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

Abstract

We study the scalar one-component two-dimensional (2D) ${\varphi }^4$ model by computer simulations, with local Metropolis moves. The equilibrium exponents of this model are well-established, e.g. for the 2D ${\varphi }^4$ 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 ${\varphi }^4$ model using two independent methods: (a) by calculating the relative terminal exponential decay time $\tau$ for the correlation function $⟨\varphi (t)\varphi (0)⟩$, 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) $⟨\Delta {\varphi }^2(t)⟩\sim t^c$ 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 $z_c=2.17\pm 0.03$ and $z_c=2.19\pm 0.03$ 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.

arXiv comment14 pages, 4 figures, 6 figure files, to appear in Phys. Rev. E

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