George Tsitsishvili

Edge States in 2D Lattices with Hopping Anisotropy and Chebyshev Polynomials

Analytic technique based on Chebyshev polynomials is developed for studying two-dimensional lattice ribbons with hopping anisotropy. In particular, the tight-binding models on square and triangle lattice ribbons are investigated with anisotropic nearest neighbouring hoppings. For special values of hopping parameters the square lattice becomes topologically equivalent to a honeycomb one either with zigzag or armchair edges. In those cases as well as for triangle lattices we perform the exact analytic diagonalization of tight-binding Hamiltonians in terms of Chebyshev polynomials. Deep inside the edge state subband the wave functions exhibit exponential spatial damping which turns into power-law damping at edge-bulk transition point. It is shown that strong hopping anisotropy crashes down edge states, and the corresponding critical conditions are found.

Spin supercurrent in the canted antiferromagnetic phase

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

The quantum group, Harper equation and structure of Bloch eigenstates on a honeycomb lattice

\ensuremath{\nu}=2

Boundary conditions and formation of pure spin currents in magnetic field

. A complex Goldstone mode emerges describing such a combined degree of freedom. In the zero tunneling-interaction limit (

Edge States of a Periodic Chain with Four-Band Energy Spectrum

{\ensuremath{\Delta}}_{\text{SAS}}\ensuremath{\rightarrow}0

Josephson inplane and tunneling currents in bilayer quantum Hall system

), 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

Nambu–Goldstone modes and the Josephson supercurrent in the bilayer quantum Hall system

\ensuremath{\nu}=1

Goldstone Modes in Bilayer Quantum Hall Systems at ν = 2

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.

Skyrmion and Bimeron Excitations in Imbalanced Bilayer Quantum Hall Systems

The tight-binding model of quantum particles on a honeycomb lattice is investigated in the presence of a homogeneous magnetic field. Provided the magnetic flux per unit hexagon is a rational of the elementary flux, the one-particle Hamiltonian is expressed in terms of the generators of the quantum group Uq(sl2). Employing the functional representation of the quantum group Uq(sl2), the Harper equation is rewritten as a system 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 a certain symmetry allowing us to resolve the entanglement, and a basic single equation determining the eigenvalues and eigenstates (polynomials) is obtained. Equations specifying the 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 and diagrams exhibiting the ordered structure of one-particle eigenstates are depicted.

Interlayer phase coherence and Josephson effects in bilayer quantum Hall systems

Abstract Schrodinger equation for an electron confined to a two-dimensional strip is considered in the presence of homogeneous orthogonal magnetic field. Since the system has edges, the eigenvalue problem is supplied by the boundary conditions (BC) aimed in preventing the leakage of matter away across the edges. In the case of spinless electrons the Dirichlet and Neumann BC are considered. The Dirichlet BC result in the existence of charge carrying edge states. For the Neumann BC each separate edge comprises two counterflow sub-currents which precisely cancel out each other provided the system is populated by electrons up to certain Fermi level. Cancelation of electric current is a good starting point for developing the spin-effects. In this scope we reconsider the problem for a spinning electron with Rashba coupling. The Neumann BC are replaced by Robin BC. Again, the two counterflow electric sub-currents cancel out each other for a separate edge, while the spin current survives thus modeling what is known as pure spin current – spin flow without charge flow.