Andreev reflection from a topological superconductor with chiral symmetry

It was pointed out by Tewari and Sau that chiral symmetry (H -> -H if e <-> h) of the Hamiltonian of electron-hole (e-h) excitations in an N-mode superconducting wire is associated with a topological quantum number Q is an element of Z (symmetry class BDI). Here we show that Q = Tr r(he) equals the trace of the matrix of Andreev reflection amplitudes, providing a link with the electrical conductance G. We derive G = (2e(2)/h)vertical bar Q vertical bar for vertical bar Q vertical bar = N, N - 1, and more generally provide a Q-dependent upper and lower bound on G. We calculate the probability distribution P(G) for chaotic scattering, in the circular ensemble of random-matrix theory, to obtain the Q dependence of weak localization and mesoscopic conductance fluctuations. We investigate the effects of chiral symmetry breaking by spin-orbit coupling of the transverse momentum (causing a class BDI-to-D crossover), in a model of a disordered semiconductor nanowire with induced superconductivity. For wire widths less than the spin-orbit coupling length, the conductance as a function of chemical potential can show a sequence of 2e(2)/h steps-insensitive to disorder.