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Correlation effects on topological crystalline insulators

We study interaction effects on the topological crystalline insulators protected by time-reversal ($T$) and reflection symmetry ($R$) 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 SPT phases protected by $Z_2\times[\mbox{U(1)}\rtimes T]$ 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 $R\times[\mbox{U(1)}\rtimes T]$ symmetry is reduced from $\mathbb{Z}$ to $\mathbb{Z}_8$ by interactions.