Title:
Numerical Analysis of Finite Difference Methods Defined on Moiré Lattices
Abstract: Placing a two-dimensional lattice with a small rotation gives rise to almost periodic “moire” patterns on a superlattice scale much larger than the original lattice. We have derived continuum models for finite difference (tight-binding) models defined on these incommensurate lattices by extending classical methods used to derive continuum models from finite difference methods defined on lattices (numerical diffusion, dispersion, shadow equations, etc.).
Twisted bilayer graphene (TBG) has drawn significant interest due to recent experiments which show that TBG can exhibit strongly correlated behavior such as the superconducting andcorrelated insulator phases. We have derived a continuum model which systematically accounts for the effects of structural relaxation. In particular, we model structural relaxation by coupling linear elasticity to a stacking energy that penalizes disregistry. We compare the impact of the two relaxation models on the corresponding many-body model by defining an interacting model projected to the flat bands. We perform tests at charge neutrality at the Hartree–Fock level of theory and find that our systematic relaxation model gives quantitative differences from earlier simplified models.