Quantum Hall states as matrix Chern-Simons theory

TL;DR

The paper introduces a finite matrix Chern-Simons model on the plane to describe fractional quantum Hall droplets of finite extent and demonstrates that the inverse filling fraction $1/\nu$ and the quasihole charge are quantized quantum mechanically, in agreement with Laughlin theory. It establishes an exact equivalence to the Calogero model, mapping matrix degrees of freedom to one-dimensional particle coordinates with a $1/(x_n-x_m)^2$ interaction; the ground state forms a circular droplet with radius $R^2\sim2N\theta$ and density $\rho_0\sim1/(2\pi\theta)$, while boundary and quasihole/quasiparticle excitations arise naturally from the finite-$N$ dynamics. Quasihole charge is quantized in units of $\nu$, and the quantum mechanics induces a shift of the Calogero coupling to $k(k+1)$ with $k=1/\nu$, ensuring consistent statistics and level quantization. The work also discusses potential phase transitions to a Wigner crystal and highlights the need to extend the dictionary to compute densities and correlations, pointing to future directions.

Abstract

We propose a finite Chern-Simons matrix model on the plane as an effective description of fractional quantum Hall fluids of finite extent. The quantization of the inverse filling fraction and of the quasiparticle number is shown to arise quantum mechanically and to agree with Laughlin theory. We also point out the effective equivalence of this model, and therefore of the quantum Hall system, with the Calogero model.