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

In order to point out experimentally accessible signatures of spin-orbit interaction, 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 nontopological behaviors. In the nontopological case (square lattice with Rashba interactions), we find that the Kramers degeneracy of Andreev levels is lifted by a phase difference between the S reservoirs except at multiples of pi, when the normal quantum wires can host several conduction channels. The level crossings at these points invariant by time-reversal symmetry are not lifted by disorder. Whereas the dc Josephson current is insensitive to these level crossings, the high-frequency admittance (susceptibility) at finite temperature reveals these level crossings and the lifting of their degeneracy at pi by a small Zeeman field. We have also investigated the hexagonal lattice with intrinsic spin-orbit interaction in the range of parameters where it is a two-dimensional topological insulator with one-dimensional helical edges protected against disorder. Nontopological superconducting contacts can induce topological superconductivity in this system characterized by zero-energy level crossing of Andreev levels. Both Josephson current and finite-frequency admittance carry then very specific signatures at low temperature of this disorder-protected Andreev level crossing at pi and zero energy.