Twisted Bilayer Graphene in Commensurate Angles
Tal Malinovitch
TL;DR
This paper develops a fundamental continuum description of twisted bilayer graphene at commensurate angles without relying on the BM model. By constructing a two-layer continuum operator $H^{\theta}(\lambda)=-\Delta+\lambda W^{\theta}$ with AA and AB stacking potentials derived from a honeycomb base, it shows that the twisted system is periodic on a scaled lattice $\Lambda^{\theta}=N\Lambda$ or $N\Lambda^{*}$ for angles in the commensurate set $\mathcal{C}$. The authors prove the existence of Dirac cones at high-symmetry points for broad admissible potentials and establish a perturbative bound $|v_{d}(\lambda)|\lesssim C/N$ for small coupling, indicating Dirac cones flatten as the scaling grows. This work lays a rigorous foundation for understanding magic-angle phenomena beyond BM by focusing on commensurate angles in a continuum setting and outlines a program to bridge to incommensurate cases, with clear pathways via symmetry and perturbation theory. Overall, it provides a rigorous, model-agnostic route to Dirac-cone physics in TBG and sets up a sequence of results toward the larger goal of a first-principles account of magic angles.
Abstract
The recent discovery of ``magic angles" in twisted bilayer graphene (TBG) has spurred extensive research into its electronic properties. The primary tool for studying this thus far has been the famous Bistritzer-MacDonald model (BM model), which relies on several approximations. This work aims to build the first steps in studying magic angles without using this model. Thus, we study a 2d model for TBG in both AA and AB stacking \emph{without} the approximations of the BM model in the continuum setting, using two copies of a potential with the symmetries of graphene, either sharing a common origin (in AA stacking) or with shifted origins (in AB stacking), and twisted with respect to each other. Our results hold for a wide class of potentials in both stacking types. We describe the angles for which the two twisted lattices are commensurate and prove the existence of Dirac cones in the vertices of the Brillouin zone for such angles. Furthermore, we show that for small potentials, the slope of the Dirac cones is small for commensurate angles that are close to incommensurate angles. This work is the first to establish the existence of Dirac cones for twisted bilayer graphene (for either stacking) in the continuum setting without relying on the BM model. This work is the first in a series of works to build a more fundamental understanding of the phenomenon of magic angles.