1 Lattice Planar QED in external magnetic field

Physical Review Letters, 2005

Monolayer graphite films, or graphene, have quasiparticle excitations that can be described by 2 + 1 dimensional Dirac theory. We demonstrate that this produces an unconventional form of the quantized Hall conductivity σxy = −(2e 2 /h)(2n + 1) with n = 0, 1, . . ., that notably distinguishes graphene from other materials where the integer quantum Hall effect was observed. This unconventional quantization is caused by the quantum anomaly of the n = 0 Landau level and was discovered in recent experiments on ultrathin graphite films.

2007

We show that the recently discovered double-valley splitting of the low-lying Landau level(s) in the Quantum Hall Effect in graphene can be explained as perturbative orbital interaction of intra-and inter-valley microscopic orbital currents with a magnetic field. This effect is provided by the translational-non-invariant terms corresponding to graphene's crystallographic honeycomb symmetry but do not exist in the relativistic theory of massless Dirac Fermions in Quantum Electrodynamics. We discuss recent data in view of these results.

PRB 69, 075104 (04); V.P. Gusynin, S.G. Sharapov, PRB 71, 125124 (05), PRL 95, 146801 (05); cond-mat/0512157; V.P. Gusynin, S.G. Sharapov, J.P. Carbotte, cond-mat/0603267.

The effects of gauge interactions in graphene have been analyzed up to now in terms of effective models of Dirac fermions. However, in several cases lattice effects play an important role and need to be taken consistently into account. In this paper we introduce and analyze a lattice gauge theory model for graphene, which describes tight binding electrons hopping on the honeycomb lattice and interacting with a three-dimensional quantum U (1) gauge field. We perform an exact Renormalization Group analysis, which leads to a renormalized expansion that is finite at all orders. The flow of the effective parameters is controlled thanks to Ward Identities and a careful analysis of the discrete lattice symmetry properties of the model. We show that the Fermi velocity increases up to the speed of light and Lorentz invariance spontaneously emerges in the infrared. The interaction produces critical exponents in the response functions; this removes the degeneracy present in the non interacting case and allow us to identify the dominant excitations. Finally we add mass terms to the Hamiltonian and derive by a variational argument the correspondent gap equations, which have an anomalous non-BCS form, due to the non trivial effects of the interaction.

Electronic properties of materials are commonly described by quasiparticles that behave as nonrelativistic electrons with a finite mass and obey the Schrödinger equation. Here we report a condensed matter system where electron transport is essentially governed by the Dirac equation and charge carriers mimic relativistic particles with zero mass and an effective "speed of light" c * ≈10 6 m/s. Our studies of graphene -a single atomic layer of carbon -have revealed a variety of unusual phenomena characteristic of two-dimensional (2D) Dirac fermions. In particular, we have observed that a) the integer quantum Hall effect in graphene is anomalous in that it occurs at halfinteger filling factors; b) graphene's conductivity never falls below a minimum value corresponding to the conductance quantum e 2 /h, even when carrier concentrations tend to zero; c) the cyclotron mass m c of massless carriers with energy E in graphene is described by equation E =m c c * 2 ; and d) Shubnikov-de Haas oscillations in graphene exhibit a phase shift of π due to Berry's phase.

Physical Review B, 2008

Motivated by recent experiments and a theoretical analysis of the gap equation for the propagator of Dirac quasiparticles, we assume that the physics underlying the recently observed removal of sublattice and spin degeneracies in graphene in a strong magnetic field is connected with the generation of both Dirac masses and spin gaps. The consequences of such a scenario for the existence of the gapless edge states with zigzag and armchair boundary conditions are discussed. In the case of graphene on a half-plane with a zigzag edge, there are gapless edge states in the spectrum only when the spin gap dominates over the mass gap. In the case of an armchair edge, however, the existence of the gapless edge states depends on the specific type of mass gaps.

Physical Review Letters, 2008

We show that the recently discovered double-valley splitting of the low-lying Landau level(s) in the Quantum Hall Effect in graphene can be explained as perturbative orbital interaction of intra-and inter-valley microscopic orbital currents with a magnetic field. This effect is provided by the translational-non-invariant terms corresponding to graphene's crystallographic honeycomb symmetry but do not exist in the relativistic theory of massless Dirac Fermions in Quantum Electrodynamics. We discuss recent data in view of these results.

Journal of Physics A: Mathematical and Theoretical, 2007

We study the Euclidean effective action per unit area and the charge density for a Dirac field in a two-dimensional spatial region, in the presence of a uniform magnetic field perpendicular to the 2D-plane, at finite temperature and density. In the limit of zero temperature we reproduce, after performing an adequate Lorentz boost, the Hall conductivity measured for different kinds of graphene samples, depending upon the phase choice in the fermionic determinant.

Solid State Communications, 2009

There are two types of edge states in graphene with/without magnetic field. One is a quantum Hall edge state, which is topologically protected against small perturbation. The other is a chiral zero mode that is localized near the boundary with/without magnetic field. The latter is also topological but is guaranteed to be zero energy by the chiral symmetry, which is also responsible for massless Dirac like dispersion. Conceptual roles of the edge states are stressed and reviewed from a view point of the bulk-edge correspondence and the topological order.

Low Temperature Physics, 2008

We review recent results concerning the spectrum of edge states in the quantum Hall effect in graphene. In particular, a special attention is payed to the derivation of the conditions under which gapless edge states exist in the spectrum of graphene with zigzag and armchair edges. We find that in the case of a half-plane or a ribbon with a zigzag edges, there are gapless edge states only when a spin gap dominates over a Dirac mass gap. In the case of a half-plane with an armchair edge, the existence of the gapless edge states depends on the specific type of Dirac mass gaps. The implications of these results for the dynamics in the quantum Hall effect in graphene are discussed.