Critical Exponents of the 3-D Ising Model

TL;DR

This paper reports on advanced Monte Carlo and finite size scaling analyses of the 3D Ising model, providing updated estimates of critical exponents and exploring scaling behaviors on large lattices.

Contribution

It offers new estimates of critical exponents and correction-to-scaling exponent for the 3D Ising model using large-scale Monte Carlo Renormalization Group and finite size scaling methods.

Findings

Estimated critical coupling $K_{nn}^c=0.221655(1)(1)$

Critical exponent $ u=0.625(1)$

Correction-to-scaling exponent $oxed{ ext{approximately }0.7}$

Abstract

We present a status report on the ongoing analysis of the 3D Ising model with nearest-neighbor interactions using the Monte Carlo Renormalization Group (MCRG) and finite size scaling (FSS) methods on $64^3$, $128^3$, and $256^3$ simple cubic lattices. Our MCRG estimates are $K_{nn}^c=0.221655(1)(1)$ and $\nu=0.625(1)$. The FSS results for $K^c$ are consistent with those from MCRG but the value of $\nu$ is not. Our best estimate $\eta = 0.025(6)$ covers the spread in the MCRG and FSS values. A surprise of our calculation is the estimate $\omega \approx 0.7$ for the correction-to-scaling exponent. We also present results for the renormalized coupling $g_R$ along the MCRG flow and argue that the data support the validity of hyperscaling for the 3D Ising model.