Multi-band Eilenberger theory of superconductivity: Systematic low-energy projection

We propose the general multi-band quasiclassical Eilenberger theory of superconductivity to describe quasiparticle excitations in inhomogeneous systems. With the use of low-energy projection matrix, the $M$-band quasiclassical Eilenberger equations are systematically obtained from $N$-band Gor'kov equations. Here $M$ is the internal degrees of freedom in the bands crossing the Fermi energy and $N$ is the degree of freedom in a model. Our framework naturally includes inter-band off-diagonal elements of Green's functions, which have usually been neglected in previous multi-band quasiclassical frameworks. The resultant multi-band Eilenberger and Andreev equations are similar to the single-band ones, except for multi-band effects. The multi-band effects can exhibit the non-locality and the anisotropy in the mapped systems. Our framework can be applied to an arbitrary Hamiltonian (e.g. a tight-binding Hamiltonian derived by the first-principle calculation). As examples, we use our framework in various kinds of systems, such as noncentrosymmetric superconductor CePt$_{3}$Si, three-orbital model for Sr$_{2}$RuO$_{4}$, heavy fermion CeCoIn$_{5}$/YbCoIn$_{5}$ superlattice, a topological superconductor with the strong spin-orbit coupling Cu$_{x}$Bi$_{2}$Se$_{3}$, and a surface system on a topological insulator.