# Is the composite fermion state of Graphene a doped Chern insulator?

## TL;DR

This paper shows that accounting for next-nearest-neighbor hopping in doped graphene under strong magnetic fields reveals a composite fermion state akin to a doped Chern insulator, with a measurable energy gap.

## Contribution

It introduces a microscopic derivation of a topological energy gap in doped graphene's fractional quantum Hall states, linking it to a doped Chern insulator and particle-hole symmetric theories.

## Key findings

- Next-nearest-neighbor hopping redistributes magnetic flux in graphene.
- A Haldane gap proportional to hopping and doping is predicted.
- The gap persists after projecting to the lowest Landau level.

## Abstract

Graphene in the presence of a strong external magnetic field is a unique attraction for investigations of the fractional quantum Hall (fQH) states with odd and even denominators of the fraction. Most of the attempts to understand Graphene in the strong-field regime were made through exploiting the universal low-energy effective description of Dirac fermions emerging from the nearest neighbor hopping model of electrons on a honeycomb lattice. We highlight that accounting for the next-nearest-neighbor hopping terms in doped Graphene can lead to a unique redistribution of magnetic fluxes within the unit cell of the lattice. While this affects all the fQH states, it has a striking effect at a half-filled Landau-level state: it leads to a composite fermion state that is equivalent to the doped topological Chern insulator on a honeycomb lattice. At energies comparable to the Fermi energy, this state possesses a Haldane gap in the bulk proportional to the next-nearest-neighbor hopping and density of dopants. We argue that this microscopically derived energy gap survives the projection to the lowest band. We also conjecture that the gap should be present in a microscopic theory giving the recently proposed particle-hole symmetric Dirac composite fermion scenario of the half-filled Landau-level. The proposed gap is lower than the chemical potential, and is predicted to be parametrically separated from the Dirac point in the latter description. Finally we conclude by proposing experiments to detect this gap; the associated boundary mode; and encourage cold-atom setups to test other predictions of the theory.