# Peierls substitution and Hall motion in exotic Carroll dynamics

H.-X Zeng, Q.-L Zhao, P.-M Zhang, Peter Horvathy

► To cite this version:

H.-X Zeng, Q.-L Zhao, P.-M Zhang, Peter Horvathy. Peierls substitution and Hall motion in exotic Carroll dynamics. Phys.Rev.D, 2025, 111 (4), pp.044025. 10.1103/PhysRevD.111.044025. hal-04816747

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Distributed under a Creative Commons Attribution 4.0 International License
Peierls substitution and Hall motion
in exotic Carroll dynamics

H-X. Zeng¹*, Q.-L. Zhao¹†, P.-M. Zhang¹‡, and P. A. Horvathy²§

¹ *School of Physics and Astronomy,*
*Sun Yat-sen University, Zhuhai, China*

² *Institut Denis-Poisson CNRS/UMR 7013 - Université de Tours - Université d'Orléans Parc de Grammont, 37200; Tours, France*

(Dated: May 15, 2025)

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 two-parameter centrally ex-
tended Galilean and Carroll systems. In the latter case the extension parameters κexo and κmag
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 im-
mobility, one of the Carroll boost generators is broken but the other remains a symmetry. Switching
off κexo, 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.

Phys. Rev. D **111** (2025) no.4, 044025 doi:10.1103/PhysRevD.111.044025 [arXiv:2411.14329 [hep-th]].

Key words: Peierls substitution; Dunne-Jackiw-Trugenberger system, anomalous Hall effect; (im)mobility of centrally extended Carroll particles; motion on the horizon of a black hole.

<footnote>* mailto:zenghx53@mail2.sysu.edu.cn</footnote>

<footnote>† mailto: zhaoqliang@mail2.sysu.edu.cn</footnote>

<footnote>‡ corresponding author. mailto:zhangpm5@mail.sysu.edu.cn</footnote>

<footnote>§ mailto:horvathy@univ-tours.fr</footnote>
# Contents

<table><tr><td>I. Introduction</td><td>2</td></tr><tr><td>II. The Dunne-Jackiw-Trugenberger (DJT) system</td><td>4</td></tr><tr><td>III. Exotic Galilean dynamics</td><td>5</td></tr><tr><td>IV. Exotic Carroll particle</td><td>9</td></tr><tr><td>A. The singular Carroll case</td><td>12</td></tr><tr><td>B. Chiral decomposition of exotic Carroll dynamics</td><td>13</td></tr><tr><td>V. Symmetry and immobility</td><td>15</td></tr><tr><td>A. Boost symmetry of DJT</td><td>16</td></tr><tr><td>B. Symmetries of exotic particles</td><td>18</td></tr><tr><td>VI. Motion on the Black Hole horizon</td><td>19</td></tr><tr><td>VII. Conclusion</td><td>21</td></tr><tr><td>Acknowledgments</td><td>22</td></tr><tr><td>References</td><td>22</td></tr><tr><td>A. The Noether theorem in Souriau's framework</td><td>28</td></tr></table>

# I. INTRODUCTION

The fundamental property which has long delayed interest in Carrollian physics is that particles with Carroll symmetry can not move [1, 2]. In this paper we show that a particle associated with the recently discovered “exotic” two-parameter central extension of the planar Carroll group [3–7] analogous to the one of the planar Galilei group in [8] does move, namely by following the Hall law. However we confirm that turning off the exotic extension a massive Carroll particle does indeed not move.

Carroll symmetry is highlighted by the curious action of Carroll (or C-) boosts which
leave the position fixed and act only on “Carroll time” we denote here by s [1, 2],

$$x \to x, \quad s \to s - b \cdot x, \qquad (I.1)$$

to be compared with usual Galilei boosts,

$$x \to x + bt, \quad t \to t. \qquad (I.2)$$

The sign difference between Galilei and Carroll boosts is conveniently explained in the Kaluza-Klein-type “Bargmann” framework [9–11]. The extended space time has coordinates $x, t, s$ where $t$ and $s$ are lightlike. Galilean physics is a projection onto non-relativistic spacetime with coordinates $(x, t)$, with $s$ corresponding to its central extension [12]. The arena of Carrollian physics is in turn the restriction of extended space to $t = \text{const.}$ and has thus coordinates $(x, s)$.

Carrollian physics has become lately a much discussed topic [7, 11, 13–18], justified by its applications to gravitational waves, dark matter and to black hole physics, see [19–28] and many further references. Our new results in this paper complete previous work [1, 13, 29, 30].

We start with recalling the Peierls substitution proposed many years ago in many-body physics [31–35] : for a charged particle in the plane subjected to perpendicular magnetic and electric fields and constrained to stay in the lowest Landau level (i) the components of quantum position operator should be non-commuting and (ii) its Hamiltonian should reduce to the mere potential and have no kinetic term,

$$[\hat{\xi}_m, \hat{\xi}_n] = \frac{i}{eB} \epsilon_{mn} \quad \text{and} \quad \hat{\mathcal{H}} = eV(\hat{\xi}). \qquad (I.3)$$

Half of a century later, Peierls’ idea was resurrected by Dunne, Jackiw and Trugenberger (DJT) [36, 37]. A remarkable feature of their first-order equations is that the motion obeys the Hall law, (II.2) below.

In [38] we re-derived the DJT system from the non-commutative mechanics associated with the exotic (double) extension of the planar Galilei group noticed by Lévy-Leblond [8] and further investigated in [38–47] by Hamiltonian reduction [48, 49]. Our new paper carries out a similar study in the hitherto unexplored exo-Carrollian case, recalling occasionally the Galilean one for the sake of comparison.

We proceed in three steps. After recalling the DJT system (sec.II) we show that it can in fact be derived from both exotic Galilei and exotic Carroll particles. Then focusing our
attention at the latter, we show that a charged exotic Carroll particle *moves* by following an anomalous Hall law (IV.7) [50, 51] both in the presence, but also in the absence of an external magnetic field, reminiscent of the anomalous Hall effect [52, 53]. Turning off the exotic extension we recover an usual Carroll particle with its celebrated immobility.

For simplicity, we limit our attention to constant electromagnetic fields $B$, $E$ in the plane. Since we work in flat euclidean space, vectors will have lower indices.

