XYmodel and the superfluid density in two dimensions

Abstract

By computing the incremental free energy of the two-dimensional $\mathrm{XY}$ model in a state with a long-wavelength twist of the local short-range order, we evaluate for all $T<\mathrm{}{T}_c$ the superfluid density ${\rho }_s$ for superfluid systems. Spin-wave excitations in the $\mathrm{XY}$ model imply a substantial (nonuniversal) depletion of ${\rho }_s$ at low temperatures. We find that these contributions do not affect the asymptotic universality; as $T\to T_{c^{-}}$, $\frac{{\rho }_s}{T}$ becomes universal and we recover the previous results of Nelson and Kosterlitz. In the Appendix we derive the Kosterlitz recursion relations by transforming to a sine-Gordon system and using conventional momentum shell integration techniques.