Fermionic Hopf solitons and Berry phase in topological surface superconductors

An interesting phenomenon in many-body physics is that quantum statistics may be an emergent property. This was first noted in the Skyrme model of nuclear matter, where a theory of a bosonic order parameter field contains fermionic excitations. These excitations are smooth field textures and are believed to describe neutrons and protons. We argue that a similar phenomenon occurs in topological insulators when superconductivity gaps out their surface states. Here, a smooth texture is naturally described by a three-component vector. Two components describe superconductivity, while the third captures the band topology. Such a vector field can assume a "knotted" configuration in three-dimensional space-the Hopf texture-that cannot smoothly be unwound. Here we show that the Hopf texture is a fermion. To describe the resulting state, the regular Landau-Ginzburg theory of superconductivity must be augmented by a topological Berry phase term. When the Hopf texture is the cheapest fermionic excitation, unusual consequences for tunneling experiments on mesoscopic samples are predicted. This framework directly generalizes the phenomenon of period doubling of Josephson effect to three-dimensional topological insulators with surface superconductivity.