Anisotropic harmonic oscillator, non-commutative Landau problem and exotic Newton-Hooke symmetry

TL;DR

This paper explores a non-commutative anisotropic harmonic oscillator in 2D, revealing its symmetry properties and phase behavior, and connecting it to the exotic Newton-Hooke symmetry and special cases like the Landau problem.

Contribution

It introduces a unified model with anisotropic and isotropic limits, analyzing its symmetry structure and phase transitions, extending understanding of non-commutative quantum systems.

Findings

The system exhibits three distinct phases based on central charges.

Special cases show additional Lie symmetries, so(3) or so(2,1).

The model generalizes the exotic Newton-Hooke symmetry to anisotropic settings.

Abstract

We investigate the planar anisotropic harmonic oscillator with explicit rotational symmetry as a particle model with non-commutative coordinates. It includes the exotic Newton-Hooke particle and the non-commutative Landau problem as special, isotropic and maximally anisotropic, cases. The system is described by the same (2+1)-dimensional exotic Newton-Hooke symmetry as in the isotropic case, and develops three different phases depending on the values of the two central charges. The special cases of the exotic Newton-Hooke particle and non-commutative Landau problem are shown to be characterized by additional, so(3) or so(2,1) Lie symmetry, which reflects their peculiar spectral properties.