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arXiv:2507.16168

Zurek's scaling-law in the Ising model out of thermal equilibrium

José Armando Pérez-Loera, Wolfgang Bietenholz A

Published Jul 22, 2025• 0 citations
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Abstract

The Kibble mechanism plays a prominent role in the theory of the early Universe, as an explanation of the possible formation of cosmic strings. Zurek suggested the analogous effect in liquid helium under rapid cooling, and he conjectured - together with del Campo and Kibble - a scaling-law for the relation between the density of remnant topological defects and the cooling rate, when a system passes through its critical temperature. Such scaling-laws were indeed observed in condensed matter experiments. Here we test the validity of Zurek's scaling-law in a different framework. We numerically study the behavior of the Ising model (with classical spins) in 1, 2 and 3 dimensions under rapid cooling. This model does not have topological defects, so we consider the dynamics of domains instead, i.e. we measure their evolution during a cooling process down to zero temperature. For several Markov chain Monte Carlo algorithms, we consistently observe scaling-laws along the lines of Zurek's conjecture, in all dimensions under consideration, which shows that this feature holds more generally than expected. It is highly remarkable that even the exponents of these scaling-laws are consistent for three different algorithms, which hints at a physical meaning.

DOI
10.48550/arxiv.2507.16168

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The Kibble mechanism plays a prominent role in the theory of the early Universe, as an explanation of the possible formation of cosmic strings. Zurek suggested the analogous effect in liquid helium under rapid cooling, and he conjectured - together with del Campo and Kibble - a scaling-law for the relation between the density of remnant topological defects and the cooling rate, when a system passes through its critical temperature. Such scaling-laws were indeed observed in condensed matter experiments. Here we test the validity of Zurek's scaling-law in a different framework. We numerically study the behavior of the Ising model (with classical spins) in 1, 2 and 3 dimensions under rapid cooling. This model does not have topological defects, so we consider the dynamics of domains instead, i.e. we measure their evolution during a cooling process down to zero temperature. For several Markov chain Monte Carlo algorithms, we consistently observe scaling-laws along the lines of Zurek's conjecture, in all dimensions under consideration, which shows that this feature holds more generally than expected. It is highly remarkable that even the exponents of these scaling-laws are consistent for three different algorithms, which hints at a physical meaning.
We present a study of the 3d O(2) non-linear σ\sigmaσ-model on the lattice, which exhibits topological defects in the form of vortices. They tend to organize into vortex lines that bear close analogies with global cosmic strings. Therefore, this model serves as a testbed for studying the dynamics of topological defects. It undergoes a second order phase transition, hence it is appropriate for investigating the Kibble-Zurek mechanism. In this regard, we explore the persistence of topological defects when the temperature is rapidly reduced from above to below the critical temperature; this cooling (or “quenching”) process takes the system out of thermal equilibrium. We probe a wide range of inverse cooling rates τQ\tau_\mathrm{Q}τQ​ and final temperatures, employing distinct Monte Carlo algorithms. The results consistently show that the density of persisting topological defects follows a power-law in τQ\tau_\mathrm{Q}τQ​, in agreement with Zurek’s conjecture. On the other hand, at this point, our results do not confirm Zurek’s prediction for the exponent in this power-law, but its final test is still under investigation.
The crossing of a continuous phase transition gives rise to the formation of topological defects described by the Kibble-Zurek mechanism (KZM) in the limit of slow quenches. The KZM predicts a universal power-law scaling of the defect density as a function of the quench time. We focus on the deviations from KZM experimentally observed in rapid quenches and establish their universality. While KZM scaling holds below a critical quench rate, for faster quenches the defect density and the freeze-out time become independent of the quench rate and exhibit a universal power-law scaling with the final value of the control parameter. These predictions are verified in several paradigmatic scenarios in both the classical and quantum domains.
