Toroidal Multipole Order

• Toroidal multipole order is defined by symmetry-adapted electric-toroidal and magnetic-toroidal moments that capture complex current distributions beyond conventional multipoles.
• It enables magnetoelectric, optical, and transport phenomena in materials by mediating interactions between electric and magnetic fields.
• Theoretical and first-principles studies reveal its role in symmetry-complete descriptions, guiding experimental probes in advanced quantum and classical systems.

Toroidal multipole order encompasses a family of microscopic order parameters—electric-toroidal and magnetic-toroidal multipoles—distinct from conventional electric and magnetic multipoles, arising in systems where spatial-inversion and time-reversal symmetries are configured appropriately. Toroidal multipoles, which naturally appear in both quantum and classical contexts, underlie a broad variety of physical phenomena including cross-correlated transport, magnetoelectric effects, hidden ordering (especially in complex oxides and f-electron materials), electromagnetic scattering, and also play a central role in the theoretical multipole completeness for describing local order in crystalline and molecular-scale systems. They can be rigorously defined as symmetry-adapted, higher-order moments of the current distribution, capturing degrees of freedom inaccessible to pure electric or magnetic order, and are now regarded as genuine and independent order parameters in the classification of material phases and in electromagnetic response theory.

1. Formal Definitions and Operator Structure

From a symmetry and operator-theoretical viewpoint, toroidal multipoles are classified as electric-toroidal (ET, $G_{\ell m}$) or magnetic-toroidal (MT, $T_{\ell m}$), depending on their parity under spatial inversion ($\mathcal{P}$) and time reversal ($\mathcal{T}$). For a static or slowly varying charge and current distribution, the multipole expansion includes, at each rank $\ell$, all four types:

• Electric multipole ($Q_{\ell m}$): $\mathcal{P}=(-1)^\ell$, $\mathcal{T}=+$,
• Magnetic multipole ($M_{\ell m}$): $\mathcal{P}=(-1)^{\ell+1}$, $\mathcal{T}=-$,
• Electric-toroidal multipole ($G_{\ell m}$): $\mathcal{P}=(-1)^{\ell+1}$, $\mathcal{T}=+$,
• Magnetic-toroidal multipole ($T_{\ell m}$): $\mathcal{P}=(-1)^{\ell}$, $\mathcal{T}=-$.

In Cartesian tensor notation, the general formula for the $\ell$th-rank toroidal moment in terms of the current density $\mathbf{j}(\mathbf{r},t)$ is

$$T_{i_1\cdots i_\ell}(t)=\frac{1}{2\,c\,(\ell+1)(2\ell+3)}\int d^3r\,\Big[2(\mathbf{r}\cdot\mathbf{j})\,r_{i_1}\cdots r_{i_\ell}-(\ell+3)r^2 j_{i_1} r_{i_2}\cdots r_{i_\ell}\Big]$$

with $c$ the speed of light (Krynytskyi et al., 2018, Gurvitz et al., 2018). Dipole, quadrupole, and higher-order explicit forms are tabulated and analyzed in the literature (Gurvitz et al., 2018, Li et al., 2018). Operator constructions for electronic systems invoke both one-body and many-body irreducible tensor decompositions (Kusunose et al., 2020, Kuniyoshi et al., 11 Mar 2026).

In lattice or atomic systems, second-quantized analogues couple orbital and spin degrees of freedom using Clebsch–Gordan algebra, resulting in systematic classification tables for atomic/molecular, site-cluster, bond-cluster, and collective toroidal multipole operators (Kusunose et al., 2020, Hayami et al., 2017, Xie et al., 19 Dec 2025). For instance, the spinful rank-1 ET operator is $$\frac{1}{\sqrt{2}}(\boldsymbol{\sigma}\times\hat{\mathbf{l}})$$ (Kusunose et al., 2020).

