---
title: 'Electrotoroidicity: Electric Vortex Order'
url: https://www.emergentmind.com/topics/electrotoroidicity
type: topic
---

# Electrotoroidicity: Electric Vortex Order

Electrotoroidicity denotes the presence of an electric toroidal moment, equivalently electric vortices in the polarization field, and the ordered phenomena associated with that moment in condensed matter and structured electromagnetic systems. In a continuum description, the electrical toroidal moment is written as
\[
\mathbf{G}=\frac{1}{2V}\int \big(\mathbf{r}\times \mathcal{P}(\mathbf{r})\big)\,d^3r,
\]
with \(\mathcal{P}(\mathbf{r})\) the local polarization field and \(V\) the volume. Across the literature, the term spans several closely related usages: vortex states in nanoscale ferroelectrics, electric-toroidal dipole order in ferroaxial materials, and electrodynamic toroidal excitations in metamaterials and pulsed fields. These usages are unified by the same toroidal geometry, but they do not always carry the same symmetry assumptions or thermodynamic status [1306.2189, 2403.09492, 1504.04958].

## 1. Definitions, symmetry, and scope

The electric toroidal moment belongs to the family of toroidal multipoles, distinct from conventional electric and magnetic multipoles. In the symmetry-based classification of electronic multipoles, electric-toroidal (ET) multipoles \(G_{lm}\) are associated with vortex-like arrangements of electric moments, while magnetic-toroidal (MT) multipoles \(T_{lm}\) correspond to vortex-like arrangements of magnetic moments. For dipoles, the cluster expressions are
\[
\mathbf{G}\propto \sum_j \mathbf{R}_j\times \mathbf{Q}_j,\qquad
\mathbf{T}\propto \sum_j \mathbf{R}_j\times \mathbf{M}_j,
\]
and the atomic-scale ET dipole operator can be written as \(\mathbf{G}\propto \mathbf{l}\times \boldsymbol{\sigma}\) [2403.09492].

A central terminological issue is that “electrotoroidicity” is not used identically in all subfields. In the nanoscale ferroelectric and optical-activity literature, it refers directly to the existence of an electrical toroidal moment or electric vortices [1306.2189]. In the ferroaxial literature, the electric toroidal dipole is treated as an axial vector that is even under both inversion \((\mathcal{I})\) and time reversal \((\mathcal{T})\), and it serves as the order parameter for ferroaxial transitions in materials that preserve global inversion symmetry [2407.08369]. By contrast, the thermodynamic review literature emphasizes that, in bulk, electric toroidization does not by itself define an ordinary symmetry-breaking bulk phase transition, whereas confined nanostructures can stabilize electric-vortex states and associated toroidal moments [1504.04958]. This divergence suggests that the word designates a family of toroidal-electric phenomena rather than a single universally standardized order parameter.

The distinction from magnetic toroidicity is also fundamental. Magnetic toroidal order is a composite of electric configuration and magnetic moments, breaks both spatial inversion and time reversal, and underlies linear magnetoelectric effects in canonical ferrotoroidic materials. Electric toroidal order, in the modern ETD/ferroaxial sense, is instead time-reversal even and can persist in globally centrosymmetric systems through locally inversion-broken sub-units [2403.09492, 2407.08369].

## 2. Microscopic formulations and order parameters

Two complementary formulations recur in the literature. The first is the classical vortex expression,
\[
\vec{G}^{(c)}\sim \sum_i \vec{R}_i\times \vec{P}_i,
\]
which encodes a net toroidal dipole generated by a vortex arrangement of local electric dipole moments \(\vec{P}_i\). The second is an atomic-scale quantum expression,
\[
\vec{G}\sim \vec{l}\times \vec{s},
\]
used in first-principles multipole analyses to extract electric-toroidal dipole moments from the density matrix. In the latter framework, the tensor component \(w^{111}_t\) is proportional to \(\vec{l}\times \vec{s}\), while \(w^{101}_t\) is used for local electric dipole moments entering the classical vortex construction [2407.08369].

