Construction of Hall states by means of a model quantum matrix model

Eddy Ardonne

Ahmed Jellal, Michael Schreiber

We develop a matrix model to describe bilayered quantum Hall fluids for a series of filling factors. Considering two coupling layers, and starting from a corresponding action, we construct its vacuum configuration at ν = q i K −1 ij q j , where K ij is a 2 × 2 matrix and q i is a vector. Our model allows us to reproduce several well-known wavefunctions. We show that the wavefunction Ψ ( m , m , n ) constructed years ago by Yoshioka, MacDonald and Girvin for the fractional quantum Hall effect at filling factor and in particular Ψ (3,3,1) at filling can be obtained from our vacuum configuration. The unpolarized Halperin wavefunction and especially that for the fractional quantum Hall state at filling factor can also be recovered from our approach. Generalization to more than two layers is straightforward.

Yi-Xin Chen, M. D. Gould, Yao-Zhong Zhang

Based on Haldane's spherical geometrical formalism of the two-dimensional quantum Hall fluids, the relation between the noncommutative geometry of S2 and the two-dimensional quantum Hall fluids is exhibited. A finite matrix model on the two-sphere is explicitly constucted as an effective description of the fractional quantum Hall fluids of finite extent, and the complete sets of physical quantum states of this matrix model are determined. We also describe how the low-lying excitations in the model are constructed in terms of the quasi-particle and quasi-hole excitations. It is shown that there exists a Haldane hierarchical structure in the two-dimensional quantum Hall fluid states of the matrix model. These hierarchical fluid states are generated by the parent fluid state by condensing the quasi-particle and quasi-hole excitations level by level.

Johannes N. Kriel

Ahmed Jellal, Michael Schreiber

We develop a matrix model to describe bilayered quantum Hall fluids for a series of filling factors. Considering two coupling layers, and starting from a corresponding action, we construct its vacuum configuration at \\nu=q_iK_{ij}^{-1}q_j, where K_{ij} is a 2\\times 2 matrix and q_i is a vector. Our model allows us to reproduce several well-known wave functions. We show that the wave function \\Psi_{(m,m,n)} constructed years ago by Yoshioka, MacDonald and Girvin for the fractional quantum Hall effect at filling factor {2\\over m+n} and in particular \\Psi_{(3,3,1)} at filling {1\\over 2} can be obtained from our vacuum configuration. The unpolarized Halperin wave function and especially that for the fractional quantum Hall state at filling factor {2\\over 5} can also be recovered from our approach. Generalization to more than 2 layers is straightforward.

Ivan Dario Jimenez Rodriguez