Chiral perturbation theory, finite size effects and the three-dimensional $XY$ model

We study finite size effects of the d=3 $XY$ model in terms of the chiral perturbation theory. We calculate by Monte Carlo simulations physical quantities which are, to order of $(1/L)^2$, uniquely determined only by two low energy constants. They are the magnetization and the helicity modulus (or the Goldstone boson decay constant) in infinite volume. We also pay a special attention to the region of the validity of the two possible expansions in the theory.