Critical Dynamical Exponent of the Two-Dimensional Scalar $\phi^4$ Model with Local Moves

TL;DR

This study determines the critical dynamical exponent $z_c$ of the 2D $\,\phi^4$ model using simulations, finding it consistent with the 2D Ising model and demonstrating the applicability of GLE with memory kernel.

Contribution

The paper provides the first precise numerical estimates of $z_c$ for the 2D $\,\phi^4$ model and shows its independence from coupling strength, aligning it with the Ising universality class.

Findings

$z_c$ is approximately 2.17-2.19 for the 2D $\,\phi^4$ model.

$z_c$ appears independent of the coupling constant $\,\lambda$.

GLE with memory kernel accurately models the observed anomalous diffusion.

Abstract

We study the scalar one-component two-dimensional (2D) $\phi^4$ model by computer simulations, with local Metropolis moves. The equilibrium exponents of this model are well-established, e.g. for the 2D $\phi^4$ model $\gamma= 1.75$ and $\nu= 1$. The model has also been conjectured to belong to the Ising universality class. However, the value of the critical dynamical exponent $z_c$ is not settled. In this paper, we obtain $z_c$ for the 2D $\phi^4$ model using two independent methods: (a) by calculating the relative terminal exponential decay time $\tau$ for the correlation function $\langle \phi(t)\phi(0)\rangle$, and thereafter fitting the data as $\tau \sim L^{z_c}$, where $L$ is the system size, and (b) by measuring the anomalous diffusion exponent for the order parameter, viz., the mean-square displacement (MSD) $\langle \Delta \phi^2(t)\rangle\sim t^c$ as $c=\gamma/(\nu z_c)$, and from the numerically obtained value $c\approx 0.80$, we calculate $z_c$. For different values of the coupling constant $\lambda$, we report that $z_c=2.17\pm0.03$ and $z_c=2.19\pm0.03$ for the two methods respectively. Our results indicate that $z_c$ is independent of $\lambda$, and is likely identical to that for the 2D Ising model. Additionally, we demonstrate that the Generalised Langevin Equation (GLE) formulation with a memory kernel, identical to those applicable for the Ising model and polymeric systems, consistently capture the observed anomalous diffusion behavior.