Construction and accuracy of electronic continuum models of incommensurate bilayer 2D materials

Abstract Single-particle continuum models such as the popular Bistritzer–MacDonald model have become powerful tools for predicting electronic phenomena of incommensurate 2D materials and developing many-body models aimed at modeling unconventional superconductivity and correlated insulators. In this work, we introduce a procedure to construct continuum models of arbitrary accuracy relative to tight-binding models for moiré incommensurate bilayers. This is done by recognizing the continuum model as arising from Taylor expansions of a high accuracy momentum space approximation of the tight-binding model. We apply our procedure in full detail to two models of twisted bilayer graphene and demonstrate both admit similar Bistritzer–MacDonald models as the leading order continuum model, while higher order expansions reveal qualitative spectral differences.