Topological Solitons in Noncommutative Plane and Quantum Hall Skyrmions

TL;DR

This paper investigates topological solitons, specifically skyrmions, in a noncommutative quantum Hall system, revealing their charge properties and constructing explicit soliton states with energy calculations under specific interactions.

Contribution

It introduces a detailed analysis of noncommutative solitons in quantum Hall systems, including explicit solutions and energy minimization methods.

Findings

A topological soliton induces a localized electron density excitation.

A noncommutative soliton carries an electric charge proportional to its topological charge.

Explicit soliton solutions and energies are derived for specific interactions.

Abstract

We analyze topological solitons in the noncommutative plane by taking a concrete instance of the quantum Hall system with the SU(N) symmetry, where a soliton is identified with a skyrmion. It is shown that a topological soliton induces an excitation of the electron number density from the ground-state value around it. When a judicious choice of the topological charge density $J_{0}(\mathbf{x})$ is made, it acquires a physical reality as the electron density excitation $\Delta \rho ^{\text{cl}}(\mathbf{x})$ around a topological soliton, $\Delta \rho ^{\text{cl}}(\mathbf{x})=-J_{0}(% \mathbf{x})$. Hence a noncommutative soliton carries necessarily the electric charge proportional to its topological charge. A field-theoretical state is constructed for a soliton state irrespectively of the Hamiltonian. In general it involves an infinitely many parameters. They are fixed by minimizing its energy once the Hamiltonian is chosen. We study explicitly the cases where the system is governed by the hard-core interaction and by the noncommutative CP$^{N-1}$ model, where all these parameters are determined analytically and the soliton excitation energy is obtained.