Collective Field Theory of the Fractional Quantum Hall Edge State and the Calogero-Sutherland Model

TL;DR

This paper uses collective field theory to analyze the one-particle density matrix of fractional quantum Hall edge states, revealing a continuous interpolation between chiral Luttinger liquid and Calogero-Sutherland model behaviors, and connecting quantum Hall droplets with one-dimensional models.

Contribution

It introduces a hydrodynamic collective field theory approach to interpolate between different edge state behaviors and links quantum Hall droplets with Calogero-Sutherland models.

Findings

Density matrix interpolates between chiral Luttinger liquid and Calogero-Sutherland behaviors.

Low-energy excitations described by a c=1 conformal field theory of a compact boson.

Reveals a relation between 2D quantum Hall droplets and 1D Calogero-Sutherland model.

Abstract

\noindent Using hydrodynamic collective field theory approach we show that one-particle density matrix of the $\nu=1/m$ fractional quantum Hall edge state interpolates between chiral Luttinger liquid behavior $\langle \psi^{\dagger}(r) \psi(0) \rangle \sim r^{-m} $ and Calogero-Sutherland model behavior $\langle \psi^{\dagger}(r) \psi(0) \rangle \sim r^{-(m+1/m)/2} $ as the droplet width is varied continuously. Low-energy excitations are described by $c=1$ conformal field theory of a compact boson of radius $\sqrt m$. The result suggests complementary relation between the two-dimensional quantum Hall droplet and the one-dimensional Calogero-Sutherland model.