We have performed Monte Carlo studies of the three-dimensional $XY$ model with random uniaxial anisotropy, which is a model for randomly pinned spin-density waves. We study $L\times L\times L$ simple cubic lattices using $L$ values in the range 16–64, and with random anisotropy strengths of $D/2J=1$, 2, 3, 6, and $\infty$. There is a well-defined finite-temperature critical point, $T_c$, for each of these values of $D/2J$. We present results for the angle-averaged magnetic structure factor $S\left(k\right)$ at $T_c$ for $L=64$. We also use finite-size scaling analysis to study scaling functions for the critical behavior of the specific heat, the magnetization, and the longitudinal magnetic susceptibility. Good data collapse of the scaling functions over a wide range of $T$ is seen for $D/2J=6$ and $\infty$. For our finite values of $D/2J$ the scaled magnetization function increases with $L$ below $T_c$ and appears to approach an $L$-independent limit for large $L$. This suggests that the system is ferromagnetic below $T_c$.

• Received 23 February 2009

DOI:https://doi.org/10.1103/PhysRevB.79.214429