Boundary e ects in super uid

We have studied the superruid density and the speciic heat of the x ? y model on lattices L L H with L H (i.e. on lattices representing a lm 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 nd that the system exhibits a Kosterlitz-Thouless phase transition at the H-dependent temperature T 2D c below the critical temperature T of the bulk system. However, right at the critical temperature the ratio of the areal superruid density to the critical temperature is H-dependent in the range of lm thicknesses considered here. We do not nd satisfactory nite-size scaling of the superruid 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 eeective thickness H eff = H +D (where D is a constant) into the corresponding scaling relations. We argue that the eeective thickness depends on the type of boundary conditions. Scaling of the speciic heat does not require an eeective thickness (within error bars) and we nd good agreement between the scaling function f 1 calculated from our Monte Carlo results, f 1 calculated by renormalization group methods, and 1 the experimentally determined function f 1 .