Scaling Behavior in the 3D Random Field $XY$ Model

TL;DR

This study investigates the scaling behavior of the 3D random field XY model using large-scale Monte Carlo simulations, revealing a logarithmic divergence in the structure factor and suggesting the lower critical dimension is three.

Contribution

The paper provides new numerical evidence on the critical behavior of the 3D random field XY model, highlighting the nature of its long-range order and critical dimension.

Findings

Logarithmic divergence of the structure factor as k approaches zero.

Indications that the lower critical dimension for order is three.

Evidence of phase transition behavior at specific temperature.

Abstract

We have performed studies of the 3D random field $XY$ model on $L \times L \times L$ simple cubic lattices with periodic boundary conditions, with a random field strength of $h_r$ = 1.875, for $L = 64$ and $L = 96$, using a parallelized Monte Carlo algorithm. We present results for the angle-averaged magnetic structure factor, $S ( k )$ at $T = 1.00$, which appears to be the temperature at which small jumps in the magnetization per spin and the energy per spin occur. The results indicate the existence of an approximately logarithmic divergence of $S ( k )$ as $k \to 0$. This suggests that the lower critical dimension for long range order in this model is three.