Topological field theories of three-dimensional rotation symmetric insulators: Coupling curvature and electromagnetism

Quantized responses are important tools for understanding and characterizing the universal features of topological phases of matter. In this work, we show that three-dimensional (3D) insulators with Cn rotation symmetry along a fixed axis can possess a mixed geometry-charge response, in which disclination lines of the 3D lattice carry electric polarization. These disclinations bind a fractional charge at the gapped surfaces of the insulator because of the surface-charge-polarization theorem. This response is accompanied by a dual response that binds fractional angular momentum to magnetic monopoles in the bulk (analogously to the Witten effect) and to magnetic fluxes on gapped surfaces. We show that these responses are described by a 3D topological response term that couples the lattice curvature to the electromagnetic field strength. Additional mirror or particle-hole symmetry quantizes the mixed geometry-charge responses and defines a new class of rotation-invariant topological crystalline insulators (rTCIs). Notably, the surface charge bound to disclinations of the rTCIs is half the minimal amount that can occur in purely two-dimensional insulators. We construct lattice models of these rTCIs and numerically verify that they exhibit mixed geometry-charge responses. We also demonstrate that the particle-hole symmetric rTCI supports anomalous surface topological order and that the mirror symmetric rTCI can be smoothly deformed into a higher order octopole insulator with quantized corner charges. Additionally, we construct symmetry indicators for diagnosing the mirror symmetric rTCIs.