Non-Abelian Quantum Hall States and their Quasiparticles: from the Pattern of Zeros to Vertex Algebra

TL;DR

This paper enhances the pattern-of-zeros framework for quantum Hall states by introducing an expanded data set that uniquely characterizes states and reveals their topological properties, including quasiparticle statistics.

Contribution

It introduces an extended set of data {n;m;S_a;c} that fully characterizes quantum Hall states, combining pattern of zeros with vertex algebra methods.

Findings

Expanded data set {n;m;S_a;c} uniquely characterizes quantum Hall states.

The approach determines topological properties like central charge and quasiparticle statistics.

Provides conditions for the validity of wave functions based on pattern of zeros.

Abstract

In the pattern-of-zeros approach to quantum Hall states, a set of data {n;m;S_a|a=1,...,n; n,m,S_a in N} (called the pattern of zeros) is introduced to characterize a quantum Hall wave function. In this paper we find sufficient conditions on the pattern of zeros so that the data correspond to a valid wave function. Some times, a set of data {n;m;S_a} corresponds to a unique quantum Hall state, while other times, a set of data corresponds to several different quantum Hall states. So in the latter cases, the patterns of zeros alone does not completely characterize the quantum Hall states. In this paper, We find that the following expanded set of data {n;m;S_a;c|a=1,...,n; n,m,S_a in N; c in R} provides a more complete characterization of quantum Hall states. Each expanded set of data completely characterize a unique quantum Hall state, at least for the examples discussed in this paper. The result is obtained by combining the pattern of zeros and Z_n simple-current vertex algebra which describes a large class of Abelian and non-Abelian quantum Hall states \Phi_{Z_n}^sc. The more complete characterization in terms of {n;m;S_a;c} allows us to obtain more topological properties of those states, which include the central charge c of edge states, the scaling dimensions and the statistics of quasiparticle excitations.