The specific heat of superfluids near the transition temperature

The specific heat of the $x-y$ model is studied on cubic lattices of sizes $L \times L \times L$ and on lattices $L \times L \times H$ with $L \gg H$ (i.e. on lattices representing a film geometry) using the Cluster Monte Carlo method. Periodic boundary conditions were applied in all directions. In the cubic case we obtained the ratio of the critical exponents $\alpha/\nu$ from the size dependence of the energy density at the critical temperature $T_{\lambda}$. Using finite--size scaling theory, we find that while for both geometries our results scale to universal functions, these functions differ for the different geometries. We compare our findings to experimental results and results of renormalization group calculations.