Effective vortex interaction in the two-dimensional XY model

Expressions for the linearly screened potential Vₗ for the classical two-dimensional XY model in both ordinary space and Fourier space are derived by means of the concept of fractional vortices. In the limit of low temperature it is shown that Vₗ is proportional to the lattice Green's function in accordance with previous results. The dependence on Vₗ of different boundary conditions is also examined. For finite temperatures Vₗ is determined by means of Monte Carlo calculations. The properties of Vₗ(r) as a function of temperature verifies the qualitative vortex-unbinding picture, and close to Tc the results are consistent with the universal jump condition. It is further shown that the expression for the dielectric function in the k{→}0 limit reduces to a known expression for the helicity modulus. The dielectric function is also shown to be equal in form with a result from standard linear screening for the two-dimensional Coulomb gas. This analogy leads to an identification of the bare vortex interaction and an expression for vorticity in terms of the rotation of the current. An interesting consequence of these connections is that the equivalent of a Coulomb gas charge in the two-dimensional XY model is an extended object with part of the charge off the central plaquette. Determinations of the screening length and the density of free charges are also presented.