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

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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.