Twisted Bilayer Graphene in Commensurate Angles

TL;DR

This paper rigorously analyzes the electronic properties of twisted bilayer graphene at commensurate angles using an exact continuum model, establishing the existence of Dirac cones and their behavior near incommensurate angles.

Contribution

It provides a rigorous mathematical framework for understanding Dirac cones in twisted bilayer graphene without relying on the Bistritzer-MacDonald approximation.

Findings

Identifies exact commensurate angles for twisted lattices.

Proves existence of Dirac cones at these angles.

Shows Dirac cone slopes flatten near incommensurate angles for small potentials.

Abstract

We study a 2D continuum model of electronic transport in twisted bilayer graphene (TBG) at commensurate angles. We use two honeycomb potentials with the symmetries of graphene, either sharing a common origin (AA stacking) or shifted by a half-lattice spacing (AB stacking), and twisted relative to each other. While the electronic properties of TBG are most commonly studied via the approximate Bistritzer-MacDonald (BM) model, our approach studies the exact continuum Schr\"{o}dinger operator without these approximations. Our results hold for a wide class of potentials in both stacking types. We describe the exact angles for which the two twisted lattices are commensurate and prove the existence of Dirac cones at the vertices of the Brillouin zone for such angles. Additionally, we establish quantitative bounds showing that, for small potentials, the slope of the Dirac cones flattens at commensurate angles near incommensurate angles. This work is the first to rigorously establish the existence of Dirac cones for twisted bilayer graphene in the continuum setting, without the BM approximations.