A model realization of electric-toroidal order appears in the two-dimensional three-orbital square-octagon model introduced for antisymmetric thermopolarization. There, each site hosts \(s\), \(p_x\), and \(p_y\) orbitals, with local dipoles
\[
\mathbf{P}_i=(P_i^x,P_i^y),\qquad P_i^\gamma \propto s_i p_{\gamma,i}+\mathrm{H.c.},
\]
and dipole-dipole interactions
\[
\mathcal{H}_{\mathrm{int}}=\sum_{ij}J_{ij}\Big[\mathbf{P}_i\cdot \mathbf{P}_j-3(\mathbf{P}_i\cdot \mathbf{e}_{ij})(\mathbf{P}_j\cdot \mathbf{e}_{ij})\Big].
\]
The ferro-toroidal order parameter is defined by
\[
G=\frac{1}{N}\sum_{p=1}^4 \hat{\mathbf e}_p\cdot \sum_{i\in p}\mathbf P_i,
\]
and the mean-field ground state exhibits a vortex-like configuration of local electric dipoles [2112.06387].

First-principles studies of ferroaxial materials sharpen the microscopic picture. In NiTiO\(_3\) and K\(_2\)Zr(PO\(_4\))\(_2\), the electric toroidal dipole acts as the order parameter for ferroaxial transitions. Both materials preserve global inversion symmetry, yet contain inversion-symmetry-broken sub-units that generate vortices of local electric dipole moments. In NiTiO\(_3\), the low-temperature ferroaxial phase exhibits nonzero and oppositely signed atomic-site ETDs at Ni and Ti sites with unequal magnitudes, producing a net ferri-type ETD; in K\(_2\)Zr(PO\(_4\))\(_2\), ETD vanishes in the high-temperature non-ferroaxial phase and appears with the structural rotation in the low-temperature phase. In both systems, spin-orbit coupling is essential: when SOC is turned off, the atomic-site ETD \(w^{111}_0\) vanishes at all sites [2407.08369].

## 3. Coupling to polarization, strain, and natural optical activity

A defining result for electrotoroidic media is that natural optical activity can emerge from the linear response of the electric toroidal moment to an applied electric field. The gyrotropy tensor is related to the spatial-dispersion tensor by
\[
g_{mk}=\frac{\omega}{2c}e_{ijm}\gamma_{ijk},
\]
and, in the absence of spontaneous magnetization and with negligible quadrupole contributions, it reduces to
\[
g_{mk}=\frac{4\pi\omega}{c}\left(\frac{dG_m}{dE_k}-\frac{dG_l}{dE_l}\delta_{mk}\right).
\]
Optical activity therefore exists when the electrical toroidal moment responds linearly to the applied electric field [1306.2189].

The corresponding phenomenology is encoded in the free energy
\[
F=F_0+\zeta_{ijkl}G_iG_j\eta_{kl}+\lambda_{ijkl}G_iG_jP_kP_l+q_{ijkl}P_iP_j\eta_{kl}-h_iG_i,
\]
where \(h_i=(\nabla\times \mathbf{E})_i\) is the field conjugate to \(G_i\). The analysis shows that linear response and gyrotropy occur only if the system breaks further symmetry either by possessing a spontaneous polarization coupled to the toroidal moment through the \(G_iG_jP_kP_l\) term, or by being piezoelectric with coupling between toroidal moment and strain. For the BaTiO\(_3\):SrTiO\(_3\) nanocomposite studied atomistically, the relevant mechanism is the coupling between electric vortices and a spontaneous polarization along the nanowire axis [1306.2189].

Effective-Hamiltonian plus molecular-dynamics simulations provide the microscopic mechanism. The system consists of BaTiO\(_3\) nanowires with electric vortices in the \(xy\) plane embedded in a (Ba,Sr)TiO\(_3\) medium, with spontaneous polarization along \(z\). An \(ac\) electric field along \(z\) causes a linear decrease or increase of \(G_z\), tightly correlated with changes in \(P_z\). As \(E_z\) increases in the positive direction, the \(x\) and \(y\) dipole components responsible for the vortices are suppressed, \(P_z\) increases, and antivortices in the medium are eliminated. Reversing the field switches the polarization and reverses the sign of the gyrotropic coefficients \(g_{11}\) and \(g_{22}\), thereby inducing a reversible transition between dextrorotatory and laevorotatory optical activity. The temperature dependence of \(g_{11}\) is fitted by
\[
g_{11}(T)\sim \frac{A}{\sqrt{(T_C-T)(T_G-T)}},
\]
with \(T_C\) the temperature where polarization vanishes and \(T_G\) the temperature where the toroidal moment vanishes [1306.2189].

