Modular Invariants in the Fractional Quantum Hall Effect

TL;DR

This paper explores the modular properties of characters in nonabelian fractional quantum Hall states, verifying invariance in some cases and revealing broken invariance in others, thereby linking modular behavior to topological order.

Contribution

It provides a detailed analysis of the modular invariance of extended characters in nonabelian FQH states, including explicit verification for several key states and identifying cases where invariance breaks down.

Findings

Modular invariance verified for spinon-holon, Pfaffian, and 331 states.

Extended characters do not form a modular representation for Haldane-Rezayi state.

The study links modular properties to the topological order in nonabelian FQH states.

Abstract

We investigate the modular properties of the characters which appear in the partition functions of nonabelian fractional quantum Hall states. We first give the annulus partition function for nonabelian FQH states formed by spinon and holon (spinon-holon state). The degrees of freedom of spin are described by the affine SU(2) Kac-Moody algebra at level $k$. The partition function and the Hilbert space of the edge excitations decomposed differently according to whether $k$ is even or odd. We then investigate the full modular properties of the extended characters for nonabelian fractional quantum Hall states. We explicitly verify the modular invariance of the annulus grand partition functions for spinon-holon states, the Pfaffian state and the 331 states. This enables one to extend the relation between the modular behavior and the topological order to nonabelian cases. For the Haldane-Rezayi state, we find that the extended characters do not form a representation of the modular group, thus the modular invariance is broken.