Two-dimensionalXYmodel in a random uniaxial field

We present a low-temperature analysis of the two-dimensional XY model in the presence of a random uniaxial field, which points in the ±{}x directions with equal probability. We first construct a domain argument, valid at zero temperature in all dimensions, which indicates that in all dimensions the ground state is ferromagnetic, for small field, and the magnetization points in the y direction. We then construct Kosterlitz-style recursion relations for this model in two dimensions. Analyzing these equations, we find an ordered phase with Ising symmetry at nonzero temperature and small values of the random-field strength. Thus, the random field induces long-range order in a system which exhibits only quasiorder in its pure state. We also consider the p-state generalization of the model, and find an ordered phase with 2p-fold symmetry at nonzero temperature. Our results are in disagreement with earlier work by Dotsenko and Feigelman, who found a paramagnetic phase at low temperatures.