## II. THE DUNNE-JACKIW-TRUGENBERGER (DJT) SYSTEM

In the early nineties Dunne, Jackiw, and Trugenberger (DJT) [36, 37] reconsidered the Peierls Ansatz [35], arguing that for a charged particle in the plane, letting the mass go to zero turns off the kinetic term, leaving us with the first-order-in-time-derivative Lagrangian which will hence be called that of a DJT particle,

$$ \mathcal{L}_{DJT} = \frac{eB}{2} \epsilon_{ij} \xi_i \xi_j' - eV(\xi), \qquad (II.1) $$

where the prime means derivation w.r.t. Galilean time $t$, $\{\cdot\}' = d/dt$. $B = \text{const.} \neq 0$ and $E = \text{const.}$ will hence be assumed for simplicity. The electric charge drops out from the equations of motion as long as it does not vanish, providing us with:

**Proposition II.1.** The DJT particle moves by following the Hall law [38, 44],

$$ \xi_i' = \epsilon_{ij} \frac{E_j}{B}. \qquad (II.2) $$

Legendre transformation of (II.1) then yields the Poisson brackets and Hamiltonian,

$$ \{\xi_i, \xi_j\} = -\frac{1}{eB} \epsilon_{ij} \quad \text{and} \quad \mathcal{H} = eV(\xi), \qquad (II.3) $$

respectively. Here $B$ is viewed as a fixed external quantity. The Poisson bracket corresponds to the symplectic form [38, 44],

$$ \Omega_{DJT} = \frac{1}{2} eB \epsilon_{ij} d\xi_i \wedge d\xi_j. \qquad (II.4) $$

To sum up,

* The DJT Lagrangian (II.1) has no mass term and is of the first order in the velocity, implying first-order equations of motion, (II.2);
* The coordinates do not commute, (II.3), and the Hamiltonian is the potential with no kinetic term;

* The Hall law (II.2) implies partial immobility: the particle moves perpendicularly to the electric field, $\mathbf{E} = -\nabla V$. When the electric field vanishes, $\mathbf{E} = 0$, then the DJT particle does not move at all, $\xi = \xi_0 = \text{const.}$

to be compared with the usual circular motions of a charged massive particle in a magnetic field.

We note for later use that the motions can also be obtained by finding the kernel of the Souriau 2-form [7, 54],

$$ \sigma_{DJT} = d\mathcal{L}_{DJT} \wedge dt = \frac{1}{2} eB\epsilon_{ij}d\xi_i \wedge d\xi_j + eE_id\xi_i \wedge dt, \qquad (II.5) $$

see the Appendix A.

In the next section we show that the DJT system can be obtained by reduction from two different, centrally extended particle models. After recalling some relevant facts from the Galilean case for comparison, we carry out a similar but more detailed analysis for Carroll.

### III. EXOTIC GALILEAN DYNAMICS

As shown in [38, 44–47], the DJT system is obtained from “exotic” Galilean mechanics [39–43] associated with the two-parameter central extension of the planar Galilei group [8] by Hamiltonian reduction [48, 49]. One of the parameters present in any dimension is the mass, $m$ [12]. The other, “exotic” one we denote here by $\kappa_{exo}$ is specific for planar physics [8]. In terms of the non-commutativity parameter and the effective mass,

$$ \theta = \frac{\kappa_{exo}}{m^2} \quad \text{and} \quad m^* = m(1 - e\theta B), \qquad (III.1) $$

the motion in a planar electromagnetic field $\mathbf{E}, \mathbf{B}$ [assumed constant for simplicity] is described by the equations [38, 44–47, 55, 56],

$$ m^*x'_i = p_i - m\theta\epsilon_{ij}E_j, \quad p'_i = eB\epsilon_{ij}x'_j + eE_i. \qquad (III.2) $$

When $m^* \neq 0$ the system is regular; the eqns. (III.2) derive from the Hamiltonian and
symplectic form,

$$ \mathcal{H}_{exoG} = \frac{\mathbf{p}^2}{2m} + eV(\mathbf{x}), \qquad (III.3a) $$

$$ \Omega_{exo} = dp_i \wedge dx_i + \frac{\theta}{2} \epsilon_{ij} dp_i \wedge dp_j + \frac{eB}{2} \epsilon_{ij} dx_i \wedge dx_j, \qquad (III.3b) $$

by bracketing with the exotic Poisson brackets,

$$ \{x_i, x_j\} = \frac{\theta}{1 - e\theta B} \epsilon_{ij}, \quad \{x_i, p_j\} = \frac{1}{1 - e\theta B} \delta_{ij}, \quad \{p_i, p_j\} = \frac{eB}{1 - e\theta B} \epsilon_{ij} \qquad (III.4) $$

or alternatively, from the kernel of the exotic Souriau form $\sigma = \Omega - d\mathcal{H} \wedge dt$ cf. (A.4),

$$ \sigma_{exoG} = dp_i \wedge dx_i + \frac{\theta}{2} \epsilon_{ij} dp_i \wedge dp_j + \frac{eB}{2} \epsilon_{ij} dx_i \wedge dx_j + eE_i dx_i \wedge dt - \frac{p_i}{m} dp_i \wedge dt. \qquad (III.5) $$

Eqns (III.2) can be solved analytically [44, 47]:

**Proposition III.1.** When the effective mass does not vanish, $m^* \neq 0$, the motions combine rotation and Hall drift,

$$ x_1(t) = \frac{c_4}{eB} \cos \left( \frac{eB}{m^*} t \right) + \frac{c_3}{eB} \sin \left( \frac{eB}{m^*} t \right) + \frac{E_2}{B} t + c_2, \qquad (III.6) $$

$$ x_2(t) = \frac{c_3}{eB} \cos \left( \frac{eB}{m^*} t \right) - \frac{c_4}{eB} \sin \left( \frac{eB}{m^*} t \right) - \frac{E_1}{B} t + \frac{m E_2}{e B^2} + c_1, \qquad (III.7) $$

$$ p_1(t) = \cos \left( \frac{eB}{m^*} t \right) c_3 - \sin \left( \frac{eB}{m^*} t \right) c_4 + \frac{m E_2}{B}, \qquad (III.8) $$

$$ p_2(t) = -\cos \left( \frac{eB}{m^*} t \right) c_4 - \sin \left( \frac{eB}{m^*} t \right) c_3 - \frac{m E_1}{B}. \qquad (III.9) $$

The arbitrary constants $c_i$ fix the initial position, and the radii of the circles.

It was shown in earlier work [38, 44–46] that when the magnetic field takes the critical value

$$ B = B_{crit} = \frac{1}{e\theta}, \qquad (III.10) $$

or equivalently when the effective mass vanishes, $m^* = 0$, we have:

**Proposition III.2.** In the critical case (III.10) the momentum is determined by the field. Both the position coordinate $\mathbf{x}$ moves according to the Hall law.

$$ p_i = m \theta \epsilon_{ij} e E_j = \text{const.} \quad \text{and} \quad x'_i = \epsilon_{ij} \frac{E_j}{B_{crit}}. \qquad (III.11) $$
More insight is obtained by combining the position and the momentum into the guiding centre [47, 50, 51],

$$Q_i = x_i + \frac{1}{eB}\epsilon_{ij}p_j, \qquad (III.12)$$

for which Proposition III.1 allows us to deduce:

**Proposition III.3.** When $m^* \neq 0$, then the guiding centre moves, *independently* of the non-commutativity parameter $\theta$ and even outside the critical value (III.10), according to the Hall law,

$$Q_i = \epsilon_{ij} \frac{E_j}{B} t + C_i \qquad C_i = \text{const.} \qquad (III.13)$$

The trajectories shown in FIG. 1 are circles which drift along the guiding centre which itself follows the Hall law (III.13) consistently with FIG.s #3 and #4 of [47].

