Fractional Chern Insulator States in Twisted Bilayer Graphene: An Analytical Approach

TL;DR

This paper demonstrates that twisted bilayer graphene at the magic angle can host fractional Chern insulator states due to favorable quantum geometry, using an analytical approach in the chiral limit and continuum real space methods.

Contribution

It provides an analytical framework showing the conditions for fractional Chern insulator stability in twisted bilayer graphene, emphasizing continuum methods over tight-binding models.

Findings

Flat bands have analytic wavefunctions in the chiral limit.

Quantum metric and Berry curvature conditions for FCI are satisfied.

Real-space Laughlin wavefunction can be constructed as a zero-energy ground state.

Abstract

Recent experiments on bilayer graphene twisted near the magic angle have observed spontaneous integer quantum Hall states in the presence of an aligned hexagonal boron nitride (hBN) substrate. These states arise from valley ferromagnetism, and the complete filling of Chern bands. A natural question is whether fractional filling of the same bands would lead to fractional quantum Hall states, i.e. to fractional Chern insulators (FCIs). Here, we argue that the magic angle graphene bands have favorable quantum geometry for realizing FCI phases. We show that in the tractable `chiral' limit, the flat bands wavefunctions are an analytic function of the crystal momentum. This remarkable property fixes the quantum metric up to an overall momentum dependent scale factor, the local Berry curvature, whose variation is itself small. Thus the three conditions associated with FCI stability (i) narrow bands (ii) quantum metric satisfying the isotropic ideal droplet condition and (iii) relatively uniform Berry curvature are all satisfied. Our work emphasizes continuum real space approaches to FCIs in contrast to earlier works which mostly focused on tight binding models. This enables us to construct a Laughlin wavefunction on a real-space torus that is a zero energy ground state of pseudopotential-like interactions. The latter can be realized for Coulomb interaction in the limit of very short screening length. We also map the problem onto an equivalent one of Dirac particles in an inhomogeneous but periodic magnetic field and no crystal potential. Finally we discuss evolution of the band geometry on tuning away from the chiral limit and show numerically that some of the desirable properties continue to hold at a quantitative level for realistic parameter values.