Quantum Hall bilayers and the chiral sine-Gordon equation

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 ( β ̂ ). We obtain exact solutions for the spectra and correlation functions of the chiral sine-Gordon theory at the two values of β ̂ at which electron tunneling in bilayers is not irrelevant. Of these, the marginal case ( β ̂ 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 ( β ̂ 2∈ Z + ) by the introduction of auxiliary degrees of freedom that could prove useful in other problems involving quantum Hall multi-layers.