Superconductivity in quantum wires: A symmetry analysis

We study properties of quantum wires with spin-orbit coupling and time reversal symmetry breaking, in normal and superconducting states. Electronic band structures are classified according to quasi-one-dimensional magnetic point groups, or magnetic classes. The latter belong to one of three distinct types, depending on the way the time reversal operation appears in the group elements. The superconducting gap functions are constructed using antiunitary operations and have different symmetry properties depending on the type of the magnetic point group. We obtain the spectrum of the Andreev boundary modes near the end of the wire in a model-independent way, using the semiclassical approach with the boundary conditions described by a phenomenological scattering matrix. Explicit expressions for the bulk topological invariants controlling the number of the boundary zero modes are presented in the general multiband case for two types of the magnetic point groups with real order parameters, corresponding to DIII and BDI symmetry classes.