Critical dynamical exponent of the two-dimensional scalar ϕ4 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 ⟨ Φ ( t ) Φ ( 0 ) ⟩ , and thereafter fitting the data as τ ∼ 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 ⟨ Δ Φ 2 ( t ) ⟩ ∼ t c as c = γ / ( ν z c ) , and from the numerically obtained value c ≈ 0.80 , we calculate z c . For different values of the coupling constant λ , we report that z c = 2.17 ± 0.03 and z c = 2.19 ± 0.03 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 generalized Langevin equation formulation with a memory kernel, identical to those applicable for the Ising model and polymeric systems, consistently captures the observed anomalous diffusion behavior.