  Based on an attractive $U$ Hubbard model on a lattice with up to second
neighbor hopping we derive an effective Hamiltonian for phase fluctuations. The
superconducting gap is assumed to have s-wave symmetry. The effective
Hamiltonian we finally arrive at is of the extended XY type. While it correctly
reduces to a simple XY in the continuum limit, in the general case, it contains
higher neighbor interaction in spin space. An important feature of our
Hamiltonian is that it gives a much larger fluctuation region between the
Berezinskii-Kosterlitz-Thouless transition temperature identified with $T_{c}$
for superconducting and the mean field transition temperature identified with
the pseudogap temperature.
