Antitoroidal Order Fundamentals

• Antitoroidal order is a staggered arrangement of vortex-like toroidal moments that cancel out globally while preserving key local symmetries.
• It manifests in metallic magnets, dielectric metamaterials, and artificial spin ice, each revealing unique experimental and simulation-based insights.
• Studies of antitoroidal order uncover novel exchange mechanisms, high-Q resonances, and topologically protected transport phenomena across diverse systems.

Searching arXiv for relevant papers on antitoroidal order and closely related toroidal/antitoroidic phenomena. Antitoroidal order denotes an ordered state in which local toroidal moments form a staggered pattern with vanishing net toroidal moment over the relevant unit cell or supercell, while retaining nontrivial broken-symmetry structure at the sublattice or plaquette level. Across the literature, the term appears in distinct but related settings: metallic magnetic systems with staggered vortex-like spin-plaquette moments, electromagnetic metamaterials whose neighboring trimers carry antiparallel toroidal dipoles, and artificial spin-ice arrays in which adjacent triangular loops alternate the sign of their local toroidal moment (Zyuzin, 26 Sep 2025, Dmitriev et al., 2021, Yue et al., 2024). In each case, the central feature is not the existence of a uniform toroidal vector, but rather an antiferro-like arrangement of toroidal building blocks. This places antitoroidal order at the intersection of magnetic symmetry, multipolar order, and topological or resonant phenomena.

1. Definition and symmetry content

In magnetic systems, a conventional toroidal order is associated with the vector

$T = \sum_i r_i\times m_i,$

which breaks time-reversal symmetry $\Theta$ and can coexist with inversion in certain patterns (Zyuzin, 26 Sep 2025). Antitoroidal order differs in that the local vortex-like moments are arranged so that the net toroidal moment vanishes within each magnetic unit cell, even though the pattern still breaks $\Theta$ on each sublattice. In the square-lattice example of Zyuzin, the two sublattices carry opposite vorticities of the local moment circulation; consequently, $\Theta: H_{\rm at}\to -H_{\rm at}$, while the combined symmetry $\tau\cdot\Theta$, with $\tau$ a one-site translation, restores the Hamiltonian (Zyuzin, 26 Sep 2025). No net magnetization or toroidal moment appears, but an odd-in-$k$ spin–momentum locking is symmetry-allowed.

In electromagnetic metamaterials, antitoroidal order is defined at the level of collective toroidal dipoles of resonant subunits. For a current distribution $j(r)$ in a volume $V$, the toroidal dipole moment is

$$\mathbf{T} \;=\;\frac{1}{10\,c}\int_V\bigl[(\mathbf{r}\cdot\mathbf{j})\,\mathbf{r}\;-\;2\,r^2\,\mathbf{j}\bigr]\,d^3r.$$

When six dielectric trimers are arranged in a hexagonal supercell, antitoroidal order corresponds to neighboring trimers carrying antiparallel $\Theta$0, so that the total toroidal moment of the supercell cancels and the collective mode is staggered (Dmitriev et al., 2021).

In direct-kagome artificial spin ice, the relevant local object is the toroidal moment of a triangular plaquette. Yue et al. define

$\Theta$1

with $\Theta$2, where the sign of $\Theta$3 distinguishes clockwise from counterclockwise circulation. The antitoroidal state, termed paratoroidic or antiferrotoroidic in the summary, consists of neighboring triangles alternating their vortex sign in a two-sublattice pattern on the dual honeycomb of loops (Yue et al., 2024).

A common misconception is to equate antitoroidal order with the absence of toroidal physics because the net toroidal moment vanishes. The cited works instead show that cancellation at the unit-cell level can coexist with robust local toroidal structure, nontrivial selection rules, and, in the metallic case, topological transport (Zyuzin, 26 Sep 2025, Dmitriev et al., 2021, Yue et al., 2024).

