We study interaction effects on the topological crystalline insulators protected by time-reversal $\left(T\right)$ and reflection symmetry $\left(R\right)$ in two and three spatial dimensions. From the stability analysis of the edge states with bosonization, we find that the classification of the two-dimensional symmetry-protected topological (SPT) phases protected by $Z_2\times [\text{U(1)}⋊T]$ symmetry is reduced from $\mathbb{Z}$ to ${\mathbb{Z}}_4$ by interactions where the $Z_2$ symmetry denotes the reflection whose mirror plane is the two-dimensional plane itself. By extending the approach recently proposed by Isobe and Fu, we show that the classification of the three-dimensional SPT phases (i.e., topological crystalline insulators) protected by $R\times [\text{U(1)}⋊T]$ symmetry is reduced from $\mathbb{Z}$ to ${\mathbb{Z}}_8$ by interactions.

• Received 25 May 2015

DOI:https://doi.org/10.1103/PhysRevB.92.085114