Modular Invariant Partition Functions in the Quantum Hall Effect

TL;DR

This paper constructs modular invariant partition functions for edge theories in the quantum Hall effect, linking topological order, symmetry algebras, and experimental Hall conductivities, especially beyond Jain series.

Contribution

It introduces new non-diagonal modular invariants for edge theories with extended symmetry, connecting topological order to modular transformations and experimental data.

Findings

Derived modular invariant partition functions for U(1)xSU(m) models

Linked Wen topological order to modular and fusion algebra

Identified invariants matching Hall conductivities beyond Jain series

Abstract

We study the partition function for the low-energy edge excitations of the incompressible electron fluid. On an annular geometry, these excitations have opposite chiralities on the two edges; thus, the partition function takes the standard form of rational conformal field theories. In particular, it is invariant under modular transformations of the toroidal geometry made by the angular variable and the compact Euclidean time. The Jain series of plateaus have been described by two types of edge theories: the minimal models of the W-infinity algebra of quantum area-preserving diffeomorphisms, and their non-minimal version, the theories with U(1)xSU(m) affine algebra. We find modular invariant partition functions for the latter models. Moreover, we relate the Wen topological order to the modular transformations and the Verlinde fusion algebra. We find new, non-diagonal modular invariants which describe edge theories with extended symmetry algebra; their Hall conductivities match the experimental values beyond the Jain series.