Noncommutative Geometry, Extended W(infty) Algebra and Grassmannian Solitons in Multicomponent Quantum Hall Systems

TL;DR

This paper explores the role of noncommutative geometry in quantum Hall systems with multiple components, revealing how SU(N) coherence and Grassmannian solitons emerge from the algebraic structure of the system.

Contribution

It introduces a novel approach using Weyl ordering to analyze multicomponent quantum Hall systems, connecting noncommutative geometry with SU(N) algebra and Grassmannian sigma models.

Findings

SU(N)-extended W_infinity algebra governs the system dynamics.

Spontaneous SU(N) quantum coherence arises from exchange Coulomb interactions.

Effective description involves Grassmannian G_{N,k} sigma model and solitons.

Abstract

Noncommutative geometry governs the physics of quantum Hall (QH) effects. We introduce the Weyl ordering of the second quantized density operator to explore the dynamics of electrons in the lowest Landau level. We analyze QH systems made of $N$-component electrons at the integer filling factor $\nu=k\leq N$. The basic algebra is the SU(N)-extended W$_{\infty}$. A specific feature is that noncommutative geometry leads to a spontaneous development of SU(N) quantum coherence by generating the exchange Coulomb interaction. The effective Hamiltonian is the Grassmannian $G_{N,k}$ sigma model, and the dynamical field is the Grassmannian $G_{N,k}$ field, describing $k(N-k)$ complex Goldstone modes and one kind of topological solitons (Grassmannian solitons).