Construction and Accuracy of Electronic Continuum Models of Incommensurate Bilayer 2D Materials

TL;DR

This paper presents a systematic method to construct highly accurate continuum models for incommensurate bilayer 2D materials, improving upon existing models like Bistritzer-MacDonald by including higher-order effects.

Contribution

The authors introduce a general procedure to derive continuum models of arbitrary accuracy from tight-binding models using Taylor expansions in momentum space.

Findings

Higher-order expansions show qualitative spectral differences from leading order models.

The method is applied to twisted bilayer graphene, confirming the Bistritzer-MacDonald model as the leading order.

The approach allows systematic improvement of continuum models for 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 the development of many-body models aimed to model 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 moir\'{e} 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 the Bistritzer-MacDonald model as the leading order continuum model, while higher order expansions reveal qualitative spectral differences.