Long range $p$-wave proximity effect into a disordered metal

We use quasiclassical methods of superconductivity to study the superconducting proximity effect from a topological $p$-wave superconductor into a disordered one-dimensional metallic wire. We demonstrate that the corresponding Eilenberger equations with disorder reduce to a closed non-linear equation for the superconducting component of the matrix Green’s function. Remarkably, this equation is formally equivalent to a classical mechanical system (i.e., Newton’s equations), with the Green function corresponding to a coordinate of a fictitious particle and the coordinate along the wire corresponding to time. This mapping allows to obtain exact solutions in the disordered nanowire in terms of elliptic functions. A surprising result that comes out of this solution is that the $p$-wave superconductivity proximity-induced into the disordered metal remains long-range, decaying as slowly as the conventional $s$-wave superconductivity. It is also shown that impurity scattering leads to the appearance of a zero-energy peak.