Electronic Observables for Relaxed Bilayer 2D Heterostructures in Momentum Space

We generalize the transformations and duality found in incommensurate 2D systems between real space, configuration space, momentum space, and reciprocal space to study electronic observables of incommensurate bilayers in the tight-binding framework using a wide class of applicable Hamiltonians. We then apply this generalization to obtain the effects of mechanical relaxation on nearly aligned materials in momentum space, which produce in-plane incommensurate scattering. The relaxation scattering is long-ranged in this case, which likewise changes the momentum space numerical scheme convergence rate. We study this convergence theoretically, and perform a numerical study on twisted bilayer graphene at small angles with mechanical relaxation.