Magnetic Multipole Currents in Electrodynamics

• Magnetic multipole currents are current-based representations of magnetic moments derived from circulating current distributions and higher-order tensor formulations.
• They are employed in classical electrodynamics, scattering theory, and nanophotonics, with applications ranging from magnetic resonance to optical and transport phenomena.
• Their study involves theoretical models, current-space hierarchies, and conservation laws that address ambiguities between source and field-based multipole descriptions.

Magnetic multipole currents are current-based source representations of magnetic multipole moments. In classical electrodynamics the canonical local object is the current-induced dipole density $\mathbf q_2=\mathbf r\times \mathbf j$, whose symmetrized and detraced integrals generate magnetic dipole, quadrupole, and higher Cartesian multipoles of arbitrary order (Niitsuma, 2013). In later source theories and condensed-matter transport formalisms, the same phrase is used for exact current-tensor decompositions of scatterers, for operator-valued currents carrying internal magnetic multipole moments, and for electric-current-induced magnetic multipoles in symmetry-broken media (Shevchenko et al., 22 Aug 2025, Tahir et al., 2022, Hayami, 2023).

1. Classical current-induced magnetic multipoles

A unified Cartesian source formulation starts from a bounded volume $V$ with closed surface $S$, outward normal $\mathbf n$, and a generalized polarization density $\mathbf Q(\mathbf r)$. The associated generalized polarization charge and current densities are

$$\rho_{\mathrm v}=-\nabla\cdot \mathbf Q,\qquad \rho_{\mathrm s}=\mathbf n\cdot \mathbf Q,$$

$$\mathbf j_{\mathrm v}=\nabla\times \mathbf Q,\qquad \mathbf j_{\mathrm s}=-\mathbf n\times \mathbf Q.$$

For magnetic multipole currents, the relevant branch is the current-induced one, built from

$\mathbf q_2=\mathbf r\times \mathbf j.$

The standard magnetic dipole density is $\frac12\,\mathbf q_2$, so $\mathbf q_2$ is twice the usual local magnetic dipole density before arbitrary-rank normalization is imposed (Niitsuma, 2013).

For rank $V$0, the nonsymmetric tensor $V$1 is symmetrized through

$V$2

and then detraced with the detracer operator $V$3, producing the symmetric traceless Cartesian tensor

$V$4

The resulting arbitrary-order current-induced relation is

$V$5

with

$V$6

For $V$7, this reduces to the familiar magnetic dipole identity including both volume and surface currents,

$V$8

For $V$9, the kernel becomes

$S$0

which is the Cartesian magnetic quadrupole analog (Niitsuma, 2013).

This formulation makes the physical distinction explicit. Charge-induced moments use $S$1; current-induced moments use $S$2. After integration and detracing, both constructions yield the same symmetric traceless tensor, but magnetic multipoles generated by circulating currents are represented by the current-induced branch.

2. Time-dependent sources, moving dipoles, and effective magnetic currents

In the quasistatic treatment of localized, time-dependent charge and current distributions, the magnetic field is decomposed as

$S$3

where the scalar $S$4 defines the magnetic multipole sector. The exact source term is the curl of the current density,

$S$5

and the magnetic multipole expansion is written with the formal effective magnetic charge density

$S$6

The arbitrary-order Cartesian magnetic moments are then

$S$7

At low order, the monopole vanishes,

$S$8

the magnetic dipole is

$S$9

and the magnetic quadrupole is

$\mathbf n$0

(Krynytskyi et al., 2018).

A distributional clarification appears for a uniformly moving point electric dipole. Its total current density is the convection current of the moving bound charge distribution, but it decomposes into a polarization current and a magnetization current,

$\mathbf n$1

with induced magnetic dipole moment

$\mathbf n$2

The paper shows explicitly that

$\mathbf n$3

and that the magnetic-dipole-type contribution is the divergenceless magnetization current rather than the full non-solenoidal total current. This resolves the factor-of-two ambiguity associated with applying $\mathbf n$4 to a current distribution with $\mathbf n$5 (Hnizdo, 2012).

