Topological invariants for spin-orbit coupled superconductor nanowires

TL;DR

This paper demonstrates that spin-orbit coupled superconductor nanowires belong to symmetry class BDI, allowing for an integer number of Majorana modes at each end, which can change when symmetry-breaking terms are introduced.

Contribution

It identifies the topological class of the nanowire system as BDI and reveals the possibility of multiple Majorana modes, extending the understanding beyond the commonly assumed class D.

Findings

Nanowires are in symmetry class BDI, not D.

The system can host an arbitrary number of Majorana modes.

Breaking chirality symmetry reduces the class to D, limiting the Majorana modes to 0 or 1.

Abstract

We show that a spin-orbit coupled semiconductor nanowire with Zeeman splitting and s-wave superconductivity is in symmetry class BDI (not D as is commonly thought) of the topological classification of band Hamiltonians. The class BDI allows for an integer Z topological invariant equal to the number of Majorana fermion (MF) modes at each end of the quantum wire protected by the chirality symmetry (reality of the Hamiltonian). Thus it is possible for this system (and all other d=1 models related to it by symmetry) to have an arbitrary integer number, not just 0 or 1 as is commonly assumed, of MFs localized at each end of the wire. The integer counting the number of MFs at each end reduces to 0 or 1, and the class BDI reduces to D, in the presence of terms in the Hamiltonian that break the chirality symmetry.