Abstract
The phase structure and renormalization-group behavior of the $\mathrm{XY}$ model in a random $p$-fold anisotropy is studied via time-dependent Langevin formulation. The dynamics and statics are derived in two dimensions and extended to $2+\ensuremath{\epsilon}$ dimensions. We find that a spin-glass-like phase at intermediate length scales becomes paramagnetic at large length scales as a result of the unbinding of vortices. In $2+\ensuremath{\epsilon}$ dimensions a pure $\mathrm{XY}$-type transition occurs for ${p}^{2}>8$ into a low-temperature quasi-spin-glass phase with a finite susceptibility. A random-bond interaction is generated via renormalization.
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