Narrow Results

The second virial coefficient for non-interacting particles moving in a two-dimensional noncommutative space and in the presence of a uniform magnetic field B is presented. The noncommutativity parameter θ can be chosen such that the can be interpreted as the second virial coefficient for anyons of statistics α in the presence of B and living on the commuting plane. In particular in the high temperature limit β → 0, we establish a relation between the parameter θ and the statistics α. Moreover, can also be interpreted in terms of composite fermions.

We determine some particular values of the noncommutativity parameter θ and show that the Murthy–Shankar approach is in fact a particular case of a more general one. Indeed, using the fractional quantum Hall effect (FQHE) experimental data, we give a measurement of θ. This measurement can be obtained by considering some values of the filling factor ν and other ingredients, magnetic field B and electron density ρ. Moreover, it is found that θ can be quantized either fractionally or integrally in terms of the magnetic length l₀ and the quantization is exactly what Murthy and Shankar formulated recently for the FQHE. On the other hand, we show that the mapping of the FQHE in terms of the composite fermion basis has a noncommutative geometry nature and therefore there is a more general way than the Murthy–Shankar method to do this mapping.

At the even-denominator filling factor ν=P/Q, even Q, a fermionic Quantum Hall (QH) state is shown to be developed at the lowest temperatures. In this state, formed by the composite (c-) fermions, each with an electron and the even number Q of flux quanta (fluxons), the conductivity σ≡J/E(J=current density, E=applied field) becomes a universal constant (e²/h)Q^-2 as the temperature T approaches zero while the Hall conductivity σ_H≡J/E_H(E_H=Hall field) becomes approximately equal to (e²/h)(P/Q).

We establish that the response to static perturbations of two-dimensional electron systems (2DESs), exposed to magnetic field B, in states with the Landau-level filling fractions ν_e close to one such as 1/2, is singularly different from that of 2DESs at small ‖B‖. Thus in addition to demonstrating the severe inadequacy of the treatment of composite fermions as non-interacting particles, we show, in contradiction to a recent theoretical finding, that the physical origin of the observed behavior of the dc longitudinal magneto-resistivity for ν_e≈1/2 in periodically modulated 2DESs remains to be determined.

By strictly adhering to the microscopic theory of composite fermions (CFs) for the Landau-level filling fractions ν_e=p/(2p+1), we reproduce, with remarkable accuracy, the surface-acoustic-wave (SAW)-based experimental results by Willett and co-workers concerning two-dimensional electron systems with ν_e close to ½. Our results imply that the electron band mass m_b, as distinct from the CF mass m_⋆, must undergo a substantial increase under the conditions corresponding to ν_e ≈ ½. In view of the relatively low aerial electronic densities n_e to which the underlying SAW experiments correspond, our finding conforms with the experimental results by Shashkin et al. [Phys. Rev. B 66, 073303 (2002)], concerning two-dimensional electrons in silicon, that signal sharp increase in m_b for n_e decreasing below approximately 2×10¹¹cm^-2. We further establish that a finite mean-free path ℓ₀ is essential for the observed linearity of the longitudinal conductivity σ_xx(q) as deduced from the SAW velocity shifts.

Collective spin excitations in quantum Hall systems are studied via inelastic light scattering. In the fractional quantum Hall effect regime, composite fermion spin excitations are observed in the range 1/3< ν <2/5. They reveal a transition from free to strongly interacting composite fermions. At ν=1, a shift of the spin-wave energy at finite wavevector from the bare Zeeman energy is observed. It allows us to evaluate the spin-stiffness of the quantum Hall ferromagnet.

We investigate the structure of gapless edge modes propagating at the boundary of some fractional quantum Hall states. We show how to deduce explicit trial wavefunctions from the knowledge of the effective theory governing the edge modes. In general, quantum Hall states have many edge states. Here, we discuss the case of fractions having only two such modes. The case of spin-polarized and spin-singlet states at filling fraction ν = 2/5 is considered. We give an explicit description of the decoupled charged and neutral modes. Then we discuss the situation involving negative flux acting on the composite fermions. This happens notably for the filling factor ν = 2/3 which supports two counterpropagating modes. Microscopic wavefunctions for spin-polarized and spin-singlet states at this filling factor are given. Finally, we present an analysis of the edge structure of a non-Abelian state involving also negative flux. Counterpropagating modes involve, in all cases, explicit derivative operators diminishing the angular momentum of the system.

The topological explanation of the origin of Laughlin correlations in 2D charged systems under strong magnetic fields is formulated. Formal, self-consistent mathematical model of originally identified cyclotron braid subgroups is given in order to fully describe fundamentals of fractional quantum Hall effect, retrieve Laughlin correlations and point physical conditions which stand behind mysterious composite fermion structure. The new complete implementation of composite fermion basing on the first principles, without involving any artificial constructions (with flux-tubes or vortices) supply an explanation of previous models of composite fermions. Presented approach can lead to some corrections of numerical results in energy minimizations made within the traditional formulation of composite fermion model. Authors also identify the relations of FQHE in cyclotron braid terms within newly developing area of topological insulators and optical lattices. The prerequisites needed for formation of the fractional state are identified beyond the traditionally assumed factors, like the flat band condition and the interaction presence. The role of high mobility of carriers is highlighted in agreement with the experimental observations. Description, in terms of cyclotron braid subgroups, of the nature of yet unexplained novel experiments in Hall 2D systems including graphene is provided as well.

Atomic vapors can be prepared and manipulated at very low densities and temperatures. When they are rotating, they can reach a quantum Hall regime in which there should be manifestations of the fractional quantum Hall effect. We discuss the appearance of the principal sequence of fractions ν=p/(p±1) for bosonic atoms. The termination point of this series is the paired Moore–Read Pfaffian state. Exotic states fill the gap between the paired state and the vortex lattice expected at high filling of the lowest Landau level. In fermionic vapors, the p-wave scattering typical of ultralow energy collisions leads to the hard-core model when restricted to the lowest Landau level.

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–Schrödinger and Dirac Hamiltonian's 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.