  The Berezinskii-Kosterlitz-Thouless (BKT) transition is a generic transition
describing the loss of coherence in two dimensional systems, and has been
invoked, for example, to describe the superconductor-insulator transition in
thin films. However, recent experiments have shown that the BKT transition,
driven by phase fluctuations, is not sufficient to describe the loss of
superconducting order, and amplitude fluctuations have also to be taken into
account. The standard models that are extensively used to model two-dimensional
superconductors are the Hubbard and XY models. Whereas the XY model allows only
phase fluctuations, the Hubbard model has an extra degree of freedom: amplitude
fluctuations. In this paper we compare two Hubbard models with the same
critical temperature but with different interaction, and deduce the role of the
amplitude fluctuations in the superconducting transition. For this purpose, a
novel approximation is presented and used. We derive an effective phase-only
(XY) Hamiltonian, incorporating amplitude fluctuations in an explicit
temperature dependence of the phase rigidity. We study the relation between
amplitude fluctuations and coupling strength. Our results support existing
claims about the suppression of phase rigidity due to amplitude fluctuations
not present in the XY model.