2. Microscopic and mesoscopic realizations

In metallic magnets, antitoroidal order is introduced as a theoretical model of a metallic magnetic system with a staggered arrangement of local vortex-like spin-plaquette moments. The microscopic mechanism is an indirect interaction of conducting fermions with localized spins based on tunneling processes of conducting fermions through the localized spins (Zyuzin, 26 Sep 2025). The minimal motif is a three-site problem with conducting fermions $\Theta$4 on two “black” sites $\Theta$5 of energy $\Theta$6, and localized-spin fermions $\Theta$7 on an intermediate “red” site $\Theta$8 of energy $\Theta$9 subject to a classical exchange field $\Theta$0: $\Theta$1 After integrating out the high-energy site in second-order perturbation theory, the effective $\Theta$2 tunnel amplitude contains a spin-dependent term proportional to $\Theta$3, and when embedded into a two-dimensional bipartite lattice with alternating vorticity, the spin-independent contributions from the two virtual paths cancel while the odd-in-$\Theta$4 term survives (Zyuzin, 26 Sep 2025).

In dielectric metamaterials, the realization is geometric and resonant rather than electronic. The elementary unit is a trimer of three high-$\Theta$5 dielectric disks whose circulating displacement currents produce three co-rotating in-plane magnetic dipoles forming a flux loop. This trimer supports a toroidal resonance that can dominate the far-field electric dipole via

$\Theta$6

Antitoroidal order emerges when such trimers are placed in a hexagonal supercell and the supercell symmetry is reduced so that the antisymmetric collective toroidal mode becomes excitable (Dmitriev et al., 2021).

In direct-kagome artificial spin ice, the realization is provided by nanomagnets on the edges of a kagome tiling. Each triangular plaquette can host a closed vortex with $\Theta$7, while broken loops with one spin flipped carry $\Theta$8 and are treated as defect units. The antitoroidal regime occurs at intermediate effective temperature, or equivalently weakened couplings via larger lattice constant, where all loops remain closed but neighboring triangles alternate their toroidal sign (Yue et al., 2024).

These realizations suggest that antitoroidal order is not tied to a single microscopic degree of freedom. It can arise from localized-spin geometry in metals, displacement-current vortices in photonic structures, or Ising-like macrospins in mesoscopic magnetic arrays.

3. Symmetry breaking and selection rules

The symmetry structure is especially explicit in the metamaterial setting. Starting from a hexagonal supercell of symmetry $\Theta$9, the one-dimensional irreducible representations relevant to out-of-plane polar vectors $\Theta: H_{\rm at}\to -H_{\rm at}$0 are $\Theta: H_{\rm at}\to -H_{\rm at}$1 and $\Theta: H_{\rm at}\to -H_{\rm at}$2. In the summary of Gorkunov et al., $\Theta: H_{\rm at}\to -H_{\rm at}$3 corresponds to the toroidal-order supermode, while $\Theta: H_{\rm at}\to -H_{\rm at}$4 corresponds to the antitoroidal-order supermode, which changes sign under $\Theta: H_{\rm at}\to -H_{\rm at}$5, $\Theta: H_{\rm at}\to -H_{\rm at}$6, and $\Theta: H_{\rm at}\to -H_{\rm at}$7 (Dmitriev et al., 2021). Under frontal illumination by an $\Theta: H_{\rm at}\to -H_{\rm at}$8- or $\Theta: H_{\rm at}\to -H_{\rm at}$9-polarized plane wave, the pure $\tau\cdot\Theta$0 eigenmodes are dark; to access them, the supercell symmetry must be broken to a subgroup compatible with the symmetry of the incident magnetic field.

The magnetic-group analysis introduces the 2D magnetic group $\tau\cdot\Theta$1 of the third category for the driving field $\tau\cdot\Theta$2, with symmetry elements

$\tau\cdot\Theta$3

A supermode may be excited only if all its symmetry elements are also symmetries of the combined structure and driving field. Under the reduction $\tau\cdot\Theta$4, the antitoroidal mode survives through the mapping $\tau\cdot\Theta$5, becoming weakly bright for the appropriate field orientation; further reduction to $\tau\cdot\Theta$6 allows access to both toroidal and antitoroidal orders depending on polarization (Dmitriev et al., 2021).

