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

Abstract

Numerical algorithms for computing electronic structure of incommensurate 2D materials using ab initio models is critical for predicting material properties and guiding experiment. For bilayers, momentum space and continuum models have been introduced to approximate observables of ab initio tight-binding models using a momenta description despite the lack of periodicity in the tight-binding model required for Bloch theory. A similar structure has been introduced for double-incommensurate trilayers using a continuum model, where the three lattices are all mutually incommensurate. However, this description leads to a four-dimensional lattice space, and numerical convergence of the density of states was observed to have poor convergence. In this work, we introduce a momentum space framework for double incommensurate trilayer graphene, and introduce an efficient truncation scheme of the four-dimensional lattice to drastically improve convergence of the density of states and momentum local density of states (a parallel object to classical band structure). We implement this algorithm on an ab initio model of twisted trilayer graphene and validate convergence estimates. We further verify numerically that the momentum space algorithm, inherently higher order than the continuum model as it is an exact transformation of the tight-binding model, captures altered band behavior near the flat bands at magic angles.