2. Symmetry Classification and Selection Rules

Toroidal multipole orders are classified by their transformation properties under $\mathcal{P}$, $\mathcal{T}$, and the relevant point group of the system (Kusunose et al., 2024, Hayami et al., 2017). The existence and allowed rank of toroidal multipole order are determined by the point-group decomposition:

• Electric-toroidal monopole ($G_0$) is permitted only in chiral ($\mathcal{P}$-odd) groups (11 crystallographic types) (Kusunose et al., 2024).
• Electric-toroidal dipole ($G$) is allowed in ferroaxial classes (10 crystallographic types lacking a mirror normal to a unique axis).
• The complete toroidal multipole family is essential for the symmetry-adapted completeness of the electronic order-parameter basis in materials (Kusunose et al., 2020).

In systems with magnetic symmetry, both ET and MT multipoles of arbitrary rank become active, leading to richer selection and coupling rules (Hayami et al., 2017, Kusunose et al., 2024). The multipole ranks split into irreducible representations of the point group: e.g., in $O_h$, the toroidal 16-pole ($\ell=4$) includes $T_{1g}$ and $T_{2g}$ components, which can acquire a finite expectation value under symmetry lowering or applied fields (Saito et al., 9 Jan 2026).

3. Microscopic Mechanisms and Realizations

Microscopically, toroidal multipole order arises in several archetypal settings:

• Parity-mixing and local inversion breaking: Spontaneous toroidal order can be stabilized in multiorbital models with local parity mixing, as in the extended Kondo lattice, producing odd-parity magnetic-toroidal and quadrupole phases with magnetoelectric responses (Hayami et al., 2015).
• Hybridized orbitals: In single-site or cluster models involving hybridized orbitals of angular momentum differing by one ($l \leftrightarrow l'\pm1$), toroidal multipoles constitute primary order parameters inaccessible to pure $l$-orbital manifolds (Hayami et al., 2017, Kusunose et al., 2020). Many-body toroidal monopoles, previously forbidden in noninteracting (one-body) sectors, emerge in interacting (two-body) spaces (Kuniyoshi et al., 11 Mar 2026).
• Cluster multipoles: In trimer or higher clusters, noncollinear arrangements (e.g., $120^\circ$ patterns of quadrupoles or octupoles) realize collective ET or MT multipole order, as seen in Ca$_{5}$Ir$_{3}$O$_{12}$ (Hayami et al., 2023).
• First-principles analyses: Symmetry-adapted Wannier and tight-binding decompositions allow direct extraction and quantification of toroidal multipole operators and their impact on phase transitions, e.g., identification of dominant electric-toroidal octupole order in ferroaxial K$_2$Zr(PO$_4$)$_2$ (Xie et al., 19 Dec 2025).

4. Physical Manifestations and Cross-Correlations

When a toroidal multipole order parameter condenses, it enables correlated responses and measurable signatures:

• Magnetoelectric and cross-correlation effects: Toroidal multipoles couple to electric and magnetic fields, inducing magnetoelectric, magnetoelastic, or electroelastic responses. For example, MT dipole order induces a linear magnetoelectric tensor $\alpha_{ij}$, while ET order permits cross-channel elasticity and polarization (Hayami et al., 2017, Hayami et al., 2015).
• Transport and optical phenomena: In ferromagnets and correlated metals, toroidal multipoles contribute to nonlinear Hall effects, nonreciprocal charge transport, and optical activity (e.g., second-harmonic generation, Kerr effect, directional dichroism) (Hayami, 2023, Kusunose et al., 2024, Saito et al., 9 Jan 2026).
• NMR and NQR spectral signatures: Odd-parity toroidal orders imprint unique splittings or selection rules in NMR/NQR spectra, with differentiable fingerprints between ET and MT multipoles (Yatsushiro et al., 2020).
• Hidden orders: Cryptic magnetic or non-dipolar phases, undetectable via ordinary dipolar probes but evident in advanced spectroscopy (e.g., in Ca$_5$Ir$_3$O$_{12}$), have been elucidated as coexisting toroidal multipolar ordered phases (Hayami et al., 2023).
• Neutron star deformations: In plasma physics and astrophysics, poloidal-toroidal fields of arbitrary multipole order (twisted tori) are constructed, with the toroidal component confined analytically, impacting density perturbation and ellipticity (gravitational wave emission) in neutron stars (Mastrano et al., 2015).

