Kohn condition and exotic Newton-Hooke symmetry in the non-commutative Landau problem

TL;DR

This paper studies a system of non-commutative particles in a magnetic field, revealing conditions for center-of-mass behavior, symmetries related to the exotic Newton-Hooke group, and phenomena like cyclotronic motion and the Hall effect.

Contribution

It demonstrates how Kohn's condition combined with non-commutative parameters leads to a unified center-of-mass behavior and explores the associated exotic Newton-Hooke symmetry in the non-commutative Landau problem.

Findings

Center-of-mass acts as a single exotic particle with combined properties.

System exhibits modified cyclotronic motion under certain conditions.

At critical magnetic field, particles form a static crystal and follow the Hall law.

Abstract

$N$ "exotic" [alias non-commutative] particles with masses $m_a$, charges $e_a$ and non-commutative parameters $\theta_a$, moving in a uniform magnetic field $B$, separate into center-of-mass and internal motions if Kohn's condition $e_a/m_a=\const$ is supplemented with $e_a\theta_a=\const.$ Then the center-of-mass behaves as a single exotic particle carrying the total mass and charge of the system, $M$ and $e$, and a suitably defined non-commutative parameter $\Theta$. For vanishing electric field off the critical case $e\Theta B\neq1$, the particles perform the usual cyclotronic motion with modified but equal frequency. The system is symmetric under suitable time-dependent translations which span a (4+2)- parameter centrally extended subgroup of the "exotic" [i.e., two-parameter centrally extended] Newton-Hooke group. In the critical case $B=B_c=(e\Theta)^{-1}$ the system is frozen into a static "crystal" configuration. Adding a constant electric field, all particles perform, collectively, a cyclotronic motion combined with a drift perpendicular to the electric field when $e\Theta B\neq1$. For $B=B_c$ the cyclotronic motion is eliminated and all particles move, collectively, following the Hall law. Our time-dependent symmetries are reduced to the (2+1)-parameter Heisenberg group of centrally-extended translations.