Large N Limit in the Quantum Hall Effect

TL;DR

This paper investigates the large N limit of Laughlin states in the fractional quantum Hall effect, revealing their semiclassical nature and describing their behavior as classical fluid droplets with area-preserving symmetries.

Contribution

It analytically solves the large N limit of Laughlin states using saddle point approximation, connecting quantum Hall states to classical fluid dynamics and symmetries.

Findings

Large N limit leads to semiclassical regime of Laughlin states.

Laughlin states behave as classical fluid droplets with uniform density.

Dynamical W_infinity symmetry describes area-preserving deformations.

Abstract

The Laughlin states for $N$ interacting electrons at the plateaus of the fractional Hall effect are studied in the thermodynamic limit of large $N$. It is shown that this limit leads to the semiclassical regime for these states, thereby relating their stability to their semiclassical nature. The equivalent problem of two-dimensional plasmas is solved analytically, to leading order for $N\to\infty$, by the saddle point approximation - a two-dimensional extension of the method used in random matrix models of quantum gravity and gauge theories. To leading order, the Laughlin states describe classical droplets of fluids with uniform density and sharp boundaries, as expected from the Laughlin ``plasma analogy''. In this limit, the dynamical $W_\infty$-symmetry of the quantum Hall states expresses the kinematics of the area-preserving deformations of incompressible liquid droplets.