Ising Universality in Three Dimensions: A Monte Carlo Study

TL;DR

This study uses Monte Carlo simulations to examine three three-dimensional Ising models, confirming universality and analyzing correction-to-scaling effects, which aids in accurately determining universal constants.

Contribution

It provides detailed Monte Carlo analysis of various 3D Ising models, revealing how additional interactions influence correction-to-scaling amplitudes and refining universal constant estimates.

Findings

Universal critical exponents y_t and y_h are precisely determined.

Correction-to-scaling amplitudes depend strongly on interaction range and spin state.

Universal ratio Q is accurately measured for cubic symmetry.

Abstract

We investigate three Ising models on the simple cubic lattice by means of Monte Carlo methods and finite-size scaling. These models are the spin-1/2 Ising model with nearest-neighbor interactions, a spin-1/2 model with nearest-neighbor and third-neighbor interactions, and a spin-1 model with nearest-neighbor interactions. The results are in accurate agreement with the hypothesis of universality. Analysis of the finite-size scaling behavior reveals corrections beyond those caused by the leading irrelevant scaling field. We find that the correction-to-scaling amplitudes are strongly dependent on the introduction of further-neighbor interactions or a third spin state. In a spin-1 Ising model, these corrections appear to be very small. This is very helpful for the determination of the universal constants of the Ising model. The renormalization exponents of the Ising model are determined as y_t = 1.587 (2), y_h = 2.4815 (15) and y_i = -0.82 (6). The universal ratio Q = <m^2>^2/<m^4> is equal to 0.6233 (4) for periodic systems with cubic symmetry. The critical point of the nearest-neighbor spin-1/2 model is K_c=0.2216546 (10).