Thermodynamic Potentials of Crystalline Soliton Superconducting Condensates

In many practical cases thermodynamic laws dictate that superconductivity must occur in the form of a spatially inhomogeneous state which brings about spatial modulation of the order parameter. Profound examples of this behavior are given by the Abrikosov vortex lattice state and the Fulde-Ferrell-Larkin-Ovchinnikov helical complex condensate. More generally, self-consistent Bogoliubov-de Gennes (BdG) equations of superconductivity are known to admit exact solutions in the form of spatial solitons. With a carefully chosen ansatz, we use an exact mapping from the BdG Hamiltonian to the integrable nonlinear Schrodinger equation and employ complimentary tools from the inverse scattering approach to construct crystalline soliton condensate solutions. We derive their spectral properties and analyze their energetic costs by computing their corresponding grand potentials. Examples from inhomogeneous quasi-1D superconductivity and systems with a spin-density-wave will be presented.