## 4. Ferroaxial order, local inversion breaking, and chirality

In ferroaxial materials, electrotoroidicity is tied to rotational structural order rather than to macroscopic ferroelectricity. NiTiO\(_3\) undergoes an order-disorder transition from \(R\bar{3}c\) to \(R\bar{3}\), while K\(_2\)Zr(PO\(_4\))\(_2\) undergoes a displacive transition from \(P\bar{3}m1\) to \(P\bar{3}\). In both cases the ETD reverses sign between ferroaxial domains and tracks the structural order parameter. The local dipole vortices arise in inversion-broken clusters that are paired by global inversion, so the crystals remain globally centrosymmetric even as they host a nonzero net ETD. Hidden spin polarization accompanies these local vortices: the band structure remains globally doubly degenerate because global \(\mathcal I\) and \(\mathcal T\) are preserved, but each inversion-broken sub-unit contributes an opposite local spin polarization, producing hidden spin textures whose vortex sense correlates with the ETD domain [2407.08369].

A distinct but related realization occurs in ferroelectric/dielectric superlattices containing polarization vortex arrays. Using four-dimensional scanning transmission electron microscopy with pixelated detectors, the electric toroidal moment was measured through the orbital angular momentum and torque transferred from the sample to a simple, zero-OAM electron probe. The relevant quantities are
\[
\mathbf{g}=\int \mathbf r\times \mathbf P(\mathbf r)\,d\mathbf r,
\qquad
\mathbf{L}=\langle \Psi|\mathbf r\times \mathbf p|\Psi\rangle,
\qquad
\langle \Gamma\rangle=\langle \Psi|\mathbf r\times (-\nabla V)|\Psi\rangle.
\]
The polarization patterns were found to be microscopically chiral, with a nontrivial axial component of the polarization. In that work, the coexistence of chiral order with polar toroidal order is explicitly identified as electrotoroidicity, and the axial polarization is presented as a route toward coupling ferroelectric and optical properties [2012.04134].

These two strands of literature emphasize different structural motifs. Ferroaxial compounds derive ETD from rotational structural distortions and locally inversion-broken units; vortex superlattices derive it from mesoscale polarization circulation with an axial polar component. A plausible implication is that chirality, axial polarization, and local inversion breaking are recurrent mechanisms by which electrotoroidic textures acquire experimentally accessible cross-couplings.

## 5. Transverse responses, thermopolarization, and nonlinear susceptibilities

One of the most active directions in recent work is the emergence of transverse responses from electric-toroidal order. In insulating systems with ferro-type electric-toroidal dipole order, a macroscopic electric polarization can appear perpendicular to an applied thermal gradient. The thermopolarization tensor is defined through
\[
\frac{P^x_{\nabla_y T}}{V}=\beta^{xy}(-\nabla_y T),
\]
with antisymmetric component \(\beta^{xy}=-\beta^{yx}\), and the linear-response coefficient is
\[
\beta^{xy}=-\frac{1}{V}\sum_{\mathbf k}\sum_{n=1}^{2M} c_1\big(n(\varepsilon_{\mathbf k n})\big)\Omega^{xy}_{\mathbf k n}.
\]
In the model calculation, \(\beta^{xy}\) is zero without toroidal order and nonzero only in its presence; the response is enhanced when \(p\)-orbital levels are low and local anisotropy is small, because \(p\)-orbital fluctuations strongly increase the contribution of low-energy collective modes [2112.06387].

