Ginzburg-Landau theory of superconductors with short coherence length

We consider fermions in two dimensions with an attractive interaction in the singlet $d$-wave channel of arbitrary strength. By means of a Hubbard-Stratonovich transformation a statistical Ginzburg-Landau theory is derived, which describes the smooth crossover from a weak-coupling BCS superconductor to a condensate of composite bosons. We show that the Nelson-Kosterlitz jump in the superfluid density vanishes in the BCS limit, where mean-field theory becomes exact. Adjusting the interaction strength to the observed slope of $H_{c_2}$ at $T_c$ in the optimally doped high-$T_c$ compounds Y-Ba-Cu-O and Bi-Sr-Ca-Cu-O, we determine the associated values of the Ginzburg-Landau correlation length $\xi$ and the London penetration depth $\lambda$. The resulting dimensionless ratio $k_F\xi \mathrm{}\left(0\right)\mathrm{}\approx 5-8$ and the Ginzburg-Landau parameter $\kappa =\lambda /\xi \approx 90-100$ agree well with the experimentally observed values. These parameters indicate that the optimally doped materials are still on the weak-coupling side of the crossover to a Bose regime.