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We describe a three-dimensional crystalline topological insulator (TI) phaseof matter that exhibits spontaneous polarization. This polarization resultsfrom the presence of (approximately) flat bands on the surface of such TIs.These flat bands are a consequence of the bulk-boundary correspondence ofpolarized topological media, and contrary to related nodal line semimetalphases also containing surface flat bands, they span the entire surfaceBrillouin zone. We also present an example Hamiltonian exhibiting a Lifshitztransition from the nodal line phase to the TI phase with polarization.Utilizing elasticity tetrads, we show a complete classification of 3Dcrystalline TI phases and invariants. The phase with polarization naturallyarises from this classification as a dual to the previously better-known 3D TIphase exhibiting quantum (spin) Hall effect. Besides polarization, anotherimplication of the large surface flat band is the susceptibility to interactioneffects such as superconductivity: the mean-field critical temperature isproportional to the size of the flat bands, and this type of systems may henceexhibit superconductivity with a very high critical temperature.

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