Zero-Temperature Coarsening in the Two-Dimensional Long-Range Ising Model

TL;DR

This paper studies the zero-temperature coarsening dynamics of a 2D long-range Ising model, revealing a universal growth exponent and confirming a key relation between exponents across different interaction ranges.

Contribution

It provides the first detailed estimates of nonequilibrium exponents for the 2D long-range Ising model and confirms the universality of the exponent relation across interaction ranges.

Findings

Growth exponent α ≈ 3/4 independent of σ

Fractal dimension d_f matches nearest-neighbor model at large σ

Persistence exponent θ relates to α and d_f as predicted

Abstract

We investigate the nonequilibrium dynamics following a quench to zero temperature of the non-conserved Ising model with power-law decaying long-range interactions $\propto 1/r^{d+\sigma}$ in $d=2$ spatial dimensions. The zero-temperature coarsening is always of special interest among nonequilibrium processes, because often peculiar behavior is observed. We provide estimates of the nonequilibrium exponents, viz., the growth exponent $\alpha$, the persistence exponent $\theta$, and the fractal dimension $d_f$. It is found that the growth exponent $\alpha\approx 3/4$ is independent of $\sigma$ and different from $\alpha=1/2$ as expected for nearest-neighbor models. In the large $\sigma$ regime of the tunable interactions only the fractal dimension $d_f$ of the nearest-neighbor Ising model is recovered, while the other exponents differ significantly. For the persistence exponent $\theta$ this is a direct consequence of the different growth exponents $\alpha$ as can be understood from the relation $d-d_f=\theta/\alpha$; they just differ by the ratio of the growth exponents $\approx 3/2$. This relation has been proposed for annihilation processes and later numerically tested for the $d=2$ nearest-neighbor Ising model. We confirm this relation for all $\sigma$ studied, reinforcing its general validity.