Peierls substitution and Hall motion in exotic Carroll dynamics

TL;DR

This work shows that the Dunne-Jackiw-Trugenberger (DJT) first-order system, used to justify the Peierls substitution, can be derived by Hamiltonian reduction from both exotic planar Galilean and exotic Carroll models. In the Carroll case, the two-parameter exotic extension introduces noncommutative coordinates and an internal magnetic field, yielding anomalous Hall motion that persists even without an external field, while turning off the exotic part recovers immobility. The analysis leverages Souriau’s two-form framework and includes a singular-mass reduction that produces a Hall-guiding center, as well as a chiral decomposition that clarifies how two Hall-like sectors interact. The study further connects these Carrollian dynamics to black hole horizon physics, holographic dualities, and fracton-like restricted mobility, highlighting the role of half-Carroll symmetry in the underlying structure and potential physical applications.

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 $κ_{exo}$ and $κ_{mag}$ 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 $κ_{exo}$, 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.

Figures (2)

• Figure 1: (i) Exotic Galilean motions with parameters $e=1,\, B = 1,\, \bm{E}=(0,1),\, m = 1,\, c_1=c_2=c_3=0,\, c_4=3$. The blue/ purple curves are below ($\theta=0.5$) / above ($\theta=2$) the critical value $\theta=1$. The orange curve is for large $|\theta|$ ($\theta=-2$). (ii) When $\theta\to 1$ either from the left or from the right, the frequency diverges and changes sign.
• Figure 2: (i) Hall motions for various values of the non-commutativity parameter $\theta\neq 1$, after fixing the other parameters as $e=1,\, B^* = 1,\, {\mathbf{E}}=(0,1)$ so that the motion remains perpendicular to the electric field ${\mathbf{E}}$. For $\theta=0$ we recover the unextended Carroll situation with no motion. (ii) When $\theta\to 1$ the velocity diverges. For very large $\theta$ instead, the residual velocity tends to the Hall value $\dot{x}_{1}=eE/B^*$ (to be compared with the Galilean case in FIG.\ref{['Exotic-G']}).

Theorems & Definitions (15)

• Proposition 2.1
• Proposition 3.1
• Proposition 3.2
• Proposition 3.3
• Proposition 4.1
• Proposition 4.2
• Proposition 4.3
• Proposition 4.4
• Proposition 4.5
• Proposition 4.6