When the critical case (III.10) is approached, the rotation speeds up and its angular velocity $\omega^* = eB/m^*$ diverges. Crossing the critical value the direction of the rotation is reversed.

For $|\theta|$ very large instead the effective mass diverges, $m^* \to \infty$, the frequency tends to 0 and the motion reduces to the (displaced) guiding centre,

$$x_1(t) = \frac{E_2}{B}t + \text{const.} \qquad x_2(t) = -\frac{E_1}{B}t + \text{const.} \qquad (III.14)$$

shown by the orange curve in FIG.1.

Further progress is achieved by rewriting the Souriau form (III.5) in terms of $Q_i$,

$$\sigma_{\text{exoG}} = \sigma_{\text{QG}} + \sigma_{\text{pG}} = \qquad (III.15)$$

$$\left\{ \frac{1}{2} eB \epsilon_{ij} dQ_i \wedge dQ_j + eE_i dQ_i \wedge dt \right\} + \left\{ \frac{1}{2} \left( \theta - \frac{1}{eB} \right) \epsilon_{ij} dp_i \wedge dp_j + \left( \epsilon_{ij} \frac{E_j}{B} - \frac{p_i}{m} \right) dp_i \wedge dt \right\}$$

where the first describes the (Hall) dynamics of the guiding centre (III.13), and $\sigma_{pG}$ determines that of the momentum which however intervenes in real-space dynamics.

When the magnetic field takes the critical value in (III.10) or equivalently, when the effective mass vanishes, $m^* = 0$, then Hamiltonian reduction shows [44, 45] that the phase 4-dimensional space reduces to a 2-dimensional one upon which the guiding centre (III.12) is a good coordinate. The reduced Hamiltonian structure

$$\Omega_{\text{red}} = \frac{1}{2\theta} d\mathbf{Q} \times d\mathbf{Q} \quad \Leftrightarrow \quad \{Q_1, Q_2\} = -\theta = -\frac{1}{eB_{\text{crit}}}, \qquad \mathcal{H}_{\text{red}} = eV(\mathbf{Q}) \qquad (III.16)$$
<figure><img src="image_2.png" /><figcaption>FIG. 1: (i) Exotic Galilean motions with parameters $e = 1$, $B = 1$, $E = (0,1)$, $m = 1$, $c_1 = c_2 = c_3 = 0$, $c_4 = 3$. 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.</figcaption></figure>

or with the “Chern-Simons-type” Lagrangian of [36, 37] with no kinetic term,

$$ \mathcal{L}_{red} = \frac{1}{2\theta} \mathbf{Q} \times \mathbf{Q}' - V(\mathbf{Q}), \qquad (III.17) $$

Comparing with the DJT expressions (II.4) and (II.1), we recover the DJT system in section II with the guiding centre $\mathbf{Q}$ replacing $\xi$.

For later comparison, the reduced Souriau form is

$$ \sigma_{exoG}^{red} = \frac{1}{2} eB \epsilon_{ij} dQ_i \wedge dQ_j + eE_i dQ_i \wedge dt. \qquad (III.18) $$
## IV. EXOTIC CARROLL PARTICLE

A remarkable fact which has long escaped attention and was recognised only recently [3–7] is that, unlike in $d \ge 3$ dimensions, the planar Carroll group admits also a two-parameter central extension. The associated particle will be called an exotic Carroll particle. (Our notations correspond to $\kappa_{exo} = m^2\theta = 2q_2$, $\kappa_{mag} = -2q_1$ in [7].) We assume that our particle has non-vanishing mass and electric charge, $m \neq 0$ and $e \neq 0$, respectively, and is subjected to constant planar magnetic and electric fields. Unlike as for Galilei [12], the mass $m$ is for Carroll an externally given constant and not a central-extension parameter.

In terms of the non-commutativity parameter and the effective magnetic field,

$$ \theta = \frac{\kappa_{exo}}{m^2} \quad \text{and} \quad B^* = eB + \kappa_{mag}, \qquad (IV.1) $$

respectively, Souriau’s orbit construction [54] applied to the exotic Carroll group yields the Souriau form [4–7],

$$ \sigma_{exoC} = d\mathbf{p} \wedge d\mathbf{x} + \frac{B^*}{2} \epsilon_{ij} dx_i \wedge dx_j + \frac{\theta}{2} \epsilon_{ij} dp_i \wedge dp_j + eE_i dx_i \wedge ds. \qquad (IV.2) $$

Comparison with (III.5) shows that (up to $t \rightsquigarrow s$ and $eB \rightsquigarrow B^*$) this is identical to the Galilean expression except for the missing kinetic term in the Hamiltonian which is merely the “naked” potential,

$$ \mathcal{H}_{Car} = eV. \qquad (IV.3) $$

The dynamics is described alternatively by the doubly-extended first-order-in-$\dot{x}$ phase-space Lagrangian with no kinetic term [23, 28],

$$ \mathcal{L}_{Car} = \mathbf{p} \cdot \dot{\mathbf{x}} + \frac{1}{2} B^* \epsilon_{ij} x_i \dot{x}_j + \frac{1}{2} \theta \epsilon_{ij} p_i \dot{p}_j - eV, \qquad (IV.4) $$

where the “dot” means derivation w.r.t. Carroll time, $d/ds$. The absence of the kinetic term implies that the Carroll equations of motion are decoupled for all values of the parameters [7],

$$ (1 - \theta B^*) \dot{x}_i = -e\theta \epsilon_{ij} E_j \quad \text{and} \quad (1 - \theta B^*) \dot{p}_i = eE_i \qquad (IV.5) $$

Then introducing again the effective mass,

$$ m^* = 1 - \theta B^* \qquad (IV.6) $$

cf. (III.1) we conclude:
**Proposition IV.1.** When $m^* \neq 0$ the system is regular and the coordinates follow an anomalous Hall law [50, 51]

$$ \dot{x}_i = -\left(\frac{e\theta}{1 - \theta B^*}\right) \epsilon_{ij} E_j \quad \Rightarrow \quad x_i(s) = -\left(\frac{e\theta}{1 - \theta B^*}\right) \epsilon_{ij} E_j s + x_i^0, \qquad (IV.7) $$

supplemented by

$$ \dot{p}_i = \left(\frac{1}{1 - \theta B^*}\right) e E_i \quad \Rightarrow \quad p_i(s) = \left(\frac{e E_i}{1 - \theta B^*}\right) s + p^0(s). \qquad (IV.8) $$

Switching off the exotic parameter, $\theta = 0$, the immobility of Carroll particles is duly recovered.

