Magnetic Toroidal Dipole Order

• Magnetic toroidal dipole order is a ferroic state characterized by vortex-like magnetic moment arrangements that break spatial inversion and time-reversal symmetries while preserving PT symmetry.
• The order is evidenced through experimental techniques such as SHG spectroscopy and nonreciprocal transport measurements, highlighting its distinct magnetoelectric and optical responses.
• Engineered metamaterials and molecular systems demonstrate practical applications, including tunable spintronic devices and photo-induced control of toroidal moments.

Magnetic toroidal dipole order is an emergent class of ferroic order characterized by a vortex-like arrangement of magnetic moments. Unlike conventional magnetic or electric dipole orders, toroidal order simultaneously breaks spatial inversion (P) and time-reversal (T) symmetries but preserves their combination (PT). This unique symmetry leads to a range of unconventional magnetoelectric, optical, and transport properties, underpinning current interest across condensed matter physics, molecular magnetism, and nanophotonics.

1. Definition, Symmetry, and Order Parameter

The magnetic toroidal dipole moment $\mathbf{T}$ is formally defined, in the continuum, as

$$T_i = \frac{1}{2}\int d^3 r\,\left[\mathbf{r}\times\mathbf{M}(\mathbf{r})\right]_i$$

where $\mathbf{M}(\mathbf{r})$ is the local magnetization density. On a discrete lattice, this reduces to

$$\mathbf{T} = \frac{1}{2}\sum_n (\mathbf{r}_n \times \mathbf{m}_n)$$

with $\mathbf{m}_n$ being the magnetic moment at site $n$ and $\mathbf{r}_n$ its position vector.

Symmetry Properties:

• Under spatial inversion (P): $\mathbf{r} \to -\mathbf{r}$, $\mathbf{M}(\mathbf{r}) \to \mathbf{M}(-\mathbf{r})$, thus $\mathbf{T}\to-\mathbf{T}$.
• Under time reversal (T): $\mathbf{M}\to-\mathbf{M}$, $\mathbf{T}\to-\mathbf{T}$.
• The toroidal dipole is therefore P-odd, T-odd, but even under combined PT.

Typical real-space arrangements include closed loops of magnetic moments or spin vortex states (e.g., $120^\circ$ configurations on triangular motifs), resulting in vanishing net magnetization and polarization but a nonzero macroscopic toroidization.

2. Microscopic Realizations and Theoretical Models

2.1 Localized-Spin Models and Lattice Geometry

Ferro-toroidal order arises naturally in noncentrosymmetric antiferromagnets where noncollinear or noncoplanar spin arrangements are stabilized, often by frustrated exchange and relativistic antisymmetric interactions: $$\mathcal{H} = \sum_{\langle ij\rangle} J\,\mathbf{S}_i\cdot\mathbf{S}_j + \sum_{\langle \langle ij \rangle \rangle} \mathbf{D}_{ij} \cdot (\mathbf{S}_i \times \mathbf{S}_j) + \sum_{\langle ij \rangle} \Gamma S_i^\alpha S_j^\beta$$ Here, $J$ is the Heisenberg exchange, $\mathbf{D}_{ij}$ denotes the Dzyaloshinskii-Moriya (DM) vectors (induced by antisymmetric spin-orbit coupling), and $\Gamma$ captures anisotropic exchange (e.g., easy-axis). In zigzag or honeycomb lattices, these terms can stabilize Néel antiferromagnetism while simultaneously producing a uniform toroidal dipole via sublattice and real-space offsets (Hayami et al., 2015).

2.2 Molecular and Atomic-Scale Systems

In molecular single-molecule toroics (SMTs) such as Fe$_{10}$Dy$_{10}$ rings, the operator

$$\boldsymbol{\tau} = \sum_i (\mathbf{r}_i \times \mathbf{M}_i)$$

quantifies the toroidal moment associated with a vortex-like arrangement of local spins and orbital currents (Soncini et al., 5 Sep 2025). Tight-binding models for atomic-scale toroidal dipoles employ multi-orbital frameworks where inter-orbital hybridization (e.g., $d$–$f$ or $s$–$p$ mixing) and local symmetry breaking generate nonzero expectation values of operators like

$$\hat{T}_x = -\sigma_y\tau_x,\quad \hat{T}_y = \sigma_x\tau_x,\quad \hat{T}_z = -\sigma_0\tau_y$$

where $\tau_{x,y}$ are orbital pseudospin Pauli matrices (Yatsushiro et al., 2019, Hayami et al., 2017).

2.3 Metamaterials and Nanostructures

Toroidal dipole order can also be engineered in artificial structures, including:

• Arrays of nanomagnets: Rings of uniformly magnetized bars or vortex-hosting single-domain triangles yield spontaneous long-range toroidization, accessible via micromagnetic simulations and magnetic force microscopy (Lehmann et al., 2022).
• All-dielectric metasurfaces: Clustered arrays (e.g., trimers or hexagonal supercells) can host symmetry-protected toroidal dipole resonances, with their mode structure tunable by lattice geometry and symmetry breaking (Dmitriev et al., 2020, Dmitriev et al., 2021).

3. Phenomenological Landau Theory and Free Energy

The toroidal dipole $\mathbf{T}$ is promoted to an order parameter in a Landau expansion: $$F[\mathbf{T},\mathbf{E},\mathbf{H},\epsilon] = \frac{a}{2}|\mathbf{T}|^2 + \frac{b}{4}|\mathbf{T}|^4 - \alpha_{ij}E_i T_j - \beta_{ij}H_i T_j - \gamma_{ij,kl} \epsilon_{ij}T_k + \ldots$$ Bilinear couplings to electric and magnetic fields arise due to the $P$-odd, $T$-odd character of $\mathbf{T}$:

• Magnetoelectric (ME) tensor: $$\alpha_{ij} = \partial^2 F/\partial H_i \partial E_j$$
• Key symmetry: PT-even (only off-diagonal elements remain for many $PT$-symmetric point groups, e.g., $\alpha_{yz} = -\alpha_{zy} \propto T_x$).

