Vortex lattices and critical fields in anisotropic superconductors

A method is developed to compute minimal energy vortex lattices in a general Ginzburg-Landau model of a superconductor subjected to an applied magnetic field. The model may have any number of components and may be spatially anisotropic. The novelty of this method is that it makes no assumptions about the orientation of the vortex lines or the period vectors of the lattice's unit cell: these are all determined dynamically. Methods to compute the first and second critical magnetic fields, $H_{c_1}$ and $H_{c_2}$, in this class of models are also developed. These methods are applied to a simple anisotropic single-component model, and to an anisotropic two-component model of strong current theoretical interest (a so-called $s+id$ model). It is found, in both cases, that at low applied field the vortex lines can tilt very significantly away from the direction of the applied field (by as much as $40^\circ$ for the single-component and $30^\circ$ for the $s+id$ model). The optimal lattice in the $s+id$ model is qualitatively very different from the conventional triangular Abrikosov lattice, exhibiting a phase transition from a system of Skyrmion chains when the external field is orthogonal to the basal plane to a deformed Abrikosov lattice when applied in the basal plane.