Placing a two-dimensional lattice on another with a small rotation gives rise to periodic “moiré” patterns on a superlattice scale much larger than the original lattice. The Bistritzer-MacDonald (BM) model attempts to capture the electronic properties of twisted bilayer graphene (TBG) by an effective periodic continuum model over the bilayer moiré pattern. We use the mathematical techniques developed to study waves in inhomogeneous media to identify a regime where the BM model emerges as the effective dynamics for electrons modeled as wave-packets spectrally concentrated at the monolayer Dirac points of linear dispersion, up to error that we rigorously estimate. Using measured values of relevant physical constants, we argue that this regime is realized in TBG at the first “magic" angle where the group velocity of the wave packet is zero and strongly correlated electronic phases (superconductivity, Mott insulators, etc.) are observed.
We are working to develop models of TBG which account for the effects of mechanical relaxation and to couple our relaxed BM model with interacting TBG models. We are also extending our approach to essentially arbitrary moiré materials such as twisted multilayer transition metal dichalcogenides (TMDs) or even twisted heterostructures consisting of layers of distinct 2D materials.