The metallic model uses a different symmetry mechanism. Antitoroidal order breaks time reversal but preserves the combined $\tau\cdot\Theta$7 symmetry, which protects band degeneracies when the ferromagnetic exchange $\tau\cdot\Theta$8 is absent (Zyuzin, 26 Sep 2025). This combined antiunitary symmetry is central to the existence of nodal points at $\tau\cdot\Theta$9 and to their subsequent gapping when $\tau$0 is turned on.

In artificial spin ice, the operative symmetry is expressed through a $\tau$1 toroidal inversion symmetry. The low-temperature ferrotoroidic state has uniform loop circulation, whereas the intermediate antitoroidal or paratoroidic state has zero global ferrotoroidicity while retaining fully developed local vortex integrity. The transition between these phases is reported as a spontaneous breaking of the $\tau$2 toroidal inversion symmetry, with a sharp low-temperature heat-capacity peak and behavior consistent with a continuous second-order Ising-like transition, although the paper does not report a full finite-size scaling study (Yue et al., 2024).

4. Effective Hamiltonians and order parameters

In the metallic case, the antitoroidal contribution to the Bloch Hamiltonian takes the form

$\tau$3

where $\tau$4, $\tau$5, and $\tau$6 (Zyuzin, 26 Sep 2025). The structure resembles Rashba spin-orbit coupling, but the extra factor of $\tau$7 distinguishes it from conventional Rashba SOC and ensures that under time reversal $\tau$8. The effective term therefore realizes odd-in-momentum spin–momentum locking while explicitly breaking time-reversal symmetry.

Adding a ferromagnetic exchange $\tau$9, nearest-neighbor hopping $k$0, and chemical potential $k$1 gives

$k$2

Diagonalization yields four bands

$k$3

with $k$4 (Zyuzin, 26 Sep 2025). At $k$5, Dirac crossings remain protected by $k$6; turning on $k$7 opens a full gap $k$8 at the nodal points.

In direct-kagome artificial spin ice, the minimal effective Hamiltonian is an Ising model truncated to three nearest-neighbor couplings: $k$9 For lattice constant $j(r)$0, micromagnetic simulation gives $j(r)$1 and $j(r)$2. The energy gaps to the first two excited vertex types are

$j(r)$3

with quasi-degeneracy $j(r)$4 yielding the emergent intermediate phase (Yue et al., 2024).

The same work introduces explicit order parameters for toroidicity and ferrotoroidicity. Defining a loop vorticity integer $j(r)$5 for fully closed vortices and $j(r)$6 for broken loops, one has

$j(r)$7

Here $j(r)$8 measures local vortex integrity and $j(r)$9 measures global toroidal alignment. In the antitoroidal regime, $V$0 while $V$1, cleanly separating local toroidal coherence from global cancellation (Yue et al., 2024).

5. Transport, topology, and resonant consequences

The metallic antitoroidal model predicts that interaction of conducting fermions with antitoroidal order results in odd-in-momentum spin–momentum locking and, when combined with ferromagnetism, an insulating anomalous Hall state without any intrinsic spin-orbit coupling (Zyuzin, 26 Sep 2025). In the two-dimensional insulating regime, with the chemical potential inside the gap, the anomalous Hall current is

$V$2

At $V$3 and $V$4, the Berry curvature is

$V$5

Numerical integration yields a nonzero $V$6 that vanishes as $V$7, reaches a maximum of order $V$8 when $V$9, and drops to zero once $$\mathbf{T} \;=\;\frac{1}{10\,c}\int_V\bigl[(\mathbf{r}\cdot\mathbf{j})\,\mathbf{r}\;-\;2\,r^2\,\mathbf{j}\bigr]\,d^3r.$$0, associated with gap closing and reopening (Zyuzin, 26 Sep 2025). The model is therefore described as realizing a Chern-insulator-like anomalous Hall response entirely without intrinsic SOC.

