Abstract
We derive an effective Hamiltonian for phase fluctuations in an s-wave superconductor starting from the attractive Hubbard model on a square lattice. In contrast to the common assumption, we find that the effective Hamiltonian is not the usual XY model but is of the extended XY type. The BCS to Bose-Einstein crossover reduces the ratio of the Berezinskii-Kosterlitz-Thouless to mean-field pseudogap transition temperature more as compared with the continuum limit.