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

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 $e_aθ_a=\const.$ 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 $e\Theta B\ne 1$, 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 $e\Theta B\ne 1$. 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.

12 pages, no figures. A minor error and some typos corrected, one reference added

References in corpus (4)

• (2+1)D Exotic Newton-Hooke Symmetry, Duality and Projective Phase
• Anisotropic harmonic oscillator, non-commutative Landau problem and exotic Newton-Hooke symmetry
• Kohn's Theorem, Larmor's Equivalence Principle and the Newton-Hooke Group
• Kohn's theorem and Galilean symmetry

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• Nonlinear realizations and the orbit method
• Group Theoretical Construction of Planar Noncommutative Systems
• Chiral Spin Noncommutative Space and Anomalous Dipole Moments
• Noncommutative effects on the fluid dynamics and modifications of the Freidmann equation