Generalized index theorem for topological superconductors with Yang-Mills-Higgs couplings

TL;DR

This paper derives a generalized index theorem for topological superconductors with Yang-Mills-Higgs couplings, linking the index to Higgs field behavior and predicting zero-energy states at defects.

Contribution

It introduces a new index theorem for BdG Hamiltonians with Yang-Mills-Higgs couplings, showing the index depends only on Higgs fields and applies to arbitrary dimensions.

Findings

Index determined solely by Higgs field asymptotics

Nonvanishing index when order parameter space matches spatial dimensions

Zero energy states localized at point defects like vortices and monopoles

Abstract

We investigate an index theorem for a Bogoliubov-de Gennes Hamiltonian (BdGH) describing a topological superconductor with Yang-Mills-Higgs couplings in arbitrary dimensions. We find that the index of the BdGH is determined solely by the asymptotic behavior of the Higgs fields and is independent of the gauge fields. It can be nonvanishing if the dimensionality of the order parameter space is equal to the spatial dimensions. In the presence of point defects there appear localized zero energy states at the defects. Consistency of the index with the existence of zero energy bound states is examined explicitly in a vortex background in two dimensions and in a monopole background in three dimensions.