Boundary effects in superfluid films

Abstract

We have studied the superfluid density and the specific heat of the XY model on lattices L x L x H with L >> H (i.e. on lattices representing a film geometry) using the Cluster Monte Carlo method. In the H-direction we applied staggered boundary conditions so that the order parameter on the top and bottom layers is zero, whereas periodic boundary conditions were applied in the L-directions. We find that the system exhibits a Kosterlitz-Thouless phase transition at the H-dependent temperature T_{c}^{2D} below the critical temperature T_{\lambda} of the bulk system. However, right at the critical temperature the ratio of the areal superfluid density to the critical temperature is H-dependent in the range of film thicknesses considered here. We do not find satisfactory finite-size scaling of the superfluid density with respect to H for the sizes of H studied. However, our numerical results can be collapsed onto a single curve by introducing an effective thickness H_{eff} = H + D (where D is a constant) into the corresponding scaling relations. We argue that the effective thickness depends on the type of boundary conditions. Scaling of the specific heat does not require an effective thickness (within error bars) and we find good agreement between the scaling function f_{1} calculated from our Monte Carlo results, f_{1} calculated by renormalization group methods, and the experimentally determined function f_1.