A complementary classical theory studies electric current multipole moments generated by magnetic systems. For translational motion of magnetic multipoles, the induced electric dipole moment is

$\mathbf n$6

while the induced electric current quadrupole moment receives contributions from the moving magnetic quadrupole $\mathbf n$7 and the anapole $\mathbf n$8,

$\mathbf n$9

For changing current strength or changing orientation of magnetic moments, the electric field can be described with an effective magnetic current density, used purely as a formal device rather than as a physical magnetic-monopole current (Silenko, 2014).

3. Source design, current patterns, and scattering realizations

In magnetostatics, a pure two-dimensional multipole field can be generated by a single azimuthal harmonic of longitudinal current on a cylinder. For a cylindrical sheet current with angular dependence

$\mathbf Q(\mathbf r)$0

the interior field is exactly the order-$\mathbf Q(\mathbf r)$1 multipole,

$\mathbf Q(\mathbf r)$2

In the corresponding iron-dominated idealization, the pole surfaces follow

$\mathbf Q(\mathbf r)$3

so a pure multipole of order $\mathbf Q(\mathbf r)$4 has $\mathbf Q(\mathbf r)$5 poles (Wolski, 2011).

For time-harmonic full-wave fields, a spherical current layer or current sphere proportional to a multipole electric field generates that same multipole field in all space. Magnetic-type multipoles are realized by the TE current pattern

$\mathbf Q(\mathbf r)$6

with $\mathbf Q(\mathbf r)$7. This current is tangential, $\mathbf Q(\mathbf r)$8, divergence-free in the source region, and loop-like on spherical surfaces. The corresponding electric field is

$\mathbf Q(\mathbf r)$9

with $$\rho_{\mathrm v}=-\nabla\cdot \mathbf Q,\qquad \rho_{\mathrm s}=\mathbf n\cdot \mathbf Q,$$0 inside the source-free interior and $$\rho_{\mathrm v}=-\nabla\cdot \mathbf Q,\qquad \rho_{\mathrm s}=\mathbf n\cdot \mathbf Q,$$1 outside. The $$\rho_{\mathrm v}=-\nabla\cdot \mathbf Q,\qquad \rho_{\mathrm s}=\mathbf n\cdot \mathbf Q,$$2 TE multipole is explicitly described as similar to a current loop distribution (Farhi et al., 2017).

In optical nanomaterials and magneto-dielectric scatterers, the same source-centered logic is expressed through induced polarization and magnetization densities. For magnetic scatterers, the source terms are

$$\rho_{\mathrm v}=-\nabla\cdot \mathbf Q,\qquad \rho_{\mathrm s}=\mathbf n\cdot \mathbf Q,$$3

and magnetization contributes explicitly through the curl term in the electric-field wave equation,

$$\rho_{\mathrm v}=-\nabla\cdot \mathbf Q,\qquad \rho_{\mathrm s}=\mathbf n\cdot \mathbf Q,$$4

The resulting exact spherical multipole coefficients split into polarization and magnetization pieces,

$$\rho_{\mathrm v}=-\nabla\cdot \mathbf Q,\qquad \rho_{\mathrm s}=\mathbf n\cdot \mathbf Q,$$5

and the scattering efficiency similarly decomposes into pure-$$\rho_{\mathrm v}=-\nabla\cdot \mathbf Q,\qquad \rho_{\mathrm s}=\mathbf n\cdot \mathbf Q,$$6, pure-$$\rho_{\mathrm v}=-\nabla\cdot \mathbf Q,\qquad \rho_{\mathrm s}=\mathbf n\cdot \mathbf Q,$$7, and interference terms. In ferrite-based scatterers in Faraday geometry, the magnetic circular dichroism can be decomposed into individual multipole contributions, and multipole resonances associated with magnetization currents can be even stronger than multipole contributions from conventional dielectric currents (Hernández-Sarria et al., 23 Oct 2025).

4. Conservation laws and current-space hierarchies

A distinct line of work promotes multipole currents from source descriptors to conserved transport quantities. After Lorenz gauge fixing in Maxwell theory, the conserved Noether current associated with a residual gauge parameter $$\rho_{\mathrm v}=-\nabla\cdot \mathbf Q,\qquad \rho_{\mathrm s}=\mathbf n\cdot \mathbf Q,$$8 is

$$\rho_{\mathrm v}=-\nabla\cdot \mathbf Q,\qquad \rho_{\mathrm s}=\mathbf n\cdot \mathbf Q,$$9

with continuity equation

$$\mathbf j_{\mathrm v}=\nabla\times \mathbf Q,\qquad \mathbf j_{\mathrm s}=-\mathbf n\times \mathbf Q.$$0

