  A variational approach for the free energy is used to study the
three-dimensional anisotropic XY model in the presence of a crystal field. The
magnetization and the phase diagrams as a function of the parameters of the
Hamiltonian are obtained. Some limiting results for isotropic XY and planar
rotator models in two and three dimensions are analyzed and compared to
previous results obtained from analytical approximations as well as from those
obtained from more reliable approaches such as series expansion and Monte Carlo
simulations. It is also shown that from this general variational approach some
simple assumptions can drastically simplify the self-consistent implicit
equations. The validity of the low temperature region of this approach is
analyzed and compared to Monte Carlo results as well.
