Moire Physics of Twisted Bilayers from the Fourier Transforms of Two-Center Amplitudes

The number of experimentally realized twisted bilayer systems has diversified over the last decade as exfoliation techniques have improved and novel phenomena have been observed. The theoretical progress in modeling these systems has been focused on a narrow subset of materials that have band structures or band edges near special points in momentum space, allowing an expansion of the Hamiltonian.

In twisted bilayer systems, the breaking of the mono-layer Bravais lattice translation symmetry induces off-diagonal couplings between monolayer Bloch States. In an effort to model the coarse-grained couplings we establish new integral results for the two dimensional Fourier Transforms of generalized Slater-Koster hopping amplitudes, for various radial scaling functions, in the form of a series which rapidly converges over the entire parameter space.

Applied with a first-order gradient expansion of the strain fields, we evaluate this continuum model approach in relation to commensurate band structure calculations for a variety of appropriate Van-der Waals bilayer structures.