In $$\mathbf j_{\mathrm v}=\nabla\times \mathbf Q,\qquad \mathbf j_{\mathrm s}=-\mathbf n\times \mathbf Q.$$1 form this becomes

$$\mathbf j_{\mathrm v}=\nabla\times \mathbf Q,\qquad \mathbf j_{\mathrm s}=-\mathbf n\times \mathbf Q.$$2

Here $$\mathbf j_{\mathrm v}=\nabla\times \mathbf Q,\qquad \mathbf j_{\mathrm s}=-\mathbf n\times \mathbf Q.$$3 is the multipole current density or flux density. The electric multipole charges are proportional to the usual electric multipole moments. A magnetic analog is introduced by duality,

$$\mathbf j_{\mathrm v}=\nabla\times \mathbf Q,\qquad \mathbf j_{\mathrm s}=-\mathbf n\times \mathbf Q.$$4

and in magnetostatics

$$\mathbf j_{\mathrm v}=\nabla\times \mathbf Q,\qquad \mathbf j_{\mathrm s}=-\mathbf n\times \mathbf Q.$$5

The same work states explicitly that a first-principles derivation of the magnetic charges remains open and does not provide a fully parallel magnetic hard/soft continuity equation (Seraj, 2016).

At the source-tensor level, exact current multipoles provide a different hierarchy. The exact current-multipole tensor is

$$\mathbf j_{\mathrm v}=\nabla\times \mathbf Q,\qquad \mathbf j_{\mathrm s}=-\mathbf n\times \mathbf Q.$$6

valid for arbitrary scatterer size and shape. In this framework, current multipoles are not divided into electric and magnetic types. Instead, classical magnetic multipoles arise only after linear transformation to the field-expansion coefficients $$\mathbf j_{\mathrm v}=\nabla\times \mathbf Q,\qquad \mathbf j_{\mathrm s}=-\mathbf n\times \mathbf Q.$$7. The lowest example is the magnetic dipole: $$\mathbf j_{\mathrm v}=\nabla\times \mathbf Q,\qquad \mathbf j_{\mathrm s}=-\mathbf n\times \mathbf Q.$$8

$$\mathbf j_{\mathrm v}=\nabla\times \mathbf Q,\qquad \mathbf j_{\mathrm s}=-\mathbf n\times \mathbf Q.$$9

Thus the magnetic dipole is the antisymmetric part of the current quadrupole tensor. The magnetic quadrupole is built from current octupoles, the magnetic octupole from current hexadecapoles, and, in general, every classical magnetic multipole moment can be written as a superposition of current multipole moments of order $\mathbf q_2=\mathbf r\times \mathbf j.$0. The same formalism retains nonradiating current configurations that are absent in the classical field-based expansion (Shevchenko et al., 22 Aug 2025).

5. Multipole currents in crystalline and quasiparticle systems

For Bloch quasiparticles, magnetic multipole moments can be defined as intrinsic, gauge-invariant wave-packet properties. The general Cartesian spin magnetic multipole moment of order $\mathbf q_2=\mathbf r\times \mathbf j.$1 is

$\mathbf q_2=\mathbf r\times \mathbf j.$2

In phosphorene subject to a perpendicular electric field, the low-energy valley quasiparticles host a spin magnetic octupole moment $\mathbf q_2=\mathbf r\times \mathbf j.$3. A linear electric-field response produces a nonequilibrium octupole density,

$\mathbf q_2=\mathbf r\times \mathbf j.$4

and for the parameters quoted in the paper the result is

$\mathbf q_2=\mathbf r\times \mathbf j.$5

A second-order response can drive the octupole current $\mathbf q_2=\mathbf r\times \mathbf j.$6. The same paper states that such multipole currents, similar to spin currents, are strictly speaking a conceptual tool rather than a well-defined observable, while the directly measurable consequence is corner-staggered spin accumulation (Tahir et al., 2022).

