Flat bands, Dirac cones, and higher-order band crossings in twisted multilayer graphene

TL;DR

This paper proves a mathematical classification of band crossings in twisted multilayer graphene, revealing conditions for flat bands, Dirac cones, or higher-order crossings depending on the twisting angle, challenging previous physics assumptions.

Contribution

It provides a rigorous mathematical proof of the band crossing behaviors in twisted multilayer graphene, identifying conditions for flat bands, Dirac cones, and higher-order crossings.

Findings

Disproves the belief that only higher order crossings or flat bands occur.

Identifies discrete magic angles where bands are flat.

Classifies band crossing types depending on twisting angle.

Abstract

For the chiral limit of two sheets of $n$-layer Bernal-stacked graphene established in the Physical Review Letters arXiv:2109.10325 and arXiv:2109.11514, we prove a trichotomy: depending on the twisting angle, we have either (1) generically, the band crossing of the first two bands is of order $n$; or (2) at a discrete set of magic angles, the first two bands are completely flat; or (3) for another discrete set of twisting parameters, the bands exhibit Dirac cones. This new mathematical discovery disproves the common belief in physics that such a twisted multilayer graphene model can only have higher order band crossings or flat bands, and it leads to a new type of topological phase transition.