We have studied the superfluid density $\rho_{ s}$ on various size-lattices in the geometry $L \times L \times H$ by numerical simulation of the $x-y$ model using the Cluster Monte Carlo method. Applying the Kosterlitz-Thouless-Nelson renormalization group equations for the superfluid density we have been able to extrapolate to the $L \to \infty$ limit for a given value of $H$. In the superfluid phase we find that the superfluid density faithfully obeys the expected scaling law with $H$, using the experimental value for the critical exponent $\nu=0.6705$. For the sizes of film thickness studied here the critical temperature $T_{ c}$ and the coefficient $b$ entering the equation $T/(\rho_{ s} H) \propto 1-b(1-T/T_{ c})^{ 1/2}$ are in agreement with the expected $H$-dependence deduced from general scaling ideas.
Categories: cond-mat hep-lat