In quadrupole-ordered nematic systems, an externally applied electric current $\mathbf q_2=\mathbf r\times \mathbf j.$7 acts as a polar, time-reversal-odd vector. When combined with the rank-2 electric quadrupole order parameter $\mathbf q_2=\mathbf r\times \mathbf j.$8, the tensor product $\mathbf q_2=\mathbf r\times \mathbf j.$9 yields current-induced magnetic toroidal dipoles $\frac12\,\mathbf q_2$0, magnetic quadrupoles $\frac12\,\mathbf q_2$1, and magnetic toroidal octupoles $\frac12\,\mathbf q_2$2. The paper states this explicitly: $\frac12\,\mathbf q_2$3 For example, under $\frac12\,\mathbf q_2$4 nematic order, $\frac12\,\mathbf q_2$5 induces $\frac12\,\mathbf q_2$6 and $\frac12\,\mathbf q_2$7, $\frac12\,\mathbf q_2$8 induces $\frac12\,\mathbf q_2$9 and $\mathbf q_2$0, and $\mathbf q_2$1 induces $\mathbf q_2$2 and $\mathbf q_2$3 (Hayami, 2023).

A more direct transport theory of magnetic multipole currents appears in altermagnets. For the magnetic octupole current, the local operator is

$\mathbf q_2$4

and the conductivity tensor $\mathbf q_2$5 is rank five. The symmetry analysis shows that time-reversal-even electric-type multipoles give rise to the dissipationless MO current, whereas time-reversal-odd magnetic-type multipoles generate dissipative MO current under an applied electric field. The same work emphasizes that MO conductivity can appear even in cases where spin conductivity is forbidden (Takasu et al., 13 Dec 2025).

In $\mathbf q_2$6-wave altermagnets, the magnetic octupole Hall effect is defined as a transverse current of the operator

$\mathbf q_2$7

induced by an electric field through

$\mathbf q_2$8

The paper identifies a second-type magnetic octupole Hall effect that persists even in symmetries where the spin-splitter effect is forbidden, and interprets it as arising from spin-dependent orbital hybridization rather than from ordinary spin-split band structure (Ko et al., 1 Aug 2025).

6. Equivalences, ambiguities, and diagnostic issues

The literature uses the phrase in several distinct senses. This suggests distinguishing at least three levels: current distributions that generate magnetic multipole fields, conserved multipole-current densities in gauge-theoretic transport laws, and operator-valued currents that carry internal magnetic multipole moments in solids. These levels are mathematically connected but not interchangeable.

A recurring source of ambiguity is the relation between source-side and field-side classifications. In the generalized Cartesian theory, charge-induced and current-induced constructions become equivalent only after integration, symmetrization, and detracing, even though magnetic multipoles are physically generated by currents rather than by scalar charge density (Niitsuma, 2013). In exact current-space formalisms, the ambiguity is stronger: current multipoles are not divided into electric and magnetic types, and classical magnetic multipoles appear only after projection onto the radiative coefficients $\mathbf q_2$9 (Shevchenko et al., 22 Aug 2025).

A second ambiguity is nonuniqueness of the inverse problem. In current-based nanophotonics, the magnetic dipole coefficients are generated by antisymmetric current quadrupoles,

$V$00

while certain higher-order current moments can radiate exactly like lower-order multipoles. The same framework identifies perfectly dark current configurations, including the spherically symmetric excitation $V$01, which radiates no electromagnetic field (Grahn et al., 2012). A plausible implication is that “magnetic multipole current” cannot be read off uniquely from far-field data without supplementary information about geometry or material response.

A third issue is observability. In Bloch-band transport, multipole currents are introduced in close analogy with spin currents, but the formalism itself states that they are conceptual tools rather than well-defined observables; the experimentally accessible signatures are boundary accumulations, nonreciprocal responses, dichroic line shapes, or symmetry-selected Hall coefficients (Tahir et al., 2022). In gauge-theoretic electromagnetism, the electric multipole-current continuity law is explicit, whereas the magnetic counterpart is only partial, because the first-principles derivation of magnetic charges remains open (Seraj, 2016). In octupolar transport, the symmetry literature therefore treats magnetic multipole currents operationally, through response tensors and selection rules, rather than through a universal local conservation law (Takasu et al., 13 Dec 2025).

Taken together, these formulations present magnetic multipole currents not as a single primitive object, but as a family of current-based representations for magnetic multipolarity: $V$02 and its higher-rank generalizations in classical source theory, exact current tensors in scattering theory, magnetization-current multipoles in magnetic media, and transport currents of magnetic multipole operators in crystalline matter.