Gyromoment (GM): Field-Dependent Perspectives

• Gyromoment (GM) is a term with field-dependent meanings, ranging from gyromagnetic relations in ferromagnets to gyrokinetic moments in plasma theory and other acronymic contexts.
• In magnetization dynamics, GM represents the inertial gyromagnetic relation linking magnetization and angular momentum, with corrections that capture ultrafast nutational effects.
• In plasma physics and metallic transport, GM denotes hierarchical moment expansions and intrinsic band magnetic moments, highlighting its role in accurately modeling transport phenomena.

Gyromoment (GM) has no single fixed technical meaning across current arXiv usage. In magnetization dynamics it denotes the gyromagnetic relation linking magnetization and angular momentum, typically written as $\vec M=\gamma \vec L$; in gyrokinetic plasma theory it denotes gyro-moments, namely Hermite–Laguerre coefficients of a distribution-function expansion; in metallic transport it is closest to the Bloch-electron magnetic moment that controls the gyrotropic magnetic effect; and in several other literatures “GM” is only an acronym, for example for the GM-rule of impartial games or Gushel–Mukai geometry (Wegrowe et al., 2011, Hoffmann et al., 2023, Zhong et al., 2015, Gurvich et al., 2023, Liu et al., 15 Dec 2025). The term therefore requires immediate contextual disambiguation.

1. Field-dependent meanings

The cited literature supports several distinct definitions rather than a single cross-disciplinary object. The principal usages are summarized below.

Usage	Core object	Representative source
Ferromagnetic dynamics	Gyromagnetic relation $\vec M=\gamma\vec L$	(Wegrowe et al., 2011)
Gyrokinetic plasma theory	Gyro-moments $N^{pj}$ or $M^{pj}$ in Hermite–Laguerre space	(Frei et al., 2022, Hoffmann et al., 2023)
Metallic gyrotropy	Band magnetic moment $\mathbf m_{n\mathbf k}$ and GME tensor $\alpha_{ij}$	(Zhong et al., 2015)
Acronymal GM	GM-rule, Gushel–Mukai, gravitomagnetic, gyratonic	(Gurvich et al., 2023, Liu et al., 15 Dec 2025, Virgilio et al., 2010, Carneiro et al., 2019)

A recurrent misconception is to treat “GM” as automatically meaning a gyromagnetic moment. The literature does not support that identification. In some papers the relevant object is explicitly the gyromagnetic ratio $\gamma$ or a magnetic moment; in others it is a hierarchy of velocity-space moments; in still others it is not a moment at all, but a purely acronymal label for an unrelated construction.

This non-uniformity is not merely terminological. Each usage comes with a different state space, different conserved or response quantities, and different asymptotic regimes. In ferromagnets the relevant structure is angular momentum and inertial dynamics; in gyrokinetics it is spectral representation of velocity space; in optical gyrotropy it is a Fermi-surface transport coefficient; and in the acronymal cases no common “moment” structure need exist.

2. Gyromoment as gyromagnetic relation in ferromagnets

In magnetization dynamics, the relevant gyromoment is the gyromagnetic relation

$$\vec M=\gamma \vec L,\qquad \vec L=\frac{1}{\gamma}\vec M,$$

with $\gamma$ the gyromagnetic ratio. In the elementary orbital model, $\gamma=q/(2m)$, and more generally $\vec M=\gamma\vec L$0, with the sign inherited from $\vec M=\gamma\vec L$1; for the electron, $\vec M=\gamma\vec L$2 and hence $\vec M=\gamma\vec L$3 (Wegrowe et al., 2011). Wegrowe and Ciornei formulate this relation mechanically by treating a uniformly magnetized ferromagnet as a symmetric top with diagonal inertial tensor

$\vec M=\gamma\vec L$4

and angular momentum

$\vec M=\gamma\vec L$5

The magnetization is written as $\vec M=\gamma\vec L$6 with fixed modulus $\vec M=\gamma\vec L$7. Since

$\vec M=\gamma\vec L$8

one obtains

$\vec M=\gamma\vec L$9

and therefore

$N^{pj}$0

This formula is the central correction to the naive instantaneous vector relation $N^{pj}$1: if $N^{pj}$2, then $N^{pj}$3 is not exactly collinear with $N^{pj}$4 because of the transverse inertial term proportional to $N^{pj}$5 (Wegrowe et al., 2011).

