Spectral characterization of magic angles in twisted bilayer graphene

TL;DR

This paper provides a spectral analysis of magic angles in twisted bilayer graphene, revealing their relation to eigenvalues of a non-hermitian operator and exploring the effects of interaction potentials and symmetries on flat bands.

Contribution

It introduces a new spectral characterization of magic angles in TBG, linking them to eigenvalues of a non-hermitian operator and analyzing their scaling and symmetry properties.

Findings

Magic angles correspond to eigenvalues of a non-hermitian operator.

Inverse magic angles scale equidistantly only for specific tunnelling potentials.

Protection of zero-energy states depends on particle-hole symmetry.

Abstract

Twisted bilayer graphene (TBG) has been experimentally observed to exhibit almost flat bands when the twisting occurs at certain magic angles. In this letter, we report new results on the continuum model of twisted bilayer graphene and its electronic band structure. Under we show that in the approximation of vanishing AA-coupling, the magic angles (at which there exist entirely flat bands) are given as the eigenvalues of a non-hermitian operator, and that all bands start squeezing exponentially fast as the angle $\theta$ tends to $0$. In particular, as the interaction potential changes, the dynamics of magic angles involves the non-physical complex eigenvalues. Using our new spectral characterization, we show that the equidistant scaling of inverse magic angles, is special for the choice of tunnelling potentials in the continuum model, and is not protected by symmetries. While we also show that the protection of zero-energy states holds in the continuum model as long as particle-hole symmetry is preserved, we observe that the existence of flat bands and the exponential squeezing are special properties of the chiral model.