Andreev spectrum with high spin-orbit interactions: revealing spin splitting and topologically protected crossings

Abstract: We investigate numerically the Andreev spectrum of a multichannel mesoscopic quantum wire (N) with high spin-orbit interaction coupled to superconducting electrodes (S), contrasting topological and non topological behaviors. In the non topological case, modeled by a square lattice with Rashba interactions, we find that as soon as the normal quantum wires can host several conduction channels, the spin degeneracy of Andreev levels is lifted by a phase difference between the S reservoirs which breaks time reversal symmetry in zero Zeeman field. The Andreev states remain degenerate at phases multiple of $\pi$ for which time reversal symmetry is preserved, giving rise to level crossings which are not lifted by disorder. In contrast with the dc Josephson current, the finite frequency admittance (susceptibility) is very sensitive to these level crossings and the lifting of their degeneracy by a small Zeeman field. More interesting is the case of the hexagonal lattice with next nearest neighbor spin-orbit interactions which exhibit 1D topological helical edge states \cite{Kane2005}. The finite frequency admittance carries then a very specific signature at low temperature of a protected Andreev level crossing at $\pi$ and zero energy in the form of a sharp peak split by a Zeeman field.