The ν = 0 quantum Hall state in a defect-free graphene sample is studied within the framework of quantum Hall ferromagnetism. We perform a systematic analysis of the "isospin" anisotropies, which arise from the valley and sublattice asymmetric short-range electron-electron (e-e) and electron-phonon (e-ph) interactions. The phase diagram, obtained in the presence of generic isospin anisotropy and the Zeeman effect, consists of four phases characterized by the following orders: spinpolarized ferromagnetic, canted antiferromagnetic, charge density wave, and Kekulé distortion. We take into account the Landau level mixing effects and show that they result in the key renormalizations of parameters. First, the absolute values of the anisotropy energies become greatly enhanced and can significantly exceed the Zeeman energy. Second, the signs of the anisotropy energies due to e-e interactions can change upon renormalization. A crucial consequence of the latter is that the short-range e-e interactions alone could favor any state on the phase diagram, depending on the details of interactions at the lattice scale. On the other hand, the leading e-ph interactions always favor the Kekulé distortion order. The possibility of inducing phase transitions by tilting the magnetic field is discussed.

Motivated to understand the nature of the strongly insulating ν = 0 quantum Hall state in bilayer graphene, we develop the theory of the state in the framework of quantum Hall ferromagnetism. The generic phase diagram, obtained in the presence of the isospin anisotropy, perpendicular electric field, and Zeeman effect, consists of the spin-polarized ferromagnetic (F), canted antiferromagnetic (CAF), and partially (PLP) and fully (FLP) layer-polarized phases. We address the edge transport properties of the phases. Comparing our findings with the recent data on suspended dual-gated devices, we conclude that the insulating ν = 0 state realized in bilayer graphene at lower electric field is the CAF phase. We also predict a continuous and a sharp insulator-metal phase transition upon tilting the magnetic field from the insulating CAF and FLP phases, respectively, to the F phase with metallic edge conductance 2e 2 /h, which could be within the reach of available fields and could allow one to identify and distinguish the phases experimentally. -is well-established [8][9][10][11][12][13][14][15][16][17], it is unambiguously identifying the particular order of the ν = 0 QHFM that presents a challenge. Given the rich phase diagram of the ν = 0 QHFM in MLG [10-13] (and as we show here, in BLG) and the fact that all phases but the spin-polarized one [18,19] are expected to be fully insulating [11,20,21], achieving this goal requires a more detailed both theoretical and experimental analysis.

Motivated by the recent experiment of Ref. 1, we develop a mean-field theory of the interactioninduced antiferromagnetic (AF) state in bilayer graphene at charge neutrality point at arbitrary perpendicular magnetic field B. We demonstrate that the AF state can persist at all B. At higher B, the state continuously crosses over to the AF phase of the ν = 0 quantum Hall ferromagnet, recently argued to be realized in the insulating ν = 0 state. The mean-field quasiparticle gap is finite at B = 0 and grows with increasing B, becoming quasi-linear in the quantum Hall regime, in accord with the reported behavior of the transport gap. By adjusting the two free parameters of the model, we obtain a simultaneous quantitative agreement between the experimental and theoretical values of the key parameters of the gap dependence -its zero-field value and slope at higher fields. Our findings suggest that the insulating state observed in bilayer graphene in Ref. 1 is antiferromagnetic (canted, once the Zeeman effect is taken into account) at all magnetic fields.

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We report the observation of an isolated charged impurity in graphene and present direct evidence of the close connection between the screening properties of a 2D electron system and the influence of the impurity on its electronic environment. Using scanning tunneling microscopy and Landau level spectroscopy, we demonstrate that in the presence of a magnetic field the strength of the impurity can be tuned by controlling the occupation of Landau-level states with a gate voltage. At low occupation the impurity is screened, becoming essentially invisible. Screening diminishes as states are filled until, for fully occupied Landau levels, the unscreened impurity significantly perturbs the spectrum in its vicinity. In this regime we report the first observation of Landau-level splitting into discrete states due to lifting the orbital degeneracy.