The p-equation has no effect on the dynamics of the position, though, and could actually be ignored when studying the x-dynamics. We recall for comparison that Galilean particles follow the Hall law (III.11) only in the critical case (III.10) when the effective mass vanishes.

The anomalous Hall trajectories for various values of $m^*$ are depicted in FIG.2(i).

<figure><img src="image_3.png" /><figcaption>FIG. 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.1).</figcaption></figure>
Approaching the critical value

$$m^* = 0 \Leftrightarrow (\theta B^*)_{crit} = 1, \qquad (IV.9)$$

the velocity diverges, cf. FIG.2(ii). The coefficient in (IV.7) changes sign when the denominator crosses zero, flipping over the direction of the motion, as in FIG.2ii. — to be compared with the reversal of the rotation in exotic Galilean mechanics [47] in FIG.1. The novelty is that we get Hall-like Carroll motion always, not only in the critical case.

An interesting particular case is obtained by presenting (IV.7) as,

$$\underbrace{(1 - \kappa_{mag}\theta - e\theta B)}_{0} \dot{x}_{i} = -e\theta\epsilon_{ij}E_{j}. \qquad (IV.10)$$

Then putting the underbraced quantity to zero,

$$\theta\kappa_{mag} = 1 \Leftrightarrow \kappa_{exo}\kappa_{mag} = m^2. \qquad (IV.11)$$

Then the exotic factor $e\theta$ drops out (as long as it does not vanish), yielding:

**Proposition IV.2.** When the parameters satisfy (IV.11) eqn. (IV.7) reduces to the usual Hall law,

$$\dot{x}_{i} = \epsilon_{ij} \frac{E_{j}}{B}. \qquad (IV.12)$$

Eqn. (IV.10) suggests to consider yet another remarkable particular case:

**Proposition IV.3.** When the external magnetic field is switched off,

$$B = 0, \qquad (IV.13)$$

then, assuming that $\theta \kappa_{mag} \neq 1$, the position satisfies

$$\dot{x}_{i} = - \frac{e\theta}{1 - \theta\kappa_{mag}} \epsilon_{ij} E_{j}. \qquad (IV.14)$$

Thus when $\theta \neq 0$ Hall motions persist even for $\kappa_{mag} = 0$. (IV.14) is reminiscent of “the” anomalous Hall effect observed in ferromagnetic materials in the absence of an external magnetic field [50–53]. Comparison of (IV.14) with (IV.7) underlines that $\kappa_{mag}$ behaves indeed as a proper magnetic field carried by the particle, and could be viewed as the planar analog of a magnetic monopole. It is also reminiscent of the rather mysterious magnetic field of relativistic anyons [57].

We underline that the system is regular in both of these particular cases (IV.11) and (IV.13).
A. The singular Carroll case

In the singular case (IV.9) both equations in (IV.5) are contradictory unless $e\mathbf{E} = 0$, when they are undetermined. This is unlike as in exotic Galilean mechanics, eqn. (III.2), where the p-terms on the rhs can compensate the exotic one, allowing, for vanishing effective mass, motions which follow the Hall law, (III.11) [38, 44–47].

We resort to Hamiltonian reduction [44, 45, 48, 49] along the lines as for exotic Galilei [44, 45]. We start with the regular case $m^* \neq 0$ and introduce new, canonical coordinates [44],

$$Q_i = x_i + \frac{1}{B^*} \left[ 1 - \sqrt{\frac{m^*}{m}} \right] \epsilon_{ij} p_j, \qquad (IV.15a)$$

$$P_i = \sqrt{\frac{m^*}{m}} p_i - \frac{1}{2} B^* \epsilon_{ij} Q_j, \qquad (IV.15b)$$

in terms of which the symplectic form and Hamiltonian are,

$$\Omega_{\text{exo}} = dP_i \wedge dQ_i, \qquad (IV.16a)$$

$$\mathcal{H}_{\text{exoC}} = -\frac{1}{2} \left( \sqrt{\frac{m}{m^*}} + 1 \right) eE_i dQ_i + \frac{1}{B^*} \left( \sqrt{\frac{m}{m^*}} - 1 \right) \epsilon_{ij} eE_i dP_j. \qquad (IV.16b)$$

The corresponding Poisson brackets and equations of motion are

$$\{Q_i, P_j\} = \delta_{ij}, \quad \{Q_i, Q_j\} = 0, \quad \{P_i, P_j\} = 0 \qquad (IV.17)$$

and

$$\dot{P}_i = \frac{1}{2} \left( \sqrt{\frac{m}{m^*}} + 1 \right) eE_i \quad \text{and} \quad \dot{Q}_i = -\frac{1}{B^*} \left( \sqrt{\frac{m}{m^*}} - 1 \right) \epsilon_{ij} eE_j. \qquad (IV.18)$$

respectively. In the critical case (IV.9) the inverse transformation,

$$x_i = \frac{1}{2} \left( \sqrt{\frac{m}{m^*}} + 1 \right) Q_i - \frac{1}{B^*} \left( \sqrt{\frac{m}{m^*}} - 1 \right) \epsilon_{ij} P_j, \qquad (IV.19)$$

becomes singular (IV.9), the velocities diverge, however the coordinates (IV.15) do have a well-defined limit. Letting $m^* \to 0$, (IV.15b) implies that $\mathbf{P}$ is rotated rigidly by $\pi/2$ w.r.t. $\mathbf{Q}$,

$$P_i = -\frac{1}{2} B_{\text{crit}}^* \epsilon_{ij} Q_j, \qquad (IV.20)$$

leaving us with the Carroll guiding centre,

$$Q_i = x_i + \frac{1}{B^*} \epsilon_{ij} p_j, \qquad (IV.21)$$
as only dynamical degree of freedom. The reduced symplectic form, Poisson brackets and reduced Hamiltonian are,

$$ \Omega = \frac{1}{2} B_{crit}^{*} \epsilon_{ij} dQ_i \wedge dQ_j, \quad \{Q_1, Q_2\} = -\frac{1}{B_{crit}^{*}}, \quad \mathcal{H}_{red} = -eE_i Q_i \quad (IV.22) $$

respectively, cf. (III.12). It follows that :

**Proposition IV.4.** In the critical case (IV.9) the guiding centre $Q_i$ follows the Hall law (III.16),

$$ \dot{Q}_i = \epsilon_{ij} \frac{E_j}{B_{crit}^*}, \quad (IV.23) $$

whereas $P_i$ is determined algebraically by $Q_i$ as in (IV.20).

