Dynamic scaling at classical phase transitions approached through non-equilibrium 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 $v \in [0, \infty)$. 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 $z_M=2.1767(5)$, for Swendsen-Wang multi-cluster updates $z_{\rm SW}=0.297(3)$, and for Wolff single-cluster updates $z_{W}=0.30(2)$. For Wolff dynamics, we find an interesting behavior with a non-analytic break-down 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.