Microscopic derivation of the Ginzburg-Landau equations for a d-wave superconductor

The Ginzburg-Landau (GL) equations for a ${\mathrm{d}}_{{\mathrm{x}}^2\mathrm{-}{\mathrm{y}}^2}$ superconductor are derived within the context of two microscopic lattice models used to describe the cuprates: the extended Hubbard model and the antiferromagnetic–van Hove model. Both models have pairing on nearest-neighbor links, consistent with theories for d-wave superconductivity mediated by spin fluctuations. Analytical results obtained for the extended Hubbard model at low electron densities and weak coupling are compared to results reported previously for a d-wave superconductor in the continuum. The variations of the coefficients in the GL equations with carrier density, temperature, and coupling constants are calculated numerically for both models. The relative importance of anisotropic higher-order terms in the GL free energy is investigated, and the implications for experimental observations of the vortex lattice are considered.