Classifications of Interacting Topological Crystalline Semimetals

Topological crystalline semimetals are a class of semimetallic materials with a nontrivial interplay between the topology in the momentum space and crystalline symmetry. In order to describe topological semimetals with strong interactions, it's desirable to develop a general framework that does not rely on the band theory. We show that the general framework for classifying crystalline symmetry protected topological phases, dubbed the topological crystal approach, is also useful for classifying the topological crystalline semimetals with strong interactions. To illustrate the main idea, we apply the topological crystal method to 3d Dirac semimetals protected by $C_n$ rotation, $z$-translation, and the combined parity and time-reversal symmetries. We show that the anomaly nature of the 3d Dirac semimetals is the filling anomaly in the one-dimensional subspace at the rotational axis, and the non-interacting $Z$ classification generally reduces to an $Z_p$ subgroup under strong interactions, where $p$ is a function of $n$. We also discuss the topological response theory and derive the classification from it through coupling to the crystalline gauge fields. Our work provides a general theoretical framework to classify and characterize interacting topological crystalline semimetals.