Superconductivity in quantum wires: A symmetry analysis

TL;DR

This paper classifies the symmetry properties of superconducting quantum wires with spin-orbit coupling and broken time reversal symmetry, analyzing their topological boundary modes and invariants in a model-independent framework.

Contribution

It introduces a symmetry classification scheme for superconducting quantum wires based on magnetic point groups and derives boundary mode spectra and topological invariants.

Findings

Classification of band structures by magnetic point groups.

Explicit formulas for topological invariants in DIII and BDI classes.

Model-independent analysis of Andreev boundary modes.

Abstract

We study properties of quantim 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, corresponding to DIII and BDI symmetry classes.