A closely related prediction is nonlinear transverse magnetic susceptibility under ETD ordering. In a five-\(d\)-orbital single-site model in a tetragonal crystalline electric field, a third-order transverse susceptibility appears below the ETD transition temperature:
\[
M_x=\chi_{xyyy}H_y^3.
\]
The linear transverse component \(\chi_{xy}\) vanishes, whereas \(\chi_{xyyy}\) becomes nonzero in the ETD-ordered phase. The response is enhanced by spin-orbital entanglement and by a low-lying first excited crystal-field level, because the dominant Kubo contribution grows when the ground and first excited states are close in energy [2212.13018].

The ferro-rotational literature pushes this theme further by proposing electrotoroidicity as a new setting for transverse electromagnetic effects under preserved inversion and time-reversal symmetries. In doped ilmenite FeTiO\(_3\), the order parameter is an electrotoroidal moment
\[
\mathbf A=\sum_i \mathbf r_i\times \mathbf P_i,
\]
and the characteristic response is an anomalous transverse susceptibility. For approximate \(C_{3z}\) symmetry, linear off-diagonal susceptibility is forbidden; the lowest allowed odd-order Hall-like term is fifth order,
\[
M_y=\chi_{yxxxxx}H_x^5.
\]
Experimentally, magnetic force microscopy and symmetry analysis reveal ATS only when the magnetic field is perpendicular to \(\mathbf A\), with sign reversal under field reversal or domain reversal, while ferro-rotational domain walls display reduced diagonal susceptibility [2510.00462]. Taken together, these results suggest that the lowest nonzero transverse response is symmetry dependent: third order in the tetragonal ETD model, fifth order in the \(C_{3z}\) ferro-rotational case.

## 6. Detection, nanoscale stabilization, and related electrodynamic realizations

Detection and control are recurrent technical constraints because electric toroidal moments do not couple directly to external linear electric fields in the same way as conventional dipoles. The 2013 optical-activity study overcame this by exploiting coupling between electric vortices and spontaneous polarization, so that an applied electric field changes the gyrotropic response through \(d\mathbf G/d\mathbf E\) [1306.2189]. The 4D-STEM work instead used orbital-angular-momentum transfer to an electron beam, providing a local, direct probe of toroidal order and chiral domains at nanometer scale [2012.04134]. The thermodynamic review literature emphasizes that nanoscale confinement, inhomogeneous fields, mechanical stress, and structural engineering are central to stabilizing and switching electric toroidal moments in ferroelectric nanodots, nanorods, disks, rings, and tori [1504.04958].

In photonics and metamaterials, the same toroidal geometry appears in dynamic rather than static form. “Toroidal optical activity” showed experimentally that optical activity in a metamaterial can be dominated by a toroidal dipole and electric quadrupole, rather than by the conventional electric–magnetic dipole pair, with circular dichroism up to \(80\%\) at the toroidal resonance [1508.06192]. “Toroidal Light Pulses” are propagating counterparts of localized toroidal dipole excitations in matter and exhibit non-transverse fields, space-time non-separability, and toroidal topology [2102.03636]. “Toroidal helical pulses” extend this class to controllable helicity while retaining non-transverse toroidal topology and space-time nonseparability [2603.09650]. In parallel, dielectric metasurface studies identify electric TO and ATO modes, establish electric-magnetic toroidal duality, and show polarization-selective excitation of electric versus magnetic toroidal families under mirror-symmetry breaking by nanorod height modulation [2605.19699].

An objective synthesis of the field must therefore keep two scales in view. At the materials level, electrotoroidicity concerns ordered electric-toroidal dipoles, vortex polar textures, ferroaxial distortions, and their cross-coupled optical, thermal, and magnetic responses. At the electrodynamic level, it also connects to toroidal multipolar excitations and propagating toroidal field topologies. The literature indicates that these domains are converging through shared observables—chirality, optical activity, orbital-angular-momentum transfer, and transverse susceptibilities—but the exact boundary between static electrotoroidic order, dynamic toroidal excitation, and the older ferrotoroidic terminology remains an active point of definition rather than a settled convention.

Source: https://www.emergentmind.com/topics/electrotoroidicity