ symmetry.
We investigate the quantum dynamics generated by quantum quenches (QQs) of the Hamiltonian parameters in many-body systems, focusing on protocols that cross first-order and continuous quantum transitions, both in finite-size systems and in the thermodynamic limit. As a paradigmatic example, we consider the quantum Ising chain in the presence of homogeneous transverse (ggg) and longitudinal (hhh) magnetic fields. This model exhibits a continuous quantum transition (CQT) at g=gcg=g_cg=gc​ and h=0h=0h=0, and first-order quantum transitions (FOQTs) driven by hhh along the line h=0h=0h=0 (g<gcg<g_cg<gc​). In the integrable limit h=0h=0h=0, the system can be mapped onto a quadratic fermionic theory; however, any nonvanishing longitudinal field breaks integrability and the spectrum of the resulting Hamiltonian is generally expected to enter a chaotic regime. We analyze QQs in which the longitudinal field is suddenly changed from a negative value hi<0h_i < 0hi​<0 to a positive value hf>0h_f>0hf​>0. We focus on values of hfh_fhf​ such that the spectrum of the post-QQ Hamiltonian H(g,hf)H(g,h_f)H(g,hf​) lies in the chaotic regime, where thermalization may emerge at asymptotically long times. We study the out-of-equilibrium dynamics for different values of ggg, finding qualitatively distinct behaviors for g>gcg > g_cg>gc​ (where the chain is in the disordered phase), for g=gcg = g_cg=gc​ (QQ across the CQT), and for g<gcg<g_cg<gc​ (QQ across the FOQT line).
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.