Symplectic Fluctuations for Electromagnetic Excitations of Hall Droplets

TL;DR

This paper develops a unified algebraic framework for integer quantum Hall systems, analyzing electromagnetic excitations via symplectic structure deformations and noncommutative gauge fields, with applications to quantum Hall droplets.

Contribution

It introduces a generalized Weyl--Heisenberg algebra approach and demonstrates the role of Moser's lemma and Seiberg--Witten map in modeling electromagnetic excitations in quantum Hall systems.

Findings

Unified algebraic scheme for quantum Hall systems.

Derivation of Hamiltonian for fermion droplets on phase space.

Emergence of noncommutative gauge fields in the model.

Abstract

We show that the integer quantum Hall effect systems in plane, sphere or disc, can be formulated in terms of an algebraic unified scheme. This can be achieved by making use of a generalized Weyl--Heisenberg algebra and investigating its basic features. We study the electromagnetic excitation and derive the Hamiltonian for droplets of fermions on a two-dimensional Bargmann space (phase space). This excitation is introduced through a deformation (perturbation) of the symplectic structure of the phase space. We show the major role of Moser's lemma in dressing procedure, which allows us to eliminate the fluctuations of the symplectic structure. We discuss the emergence of the Seiberg--Witten map and generation of an abelian noncommutative gauge field in the theory. As illustration of our model, we give the action describing the electromagnetic excitation of a quantum Hall droplet in two-dimensional manifold.