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

Abstract

We study the scalar one-component two-dimensional (2D) $ϕ^4$ model by computer simulations, with local Metropolis moves. The equilibrium exponents of this model are well-established, e.g. for the 2D $ϕ^4$ model $γ= 1.75$ and $ν= 1$. 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 $ϕ^4$ model using two independent methods: (a) by calculating the relative terminal exponential decay time $τ$ for the correlation function $\langle ϕ(t)ϕ(0)\rangle$, and thereafter fitting the data as $τ\sim L^{z_c}$, where $L$ is the system size, and (b) by measuring the anomalous diffusion exponent for the order parameter, viz., the mean-square displacement (MSD) $\langle Δϕ^2(t)\rangle\sim t^c$ as $c=γ/(νz_c)$, and from the numerically obtained value $c\approx 0.80$, we calculate $z_c$. For different values of the coupling constant $λ$, 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 $λ$, 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.