Composite fermion state of graphene as a Haldane-Chern insulator

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 gapless composite fermion state that is equivalent to the doped topological Chern insulator on a honeycomb lattice coupled to a Chern Simons gauge field. 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.

Figure 1

(a) A schematic of the dispersion of electronic states in graphene in zero field. (b) The spectrum of composite fermion states at half-filling condition based on naive application of HLR-type theory (uniform background field cancellation). (c) The composite fermion spectra based on this work. It consists of a bulk gap and an edge state. (d) Conjecture for how the Dirac composite fermion analog might look like. This is not addressed in this work. The inset represents the scenario for a parabolic band. $E_F$ is the Fermi level.

Figure 2

Flux distribution in a unit cell of graphene in uniform external field, CS field, $\mathrm{external}+\mathrm{CS}$ field. The CS field consists of a Maxwell part ${\varphi }_{\mathrm{MW}}$ and a modulated part ${\varphi }_H$ ($={\varphi }_{\mathrm{MW}}$) that does not contribute to the total flux. In the formulas, we use the dimensionless fluxes ${\varphi }_{b,h,mw}$ which are defined at appropriate places.

Figure 3

Flux distribution in a unit cell of graphene for various fQHE states given by the Jain series.

Figure 4

The Hofstadter spectrum for a CS field acting on nnn graphene with $r=0.11$ (left) and 0.22 (right). Observe the linear scaling and the flatness of the lowest energy level in the two plots.

Figure 5

Extended scheme outlining the phase accumulation in addition to $\int \mathbf{A}·d\mathbf{l}$ along each bond. The field $\mathbf{A}$ is written in symmetric gauge. The figure is split into $\left(a\right)$ and $\left(b\right)$ for clarity. The flux ${\varphi }_{\mathrm{MW}}$ adds up with area, whereas the flux ${\varphi }_H$ denotes flux modulation within the unit cell such that net contribution from ${\varphi }_H$ to the unit cell is zero. The presence of ${\varphi }_H$ does not break translational symmetry.

Figure 6

The flux distribution arising from CS field in graphene with next-nearest-neighbor hopping. The shaded region denotes the region of nonzero flux.