Maxwell-Chern-Simons Electrodynamics on a Disk

TL;DR

This paper investigates Maxwell-Chern-Simons electrodynamics on a disk, revealing how boundary conditions and a parameter λ influence edge states, degeneracies, and excitations, with implications for quantum Hall systems.

Contribution

It introduces the dependence of quantization and edge states on the parameter λ in Maxwell-Chern-Simons theory on a disk, highlighting new excitations and their potential observability.

Findings

Edge states exist for λ=0 and are described by an affine Lie group.

Degeneracy of states is infinite at λ=0, lifted for λ>0.

New excitations with continuous spectrum are predicted.

Abstract

The Maxwell-Chern-Simons (MCS) Lagrangian is the Maxwell Lagrangian augmented by the Chern-Simons (CS) term. In this paper, we study the MCS and Maxwell Lagrangians on a disk $D$. They are of interest for the quantum Hall effect, and also when the disk and its exterior are composed of different media. We show that quantization is not unique, but depends on a nonnegative parameter $\lambda$. $1/\lambda$ is the penetration depth of the fields into the medium in the exterior of $D$. For $\lambda=0$, there are edge observables and edge states localized at the boundary $\partial D$ for the MCS system. They describe the affine Lie group $\tilde{L}U(1)$. Their excitations carry zero energy signifying an infinite degeneracy of all states of the theory. There is also an additional infinity of single particle excitations of exactly the same energy proportional to $|k|$, $k$ being the strength of the CS term. The MCS theory for $\lambda=0$ has the huge symmetry group $\tilde{L}U(1) \times U(\infty)$. In the Maxwell theory, the last mentioned excitations are absent while the edge observables, which exist for $\lambda=0$, commute. Also these excitations are described by states which are not localized at $\partial D$ and are characterized by a continuous and infinitely degenerate spectrum. All these degeneracies are lifted and edge observables and their states cease to exist for $\lambda>0$. The novel excitations discovered in this paper should be accessible to observations. We will discuss issues related to observations, as