Theory of time-reversal topological superconductivity in double Rashba wires: symmetries of Cooper pairs and Andreev bound states

We study the system of double Rashba wires brought into proximity to an s-wave superconductor. Time-reversal invariant topological superconductivity is realized if the interwire pairing, corresponding to a crossed Andreev reflection, dominates over the standard intrawire pairing. We classify the symmetry of the Cooper pairs focusing on four degrees of freedom, i. e., frequency, spin, spatial parity inside wires, and spatial parity between wires. The presence of the spatial parity between wires is a unique feature of this model. The magnitude of the odd-frequency pairing is strongly enhanced in the topological state irrespective of spatial parities. We also explore the properties of junctions occurring in such double-wire systems. If one section of the junction is in a topological state and the other is in a trivial state, the energy dispersion of the Andreev bound states is proportional to similar to +/- sin phi, where phi denotes the macroscopic phase difference between the two sections. This behavior can be intuitively explained by the couplings of a Kramers pair of Majorana fermions and a spin-singlet s-wave Cooper pair and can also be understood by analyzing an effective continuum model of the s + p/s-wave superconductor hybrid system.