Dynamic scaling at classical phase transitions approached through nonequilibrium quenching

Abstract

We use Monte Carlo simulations to demonstrate generic scaling aspects of classical phase transitions approached through a quench (or annealing) protocol where the temperature changes as a function of time with velocity v. Using a generalized Kibble-Zurek ansatz, we demonstrate dynamic scaling for different types of stochastic dynamics (Metropolis, Swendsen-Wang, and Wolff) on Ising models in two and higher dimensions. We show that there are dual scaling functions governing the dynamic scaling, which together describe the scaling behavior in the entire velocity range v ∈ [0, ∞). These functions have asymptotics corresponding to the adiabatic and diabatic limit, and close to these limits they are perturbative in v and 1/v, respectively. Away from their perturbative domains, both functions cross over into the same universal power-law scaling form governed by the static and dynamic critical exponents (as well as an exponent characterizing the quench protocol). As a by-product of the scaling studies, we obtain high-precision estimates of the dynamic exponent z for the two-dimensional Ising model subject to the three variants of Monte Carlo dynamics; for single-spin Metropolis updates zM = 2.1767(5), for Swendsen-Wang multi-cluster updates zSW = 0.297(3), and for Wolff singlecluster updates zW = 0.30(2). For Wolff dynamics, we find an interesting behavior with a non-analytic breakdown of the quasi-adiabatic and diabatic scaling, instead of the generic smooth cross-over described by a power law. We interpret this disconnect between the two scaling regimes as a dynamic phase transition of the Wolff algorithm, caused by an effective sudden loss of ergodicity at high velocity.

Dynamic scaling in the two-dimensional Ising spin glass with normal-distributed couplings

“…In contrast, at T > 0 transitions, in both nonrandom and spin-glass models [12,20,21], the dynamic exponent is finite and takes the same value at equilibrium and in SA analyzed within the KZ hypothesis. Clearly the source of this difference lays in the fact that the equilibrium dynamics is nonergodic in the limit T → 0.…”

“…(1) by replacing L by ξ v . Thus, we conclude that, unlike other cases studied so far [20,44], here f 0 (vL z+1/ν ) is not Taylorexpandable but a corresponding functionf 0 (L/ξ v ) is. We do not have an explanation for this apparently different analytic form of the scaling function in this case, but empirically the evidence is compelling, as seen more directly in Fig.…”

“…(6). According to the general non-equilibrium scaling form that describes the dynamics in its full regime of velocities and sufficiently large system sizes, the order parameter q 2 can be written in the following way [20]:…”

“…(5), a physical quantity A evaluated at the critical point can be written in the following finitesize scaling form [19,20,22]:…”

“…Given the difficulties in studying the critical behavior through equilibrium simulations, the recently developed non-equilibrium approach based on generalized KibbleZurek (KZ) scaling [15][16][17][18][19][20][21][22] provides a powerful alternative method for studies of spin-glass models. KZ scaling of simulated annealing (SA) results has been successfully applied to 3D and 2D spin glass systems in order to extract the dynamic exponent z and other critical exponents [23,24].…”