## B. Chiral decomposition of exotic Carroll dynamics

Further insight is provided by chiral decomposition proposed in [42, 43]. Starting with the exotic Souriau form $\sigma_{exoC}$ (IV.2) (or symplectic form $\Omega_{exo}$ (III.3b) and Hamiltonian, $\mathcal{H}_{Car}$ (IV.3)), we introduce chiral coordinates,

$$ p_i = B^* \epsilon_{ij} X_j^-, \quad x_i = X_i^+ + X_i^- . \quad (IV.24) $$

Here $X_i^+ = Q_i$ is in fact the guiding centre, (IV.21). The symplectic form and Hamiltonian are decomposed as,

$$ \Omega = \Omega^+ + \Omega^- = \left\{ \frac{B^*}{2} \epsilon_{ij} dX_i^+ \wedge dX_j^+ \right\} - \left\{ \frac{B^*}{2} (1 - \theta B^*) \epsilon_{ij} dX_i^- \wedge dX_j^- \right\}, \quad (IV.25a) $$

$$ \mathcal{H} = \mathcal{H}^+ + \mathcal{H}^- = -eE_i X_i^+ - eE_i X_i^- . \quad (IV.25b) $$

We note the relative minus in (IV.25a) but not in (IV.25b).

In the regular case $m^* = (1 - B^*\theta) \neq 0$ the associated Poisson brackets

$$ \{X_i^+, X_j^+\} = -\frac{1}{B^*} \epsilon_{ij}, \quad \{X_i^-, X_j^+\} = 0, \quad \{X_i^-, X_j^-\} = \frac{1}{B^* (1 - \theta B^*)} \epsilon_{ij}, \quad (IV.26) $$

yield the Hamilton equations,

$$ \frac{dX_i^+}{ds} = \epsilon_{ij} \frac{eE_j}{B^*}, \quad (IV.27a) $$

$$ \frac{dX_i^-}{ds} = -\frac{1}{(1 - \theta B^*)} \epsilon_{ij} \frac{eE_j}{B^*}. \quad (IV.27b) $$

which imply :
**Proposition IV.5.** Both chiral coordinates follow a Hall law — the guiding centre, $X_i^+$, the usual one and $X_i^-$ an anomalous one which involves also the exotic parameter $\theta$.

For $m^* > 0$ $X_i^-$ moves in the opposite direction as $X_i^+$ does but switches direction beyond the critical value. For comparison, remember that in the Galilean case $X_i^-$ circles around the guiding centre $X_i^+$ [47]. Combining $X_i^+$ and $X_i^-$ the anomalous Hall dynamics (IV.7) is recovered. Eqn. (IV.27) allows us to deduce:

**Proposition IV.6.** When the non-commutative parameter is switched off then $X_i^-$ cancels the motion of the guiding centre $X_i^+$ and the particle's position coordinate, $\mathbf{x}$, stops moving,

$$ \theta = 0 \Rightarrow \dot{\mathbf{x}} = \dot{X}^+ + \dot{X}^- = 0. \qquad (IV.28) $$

In other regular cases with $\theta \neq 0$ the motion is a combination of Hall motion of $X_i^+$ deviated by that of $X_i^-$, as illustrated in FIG. 2.

The same conclusion could have been derived directly by combining $\mathbf{x}$ and $\mathbf{p}$ into

$$ \tilde{Q}_i = x_i + \theta \epsilon_{ij} p_j \qquad (IV.29) $$

which is the guiding centre $\mathbf{Q}$ in the critical case (IV.9), for which the equations (IV.5) imply, even for unrelated $B^*$ and $\theta$,

$$ \tilde{Q}_i = 0 \Rightarrow \tilde{Q}_i = \text{const.} \qquad (IV.30) $$

Coordinate-immobility (IV.28) is recovered. When the exotic parameter is turned off, $\theta \to 0$.

We briefly recall for comparison the discussion of the exotic Galilean model in chiral terms [47]. The symplectic form and thus commutation relations are still (IV.25a) and (IV.26) (up to $B^* \rightsquigarrow eB$), but the Hamiltonian has a kinetic term which comes from $\mathbf{p}^2/2m$,

$$ \mathcal{H}_{\text{exoG}} = \left\{ -e\mathbf{E} \cdot \mathbf{X}^+ \right\} + \left\{ \frac{e^2 B^2}{2m} (\mathbf{X}^-)^2 - e\mathbf{E} \cdot \mathbf{X}^- \right\}. \qquad (IV.31) $$

The system splits into two uncoupled parts, however the $X_i^-$ equation picks up a kinetic term, cf. (IV.27).

$$ \begin{gathered} \frac{dX_i^+}{dt} = \epsilon_{ij} \frac{E_j}{B}, \qquad (IV.32) \\ (1 - \theta eB) \frac{dX_i^-}{dt} = eB \epsilon_{ij} X_j^- - \epsilon_{ij} \frac{E_j}{B}. \end{gathered} $$
For $m^* \neq 0$ the guiding centre follows the Hall law and $X_i^-$ rotates around it. In the critical case $m^* = 0$, $X_i^+$ moves by following the Hall law and $X_i^-$ is frozen into $X_i^- = E_i/eB^2$.

For $m^* = 0$ both systems become singular. For Galilei it is the frequency of the rotation which diverges, FIG.1, and for Carroll it is the velocity which becomes infinite, FIG.2.

The Souriau form of the exotic Carroll system (IV.25) splits into two parts,

$$ \sigma_{exoC} = \left\{ \frac{1}{2} B^* \epsilon_{ij} dQ_i \wedge dQ_j + eE_i dQ_i \wedge ds \right\} + \left\{ \frac{1}{2} \left(\theta - \frac{1}{B^*}\right) \epsilon_{ij} dp_i \wedge dp_j + \epsilon_{ij} \frac{eE_j}{B^*} dp_i \wedge ds \right\}. \qquad (IV.33) $$

cf. (III.15) in the Galilean case. The first brace rules the motion of the guiding center, and the second that of the uncoupled momentum — for any values of $B^*$ and $\theta$. In the singular case $m^* = m(1 - \theta B^*) = 0$ (IV.33) projects, after removing the exact $dp_i \wedge ds$ term, to

$$ \sigma_{exoC}^c = \frac{1}{2} B^* \epsilon_{ij} dQ_i \wedge dQ_j + eE_i dQ_i \wedge ds \qquad (IV.34) $$

the same as (III.18) (up to $eB \leftrightarrow B^*$ and $t \leftrightarrow s$), while the second braces are somewhat different. The trajectories are in general different: spiralling for exoG, and doubly Hall for exoC. However in the critical case (III.10) the Galilean form (III.15) also decouples, and projecting into 2 + 1 dimension, we get the same reduced form $\sigma_{exoC}$, (up to $B^* \leftrightarrow eB_{crit}$ and $s \leftrightarrow t$) — which is indeed identical to the DJT form $\sigma_{DJT}$ in (II.5) (up to $eB \leftrightarrow B^*$, $t \leftrightarrow s$, $Q \leftrightarrow x$). In conclusion:

