Abstract:
The tight-binding model of quantum particles on a honeycomb lattice is investigated in the presence of homogeneous magnetic field. Provided the magnetic flux per unit hexagon is rational of the elementary flux, the one-particle Hamiltonian is expressed in terms of the generators of the quantum group $U_q(sl_2)$. Employing the functional representation of the quantum group $U_q(sl_2)$ the Harper equation is rewritten as a systems of two coupled functional equations in the complex plane. For the special values of quasi-momentum the entangled system admits solutions in terms of polynomials. The system is shown to exhibit certain symmetry allowing to resolve the entanglement, and basic single equation determining the eigenvalues and eigenstates (polynomials) is obtained. Equations specifying locations of the roots of polynomials in the complex plane are found. Employing numerical analysis the roots of polynomials corresponding to different eigenstates are solved out and the diagrams exhibiting the ordered structure of one-particle eigenstates are depicted.

Abstract:
The spin and layer (pseudospin) degrees of freedom are entangled coherently in the canted antiferromagnetic phase of the bilayer quantum Hall system at the filling factor $\nu =2$. There emerges a complex Goldstone mode describing such a combined degree of freedom. In the zero tunneling-interaction limit ($\Delta_{\text{SAS}}\rightarrow 0$), its phase field provokes a supercurrent carrying both spin and charge within each layer. The Hall resistance is predicted to become anomalous precisely as in the $\nu =1$ bilayer system in the counterflow and drag experiments. Furthermore, it is shown that the total current flowing in the bilayer system is a supercurrent carrying solely spins in the counterflow geometry. It is intriguing that all these phenomena occur only in imbalanced bilayer systems.

Abstract:
An interlayer phase coherence develops spontaneously in the bilayer quantum Hall system at the filling factor $\nu =1$. On the other hand, the spin and pseudospin degrees of freedom are entangled coherently in the canted antiferromagnetic phase of the bilayer quantum Hall system at the filling factor $\nu =2$. There emerges a complex Nambu-Goldstone mode with a linear dispersion in the zero tunneling-interaction limit for both cases. Then its phase field provokes a Josephson supercurrent in each layer, which is dissipationless as in a superconductor. We study what kind of phase coherence the Nambu-Goldstone mode develops in association with the Josephson supercurrent and its effect on the Hall resistance in the bilayer quantum Hall system at $\nu=1,2$, by employing the Grassmannian formalism.

Abstract:
The exciton ground state in strained quantum dots similar to those fabricated in article specified in the title is shortly discussed within a relevant model Hamiltonian. Some characteristics of the light--hole exciton ground state reached in a dot under the tensile biaxial strain appear to be sensitive to the strain anisotropy breaking a purity of this state. It refers in particular to a degree of the in--plane polarization of the emission and the fine structure of the ground state.

Abstract:
The relativistic model with two types of planar fermions interacting with the Chern-Simons and Maxwell fields is applied to the study of anyon superconductor. It is demonstrated, that the Meissner effect can be realized in the case of the simultaneous presence of the fermions with a different magnetic moment interactions. Under the certain conditions there occures an extra plateau at the magnetization curve. In the order under consideration the spectrum of the electromagnetic field excitations contains the long-range interaction and one massive "photon" state.

Abstract:
Magnetic moment interaction is shown to play a defining role in the magnetic properties of anyon superconductors. The necessary condition for the existence of the Meissner effect is found.

Abstract:
We consider the influence of spin effects on the inter-Landau level electron-electron scattering rate in a two-dimensional electron gas. Due to the exchange spin splitting, the Landau levels are not equidistant. This leads to the suppresion of Auger processes and a nonlinear dependence of the lifetime on the concentration of the excited electrons even at very low excitation levels.

Abstract:
We study an impact of the fine structure of the heavy--hole ground state exciton confined in semiconductor quantum dots on the photoluminescence polarization dynamics solving the relevant system of the rate equations. The presence of the dark excitons is usually ignored and the polarization decay is assumed to be caused by direct transitions within the radiative doublet. We demonstrate that in strongly confined quantum dots the dark excitons, which are energetically well below the bright excitons, have actually a decisive effect on the polarization dynamics due to their persistent nature. The linear polarization shows nonexponential decay controlled by a conversion of the dark into a bright exciton. To get quantitative answers for specific quantum dot structures, all the necessary information can be obtained already from experiments on the luminescence dynamics following nonresonant excitation in these dots.

Abstract:
By considering the area preserving geometric transformations in the configuration space of electrons moving in the lowest Landau level (LLL) we arrive at the Chern-Simons type Lagrangian. Imposing the LLL condition, we get a scheme with the complex gauge fields and transformations. Quantum theory for the matter field in LLL is considered and formal expressions for Read's operator and Laughlin wave function are presented in the second quantized form.

Abstract:
We consider the parity invariant (2+1)-dimensional QED where the matter is represented as a mixture of fermions with opposite spins. It is argued that the perturbative ground state of the system is unstable with respect to the formation of magnetized ground state. Carrying out the finite temperature analysis we show that the magnetic instability disappears in the high temperature regime.