Band theory for heterostructures with interface superlattices

Motivated by recent experiments demonstrating the creation of atomically sharp interfaces between hexagonal sapphire and cubic ${\mathrm{SrTiO}}_{3}$ with finite twist, we here develop and study a general electronic band theory for this novel class of moir\'e heterostructures. We take into account the three-dimensional nature of the two crystals, allow for arbitrary combinations of Bravais lattices, finite twist angles, and different locations in momentum space of the low-energy electronic bands of the constituent materials. We analyze the general condition for a well-defined crystalline limit in the interface electron system and classify the associated ``crystalline reference points.'' We discuss this in detail for the example of the two-dimensional lattice planes being square and triangular lattices on the two sides of the interface; this reveals nontrivial reference points at finite twist angle and lattice mismatch, leading to a unique form of magic angles, which we refer to as ``geometric magic angles.'' We further show that band structures of mixed dimensionality naturally emerge, where quasi-one- and two-dimensional pockets coexist. Explicit computations for different bulk Bloch Hamiltonians yield a collection of interesting features, such as isolated bands localized at interfaces of nontopological insulators, Dirac cones, van Hove singularities, a nontrivial evolution of the band structures with Zeeman field, and topological interface bands. Our work illustrates the potential of these heterostructures and is anticipated to provide the foundation for moir\'e interface design and for the analysis of correlated physics in these systems.