Andreev reflection from a topological superconductor with chiral symmetry

Abstract

It was pointed out by Tewari and Sau that chiral symmetry ($H\ensuremath{\mapsto}\ensuremath{-}H$ if $\mathrm{e}\ensuremath{\leftrightarrow}\mathrm{h}$) of the Hamiltonian of electron-hole (e-h) excitations in an $N$-mode superconducting wire is associated with a topological quantum number $Q\ensuremath{\in}\mathbb{Z}$ (symmetry class BDI). Here we show that $Q=\mathrm{Tr}\phantom{\rule{0.16em}{0ex}}{r}_{\mathrm{he}}$ equals the trace of the matrix of Andreev reflection amplitudes, providing a link with the electrical conductance $G$. We derive $G=(2{e}^{2}/h)|Q|$ for $|Q|=N,N\ensuremath{-}1$, 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 $2{e}^{2}/h$ steps---insensitive to disorder.