Symmetry enforced non-Abelian topological order at the surface of a topological insulator

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 $\text{U}\left(1\right)$ charge conservation symmetry. Here, we discuss a 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 can not be realized in a two-dimensional (2D) system with the same symmetries. We discuss candidate surface states, non-Abelian quantum Hall states which, when realized in 2D, have ${\sigma }_{xy}=\frac{1}{2}$ 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.

Figure 1

Braiding and fusion moves on the string-net configurations.

Figure 2

The anyonic excitations on the surface are created by open strings. The end of strings of type $\alpha$ are anyonic excitations of type $\alpha$ 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

If an anyon $\alpha$ has nontrivial braiding with some other anyon $\beta$, then the open string of type $\alpha$ in the bulk can change the quantum fluctuation phase factors of $\beta$ loops along its length, which costs finite energy. Therefore, the end of strings of type $\alpha$ are confined in the bulk. Otherwise, the end of string of type $\alpha$ is a deconfined point excitation, which must be either bosonic or fermionic in a 3D system.

Figure 4

Sample spin configuration in the decorated Walker-Wang model. Black dots represent spin $\frac{1}{2}$'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

A trivalent 3D lattice obtained by splitting the vertices in a cubic lattice (taken from [25]).

Figure 6

Plaquette term in Walker-Wang Hamiltonian (taken from [25]).