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

TL;DR

The paper demonstrates that symmetry-preserving gapped surfaces of 3D topological insulators can host non-Abelian surface topological orders, specifically the T-Pfaffian and Pfaffian-antisemion states, realized via soluble 3D Walker-Wang constructions. It analyzes how time-reversal symmetry can act on these anyons, showing two consistent realizations (η=±1) related by bosonic SPTs, and argues that one realization corresponds to the free-fermion TI surface while the other differs by an eTmT bosonic topological superconductor. The work also shows how two copies of these surface states can become trivial and how breaking either TR or charge conservation can connect these orders to familiar TI surface physics, including a superconducting TI surface and the θ = π magnetoelectric response. By constructing explicit 3D lattice models, the authors establish concrete routes to realize and manipulate symmetric, gapped TI surfaces with non-Abelian anyons, advancing the understanding of strongly correlated topological phases. These results illuminate the landscape of interacting TI surface states and their connections to bosonic SPTs and non-Abelian anyon theories.

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 $σ_{xy}=1/2$ 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 $θ=π$ 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.

Figures (6)

• Figure 1: (color online) Braiding and fusion moves on the string-net configurations.
• Figure 2: (color online) The anyonic excitations on the surface are created by open strings. The end of strings of type $\al$ are anyonic excitations of type $\al$ with the expected statistics. This can be seen from the braiding statistics of the strings generating (a) the exchange and (b) the braiding of the end of strings.
• Figure 3: (color online) If an anyon $\al$ has nontrivial braiding with some other anyon $\bt$, then the open string of type $\al$ in the bulk can change the quantum fluctuation phase factors of $\bt$ loops along its length, which costs finite energy. Therefore, the end of strings of type $\al$ are confined in the bulk. Otherwise, the end of string of type $\al$ is a deconfined point excitation, which must be either bosonic or fermionic in a 3D system.
• Figure 4: (color online) Sample spin configuration in the decorated Walker-Wang model. Black dots represent spin $1/2$'s, and blue ellipses represent spin singlets. The leftmost vertex $\{I_2,\psi_2,\psi_4\}$ prefers a down spin while the rightmost vertex $\{\psi_2,I_2,\psi_4\}$ prefers an up spin.
• Figure 5: (color online) A trivalent 3D lattice obtained by splitting the vertices in a cubic lattice (taken from Walker2012).