Momentum Space Algorithm for Electronic Structure of Double-Incommensurate Twisted Trilayer Graphene

Moiré 2D materials are highly tunable through variables including twist angle, species of layers, and number of layers. Various configurations lead to useful physical phenomena and possible applications, including many-body physics such as correlated insulators and superconductivity. To understand many-body models, a careful single-particle model must first be constructed. For example in twisted bilayer graphene, the Bistritzer-MacDonald model is frequently used to capture magic-angle physics in twisted bilayer graphene.

More complex geometries including double-incommensurate trilayers however become difficult to accurately quantify even in the single-particle regime, particularly in a momentum framework where one also aims to obtain quasi-band structure. Current methodologies suffer from a dimensional bottleneck due to the complex scattering description. Here we present a momentum space algorithm for computing observables for double-incommensurate trilayers with rigorous error analysis compared to the real space ab initio tight-binding model. We implement a careful truncation scheme to reduce the dimensionality problem by two orders of magnitude exploiting an energy-momentum confinement provided by the Dirac cones of the graphene layers. We include computations of the closest equivalent observable to band structure that this structure seems to admit, the momentum local density of states.