Graphene integer quantum Hall effect in the ferromagnetic and paramagnetic regimes

Starting from the graphene lattice tight-binding Hamiltonian with an on-site $U$ and long-range Coulomb repulsion, we derive an interacting continuum Dirac theory governing the low-energy behavior of graphene in an applied magnetic field. Initially, we consider a clean graphene system within this effective theory and explore integer quantum Hall ferromagnetism stabilized by exchange from the long-range Coulomb repulsion. We study in detail the ground state and excitations at $\nu =0$ and $\nu =\pm 1$, taking into account small symmetry-breaking terms that arise from the lattice-scale interactions, and also explore the ground states selected at $\nu =\pm 3$, $\pm 4$, and $\pm 5$. We argue that the ferromagnetic regime may not yet be realized in current experimental samples, which at the above filling factors perhaps remain paramagnetic due to strong disorder. In an attempt to access the latter regime where the role of exchange is strongly suppressed by disorder, we apply Hartree theory to study the effects of interactions. Here, we find that Zeeman splitting together with symmetry-breaking interactions can in principle produce integer quantum Hall states in a paramagnetic system at $\nu =0$, $\pm 1$, and $\pm 4$, but not at $\nu =\pm 3$ or $\pm 5$, consistent with recent experiments in high magnetic fields. We make predictions for the activation energies in these quantum Hall states which will be useful for determining their true origin.

Figure 1

(Color online) Schematic lattice-scale order in the $\nu =-1$ integer quantum Hall state. “1” and “2” label the two sublattices of the honeycomb. Electrons in the highest-occupied Landau level reside only on sublattice 1 (for instance) and undergo circulating currents around second-neighbor plaquettes oriented along the arrows.

Figure 2

Schematic effective single-particle energy levels obtained within a Hartree analysis of the interacting theory at (a) $\nu =-1$, (b) $\nu =0$, and (c) $\nu =4$. Here ${\Delta }_s=g{\mu }_{B}B+2u_0{\rho }_0$, while ${\Delta }_{u_1}=4u_1{\rho }_0$. For $\nu =-1$ and 0, the levels on the left-side correspond to ${\Delta }_s>{\Delta }_{u_1}$, while the right side corresponds to ${\Delta }_s<{\Delta }_{u_1}$.