Critical exponent $\nu$ of the Ising model in three dimensions with long-range correlated site disorder analyzed with Monte Carlo techniques

TL;DR

This study uses Monte Carlo simulations to analyze how long-range correlated site disorder affects the critical exponent of the 3D Ising model, providing new insights into disorder correlations and critical behavior.

Contribution

It offers the first comprehensive analysis of the critical exponent $ u$ in the 3D Ising model with long-range correlated disorder across a wide range of correlation exponents.

Findings

Critical exponent $ u$ depends on the correlation decay parameter $a$.

Deviations from the Weinrib-Halperin conjecture $ u=2/a$ are observed.

Critical temperatures vary with disorder correlation and defect concentration.

Abstract

We study the critical behavior of the Ising model in three dimensions on a lattice with site disorder by using Monte Carlo simulations. The disorder is either uncorrelated or long-range correlated with correlation function that decays according to a power-law $r^{-a}$. We derive the critical exponent of the correlation length $\nu$ and the confluent correction exponent $\omega$ in dependence of $a$ by combining different concentrations of defects $0.05 \leq p_d \leq 0.4$ into one global fit ansatz and applying finite-size scaling techniques. We simulate and study a wide range of different correlation exponents $1.5 \leq a \leq 3.5$ as well as the uncorrelated case $a = \infty$ and are able to provide a global picture not yet known from previous works. Additionally, we perform a dedicated analysis of our long-range correlated disorder ensembles and provide estimates for the critical temperatures of the system in dependence of the correlation exponent $a$ and the concentrations of defects $p_d$. We compare our results to known results from other works and to the conjecture of Weinrib and Halperin: $\nu = 2/a$ and discuss the occurring deviations.