Crystalline-electromagnetic responses of Dirac semimetal-charge density wave insulators

We show that insulators arising from coupling a three-dimensional Dirac semimetal to charge density wave (CDW) order can exhibit a three-dimensional generalization of the Wen-Zee response. This three-dimensional Wen-Zee response binds a quantized change per length to disclination defects in an amount proportional to the separation of the Dirac nodes in momentum space. Furthermore, when the CDW respects the inversion symmetry of the underlying Dirac semimetal, the Dirac-CDW insulator exhibits a disclination filling anomaly that manifests as a quantized difference in the charge bound to a disclination for systems with periodic and open boundary conditions. We construct an effective topological response theory that captures these crystalline responses of Dirac-CDW insulators and confirm our predictions numerically with lattice model calculations.