An abstract formulation of the flat band condition

TL;DR

This paper provides abstract conditions ensuring flat bands in periodic Hamiltonians, motivated by twisted bilayer graphene, and demonstrates these conditions with a scalar operator model.

Contribution

It introduces abstract criteria for flat band existence and applies them to a scalar operator related to the chiral TBG model.

Findings

Flat bands occur at discrete parameter sets in the models studied.

Abstract conditions can predict flat band existence in periodic Hamiltonians.

Application to a scalar operator confirms the theoretical criteria.

Abstract

Motivated by the study of flat bands in models of twisted bilayer graphene (TBG), we give abstract conditions which guarantee the existence of a discrete set of parameters for which periodic Hamiltonians exhibit flat bands. As an application, we show that a scalar operator derived from the chiral model of TBG has flat bands for a discrete set of parameters.