Current Oscillations, Interacting Hall Discs and Boundary CFTs

TL;DR

This paper explores the behavior of conformal field theories at a single interaction point, revealing current oscillations in quantum Hall edge states and linking boundary interactions to Chern-Simons gauge theories.

Contribution

It introduces a twisted current algebra representation for interacting quantum Hall edge states and analyzes the resulting current oscillations and boundary conditions.

Findings

Current oscillations occur between two touching quantum Hall discs.

The oscillation frequency is determined by an effective interaction parameter.

Boundary conditions on fields are derived from chiral conformal field theory principles.

Abstract

In this paper, we discuss the behavior of conformal field theories interacting at a single point. The edge states of the quantum Hall effect (QHE) system give rise to a particular representation of a chiral Kac-Moody current algebra. We show that in the case of QHE systems interacting at one point we obtain a ``twisted'' representation of the current algebra. The condition for stationarity of currents is the same as the classical Kirchoff's law applied to the currents at the interaction point. We find that in the case of two discs touching at one point, since the currents are chiral, they are not stationary and one obtains current oscillations between the two discs. We determine the frequency of these oscillations in terms of an effective parameter characterizing the interaction. The chiral conformal field theories can be represented in terms of bosonic Lagrangians with a boundary interaction. We discuss how these one point interactions can be represented as boundary conditions on fields, and how the requirement of chirality leads to restrictions on the interactions described by these Lagrangians. By gauging these models we find that the theory is naturally coupled to a Chern-Simons gauge theory in 2+1 dimensions, and this coupling is completely determined by the requirement of anomaly cancellation.