  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.
