Area Preserving Transformations in Non-commutative Space and NCCS Theory

We propose an heuristic rule for the area transformation on the non-commutative plane. The non-commutative area preserving transformations are quantum deformation of the classical symplectic diffeomorphisms. Area preservation condition is formulated as a field equation in the non-commutative Chern-Simons gauge theory. The higher dimensional generalization is suggested and the corresponding algebraic structure - the infinite dimensional $\sin$-Lie algebra is extracted. As an illustrative example the second-quantized formulation for electrons in the lowest Landau level is considered.Comment: revtex, 9 pages, corrected typo

Magnetic Instability in a Parity Invariant 2D Fermion System

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.Comment: 7 pages, RevTe

Geometric Transformations and NCCS Theory in the Lowest Landau Level

Chern-Simons type gauge field is generated by the means of the singular area preserving transformations in the lowest Landau level of electrons forming fractional quantum Hall state. Dynamics is governed by the system of constraints which correspond to the Gauss law in the non-commutative Chern-Simons gauge theory and to the lowest Landau level condition in the picture of composite fermions. Physically reasonable solution to this constraints corresponds to the Laughlin state. It is argued that the model leads to the non-commutative Chern-Simons theory of the QHE and composite fermions.Comment: Latex, 13 page

Low Energy States in the SU(N) Skyrme Models

We show that any solution of the SU(2) Skyrme model can be used to give a topologically trivial solution of the SU(4) one. In addition, we extend the method introduced by Houghton et al. and use harmonic maps from S2 to CP(N-1) to construct low energy configurations of the SU(N) Skyrme models. We show that one of such maps gives an exact, topologically trivial, solution of the SU(3) model. We study various properties of these maps and show that, in general, their energies are only marginally higher than the energies of the corresponding SU(2) embeddings. Moreover, we show that the baryon (and energy) densities of the SU(3) configurations with baryon number B=2-4 are more symmetrical than their SU(2) analogues. We also present the baryon densities for the B=5 and B=6 configurations and discuss their symmetries.Comment: latex : 25 pages, 9 Postscript figures, uses eps

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

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.Comment: 11 pages, 4 figure