**Proposition IV.7.** *Carrollian and critical Galilean motions project onto the same Hall motion, namely that for DJT in (II.2) (after identifying the guiding center $Q$ with the DJT position).*

## V. SYMMETRY AND IMMOBILITY

This section is devoted to the symmetries of the various models and to clarify the intimate but subtle relation between boost symmetry and (im)mobility properties of Carroll particles. Let us keep in mind that the Hall law (II.2) and its anomalous exotic extension (IV.7) both imply *partial immobility*: the only allowed motions are perpendicular to the electric field.
A. Boost symmetry of DJT

This section is devoted to the curious double symmetry of the DJT model. We start with Galilei boosts,

$$ \boldsymbol{x} \rightarrow \boldsymbol{x} + \mathbf{b}t, \quad t \rightarrow t, \qquad (V.1) $$

and observe that the rhs of the Hall law (II.2) is shifted, $(x_i)' \rightarrow (x_i)' + b_i$, whereas the rhs of (II.2) is left invariant. Thus:

**Proposition V.1.** Galilei boost symmetry is broken for the DJT system by the electromagnetic field.

The statement is readily confirmed also in either the Lagrangian or in the symplectic framework:

$$ \mathcal{L}_{\text{D}JT} \rightarrow \mathcal{L}_{\text{D}JT} + \left( -\frac{1}{2} e B \epsilon_{ij} x_i b_j t + \frac{1}{2} e E_i b_i t^2 \right)' + e B \epsilon_{ij} x_i b_j, \qquad (V.2) $$

$$ \sigma_{\text{D}JT} \rightarrow \sigma_{\text{D}JT} - eB\epsilon_{ij}b_i dx^j \wedge dt, \qquad (V.3) $$

respectively.

The unexpected result is that *Carroll boosts*, (I.1), behave better: a C-boost takes the Souriau form (II.5) [up to trading Galilean time $t$ for Carroll time, $s$] into

$$ \sigma_{\text{D}JT} \rightarrow \sigma_{\text{D}JT} - e\epsilon_{ij}E_i b_j \, dx_1 \wedge dx_2 \qquad (V.4) $$

and the extra term vanishes when the boost is along the electric field. Alternatively, the DJT Lagrangian changes as,

$$ \mathcal{L}_{\text{D}JT} ds \rightarrow \mathcal{L}_{\text{D}JT} ds + d\left(\frac{1}{2} e E_i x_i^2 b_i\right) - e\left(E_1 b_2 x_1 dx_2 + E_2 b_1 x_2 dx_1\right), \qquad (V.5) $$

where the last term is a total derivative if $\epsilon_{ij}E_i b_j = 0$. Thus, in contrast with the Galilean case:

**Proposition V.2.** The DJT system has a “half-Carroll symmetry” : boosts which are perpendicular to the electric field are broken but those along the electric field remain unbroken.

Conserved quantities are readily derived by the symplectic Noether theorem [54] recalled in Appendix A. Contracting the Souriau form (II.5) by an (infinitesimal) Carroll boost $\beta$,

$$ X_{\text{Carr}}: \quad \delta \boldsymbol{x} = 0 \quad \text{and} \quad \delta s = -x_i \beta_i, \qquad (V.6) $$
we get, choosing again $\mathbf{E} = (0, E)$ for simplicity,

$$ \sigma(X, \cdot) = eE \left\{ (x_1 dx_2) \beta_1 + d\left(\frac{x_2^2}{2}\right) \beta_2 \right\}. \qquad (V.7) $$

The 1st term here, obtained by boosting perpendicularly to $\mathbf{E}$ i.e. in the direction of the (Hall) motion, is not closed: $d(x_1 dx_2)\beta_1 = (dx_1 \wedge dx_2)\beta_1 \neq 0$. Thus:

**Proposition V.3.** Boosting perpendicularly to $\mathbf{E}$ is not a symmetry, but boosting along $\mathbf{E}$ is a symmetry, with associated conserved quantity

$$ \mathcal{K}_{\parallel} = \frac{eE}{2} x_2^2 \Rightarrow x_2 = x_2^0 \text{ fixed,} \qquad (V.8) $$

consistently with the half-broken Carroll symmetry in Prop. V.2.

The conservation of $\mathcal{K}_{\parallel}$ can be checked also directly by using the equations of motion.

Rotational symmetry is broken when $\mathbf{E} \neq 0$, however the system is invariant w.r.t. space and Carroll-time translations,

$$ X_C = \gamma_i \frac{\partial}{\partial x_i} + \varepsilon \frac{\partial}{\partial s}, \qquad (V.9) $$

which generate, consistently with the equations of motion, the conserved quantities

$$ p_i = eB\epsilon_{ij}x_j + eE_i s \quad \text{Carroll momentum} \qquad (V.10) $$

$$ \mathcal{H} = eE_i x_i = eV(\mathbf{x}) \quad \text{Carroll energy} $$

Conversely, momentum conservation implies the Hall law (II.2) (with Carroll time), while that of the Carroll energy implies that the motion is on an equipotential.

These results could also be obtained in Hamiltonian terms. For example, $f_2$ Poisson-commutes with $\mathcal{H}$, $\{f_2, \mathcal{H}\} = 0$. Conversely, $f_2$ generates the Hamiltonian vectorfield $X_1 = E/B$, $X_2 = 0$. More generally, if $f = f(x_1, x_2)$ then $\{f, \mathcal{H}\} = \frac{1}{B}\partial_1 f$, which is conserved for an arbitrary function of $x_2$, $f = f(x_2)$.

It is instructive to compare the Carroll results with those in the Galilean case: the Galilei boost symmetry is fully broken by the magnetic field, as we have seen. Accordingly, inserting the Galilean generators into the Souriau form yields,

$$ \sigma_{DJT}(X, \cdot) = d\left\{\gamma_i e(B\epsilon_{ij}x_j + E_i t) - \epsilon(eE_i x_i) + \beta_i \left(\frac{1}{2}eE_i t^2\right)\right\} + \beta_i eBt, \epsilon_{ij} dx_j : \qquad (V.11) $$

the last term is not closed, breaking the Galilei boost symmetry, consistently with Prop. V.1.
## B. Symmetries of exotic particles

* Lifting (V.1) to the evolution space of an exotic Galilean particle,

$$ \boldsymbol{x} \rightarrow \boldsymbol{x} + \boldsymbol{b}t, \quad \boldsymbol{p} \rightarrow \boldsymbol{p} + m\boldsymbol{b}, \quad t \rightarrow t, \qquad (V.12) $$

the Souriau form (III.5) changes as for DJT in (V.3) [with $\sigma_{DJT} \rightsquigarrow \sigma_{exoG}$], leading to:

**Proposition V.4.** A nonzero magnetic field $B \neq 0$ breaks Galilean boost symmetry of an exotic Galilean particle (as seen also directly from in Prop. III.1: the rhs is invariant w.r.t. (V.1)).

