3D Topological crystalline insulators protected by mirror reflection and time-reversal symmetry have Z classification, with the corresponding topological invariant being mirror Chern number. Under interaction, this topological classification is changed. To show this modification, we first consider 2D system with reflection as onsite Z2 symmetry and bosonize its 1D helical edge modes. We find four pairs of helical edge modes can be gapped by interacting terms, therefore the classification is reduced from Z to Z4 in 2D. With the 2D result, we extend to 3D system and show that the classfication is reduced from Z to Z8.

References:

[1]Yoshida, T., & Furusaki, A. (2015). Correlation effects on topological crystalline insulators. Physical Review B, 92(8), 085114.

[2]Isobe, H., & Fu, L. (2015). Theory of interacting topological crystalline insulators. Physical Review B, 92(8), 081304.