Clustering properties, Jack polynomials and unitary conformal field theories

TL;DR

This paper explores the connection between Jack polynomials and unitary conformal field theories, demonstrating that unitary theories can produce wavefunctions with similar clustering properties to non-unitary cases, revealing new mathematical structures.

Contribution

It shows that unitary CFTs, related to Jack polynomials, can generate trial wavefunctions with clustering properties akin to non-unitary theories, expanding understanding of fractional quantum Hall states.

Findings

Unitary CFTs produce wavefunctions with clustering properties similar to non-unitary theories.

Mathematical properties of Jack polynomials are relevant even when wavefunctions are not expressible as a single Jack polynomial.

Unitary solutions provide new trial wavefunctions for non-Abelian fractional quantum Hall systems.

Abstract

Recently, Jack polynomials have been proposed as natural generalizations of Z_k Read-Rezayi states describing non-Abelian fractional quantum Hall systems. These polynomials are conjectured to be related to correlation functions of a class of W-conformal field theories based on the Lie algebra A_{k-1}. These theories can be considered as non-unitary solutions of a more general series of CFTs with Z_k symmetry, the parafermionic theories. Starting from the observation that some parafermionic theories admit unitary solutions as well, we show, by computing the corresponding correlation functions, that these theories provide trial wavefunctions which satisfy the same clustering properties as the non-unitary ones. We show explicitly that, although the wavefunctions constructed by unitary CFTs cannot be expressed as a single Jack polynomial, they still show a fine structure where the mathematical properties of the Jack polynomials play a major role.