Topological Order and Symmetry Anomaly on the Surface of Topological Crystalline Insulators

We present microscopic Hamiltonians that gap the Dirac fermions on the surface of topological crystalline insulators (TCIs) protected by reflection symmetry and create symmetry-preserving states with Abelian topological orders. For TCIs with $n = 2$ copies of surface Dirac fermions, we find a fermionic $\mathbb{Z}_4$ topological order with charge-spin self duality and with a novel symmetry action on the anyons, whereby reflections with respect to two parallel planes act differently. We also analyze TCIs with $n = 1$ and $4$ copies of Dirac fermions, which taken together covers every nontrivial case in the $\mathbb{Z}_8 \oplus \mathbb{Z}_2$ classification of interacting TCIs. Our work reveals a fundamental difference between reflection and charge-conjugation-time-reversal symmetry anomalies.