Collective Field Theory for Quantum Hall States

TL;DR

This paper develops a collective field theory for fractional quantum Hall states, revealing connections to Gaussian free fields, gravitational anomalies, and Liouville quantum gravity, simplifying correlation function calculations.

Contribution

It introduces a Gaussian free field framework with background charge for FQH states, incorporating gradient corrections linked to gravitational anomalies and Liouville theory.

Findings

Captures Laughlin state properties with Gaussian free fields

Identifies gradient corrections as gravitational anomalies

Simplifies correlation function computations in FQH states

Abstract

We develop a collective field theory for fractional quantum Hall (FQH) states. We show that in the leading approximation for a large number of particles, the properties of Laughlin states are captured by a Gaussian free field theory with a (filling fraction dependent) background charge. Gradient corrections to the Gaussian field theory arise from ultraviolet regularization. They are the origin of the gravitational anomaly and are described by the Liouville theory of quantum gravity. The field theory simplifies the computation of correlation functions in FQH states and makes manifest the effect of quantum anomalies.