From Composite Fermions to Calogero-Sutherland Model: Edge of Fractional Quantum Hall Liquid and the Dimension Reduction

We derive a microscopic model describing the low-lying edge excitations in the fractional quantum Hall liquid with $\nu=\frac{\nu^*} {\tilde\phi\nu^*+1}$. For $\nu^*>0$, it is found that the composite fermion model reduces to an SU$(\nu^*)$ Calogero-Sutherland model in a dimension reduction, whereas it is not exact soluble for $\nu^*<0$. However, the ground states in both cases can be found and the low-lying excitations can be shown the chiral Luttinger liquid behaviors. On the other hand, we shows that the finite temperature behavior of $G-T$ curve will deviate from the prediction of the chiral Luttinger liquid. We also point out that the suppression of the `spin' degrees of freedom agrees with very recent experiments by Chang et al. The two-boson model of Lee and Wen is described microscopically.