With Gilbert damping introduced through a Rayleigh function, the generalized magnetization dynamics becomes

$N^{pj}$6

Relative to the usual Gilbert equation, the new term is the inertial or nutational correction

$N^{pj}$7

The paper’s conceptual point is that once $N^{pj}$8 is defined through a genuine inertial tensor, inertia cannot be absent from magnetization dynamics except as a long-time approximation (Wegrowe et al., 2011).

The ordinary Landau–Lifshitz–Gilbert regime is recovered only in the kinetic limit

$N^{pj}$9

or equivalently $M^{pj}$0 at fixed observation time. In that limit,

$M^{pj}$1

This makes the standard gyromagnetic proportionality a kinetic-limit relation rather than an exact instantaneous identity of the full inertial theory. The same framework identifies the short-time correction as an ultrafast nutational modification of ordinary gyromagnetic precession, with characteristic time scale $M^{pj}$2 and atomic estimates of order a femtosecond for plausible parameters (Wegrowe et al., 2011).

3. Gyro-moments in gyrokinetic plasma models

In plasma physics, GM usually means gyro-moments: coefficients of a Hermite–Laguerre expansion of a gyrokinetic or gyrocenter distribution function. In the linear electromagnetic flux-tube model,

$M^{pj}$3

with coefficients defined by projection in $M^{pj}$4 space. Low-order moments recover density, parallel flow, and parallel and perpendicular temperatures; higher moments encode heat fluxes and progressively finer velocity-space structure. The corresponding hierarchy remains fully gyrokinetic in content if all $M^{pj}$5 are retained, and trapped-particle, FLR, mirror-force, and collision effects are represented as structured couplings in moment space (Frei et al., 2022).

A closely related formulation is used in moment-based simulations of the cyclone base case and the Dimits shift, where the perturbed distribution is expanded as

$M^{pj}$6

Here the coefficients

$M^{pj}$7

are explicitly called gyromoments. This hierarchy is obtained by exact projection of the gyrokinetic equation and is therefore not a low-order phenomenological gyrofluid closure. The finite-$M^{pj}$8 system is a systematic truncation, with numerical closure

$M^{pj}$9

The same work reports that the GM approach converges more rapidly in capturing CBC nonlinear dynamics than continuum GENE when one compares the number of velocity-space unknowns, that about $\mathbf m_{n\mathbf k}$0 Hermite–Laguerre modes already give a good transport estimate for CBC turbulence, that GENE requires roughly $\mathbf m_{n\mathbf k}$1 velocity grid points for comparably reliable saturated heat flux, and that excessive numerical velocity-space dissipation can bias the saturated heat flux by about $\mathbf m_{n\mathbf k}$2 (Hoffmann et al., 2023).

The moment representation is especially natural because Hermite order resolves parallel phase mixing and Landau damping, while Laguerre order resolves perpendicular-energy structure and FLR physics. The Bessel gyroaverage is expanded algebraically in Laguerre space, and the field equations couple directly to $\mathbf m_{n\mathbf k}$3 moments in quasineutrality and to $\mathbf m_{n\mathbf k}$4 moments in Ampère’s law. In the collisionless linear benchmarks, the hierarchy recovers ITG, TEM, KBM, microtearing modes, and collisionless zonal-flow damping; in the high-collisionality regime the number of moments needed for convergence decreases, and the requirement is lower for pressure-gradient-driven modes than for trapped-particle and magnetic-drift-driven modes (Frei et al., 2022).

The same GM philosophy extends to full-$\mathbf m_{n\mathbf k}$5 linear-device turbulence. In a simplified LAPD-like model, the full ion gyrocenter distribution is expanded on a flow-shifted Hermite–Laguerre basis, yielding a hierarchy for coefficients $\mathbf m_{n\mathbf k}$6 with a nonlinear Dougherty collision operator, localized sources, and Bohm sheath boundary conditions. In that setting, higher-order GMs are damped by collisions in the high-collisionality regime, with roughly $\mathbf m_{n\mathbf k}$7 sufficient there, while $\mathbf m_{n\mathbf k}$8 may be needed in the low-collisionality regime (Frei et al., 2023).

