Ginzburg-Landau theory of the Abrikosov lattice in a d-wave superconductor

We consider the Ginzburg-Landau free energy for the case of coexisting s- and d-wave order parameters, coupled to a gauge field, for a system in which the minimum energy state in the absence of magnetic fields is uniform d-wave. The eigenvalue equation arising from the coupled Ginzburg-Landau equations, in the presence of a uniform magnetic field, is implicitly non-linear and has no obvious closed-form solution. We employ a simple variational two-component wave function to minimize the quadratic free energy, and use linear combinations of these wave functions to construct variational Abrikosov lattice solutions which minimize the total free energy. The resulting lattice has an oblique structure, similar to that observed by small angle neutron scattering.