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

TL;DR

The paper develops a unified framework to classify non-Abelian quantum Hall states by extending the pattern-of-zeros description with a real central charge c, linking the data to Z_n simple-current vertex algebras. It derives sufficiency conditions for pattern data via generalized Jacobi identities and exploits associativity to compute central charges, quasiparticle scaling dimensions, and fusion, providing explicit examples including Z_n parafermion states, Z_n|Z_n and Gaffnian-like constructions. The expanded data {n;m;S_a;c} completely characterizes the described states in the examined cases, enabling a direct route from pattern data to edge properties and bulk quasiparticle content, while also revealing that some patterns correspond to gapless or non-unique theories. Overall, the work offers a constructive CFT-based methodology to determine topological order and quasiparticle statistics from a compact set of algebraic data, with clear implications for identifying and understanding non-Abelian FQH states.

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 Φ_{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.