Peierls substitution and Hall motion in exotic Carroll dynamics

TL;DR

This paper explores the dynamics of particles with first-order motion in Carroll and Galilean systems, revealing non-commutative coordinates, anomalous Hall motions, and dualities with black hole horizon anyons, with implications for physical phenomena like the spin-Hall effect.

Contribution

It introduces a unified framework for Carroll and Galilean particles with non-commutative geometry and anomalous Hall motion, extending the understanding of Peierls substitution in exotic dynamics.

Findings

Particles exhibit non-commutative coordinates due to extension parameters.

Uncoupled anomalous Hall motions are observed in the systems.

The Carroll system is dual to a black hole horizon anyon showing spin-Hall effect.

Abstract

The particle with first-order dynamics proposed by Dunne, Jackiw and Trugenberger (DJT) to justify the ``Peierls substitution" is obtained by reduction from both of the planar two-parameter centrally extended Galilean and Carroll systems. In the latter case the extension parameters $\kappa_{exo}$ and $\kappa_{mag}$ generate non-commutativity of the coordinates resp. behave as an internal magnetic field. The position and momentum follow uncoupled anomalous Hall motions. Consistently with partial immobility, one of the Carroll boost generators is broken but the other remains a symmetry. Switching off $\kappa_{exo}$, the immobility of unextended Carroll particles is recovered. The Carroll system is dual to an uncharged anyon on the horizon of a black hole which exhibits the spin-Hall effect. Physical applications are shortly reviewed.