$W_{1+\infty}$ Dynamics of Edge Excitations in the Quantum Hall Effect

TL;DR

This paper demonstrates that the $W_{1+ abla}$ symmetry governs the dynamics of edge excitations in quantum Hall systems, providing algebraic spectra and connecting to experimental observations.

Contribution

It introduces a $W_{1+ abla}$ symmetry-based framework to analyze edge excitations, including algebraic spectra and bosonization, extending to fractional fillings.

Findings

Algebraic spectra for Coulomb and short-range interactions

Logarithmic enhancement in Coulomb spectrum matches experiments

Extension of results to fractional quantum Hall states

Abstract

Quantum Hall universality classes can be classified by $W_{1+\infty}$ symmetry. We show that this symmetry also governs the dynamics of quantum edge excitations. The Hamiltonian of interacting electrons in the fully-filled first Landau level is expressed in terms of $W_{1+\infty}$ generators. The spectra for both the Coulomb and generic short-range interactions are thus found algebraically. We prove the one-dimensional bosonization of edge excitations in the limit of large number of particles. Moreover, the subleading corrections are given by the higher-spin $W_{1+\infty}$ generators, which measure the radial fluctuations of the electron density. The resulting spectrum for the Coulomb interaction contains a logarithmic enhancement, in agreement with experimental observations. The spectrum for generic short-range interactions is subleading and reproduces the classical capillary frequencies. These results are also extended to fractional filling by using symmetry arguments.