Abstract
We resolve several puzzles related to the electromagnetic response of topological superconductors in 3+1 dimensions. In particular we show by an analytical calculation that the interface between a topological and normal superconductor does not exhibit any quantum Hall effect as long as time reversal invariance is preserved. We contrast this with the analogous case of a topological insulator to normal insulator interface. The difference is that in the topological insulator the electromagnetic vector potential couples to a vector current in a theory with a Dirac mass, while in the superconductor a pair of Weyl fermions are gapped by Majorana masses and the electromagnetic vector potential couples to their axial currents.
Highlights
The tenfold way to classify topological matter, and in particular topological insulators (TIs) and topological superconductors (TSCs), was originally derived using noninteracting fermions [1,2]
In particular we show by an analytical calculation that the interface between a topological and normal superconductor does not exhibit any quantum Hall effect as long as time reversal invariance is preserved
Because topological effects are often related to surfaces or defects, the characteristic features of TSCs may yet be coded into an effective topological field theory (TFT) for the electromagnetic field
Summary
The tenfold way to classify topological matter, and in particular topological insulators (TIs) and topological superconductors (TSCs), was originally derived using noninteracting fermions [1,2]. Such a TFT was proposed some time ago by Qi, Witten, and Zhang [4], who argued that the action governing the low-energy electromagnetic response of a TSC contains a topological term ∼ θ F F , where F is the electromagnetic field tensor and Fits dual In appearance this term is similar to the θ term in the effective action for a TI, it differs in that, rather than being an external parameter that distinguishes the trivial from the nontrivial phase, θ is a dynamical field, namely the phase of the SC order parameter. The TSC surface exhibits a neutral 2D Majorana mode which appears incapable of supporting a Hall current These two problems are clearly related as we could model a vortex in a TSC as a cylinder of an ordinary superconductor containing an Abrikosov vortex, in which case the putative charge transport would occur at the surface of the cylinder.
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