Monte-Carlo study of anisotropic scaling generated by disorder

TL;DR

This study uses Monte Carlo simulations to analyze the anisotropic critical behavior of a 3D Ising model with line defects, confirming previous renormalization group predictions about critical exponents and anisotropy.

Contribution

First Monte Carlo simulation study of anisotropic scaling in 3D Ising model with line defects, providing quantitative estimates of critical exponents and validating RG predictions.

Findings

Estimated anisotropy exponent $ heta$ consistent with RG predictions.

Obtained susceptibility exponent $\gamma$ for the system.

Confirmed algebraic divergence of correlation lengths in different directions.

Abstract

We analyze the critical properties of the three-dimensional Ising model with linear parallel extended defects. Such a form of disorder produces two distinct correlation lengths, a parallel correlation length $\xi_\parallel$ in the direction along defects, and a perpendicular correlation length $\xi_\perp$ in the direction perpendicular to the lines. Both $\xi_\parallel$ and $\xi_\perp$ diverge algebraically in the vicinity of the critical point, but the corresponding critical exponents $\nu_\parallel$ and $\nu_\perp$ take different values. This property is specific for anisotropic scaling and the ratio $\nu_\parallel/\nu_\perp$ defines the anisotropy exponent $\theta$. Estimates of quantitative characteristics of the critical behaviour for such systems were only obtained up to now within the renormalization group approach. We report a study of the anisotropic scaling in this system via Monte Carlo simulation of the three-dimensional system with Ising spins and non-magnetic impurities arranged into randomly distributed parallel lines. Several independent estimates for the anisotropy exponent $\theta$ of the system are obtained, as well as an estimate of the susceptibility exponent $\gamma$. Our results corroborate the renormalization group predictions obtained earlier.