Topological invariants for spin-orbit coupled superconductor nanowires

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.