Non-Abelian Topological Order on the Surface of a 3D Topological Superconductor from an Exactly Solved Model

TL;DR

This paper demonstrates that strongly interacting three-dimensional topological superconductors can have gapped, symmetric surfaces with non-Abelian topological order, challenging the belief that such surfaces must be gapless.

Contribution

It introduces exactly solvable models showing how surface topological order can realize time reversal symmetry anomalously in 3D topological superconductors, including non-Abelian states.

Findings

Constructed an exactly solvable Walker-Wang model for non-Abelian surface topological order.

Proposed the SO(3)_6 non-Abelian topological order as a surface state for odd n TScs.

Linked surface topological orders to bulk TSc classifications, including the trivialization at n=16.

Abstract

Three dimensional 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 n, of which n=1 is realized by the B-phase of superfluid Helium-3. 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 n TScs, the surface topological order must be non-Abelian. We propose the simplest non-Abelian topological order that contains electron like excitations, SO(3)_6, with four quasiparticles, as a candidate surface state. Remarkably, this theory has a hidden T invariance which however is broken in any 2D 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 trasforming as Kramers pairs, i.e. T^2=-1 under time reversal. We also propose an Abelian theory, the semion-fermion topological order, to realize an even n TSc surface, for which an explicit model is derived using a coupled layer construction. We argue that this is related to the n=2 TSc, and use this to build candidate surface topological orders for n=4 and n=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 is that an n=16 TSc admits a trivially gapped T-symmetric surface.