* Let us now turn to exotic Carroll particles. Assuming that $\theta B^* \neq 1$, the system is regular, described by the Souriau form (IV.2) on the evolution space $\{\boldsymbol{x}, \boldsymbol{p}, s\}$, upon which Carroll boosts act as [7],

$$ \boldsymbol{x} \rightarrow \boldsymbol{x}, \quad \boldsymbol{p} \rightarrow \boldsymbol{p} + m\boldsymbol{b}, \quad s \rightarrow s - b_i x_i, \qquad (V.13) $$

to be compared with (V.12). Inserting into (IV.2) we find again (V.4) [up to $\sigma_{DJT} \rightsquigarrow \sigma_{exoC}$], implying:

**Proposition V.5.** For an exotic Carroll particle, boosts perpendicular to $\boldsymbol{E}$ are broken by the electric field, but the parallel ones remain unbroken: we get a “half-Carroll symmetry” (as does its DJT reduction).

Conserved quantities are readily found by contracting the Souriau form $\sigma_{exoC}$ with the infinitesimal boost generator (V.6) lifted to the evolution space,

$$ X_b : \delta\boldsymbol{x} = 0, \quad \delta\boldsymbol{p} = m\boldsymbol{\beta}, \quad \delta s = -\boldsymbol{\beta} \cdot \boldsymbol{x} \qquad (V.14) $$

cf. eqn. (3.3) of [7], yields

$$ \sigma(X_b, \cdot) = d\{\beta_i(x_i + \theta\epsilon_{ij}p_j)\} + eE_i dx_i (\beta_k x_k), \qquad (V.15) $$

which is closed when the last term is closed. But for $\boldsymbol{E} = (0, E)$,

$$ \left\{ \begin{array}{llll} \text{for } \beta_1 = 0 & d\left(\frac{1}{2}eEx_2^2\right)\beta_2 & \text{closed} & \Rightarrow \text{symmetry} \\ \text{for } \beta_2 = 0 & eEx_1dx_2\beta_1 & \text{not closed} & \Rightarrow \text{broken} \end{array} \right. . \qquad (V.16) $$

Thus, consistently with (V.8):
**Proposition V.6.** Boosting the exotic Carroll particle along the electric field **E** = (0, **E**) generates the conserved quantity

$$ K_{\parallel} = (x_2 - \theta p_1) + \frac{1}{2} eE x_2^2 = \tilde{Q}_2 + \frac{1}{2} eE x_2^2. \qquad (V.17) $$

with the notation in (IV.30).

The conservation of $K_{\parallel}$ is readily confirmed by using (after replacing $eB$ by $B^*$) the commutation relations (III.4). For $\mathcal{H}_{exo} = \mathcal{H} = -eEx_2$ we find,

$$ (1 - \theta B^*) \{K_{\parallel}, \mathcal{H}\} \propto \{x_2 - \theta p_1 + \frac{1}{2} eE x_2^2, eEx_2\} \propto -\theta \{p_1, x_2\} = 0. $$

Contracting $\sigma$ with $s$-translations, $\delta s = \varepsilon$, yields $\sigma(X_s, \cdot) = d(E_ix_i)$, which implies that the projection of the position onto the electric direction is a constant of the motion,

$$ \mathcal{E} = eE_ix_i. \qquad (V.18) $$

Thus the two terms in (V.17) are separately conserved.

At last, contracting $\sigma$ with translations,

$$ X_{\gamma}: \quad \delta x = \gamma, \quad \delta p_i = 0, \quad \delta s = 0, \qquad (V.19) $$

we get two conserved quantities,

$$ \left\{ \begin{array}{l} \mathcal{T}_1 = B^* \left( x_2 - \frac{1}{B^*} p_1 \right) = B^* Q_2 \\ \mathcal{T}_2 = -B^* \left( x_1 + \frac{1}{B^*} p_2 - \frac{eE}{B^*} s \right) = -B^* \left( Q_1 - \frac{eE}{B^*} s \right) \end{array} \right. \qquad (V.20) $$

which imply again that $Q_2$ is fixed and $Q_1$ follows the Hall law. This should be compared with (III.13) for Galilei.

DJT is the commun projection of both the exo-Galilei and exo-Carroll systems (up to $t \leftrightarrow s$) which thus and inherits both of their symmetries: Galilei boost symmetry is broken, however Carroll boosts along the electric field are symmetries.

## VI. MOTION ON THE BLACK HOLE HORIZON

Our results above are analogous to but different from those we had found before on the horizon of a black hole for an exotic Carroll particle with parameters $\kappa_{exo}$ and $\kappa_{mag}$, with zero
<table><caption>TABLE I: *Motions and boosts symmetries.*</caption><thead><tr><th>particle type</th><th>position <b>x</b></th><th>guiding centre <b>Q</b></th><th>Gal-boost</th><th>Carr-boost</th></tr></thead><tbody><tr><td>DJT</td><td>Hall motion</td><td>—</td><td>broken</td><td>half broken</td></tr><tr><td>Galilei</td><td>screw precession</td><td>Hall motion</td><td>broken</td><td>—</td></tr><tr><td>Exotic-Galilei</td><td>anomalous screw</td><td>Hall motion</td><td>broken</td><td>—</td></tr><tr><td>Reduced-Galilei</td><td>—</td><td>Hall motion</td><td>broken</td><td>half broken</td></tr><tr><td>Carroll</td><td>no motion</td><td>Hall motion</td><td>—</td><td>half broken</td></tr><tr><td>Exotic-Carroll</td><td>anomalous Hall motion</td><td>Hall motion</td><td>—</td><td>half broken</td></tr><tr><td>Reduced-Carroll</td><td>—</td><td>Hall motion</td><td>broken</td><td>half broken</td></tr></tbody></table>

electric charge but with a magnetic moment $\mu$ and anyonic spin $\chi$ [7, 23, 24]. Its Hamiltonian is $\mathcal{H} = -(\mu\chi)B$, where $B$ may not be a constant (as it happens for a Kerr-Newman black hole). The equations of motion are,

$$ \dot{x}_i = (\mu\chi) \frac{\kappa_{\text{exo}}}{\kappa_{\text{exo}}\kappa_{\text{mag}} - m^2} \epsilon_{ij} \partial_j B, \qquad (\text{VI.1}) $$

cf. eqn. (2.4) of [23]. The product $(\mu\chi)$ behaves as an electric charge and the magnetic field, $B$, plays a role analogous to that of an electric potential, cf. (IV.3).

