Symmetries as the guiding principle for flattening bands of Dirac fermions

TL;DR

This paper develops a symmetry-based criterion to determine when Dirac bands in materials like twisted bilayer graphene can be flattened, leading to perfectly-flat bands under specific symmetry conditions and parameter tuning.

Contribution

It introduces a symmetry-guided method to identify parameter conditions for Dirac band flattening and constructs models with perfectly-flat bands based on this principle.

Findings

Derived a criterion linking symmetry groups to Dirac velocity vanishing

Applied the criterion to twisted bilayer graphene and topological insulators

Constructed models exhibiting perfectly-flat bands at specific parameters

Abstract

Since the discovery of magic-angle twisted bilayer graphene (TBG), flat bands in Dirac materials have become a prominent platform for realizing strong correlation effects in electronic systems. Here we show that the symmetry group protecting the Dirac cone in such materials determines whether a Dirac band may be flattened by the tuning of a small number of parameters. We devise a criterion that, given a symmetry group, allows for the calculation of the number of parameters required to make the Dirac velocity vanish. This criterion is employed to study band flattening in twisted bilayer graphene and in surface states of 3D topological insulators. Following this discussion, we identify the symmetries under which the vanishing of the Dirac velocity implies the emergence of perfectly-flat bands. Our analysis allows us to construct additional model Hamiltonians that display perfectly-flat bands at certain points in the space of parameters: the first is a toy model of two coupled 3D TI surfaces, and the second is a quasi-crystalline generalization of the chiral model of TBG.