5. Electrodynamics and Nanophotonics: Pure Toroidal Sources and Anapoles

Toroidal multipoles constitute a distinct set of current distributions with unique electromagnetic properties:

• Pure toroidal sources: It is now established that current distributions exist which support purely toroidal moments, with all electric and magnetic multipoles vanishing ($\nabla\cdot\mathbf{J}=0$, $\mathbf{J}\cdot\hat{n}=0$ on boundaries, $\mathbf{r}\cdot(\nabla\times\mathbf{J})=0$) (Valero et al., 2022). Experimental verification via broadband antennas and nanostructures demonstrates their independent excitation.
• Anapole states: Interference between electric and toroidal multipoles of the same rank yields nonradiating (anapole) configurations. The general cancellation condition is $a_{E,\,\text{d}}+a_{E,\,\text{t}}=0$ for each $(\ell, m)$, or $\mathbf{p}+ik\mathbf{T}=0$ for dipoles (Li et al., 2018, Valero et al., 2022, Gurvitz et al., 2018).
• Multipole completeness: Toroidal multipoles, including their higher mean-square-radius (MSR) corrections, are required for a full expansion of quasi-static and radiative electromagnetic fields (Krynytskyi et al., 2018). This systematic approach informs the design of nonradiating or field-enhancing nanostructures, metamaterials, and near-field sensing devices (Gurvitz et al., 2018).

6. Quantum and Many-Body Aspects

In quantum contexts, the toroidal dipole operator is self-adjoint, with quantized eigenvalues forming the basis for a toroidization order parameter (Dolineanu et al., 2022). Many-body generalizations identify two-body toroidal monopoles as new symmetry-adapted collective order parameters in spinless electrons, accessible only in interacting systems and tightly controlled by Clebsch–Gordan and Wigner–Eckart algebra (Kuniyoshi et al., 11 Mar 2026). This opens routes to exploring toroidal order in correlated quantum phases and to engineering topologically protected, nonradiative quantum states.

7. Experimental Probes and Candidate Materials

Numerous experimental techniques directly probe and quantify toroidal order:

• Magnetoelectric and Kerr measurements: Detection of linear and nonlinear responses controlled by toroidal multipoles in candidate magnets and metals (Kusunose et al., 2024, Hayami, 2023).
• Optical and x-ray spectroscopy: Second-harmonic generation, Kerr and Faraday rotation, and x-ray dichroism can detect both MT and ET orders, as in RbFe(MoO$_4$)$_2$, Ca$_5$Ir$_3$O$_{12}$, and Ca$_2$RuO$_4$ (Kusunose et al., 2024, Hayami et al., 2023).
• Strain and external field control: Manipulation of ferroaxial or chiral toroidal order via pressure, uniaxial strain, and applied electrical or magnetic fields, as in ferroaxial transitions in K$_2$Zr(PO$_4$)$_2$ (Xie et al., 19 Dec 2025).
• Multipole decomposition of Hamiltonians: First-principles, Wannier-based Hamiltonian decompositions allow quantification and tuning of toroidal multipole contributions to key properties, such as the anomalous Hall effect in bcc-Fe (Saito et al., 9 Jan 2026).

8. Controversies and Conceptual Clarifications

Some prior works proposed that toroidal multipoles are not fundamentally distinct, but merely higher-order corrections to electric multipole expansions (Fernandez-Corbaton et al., 2015). However, modern analyses employing complete multipole bases, symmetry-adapted definitions, and both classical and quantum operator frameworks now conclusively establish the independent status of toroidal multipoles, both in physical principle and in experimental realization (Valero et al., 2022, Krynytskyi et al., 2018, Kusunose et al., 2020).

In summary, toroidal multipole order is a well-defined, symmetry-adapted, and physically realized class of order parameters and moments, fundamental to the thorough classification and understanding of complex electronic, magnetic, and electrodynamic phenomena across a diverse range of materials and quantum systems.