Quantum Hall Bilayers and the Chiral Sine-Gordon Equation

TL;DR

This paper analyzes the edge states of symmetric double-layer quantum Hall systems, revealing unique behaviors of the chiral sine-Gordon theory and providing exact solutions at specific interaction strengths relevant to experiments.

Contribution

It demonstrates the distinct renormalization group flow of the chiral sine-Gordon theory and provides exact solutions at key interaction values, including a fermionization method for multilayer systems.

Findings

Exact solutions for spectra and correlation functions at specific beta values.

The spectrum at beta^2=4 consists of two Majorana fermions with different velocities.

Implications for electron behavior in the experimentally observed 331 state.

Abstract

The edge state theory of a class of symmetric double-layer quantum Hall systems with interlayer electron tunneling reduces to the sum of a free field theory and a field theory of a chiral Bose field with a self-interaction of the sine-Gordon form. We argue that the perturbative renormalization group flow of this chiral sine-Gordon theory is distinct from the standard (non-chiral) sine-Gordon theory, contrary to a previous assertion by Renn, and that the theory is manifestly sensible only at a discrete set of values of the inverse period of the cosine interaction (beta). We obtain exact solutions for the spectra and correlation functions of the chiral sine-Gordon theory at the two values of beta at which the electron tunneling in bilayers is not irrelevant. Of these, the marginal case (beta^2=4) is of greatest interest: the spectrum of the interacting theory is that of two Majorana fermions with different, dynamically generated, velocities. For the experimentally observed bilayer 331 state at filling factor 1/2, this implies the trifurcation of electrons added to the edge. We also present a method for fermionizing the theory at the discrete points (integer beta^2) by the introduction of auxiliary degrees of freedom that could prove useful in other problems involving quantum Hall multilayers.