The two-dimensional classical $\mathrm{XY}$ model in a random $p$-fold symmetry-breaking field is studied using the replica method. For $p>\mathrm{}2\mathrm{}\sqrt{2}$ an $\mathrm{XY}$ like phase exists at intermediate values of temperature and weak field. For $p\le 4$ we are able to describe the transition into the low-temperature glassy continuation of the paramagnetic phase, while for $p>\mathrm{}4\mathrm{}$ the results suggest the transition may be first order, driven by the unbinding of vortices. Several new fixed points and lines are found in the replicated Kosterlitz-Thouless-type recursion relations corresponding to these various transitions. The method we use considers $n$ coupled $\mathrm{XY}$ models from which we construct a Coulomb gas with $\frac{n(n-1\mathrm{})\mathrm{}}{2}$ types of ($n-1\mathrm{}$)-dimensional vector charges.

• Received 6 July 1981

DOI:https://doi.org/10.1103/PhysRevB.25.6899