Probing vortex unbinding via dipole fluctuations

We develop a numerical method for detecting a vortex-unbinding transition in a two-dimensional system by measuring large-scale fluctuations in the total vortex dipole moment P^→ of the system. These are characterized by a quantity ℱ that measures the number of configurations in a simulation for which either P_x or P_y is half the system size. It is shown that ℱ tends to a nonvanishing constant for large system sizes in the unbound phase, and vanishes in the bound phase. The method is applied to the XY model both in the absence and in the presence of a magnetic field. In the latter case, the system size dependence of ℱ suggests that there exist three distinct phases, one unbound vortex phase, a logarithmically bound phase, and a linearly bound phase.