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

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.