U(1)xSU(m)_1 Theory and c=m W_{1+\infty} Minimal Models in the Hierarchical Quantum Hall Effect

TL;DR

This paper compares two conformal field theories used to model the Hierarchical Quantum Hall Effect, analyzing their differences and showing how one flows into the other via a specific Hamiltonian.

Contribution

It provides a detailed relation between the U(1)xSU(m)_1 theory and the c=m W_{1+ ablafty} minimal models, including character decompositions and an interpolating Hamiltonian.

Findings

Decomposition of characters from U(1)xSU(m)_1 to W_{1+ ablafty} minimal models.

Identification of a Hamiltonian with RG flow from one model to the other.

Clarification of differences in degeneracy and statistics despite similar edge spectra.

Abstract

Two classes of Conformal Field Theories have been proposed to describe the Hierarchical Quantum Hall Effect:the multi-component bosonic theory, characterized by the symmetry U(1)xSU(m)_1 and the W_{1+\infty} minimal models with central charge c=m. In spite of having the same spectrum of edge excitations, they manifest differences in the degeneracy of the states and in the quantum statistics, which call for a more detailed comparison between them. Here, we describe their detailed relation for the general case, c=m and extend the methods previously published for c < 4. Specifically, we obtain the reduction in the number of degrees of freedom from the multi-component Abelian theory to the minimal models by decomposing the characters of the U(1)xSU(m)_1 representations into those of the c=m W_{1+\infty} minimal models. Furthermore, we find the Hamiltonian whose renormalization group flow interpolates between the two models, having the W_{1+\infty} minimal models as infra-red fixed point.