Non-Abelian topological order on the surface of a 3d topological superconductor from an exactly solved model

Abstract

Three-dimensional (3D) topological superconductors (TScs) protected by time-reversal (T) symmetry are characterized by gapless Majorana cones on their surface. Free-fermion phases with this symmetry (class DIII) are indexed by an integer ν, of which ν = 1 is realized by the B phase of superfluid 3He. Previously, it was believed that the surface must be gapless unless time-reversal symmetry is broken. Here, we argue that a fully symmetric and gapped surface is possible in the presence of strong interactions, if a special type of topological order appears on the surface. The topological order realizes T symmetry in an anomalous way, one that is impossible to achieve in purely two dimensions. For odd ν TScs, the surface topological order must be non-Abelian. We propose the simplest non-Abelian topological order that contains electronlike excitations, SO(3)6, with four quasiparticles, as a candidate surface state. Remarkably, this theory has a hidden T invariance that, however, is broken in any two-dimensional realization. By explicitly constructing an exactly soluble Walker-Wang model, we show that it can be realized at the surface of a short-ranged entangled 3D fermionic phase protected by T symmetry, with bulk electrons transforming as Kramers pairs, i.e. T2=-1 under time reversal. We also propose an Abelian theory, the semion-fermion topological order, to realize an even ν TSc surface, for which an explicit model is derived using a coupled-layer construction. We argue that this is related to the ν = 2 TSc, and we use this to build candidate surface topological orders for ν = 4 and ν = 8 TScs. The latter is equivalent to the three-fermion state, which is the surface topological order of a Z2 bosonic topological phase protected by T invariance. One particular consequence of this equivalence is that a ν = 16 TSc admits a trivially gapped T -symmetric surface.