Majorana bound states in two-channel time-reversal-symmetric nanowire systems

We consider time-reversal-symmetric two-channel semiconducting quantum wires proximity coupled to a conventional s-wave superconductor. We analyze the requirements for a non-trivial topological phase, and find that necessary conditions are 1) the determinant of the pairing matrix in channel space must be negative, 2) inversion symmetry must be broken, and 3) the two channels must have different spin-orbit couplings. For the case of collinear spin-orbit directions, we find a general expression for the topological invariant by block diagonalization into two blocks with chiral symmetry only. By projection to the low-energy sector we solve for the zero modes explicitly and study the details of the gap closing, which in the general case happens at finite momenta.