Behavior of the Random Field $XY$ Model on Simple Cubic Lattices at $h_r=1.5$

We have performed studies of the 3D random field $XY$ model on 32 samples of $L\times L\times L$ simple cubic lattices with periodic boundary conditions, with a random field strength of $h_r$ = 1.5, for $L=$ 128, using a parallelized Monte Carlo algorithm. We present results for the sample-averaged magnetic structure factor, $S(\vec{\mathbf{k}})$ over a range of temperature, using both random hot start and ferromagnetic cold start initial states, and $\vec{\mathbf{k}}$ along the [1,0,0] and [1,1,1] directions. At $T=$ 1.875, $S (\vec{\bf k})$showsabroadpeaknear$|\vec{\bf k}| = 0$, with a correlation length which is limited by thermal fluctuations, rather than the lattice size. As $T$ is lowered, this peak grows and sharpens. By $T=$ 1.5, it is clear that the correlation length is larger than $L=$ 128. The lowest temperature for which $S(\vec{\mathbf{k}})$ was calculated is $T=$ 1.421875, where the hot start and cold start initial conditions are usually not finding the same local minimum in the phase space. Our results are consistent with the idea that there is a finite value of $T$ below which $S(\vec{\mathbf{k}})$ diverges slowly as $\mathrm{\mid }\vec{\mathbf{k}}\mathrm{\mid }$ goes to zero. This divergence would imply that the relaxation time of the spins is also diverging. That is the signature of an ergodicity-breaking phase transition.