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

We describe a three-dimensional crystalline topological insulator (TI) phase of matter that exhibits spontaneous polarization. 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. Utilizing elasticity tetrads, we show a complete classification of 3D crystalline TI phases and invariants. 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 systems may hence exhibit superconductivity with a very high critical temperature.