Topological polarization, dual invariants, and surface flat bands in crystalline insulators

We describe a crystalline topological insulator (TI) phase of matter that exhibits spontaneous polarization in arbitrary dimensions. The bulk polarization response is constructed by coupling the system to geometric deformations of the underlying crystalline order, represented by local lattice vectors—the elasticity tetrads. This polarization results from the presence of (approximately) flat bands on the surface of such TIs. These flat bands are a consequence of the bulk-boundary correspondence of polarized topological media, and contrary to related nodal line semimetal phases also containing surface flat bands, they span the entire surface Brillouin zone. We also present an example Hamiltonian exhibiting a Lifshitz transition from the nodal line phase to the TI phase with polarization. In addition, we discuss a general classification of three-dimensional (3D) crystalline TI phases and invariants in terms of the elasticity tetrads. The phase with polarization naturally arises from this classification as a dual to the previously better-known 3D TI phase exhibiting quantum (spin) Hall effect. Besides polarization, another implication of the large surface flat band is the susceptibility to interaction effects such as superconductivity: The mean-field critical temperature is proportional to the size of the flat bands, and this type of system may hence exhibit superconductivity with a very high critical temperature.