In the metamaterial context, the consequences are optical and resonant rather than transport-based. Antitoroidal order gives an odd staggered supermode with zero net toroidal moment per cell, but it can still be bright under symmetry-reduced excitation conditions (Dmitriev et al., 2021). Full-wave simulations show that the out-of-plane normal magnetic field $$\mathbf{T} \;=\;\frac{1}{10\,c}\int_V\bigl[(\mathbf{r}\cdot\mathbf{j})\,\mathbf{r}\;-\;2\,r^2\,\mathbf{j}\bigr]\,d^3r.$$1 loops clockwise in one trimer and counterclockwise in its neighbor, producing staggered near-field hotspots at the centers of individual trimers. Because the antitoroidal mode suppresses radiative loss, it exhibits higher quality factors than the toroidal-order mode in the reported structures (Dmitriev et al., 2021).

In artificial spin ice, the principal consequences are thermodynamic and domain-structural. The antitoroidal phase occupies an intermediate regime in which all loops remain closed while global toroidicity cancels, and it is separated from the low-temperature ferrotoroidic phase by a sharp low-$$\mathbf{T} \;=\;\frac{1}{10\,c}\int_V\bigl[(\mathbf{r}\cdot\mathbf{j})\,\mathbf{r}\;-\;2\,r^2\,\mathbf{j}\bigr]\,d^3r.$$2 anomaly in the heat capacity and from the high-temperature paramagnetic regime by a broader crossover around $$\mathbf{T} \;=\;\frac{1}{10\,c}\int_V\bigl[(\mathbf{r}\cdot\mathbf{j})\,\mathbf{r}\;-\;2\,r^2\,\mathbf{j}\bigr]\,d^3r.$$3 (Yue et al., 2024). This suggests that antitoroidal order can function as a genuine phase, not merely a fluctuating precursor.

6. Experimental and numerical manifestations

The three cited works employ different evidentiary modalities: theoretical band-structure analysis for metallic magnets, full-wave simulation and microwave experiment for dielectric metamaterials, and magnetic imaging plus Monte Carlo simulation for artificial spin ice.

System	Observable manifestation of antitoroidal order	Reported evidence
Metallic magnetic model	Odd-in-$$\mathbf{T} \;=\;\frac{1}{10\,c}\int_V\bigl[(\mathbf{r}\cdot\mathbf{j})\,\mathbf{r}\;-\;2\,r^2\,\mathbf{j}\bigr]\,d^3r.$$4 spin–momentum locking; insulating anomalous Hall effect when ferromagnetism is added	Band analysis and Berry-curvature integration (Zyuzin, 26 Sep 2025)
Dielectric trimer array	Alternating-sign $$\mathbf{T} \;=\;\frac{1}{10\,c}\int_V\bigl[(\mathbf{r}\cdot\mathbf{j})\,\mathbf{r}\;-\;2\,r^2\,\mathbf{j}\bigr]\,d^3r.$$5 in neighboring trimers; polarization-selective resonance	Full-wave simulations and microwave measurements (Dmitriev et al., 2021)
Direct-kagome ASI	Adjacent triangular loops with alternating $$\mathbf{T} \;=\;\frac{1}{10\,c}\int_V\bigl[(\mathbf{r}\cdot\mathbf{j})\,\mathbf{r}\;-\;2\,r^2\,\mathbf{j}\bigr]\,d^3r.$$6 sign	Monte Carlo and MFM imaging (Yue et al., 2024)

