Symmetry Enforced Non-Abelian Topological Order at the Surface of a Topological Insulator

TL;DR

This paper demonstrates that 3D topological insulator surfaces can host symmetry-preserving, gapped, topologically ordered states with non-Abelian anyons, expanding understanding of surface phases beyond traditional gapless Dirac cones.

Contribution

It introduces exactly soluble models showing that non-Abelian topological order can exist on TI surfaces without breaking symmetries, revealing new symmetry-preserving gapped phases.

Findings

Surface states can be gapped and symmetric with non-Abelian topological order.

Constructed exactly soluble 3D lattice models for these states.

Identified specific topological orders like T-Pfaffian and Pfaffian-antisemion.

Abstract

The surfaces of three dimensional topological insulators (3D TIs) are generally described as Dirac metals, with a single Dirac cone. It was previously believed that a gapped surface implied breaking of either time reversal $\mathcal T$ or U(1) charge conservation symmetry. Here we discuss a novel possibility in the presence of interactions, a surface phase that preserves all symmetries but is nevertheless gapped and insulating. Then the surface must develop topological order of a kind that cannot be realized in a 2D system with the same symmetries. We discuss candidate surface states - non-Abelian Quantum Hall states which, when realized in 2D, have $\sigma_{xy}=1/2$ and hence break $\mathcal T$ symmetry. However, by constructing an exactly soluble 3D lattice model, we show they can be realized as $\mathcal T$ symmetric surface states. The corresponding 3D phases are confined, and have $\theta=\pi$ magnetoelectric response. Two candidate states have the same 12 particle topological order, the (Read-Moore) Pfaffian state with the neutral sector reversed, which we term T-Pfaffian topological order, but differ in their $\mathcal T$ transformation. Although we are unable to connect either of these states directly to the superconducting TI surface, we argue that one of them describes the 3D TI surface, while the other differs from it by a bosonic topological phase. We also discuss the 24 particle Pfaffian-antisemion topological order (which can be connected to the superconducting TI surface) and demonstrate that it can be realized as a $\mathcal T$ symmetric surface state.