Symmetric-gapped surface states of fractional topological insulators

We construct the symmetric-gapped surface states of a fractional topological insulator with an electromagnetic $\theta$ angle ${\theta }_{\text{em}}=\frac{\pi }{3}$ and a discrete ${\mathbb{Z}}_3$ gauge field. They are the proper generalizations of the T-Pfaffian state and Pfaffian/antisemion state and feature an extended periodicity compared with their “integer” topological band insulator counterparts. We demonstrate that the surface states have the correct anomalies associated with time-reversal symmetry and charge conservation.