Coulomb-gas sum rules for vortex-pair fluctuations in 2D superfluids

Coulomb-gas sum rules are used to characterize thermal vortex-pair fluctuations in 2D superfluids.  Simulations of the 2D XY model have been carried out to study the net winding number of vortices at a given temperature in a circle of radius R, squared and averaged over 1000 instances.  At all temperatures the net squared winding number is found to scale as a perimeter law, linear in R, in agreement with Coulomb-gas theories [1], and at infinite temperature agrees nearly exactly with an early theory by D. Dhar [2].  The linear slope of the perimeter variation is found to display a sharp peak with temperature, starting below the critical Kosterlitz-Thouless temperature T_KT and peaking near 1.15 T_KT, very similar to the peak in specific heat.  We have also computed the vortex-vortex distribution functions, finding an asymptotic power-law variation in the vortex separation distance at all temperatures.  In conjunction with a Coulomb-gas sum rule [1] on the perimeter fluctuations, these can be used to successfully model the start of the perimeter-slope peak in the region below T_KT. 

[1] Ph. A. Martin, Sum rules in charged fluids, Rev. Mod. Phys. 60, 1075 (1988).

[2] D.Dhar, On the topological characterization of two-dimensional phase transitions, Phys. Lett. 81A, 19 (1981).