While free fermion topological crystalline insulators have been largely classified, the analogous problem in the strongly interacting case has been only partially solved, and the relationship between the free and interacting classifications is not well understood. In this paper, we develop a characterization and classification of interacting, invertible fermionic topological phases in ($2+1$) dimensions with charge conservation, discrete magnetic translation, and $M$-fold point group rotation symmetries, which form the group $G_f=\text{U}{\left(1\right)}^f{\times }_{\varphi }[{\mathbb{Z}}^2⋊{\mathbb{Z}}_M]$ for $M=1,2,3,4,\text{and}6$. $\varphi$ is the magnetic flux per unit cell. We derive a topological response theory in terms of background crystalline gauge fields, which gives a complete classification of different phases and a physical characterization in terms of quantized response to symmetry defects. We then derive the same classification in terms of a set of real-space invariants $\left\{{\mathrm{\Theta }}_{\text{o}}^{\pm }\right\}$ that can be obtained from ground state expectation values of suitable partial rotation operators. We explicitly relate these real-space invariants to the quantized coefficients in the topological response theory, and find the dependence of the invariants on the chiral central charge $c_{-}$ of the invertible phase. Finally, when $\varphi =0,$ we derive an explicit map between the free and interacting classifications.

• Received 4 October 2023
• Accepted 1 December 2023

DOI:https://doi.org/10.1103/PhysRevB.109.035168