Ahmed Jellal

Quantum Hall effect on higher dimensional spaces

We analysis the quantum Hall effect exhibited by a system of particles moving in a higher-dimensional space. This can be done by considering particles on the Bergman ball Bρd of radius ρ in the presence of an external magnetic field B and investigate its basic features. Solving the corresponding Hamiltonian to get the energy levels as well as the eigenfunctions. This can be used to study quantum Hall effect of confined particles in the lowest Landau level where density of particles and two point functions are calculated. We take advantage of the symmetry group of the Hamiltonian on Bρd to make link to the Landau problem analysis on the complex projective spaces CPd. In the limit ρ→∞, our analysis coincides with that corresponding to particles on the flat geometry Cd. This task has been done for d=1,2 and finally for the generic case, i.e. d⩾3.

THERMODYNAMIC PROPERTIES OF A QUANTUM GROUP BOSON GAS GLp,q(2)

An approach is proposed enabling to effectively describe the behaviour of a bosonic system. The approach uses the quantum group GLp,q(2) formalism. In effect, considering a bosonic Hamiltonian in terms of the GLp,q(2) generators, it is shown that its thermodynamic properties are connected to deformation parameters p and q. For instance, the average number of particles and the pressure have been computed. If p is fixed to be the same value for q, our approach coincides perfectly with some results developed recently in this subject. The ordinary results, of the present system, can be found when we take the limit p = q = 1.

Anomalous quantum Hall effect on sphere

Abstract We study the anomalous quantum Hall effect exhibited by the relativistic particles living on two-sphere S 2 and submitted to a magnetic monopole. We start by establishing a direct connection between the Dirac and Landau operators through the Pauli–Schrodinger Hamiltonian H s SP . This will be helpful in the sense that the Dirac eigenvalues and eigenfunctions will be easily derived. In analyzing H s SP spectrum, we show that there is a composite fermion nature supported by the presence of two effective magnetic fields. For the lowest Landau level, we argue that the basic physics of graphene is similar to that of two-dimensional electron gas, which is in agreement with the planar limit. For the higher Landau levels, we propose a SU ( N ) wavefunction for different filling factors that captures all symmetries. Focusing on the graphene case, i.e. N = 4 , we give different configurations those allowed to recover some known results.

Electrostatic and magnetic fields in bilayer graphene

Abstract We compute the transmission probability through rectangular potential barriers and p–n junctions in the presence of a magnetic and electric fields in bilayer graphene taking into account contributions from the full four bands of the energy spectrum. For energy E higher than the interlayer coupling γ1 ( E > γ 1 ) two propagation modes are available for transport giving rise to four possible ways for transmission and reflection coefficients. However, when the energy is less than the height of the barrier the Dirac fermions exhibit transmission resonances and only one mode of propagation is available for transport. We study the effect of the interlayer electrostatic potential denoted by δ and variations of different barrier geometry parameters on the transmission probability.

Tunneling of massive dirac fermions in graphene through time-periodic potential

Abstract The energy spectrum of a graphene sheet subject to a single barrier potential having a time periodic oscillating height and subject to a magnetic field is analyzed. The corresponding transmission is studied as function of the incident energy and potential parameters. Quantum interference within the oscillating barrier has an important effect on quasiparticles tunneling. In particular the time-periodic electrostatic potential generates additional sidebands at energies ϵ + lħω (l = 0, ±1,...) in the transmission probability originating from the photon absorption or emission within the oscillating barrier. Due to numerical difficulties in truncating the resulting coupled channel equations we limited ourselves to low quantum channels, i.e. l = 0, ± 1.

FRACTIONAL QUANTUM HALL STATES IN GRAPHENE

We quantum mechanically analyze the fractional quantum Hall effect in graphene. This will be done by building the corresponding states in terms of a potential governing the interactions and discussing other issues. More precisely, we consider a system of particles in the presence of an external magnetic field and take into account of a specific interaction that captures the basic features of the Laughlin series . We show that how its Laughlin potential can be generalized to deal with the composite fermions in graphene. To give a concrete example, we consider the SU(N) wavefunctions and give a realization of the composite fermion filling factor. All these results will be obtained by generalizing the mapping between the Pauli–Schrodinger and Dirac Hamiltonians to the interacting particle case. Meantime by making use of a gauge transformation, we establish a relation between the free and interacting Dirac operators. This shows that the involved interaction can actually be generated from a singular gauge transformation.

Factorization of Dirac equation in two space dimensions

We present a systematic approach for the separation of variables for the two-dimensional (2D) Dirac equation in polar coordinates. The three vector potential, which couple to the Dirac spinor via minimal coupling, along with the scalar potential are chosen to have angular dependence which emanate the Dirac equation to complete separation of variables. Exact solutions are obtained for a class of solvable potentials along with their relativistic spinor wavefunctions. Particular attention is paid to the situation where the potentials are confined to a quantum dot region and are of scalar, vector and pseudo-scalar type. The study of a single charged impurity embedded in a 2D Dirac equation in the presence of a uniform magnetic field was treated as a particular case of our general study.

Periodic barrier structure in AA-stacked bilayer graphene

We study the charge carriers transport in an AA-stacked bilayer graphene modulated by a lateral one-dimensional multibarrier structure. We investigate the band structures of our system, that is made up of two shifted Dirac cones, for finite and zero gap. We use the boundary conditions to explicitly determine the transmission probability of each individual cone () for single, double and finite periodic barrier structure. We find that the Klein tunneling is only possible when the band structure is gapless and can occur at normal incidence as a result of the Dirac nature of the quasiparticles. We observe that the band structure of the barriers can have more than one Dirac points for finite periodic barrier. The resonance peaks appear in the transmission probability, which correspond to the positions of new cones index like associated with . Two conductance channels through different cones () are found where the total conductance has been studied and compared to the cases of single layer and AB-stacked bilayer graphene.

Factorization of the Dirac equation and a graphene quantum dot

We consider a quantum dot described using a cylindrically symmetrical 2D Dirac equation. The potentials representing the quantum dot are taken to be of different types of potential configuration, scalar, vector, and pseudo-scalar to enable us to enrich our study. Using various potential configurations, we found that in the presence of a mass term, an electrostatically confined quantum dot can accommodate true bound states, which is in agreement with our previous work. The differential cross section associated with one specific potential configuration has been computed and discussed as a function of the various potential parameters.

NONCOMMUTATIVE DESCRIPTION OF SPIN HALL EFFECT

We propose an approach based on a generalized quantum mechanics to deal with the basic features of the intrinsic spin Hall effect. This can be done by considering two decoupled harmonic oscillators on the noncommutative plane and evaluating the spin Hall conductivity. Focusing on the high frequency regime, we obtain a diagonalized Hamiltonian. After getting the corresponding spectrum, we show that there is a Hall conductivity without an external magnetic field, which is noncommutativity parameter θ-dependent. This allows us to make contact with the spin Hall effect and also give different interpretations. Fixing θ, one can recover three different approaches dealing with the phenomenon.