For the metamaterial arrays, the unperturbed $$\mathbf{T} \;=\;\frac{1}{10\,c}\int_V\bigl[(\mathbf{r}\cdot\mathbf{j})\,\mathbf{r}\;-\;2\,r^2\,\mathbf{j}\bigr]\,d^3r.$$7 structure hosts two dark supermodes at normalized wavelengths $$\mathbf{T} \;=\;\frac{1}{10\,c}\int_V\bigl[(\mathbf{r}\cdot\mathbf{j})\,\mathbf{r}\;-\;2\,r^2\,\mathbf{j}\bigr]\,d^3r.$$8 and $$\mathbf{T} \;=\;\frac{1}{10\,c}\int_V\bigl[(\mathbf{r}\cdot\mathbf{j})\,\mathbf{r}\;-\;2\,r^2\,\mathbf{j}\bigr]\,d^3r.$$9, with no transmission features for either polarization (Dmitriev et al., 2021). In a $\Theta$00 array with $\Theta$01, the antitoroidal resonance appears at $\Theta$02 with $\Theta$03 for $\Theta$04 and no feature for $\Theta$05; as $\Theta$06, $\Theta$07, consistent with a symmetry-protected bound state in the continuum (Dmitriev et al., 2021). In a $\Theta$08 array with $\Theta$09, both orders appear, with ATO at $\Theta$10, $\Theta$11 for $\Theta$12, and TO at $\Theta$13, $\Theta$14 for $\Theta$15 (Dmitriev et al., 2021).

The microwave experiments on ceramic-disk samples report two resonances in the $\Theta$16 band: $\Theta$17 with $\Theta$18, and $\Theta$19 with $\Theta$20. Near-field scanning shows alternating phase $\Theta$21 in adjacent trimers for ATO and a hotspot centered in the hexagon for TO. These results agree well with loss-included full-wave predictions $\Theta$22, $\Theta$23, and $\Theta$24, $\Theta$25 respectively (Dmitriev et al., 2021).

In direct-kagome ASI, Monte Carlo simulations on a $\Theta$26 kagome site array reproduce three regimes: ferrotoroidic, paratoroidic or antitoroidal, and paramagnetic (Yue et al., 2024). Experimentally, varying the lattice constant from $\Theta$27 to $\Theta$28 tunes the effective coupling. Magnetic force microscopy after thermal annealing and quench shows uniform contrast for small $\Theta$29, a mosaic of adjacent red and blue loops at intermediate $\Theta$30 constituting direct real-space evidence of alternating toroidal moments, and disordered broken loops at large $\Theta$31 (Yue et al., 2024).

7. Relation to toroidal order and broader significance

Antitoroidal order is best understood as the staggered counterpart of toroidal order. In toroidal order, local toroidal moments align to produce a net toroidal vector. In antitoroidal order, the same local building blocks are arranged antiparallel or with alternating vorticity, so the cell-averaged toroidal moment vanishes. The distinction is explicit in the metamaterial language of TO versus ATO supermodes, in the artificial-spin-ice distinction between ferrotoroidicity and paratoroidicity, and in the metallic distinction between local time-reversal breaking and restoration of a combined translation–time-reversal symmetry (Dmitriev et al., 2021, Yue et al., 2024, Zyuzin, 26 Sep 2025).

The physical significance of this distinction varies by platform. In metals, antitoroidal order supplies a route to momentum-odd exchange splittings in low-$\Theta$32 magnets where intrinsic SOC is weak, and when combined with uniform magnetization it yields an insulating anomalous Hall state with chiral edge modes according to the summary (Zyuzin, 26 Sep 2025). In metamaterials, antitoroidal order supports high-$\Theta$33, polarization-sensitive resonances with strong local field confinement and staggered near-field structure (Dmitriev et al., 2021). In artificial spin ice, it provides a directly imageable intermediate phase enabled by quasi-degenerate vertex energetics and tunable interaction scales (Yue et al., 2024).

A plausible implication is that antitoroidal order serves as a unifying concept across condensed-matter, photonic, and mesoscopic magnetic systems: the essential ingredient is not a uniform toroidal dipole, but an ordered pattern of local vortices whose cancellation at the global level coexists with sharply defined symmetry, collective response, and, in some realizations, topological functionality.