This predicts experimental signatures such as the appearance of linear ME effects, with polarization components $P_i=\alpha_{ij}H_j$ nonzero only if the toroidal moment is allowed by the symmetry of the phase (Hayami et al., 2017, Wang et al., 16 Sep 2025, Hayami et al., 2022, Harris, 2010).

4. Experimental Realizations, Detection, and Domain Control

4.1 Direct Detection

Macroscopic toroidal order has been observed in molecular systems such as Fe$_{10}$Dy$_{10}$ rings, where toroidal polarization results in measurable responses to spatially inhomogeneous magnetic fields (nonzero curl) and is characterized thermodynamically by the toroidal susceptibility $\xi$: $$\xi_{\alpha\beta} = -\frac{\partial^2 F}{\partial(\nabla\times \mathbf{B})_\alpha \partial(\nabla\times \mathbf{B})_\beta}|_{\nabla\times \mathbf{B}=0}$$ Finite-temperature ground-state toroidal polarization is accessible via focused field gradients or femtosecond laser-generated current curls (Soncini et al., 5 Sep 2025).

4.2 Magnetoelectric and Optical Measurements

In van der Waals antiferromagnets like bilayer CrSBr, the layer-antiferromagnetic order simultaneously encodes a well-defined in-plane toroidal moment, which can be mapped using magnetic-field-dependent second-harmonic generation (SHG) spectroscopy. The decomposition into time-reversal-odd (toroidal) and even (polar) tensor components enables quantitative tracking of both order parameters and their field manipulation, including domain wall motion under combined in-plane electric and out-of-plane magnetic fields (Wang et al., 16 Sep 2025).

4.3 Transport and Nonreciprocal Phenomena

Spontaneous toroidal ordering modifies magnon band structures, leading to asymmetric (nonreciprocal) transport of spin and heat (e.g., shifted magnon band bottoms and valley splitting in zigzag/honeycomb antiferromagnets (Hayami et al., 2015)). Macroscopic detection includes nonreciprocal spin/thermal conductivity and optical directional dichroism.

4.4 Metamaterials and Nano-optics

Symmetry-protected toroidal dipole modes in trimer-based metasurfaces can achieve sharp Fano resonances and high field confinement. Selection rules for toroidal versus antitoroidic orders are governed by point group reductions ($C_{3v}\rightarrow C_s$, $C_{6v}\rightarrow C_{2v}/C_s$), and experimental signals include strong circular dichroism, nonreciprocal transmission, and field-enhanced sensing (Tuz et al., 2023, Dmitriev et al., 2021).

5. Material and Design Platforms

Natural materials:

• Zigzag chain compounds (e.g., $\alpha$–Cu$_2$V$_2$O$_7$), honeycomb magnets (MnPS$_3$, MnPSe$_3$), and metallic partially-disordered toroidal antiferromagnets (Ce$_3$TiBi$_5$) exhibit static or field-tunable toroidal dipole order (Hayami et al., 2015, Hayami et al., 2022).
• Chiral-lattice magnets (BaCoSiO$_4$) realize field-switchable ferritoroidal and ferrotoroidal states, enabling direct encoding and erasure of toroidal bits (Ding et al., 2021).

Engineered systems:

• Mesoscale arrays of bar magnets or vortex-hosting nanomagnets (e.g., rings or triangles) provide platforms for controlling domain formation and domain wall types (intra- and inter-toroidal) (Lehmann et al., 2022).
• Dielectric and magnetic metasurfaces can be tuned in situ from radial to toroidal order via bias magnetic fields, with clear control over field localization and resonance morphology (Tuz et al., 2023).

Molecular systems:

• Large molecule toroics—Fe$_{10}$Dy$_{10}$, prototypical Dy$_3$—offer enhanced toroidal response, substantial ground-state polarization, and quantifiable response functions ($\xi$, $|\tau|$) for quantum technological applications (Soncini et al., 5 Sep 2025).

6. Photoinduced and Floquet Engineering of Toroidal Dipole Order

Time-dependent electromagnetic fields can drive systems into nonequilibrium toroidal states. Using high-frequency Floquet formalism, it is shown that irradiating $s$–$p$ hybridized atoms with intense optical fields drives effective static couplings to the toroidal dipole, with the interaction

$$H^{(2)}_{\rm toroidal} = \frac{E B}{4 \Omega^2}\left(3\Delta T_z - \sqrt{2}\lambda T_z^{(s)}\right)$$

establishing conditions for photo-induced toroidal moments independent of the field's polarization, as long as $E\perp B$ (Hayami et al., 2023). Detection routes include optical pump-probe or magneto-optical Kerr experiments.

7. Outlook and Applications

Magnetic toroidal dipole order provides a symmetry-distinct ferroic channel intrinsically associated with antisymmetric magnetoelectric tensors and nonreciprocal effects. Applications leveraging these properties span nonvolatile toroidic memory, tunable spintronic and magnonic devices, high-sensitivity sensors, and ultrahigh-Q photonic components. Toroidal order also supports the realization of nonradiating (anapole) states and multi-band multipolar interference for advanced meta-atom design (Li et al., 2018). Recent advances in van der Waals magnets, molecular ring systems, and photonic metamaterials underscore the broad relevance and engineering flexibility of toroidal dipole ordering frameworks.