For non-zero mass $m$, putting again $\theta = \kappa_{\text{exo}}/m^2$ we get an anomalous Hall effect for the position,

$$ \dot{x}_i = -(\mu\chi) \frac{\theta}{1 - \theta\kappa_{\text{mag}}} \epsilon_{ij} \partial_j B, \qquad (\text{VI.2}) $$

which is a sort of dual to (IV.14) with the gradient of the magnetic field behaving as an electric field. Turning off the exotic structure, $\theta \to 0$, the particle *stops moving* and we recover the celebrated *immobility of massive unextended Carroll particles*.

Letting instead the mass go to zero (as suggested by DJT in the Peierls case) while keeping $\kappa_{\text{exo}} \neq 0$, the latter drops out, leaving us with,

$$ \dot{x}_i = (\mu\chi) \epsilon_{ij} \frac{\partial_j B}{\kappa_{\text{mag}}}. \qquad (\text{VI.3}) $$

Our “exotic Carroll photons” thus exhibits an anomalous spin-Hall effect on the black hole horizon, underlining the “duality” of $(\mu\chi)$ with electric charge, the gradient of the magnetic field playing the role of an electric field, and second Carroll charge $\kappa_{\text{mag}}$ behaving as an effective magnetic field,

$$ e \rightsquigarrow \mu\chi, \quad \mathbf{E} \rightsquigarrow \nabla B, \quad B \rightsquigarrow \kappa_{\text{mag}}. \qquad (\text{VI.4}) $$
Magnetic fields which differ by a constant generate identical motion. A constant magnetic field $B = \text{const.}$ would, in particular, imply immobility, — and it precisely our search for non-trivial motion that led us to Kerr-Newman BHs [23].

## VII. CONCLUSION

The Peierls-Dunne-Jackiw-Trugenberger system (II.1) [36, 37], which resurrects Peierls' historical idea [35], can be derived both from the doubly extended (“exotic”) Galilean [8, 38, 44–47], and from the recently proposed exotic Carroll model [3–7, 23] by Hamiltonian reduction [49].

The Galilean case is well-known, therefore we focus our attention at the Carroll case. Having suppressed the kinetic term from the Hamiltonian results in a substantially poorer dynamics: the equations for position and for momentum are fully decoupled and the position performs anomalous Hall motion which, for particular values of the parameters, becomes ordinary Hall motion. The extended model allows for an additional, constant “internal” magnetic field represented by our parameter $\kappa_{mag}$ which is added to the external magnetic field $B$ (IV.1) which might play a role in the anomalous Hall effect [52] with no external field.

Neither the anomalous and let alone the usual Hall effect can be obtained by an “ordinary” [meaning unextended] Carroll particle. By (IV.7) $\dot{x} \neq 0$ requires $\theta \neq 0$. The condition (IV.12) requires the constraint (IV.11) which in the massive case $m \neq 0$ can not be satisfied when either of the “exotic” parameters vanishes. When the exotic parameter is turned off, $\theta \to 0$, then eqn (IV.7) behaves as $\dot{x}_i \to 0$ confirming the general wisdom that (unextended) *Carroll particles do not move*.

Recent interest in Carroll symmetry comes also from the study of fractons in condensed matter physics [28, 58–66], whose restricted mobility was attributed to their *conserved dipole momentum* — generated by a “dual” of Carroll symmetry (I.1) familiar from the BMS context [67–70].

Putting a DJT particle into a planar electromagnetic field breaks Galilei boost symmetry (Proposition V.1). However, owing to the poorer nature of Carroll dynamics, one of the Carroll boosts remains a symmetry for DJT (Proposition V.2) : we get what we called half-Carroll symmetry.
The DJT dynamics actually follows from that of exotic Carroll particles, see Proposition V.5 : half-boost symmetry implies the conservation law (V.17) for $\mathcal{K}_2$, (Proposition V.6) which, combined with the immobility of $\tilde{Q}$ in (IV.30), implies in turn no motion along $\mathbf{E}$. $\mathbf{E} = 0$ has arbitrary direction yielding no motion at all.

The “no-motion-property” has recently reemerged, independently, in condensed matter physics in the study of quasiparticles called fractons [58–66] whose restricted mobility is attributed to their *conserved dipole momentum* – generated by a dual counterpart of Carroll symmetry [28].

Particles with no electric charge but with magnetic momentum and anyonic spin which move on the horizon of a black hole [23, 24] behave also analogously to our electrically charged Carroll particles.

Our study shows strong similarity also with vortex dynamics [71] as it will be explained in a forthcoming paper [72]. In conclusion, 450 years after Galilei [73–75] the Carrollian context rekindles once again the subtle question of boosts.

## Acknowledgments

We are indebted to X. Bekaert, J. Balog, M. Elbistan, L. Fehér, G. Gibbons, P. Iglesias and F. Ziegler for discussions and for correspondence. PMZ was partially supported by the National Natural Science Foundation of China (Grant No. 11975320).

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## Appendix A: The Noether theorem in Souriau's framework

Here we summarise Souriau's version of the Noether theorem, taken from chapter 12 of [54]. First we recall that a mechanical system is described by a closed 2-form $\sigma$ of constant rank defined on an evolution space $\mathcal{E}$. The motions $\gamma$ are the characteristic curves of $\sigma$,

$$ \sigma(\gamma', \cdot) = 0. \qquad (A.1) $$

A transformation $\Phi$ of the evolutions space is a symmetry if it leaves $\sigma$ invariant,

$$ \Phi^*\sigma = \sigma. \qquad (A.2) $$
Infinitesimally, a vector field X is a symmetry when it preserves σ; which boils down to that contracted by X is the differential of a function $f_X$,

$$ \sigma(X, \cdot) = df_X; \qquad (A.3) $$

then $f_X$ is a constant of the motion. The Souriau version is equivalent to the more conventional approaches. The Lagrangian should, for example, change by a total derivative. The relation with the Hamiltonian framework is obtained in turn by splitting the Souriau form as,

$$ \sigma = \Omega - d\mathcal{H} \wedge dt. \qquad (A.4) $$

The Poisson Bracket (PB) associated with the symplectic form $\Omega$ is:

$$ \{f, g\} = \Omega^{ij} (\partial_i f \partial_j g - \partial_j f \partial_i g), \qquad (A.5) $$

where $\Omega^{ij}$ is the inverse of the symplectic matrix $\Omega_{ij}$, $\Omega^{ij}\Omega_{jk} = \delta_k^i$. For $f$ may depend explicitly on $t$, then the symmetry condition (A.3) requires that

$$ \partial_t f + \{f, \mathcal{H}\} = 0. \qquad (A.6) $$