The Magnetization of the 3D Ising Model

TL;DR

This paper reports highly accurate Monte Carlo simulations of the 3D Ising model's magnetization using a specialized computer, providing precise critical parameters and confirming the magnetization's scaling behavior over a wide temperature range.

Contribution

The study introduces a new high-precision Monte Carlo approach with a specialized computer to analyze the 3D Ising model, yielding improved estimates of critical parameters and scaling functions.

Findings

Magnetization described by a specific scaling function with negligible corrections.

Critical temperature determined as K_c=0.2216544 with high precision.

Magnetization exponent found as β=0.3269(6).

Abstract

We present highly accurate Monte Carlo results for simple cubic Ising lattices containing up to $256^3$ spins. These results were obtained by means of the Cluster Processor, a newly built special-purpose computer for the Wolff cluster simulation of the 3D Ising model. We find that the magnetization $M(t)$ is perfectly described by $M(t)=(a_0-a_1 t^{\theta} - a_2 t) t^{\beta} $, where $t=(T_{\rm c}-T)/T_{\rm c}$, in a wide temperature range $0.0005 < t < 0.26 $. If there exist corrections to scaling with higher powers of $t$, they are very small. The magnetization exponent is determined as $\beta=0.3269$ (6). An analysis of the magnetization distribution near criticality yields a new determination of the critical point: $K_{\rm c}=J/k_B T_{\rm c}=0.2216544$, with a standard deviation of $3\cdot 10^{-7}$.