Kibble-Zurek mechanism is a theory of defect formation in a non-equilibrium continuous phase transition. So far the theory has been successfully tested by numerical simulations and condensed matter experiments in a number of systems with small thermal fluctuations. This paper reports first numerical test of the mechanism in a system with large thermal fluctuations and strongly non-mean-field behavior: the two dimensional Ising model. The theory predicts correctly the initial density of defects that survive a quench from the disordered phase. However, before the system leaves the Ginzburg regime of large fluctuations most of these defects are annihilated and the final density is determined by the dynamics of the annihilation process only.
We propose a theory to explain the experimental observed deviation from the Kibble-Zurek mechanism (KZM) scaling in rapidly quenched critical phase transition dynamics. There is a critical quench rate τQc1τ_{Q}^{c1}τQc1​ above it the KZM scaling begins to appear. Smaller than τQc1τ_Q^{c1}τQc1​, the defect density nnn is a constant independent of the quench rate but depends on the final temperature TfT_fTf​ as n∝LdεTfdνn \propto L^d ε_{T_f} ^{d ν}n∝LdεTf​dν​, the freeze out time t^\hat{t}t^ admits the scaling law t^∝εTf−νz\hat{t} \propto ε_{T_f}^{-νz}t^∝εTf​−νz​ where ddd is the spatial dimension, εTf=(1−Tf/Tc)ε_{T_f}= (1-T_f/T_c)εTf​​=(1−Tf​/Tc​) is the dimensionless reduced temperature, LLL is the sample size, ννν and zzz are spatial and dynamical critical exponents. Quench from TcT_cTc​, the critical rate is determined by the final temperature TfT_fTf​ as $τ_Q^{c1} \propto ε_{T_f}^{-(1+z ν)} $. All the scaling laws are verified in a rapidly quenched superconducting ring via the AdS/CFT correspondence.
We analyze scaling behaviors of simulated annealing carried out on various classical systems with topological order, obtained as appropriate limits of the toric code in two and three dimensions. We first consider the three-dimensional Z2{\mathbb{Z}}_{2}Z2​ (Ising) lattice gauge model, which exhibits a continuous topological phase transition at finite temperature. We show that a generalized Kibble-Zurek scaling ansatz applies to this transition, in spite of the absence of a local order parameter. We find perimeter-law scaling of the magnitude of a nonlocal order parameter (defined using Wilson loops) and a dynamic exponent $z=2.70\ifmmode\pm\else\textpm\fi{}0.03$, the latter in good agreement with previous results for the equilibrium dynamics (autocorrelations). We then study systems where (topological) order forms only at zero temperature---the Ising chain, the two-dimensional Z2{\mathbb{Z}}_{2}Z2​ gauge model, and a three-dimensional star model (another variant of the Z2{\mathbb{Z}}_{2}Z2​ gauge model). In these systems the correlation length diverges exponentially, in a way that is nonsmooth as a finite-size system approaches the zero temperature state. We show that the Kibble-Zurek theory does not apply in any of these systems. Instead, the dynamics can be understood in terms of diffusion and annihilation of topological defects, which we use to formulate a scaling theory in good agreement with our simulation results. We also discuss the effect of open boundaries where defect annihilation competes with a faster process of evaporation at the surface.
When a system is swept through a quantum critical point, the Kibble-Zurek mechanism predicts that the average number of topological defects follows a universal power-law scaling with the ramp timescale. This scaling behavior is determined by the equilibrium critical exponents of the underlying phase transition. We show that the correspondence between Kibble-Zurek scaling and quantum criticality does not hold generally. In particular, the defect density can exhibit a suppression faster than the Kibble-Zurek prediction even when the quench crosses a critical point, while conventional Kibble-Zurek scaling may persist for quenches through a noncritical point. Our results, based on models representative of a broad class of quasi-one-dimensional Fermi systems, identify the dynamical conditions under which universal defect scaling emerges and clarify the relation between defect generation and equilibrium criticality. © 2026 authors. Published by the American Physical Society.