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Heat transport in nonuniform superconductors

We calculate electronic energy transport in inhomogeneous superconductors using a fully self-consistent non-equilibrium quasiclassical Keldysh approach. We develop a general theory and apply it a superconductor with an order parameter that forms domain walls, of the type encountered in Fulde-Ferrell-Larkin-Ovchinnikov state. The heat transport in the presence of a domain wall is inherently anisotropic and non-local. Bound states in the nonuniform region play a crucial role and control heat transport in several ways: (i) they modify the spectrum of quasiparticle states and result in Andreev reflection processes, and (ii) they hybridize with impurity band and produce local transport environment with properties very different from those in uniform superconductor. As a result of this interplay, heat transport becomes highly sensitive to temperature, magnetic field and disorder. For strongly scattering impurities we find that the transport across domain walls at low temperatures is considerably more efficient than in the uniform superconducting state.