# Local incompressibility estimates for the Laughlin phase

## TL;DR

This paper establishes sharp density upper bounds for ground states of 2D Coulomb systems and related states, using a novel Thomas-Fermi-like approach, with implications for fractional quantum Hall systems' response to perturbations.

## Contribution

Introduces a new method based on an auxiliary Thomas-Fermi-like model to derive density bounds for Coulomb systems and Laughlin state perturbations.

## Key findings

- Density upper bounds for classical Coulomb ground states.
- Density bounds for low-temperature Gibbs states.
- Perturbations of Laughlin states do not increase particle density.

## Abstract

We prove sharp density upper bounds on optimal length-scales for the ground states of classical 2D Coulomb systems and generalizations thereof. Our method is new, based on an auxiliary Thomas-Fermi-like variational model. Moreover, we deduce density upper bounds for the related low-temperature Gibbs states. Our motivation comes from fractional quantum Hall physics, more precisely, the perturbation of the Laughlin state by external potentials or impurities. These give rise to a class of many-body wave-functions that have the form of a product of the Laughlin state and an analytic function of many variables. This class is related via Laughlin's plasma analogy to Gibbs states of the generalized classical Coulomb systems we consider. Our main result shows that the perturbation of the Laughlin state cannot increase the particle density anywhere, with implications for the response of FQHE systems to external perturbations.