Fractional Quantum Hall States and Jack Polynomials

TL;DR

This paper presents a polynomial wavefunction framework using Jack symmetric polynomials to model fractional quantum Hall states, capturing their occupation configurations and clustering properties.

Contribution

It introduces a novel occupation-number-like picture for FQH states using Jack polynomials, unifying various known states with a squeezing rule and clustering conditions.

Findings

Jack polynomials model bosonic FQH states effectively

The squeezing rule constrains allowed occupation configurations

Clustering conditions describe quasiparticle states

Abstract

We describe an occupation-number-like picture of Fractional Quantum Hall (FQH) states in terms of polynomial wavefunctions characterized by a dominant occupation-number configuration. The bosonic variants of single-component abelian and non-abelian FQH states are modeled by Jacks (Jack symmetric polynomials), characterized by dominant occupation-number configurations satisfying a generalized Pauli principle. In a series of well-known Quantum Hall states, including the Laughlin, Read-Moore, and Read-Rezayi, the Jack polynomials naturally implement a ``squeezing rule'' that constrains allowed configurations to be restricted to those obtained by squeezing the dominant configuration. The Jacks describing uniform FQH states satisfy a highest-weight condition, and a clustering condition which can be generalized to describe quasiparticle states.