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

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.