A recent asymptotic development derives a hot-electron-limit closure for the same moment hierarchy. For

$\mathbf m_{n\mathbf k}$9

the otherwise infinite GM system admits a four-moment closure retaining

$\alpha_{ij}$0

identified as density, parallel velocity, and parallel and perpendicular temperatures. In Z-pinch geometry this HEL–GM model is analytically equivalent to the Ivanov et al. reduced system, while in tokamak $\alpha_{ij}$1-$\alpha_{ij}$2 geometry it preserves qualitative transport dynamics but fails to capture the Dimits shift because the missing higher-order kinetic moments prevent accurate zonal-flow amplification and transport suppression (Hoffmann et al., 18 Sep 2025).

4. Bloch-electron magnetic moment and the gyrotropic magnetic effect

In metallic transport theory, the closest gyromoment-like quantity is the intrinsic magnetic moment of Bloch quasiparticles on the Fermi surface. The gyrotropic magnetic effect is defined by

$\alpha_{ij}$3

and in the uniform, low-frequency transport limit the response tensor is

$\alpha_{ij}$4

At zero temperature this becomes the Fermi-surface formula

$\alpha_{ij}$5

The effect is therefore controlled by the Fermi-surface average of $\alpha_{ij}$6, not by a bulk occupied-band integral (Zhong et al., 2015).

The relevant moment is

$\alpha_{ij}$7

with orbital part

$\alpha_{ij}$8

This is the intrinsic magnetic dipole moment of a Bloch quasiparticle. In the GME, a slowly varying magnetic field shifts the quasiparticle energy by

$\alpha_{ij}$9

thereby distorting the Fermi-surface occupation and producing a transport current. The response is $\gamma$0-even and $\gamma$1-odd, so inversion symmetry must be broken, whereas time-reversal breaking is not required. The same paper emphasizes that GME is fundamentally different from the chiral magnetic effect: CME is governed by Berry curvature and a nonequilibrium chiral chemical-potential imbalance in the static limit, whereas GME is the uniform-limit low-frequency manifestation of natural gyrotropy (Zhong et al., 2015).

The same transport framework has been extended to metallic chiral magnets in which inversion symmetry is broken not by the lattice alone but by a chiral spin texture coupled to conduction electrons through Hund’s coupling. In that setting, the spin texture imprints itself on the electronic structure, and the GME becomes a probe of magnetic chirality. Analytical expressions for the rotatory power are obtained in terms of universal scaling functions, and the paper concludes that the orbital contribution is substantial for relativistic electrons with large intrinsic Berry curvature, while almost negligible for non-relativistic electrons without inherent Berry curvature; conversely, the spin contribution can be substantial in non-relativistic metals at large Fermi energy (Paul et al., 28 Apr 2025).

The optical observable is the rotatory power

$\gamma$2

which is related to the GME tensor through the gyration tensor. In cubic or higher symmetry,

$\gamma$3

For metallic chiral magnets the predicted rotatory powers are in an experimentally viable range and can be comparable to quartz. Candidate systems explicitly discussed include MnSi, FeGe, SmAlSi, EuCuAs, MnGe, and EuIn$\gamma$4As$\gamma$5 (Paul et al., 28 Apr 2025).

5. Rotational observables and gyroscopic response

In nonequilibrium statistical mechanics, the object closest to a gyromoment is the specific angular momentum of the Brownian gyrator,

$\gamma$6

in the inertial model, and

$\gamma$7

in the overdamped model. Its steady-state mean is

$\gamma$8

which reduces in the overdamped limit to

$\gamma$9

The same quantity is directly tied to the systematic torque, heat currents, and entropy production, so it functions as the paper’s central rotational moment observable rather than an auxiliary kinematic diagnostic (Bae et al., 2020).

