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 zc , where L is the system size, and (b) by measuring the anomalous diffusion exponent for the order parameter, viz., the mean-square displacement (MSD) ∆Φ 2 (t) ∼ t c as c = γ/(νz 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 (GLE) formulation with a memory kernel, identical to those applicable for the Ising model and polymeric systems, consistently captures the observed anomalous diffusion behaviour.

Approximate dynamical eigenmodes of the Ising model with local spin-exchange moves

“…We find that line magnetisation exhibits anomalous diffusion. Our results for anomalous diffusion is consistent with a pattern that the dynamics of magnetisation at the critical temperature in spin models is anomalous [21][22][23]. Importantly, the anomalous diffusion is described by the Generalised Langevin Equation (GLE) [22,23] (and bears strong resemblance to anomalous diffusion in polymeric and membrane systems under a variety of circumstances [3,5,12,[24][25][26][27][28][29][30][31][32][33][34][35][36][37]), which we verify in Sec.…”

“…. , 10 for three different system sizes L = 120, 160, 200 (from our earlier works on spin systems [21][22][23] we expect that the data collapse would require scaling time with a prefactor (p/L) zc ). The solid line in the figure then represents…”

“…III we have demonstrated that at the intermediate time regime, the line magnetisation in the Ising model with Kawasaki dynamics exhibits anomalous diffusion. In our recent studies on the Ising and φ 4 model with Glauber dynamics [22,23], we have argued that the anomalous diffusion of the magnetization belongs to the GLE class, for which the restoring force plays an important role.…”

Classical critical dynamics in quadratically driven Kerr resonators

“…The most well-known of these rules is the single-site update rule or Metropolis algorithm, which is consistent with experimental observations of the critical dynamics in an iron film [44]. The same value of z is found in the Hohenberg-Halperin model A dynamics [45] as well as in φ 4 -theory [46].…”

Spectral functions and dynamic critical behavior of relativistic $Z_2$ theories

“…The most recent result by Zhong et al [50] for a two-dimensional scalar φ 4 model with local Metropolis updates seems to confirm the results for 2D Model A by Nightingale and Blöte from [7], as they find z A " 2.17p3q and z A " 2.19p3q for two different values of the coupling constant. Additionally, they demonstrated that quantities derived from the two-point function follow a scaling behaviour.…”