In classical mechanics on curved surfaces, the corresponding object is the axial angular momentum

$$\vec M=\gamma \vec L,\qquad \vec L=\frac{1}{\gamma}\vec M,$$0

of a spinning disk constrained to remain tangent to the surface. The spin-induced gyroscopic force is

$$\vec M=\gamma \vec L,\qquad \vec L=\frac{1}{\gamma}\vec M,$$1

where $$\vec M=\gamma \vec L,\qquad \vec L=\frac{1}{\gamma}\vec M,$$2 is the Gaussian curvature and $$\vec M=\gamma \vec L,\qquad \vec L=\frac{1}{\gamma}\vec M,$$3 is the velocity rotated by $$\vec M=\gamma \vec L,\qquad \vec L=\frac{1}{\gamma}\vec M,$$4 in the tangent plane. The paper interprets this as an exact Lorentz-force analogue in which the Gaussian curvature plays the role of a magnetic field normal to the surface and the axial spin plays the role of charge (Cox et al., 2016).

In ferromagnetic gyroscopes, the GM-like quantity is again the gyromagnetic ratio. The basic field-to-precession conversion is

$$\vec M=\gamma \vec L,\qquad \vec L=\frac{1}{\gamma}\vec M,$$5

with a spin-inertia scale

$$\vec M=\gamma \vec L,\qquad \vec L=\frac{1}{\gamma}\vec M,$$6

The crossover between precession-dominated and libration-dominated dynamics is set by

$$\vec M=\gamma \vec L,\qquad \vec L=\frac{1}{\gamma}\vec M,$$7

and in the superconducting-levitated configuration the observed precession is reduced to an effective gyromagnetic response

$$\vec M=\gamma \vec L,\qquad \vec L=\frac{1}{\gamma}\vec M,$$8

The intrinsic magnetic moment-to-angular-momentum ratio is unchanged; what is modified is the measured response because Meissner-image feedback suppresses the precession frequency (Fadeev et al., 2020).

Taken together, these works show that “gyromoment” in a mechanical or stochastic setting often denotes a rotational observable—axial angular momentum, specific angular momentum, or effective gyromagnetic response—rather than a uniquely defined magnetic dipole moment. The common element is the conversion between rotation, torque, and measurable transport or precession.

6. Acronymal collisions and unrelated GM usages

Several arXiv usages of “GM” are unrelated to any gyromagnetic or rotational moment. This is clearest in gravitational, mathematical, and game-theoretic contexts.

In gyratonic pp-wave spacetimes, the relevant quantity is not a gyromagnetic moment but the gravitational angular-momentum density carried by the wave. The metric function $$\vec M=\gamma \vec L,\qquad \vec L=\frac{1}{\gamma}\vec M,$$9 encodes the rotational character of the null source, and in the TEGR Hamiltonian formalism the distinctive signature is a nonzero radial angular-momentum density of the gravitational field. The paper explicitly states that it does not discuss electromagnetic gyromagnetic moment, spin $\gamma$0-factor, or magnetic dipole moment (Carneiro et al., 2019).

In ring-laser proposals for measuring the Earth’s Lense–Thirring signal, “GM” refers to the gravitomagnetic effect rather than to any microscopic moment. The target observable is the tiny $\gamma$1-dependent relativistic correction to the beat frequency of a ring laser,

$\gamma$2

with the gravitomagnetic contribution about $\gamma$3 times smaller than the ordinary Earth-rotation Sagnac term (Virgilio et al., 2010).

In combinatorics and impartial games, the GM-rule is a deterministic update on ordered integer vectors: keep the rightmost minimal entry divisible by $\gamma$4 and reduce the remaining $\gamma$5 entries by $\gamma$6. Its central structural theorem is

$\gamma$7

and the paper states that “GM” is simply the notation for the rule, not an expansion such as gyromoment (Gurvich et al., 2023).

In algebraic geometry, GM stands for Gushel–Mukai. EPW varieties associated with strongly smooth ordinary GM surfaces and special GM threefolds are realized as moduli spaces of semistable objects in $\gamma$8 and $\gamma$9, for example

$\gamma=q/(2m)$0

Here the acronym has no relation to gyromagnetic response or rotational moments (Liu et al., 15 Dec 2025).

The literature therefore supports a strict contextual rule: “Gyromoment (GM)” should be read as a field-specific term only when the surrounding formalism makes that meaning explicit. In magnetization dynamics it is the gyromagnetic relation; in gyrokinetics it is the moment hierarchy; in metallic optical response it is the band magnetic moment underlying the GME; and in several other literatures it is not a moment at all.