---
title: 'Single-Molecule Toroics: Magnetic Vortices'
url: https://www.emergentmind.com/topics/single-molecule-toroics-smts
type: topic
---

# Single-Molecule Toroics: Magnetic Vortices

Single-molecule toroics (SMTs) are molecular clusters whose low-energy magnetic states carry a toroidal, or anapole, moment generated by head-to-tail spin and orbital-current textures arranged around a closed loop, so that the net magnetic dipole vanishes while a magnetic vortex remains finite [2509.05424]. In lanthanide-based archetypes such as Dy\(_3\), coupled double triangles such as MDy\(_6\) (\(M=\mathrm{Al}^{3+},\mathrm{Cr}^{3+}\)), and larger mixed \(3d\)–\(4f\) wheels such as Fe\(_{10}\)Dy\(_{10}\), SMT behavior arises from strong single-ion anisotropy combined with weak, often competing, inter-ion couplings that stabilize counter-rotating ground doublets and suppress uniform-field magnetization [2504.08701]. Recent work has extended the field from indirect assignment by magnetometry and ab initio modeling to atomic-scale visualization by polarized neutron diffraction and to explicit protocols for coherent toroidal-state preparation and field-curl manipulation [2606.10077].

## 1. Definition of the toroidal degree of freedom

For a discrete molecular cluster, the toroidal moment is defined from the spatial distribution of local magnetic moments. In the Dy\(_3\) neutron-diffraction study it is written as
\[
\mathbf{T}=\sum_i \mathbf{r}_i \times \boldsymbol{\mu}_i,
\]
whereas the Fe\(_{10}\)Dy\(_{10}\) work uses the molecular toroidal-moment operator
\[
\boldsymbol{\tau}=\sum_i \mathbf{r}_i \times \mathbf{M}_i.
\]
The microwave-control study notes that standard SMT conventions sometimes include a factor \(1/2\),
\[
\bm{\tau}=\frac{1}{2}\sum_i \mathbf{r}_i \times \mathbf{m}_i,
\]
and attributes these differences to multipolar convention; in the Fe\(_{10}\)Dy\(_{10}\) analysis, \(\boldsymbol{\tau}=\sum_i \mathbf{r}_i\times\mathbf{M}_i\) is the operative definition [2509.05424].

The toroidal moment is odd under time reversal and odd under spatial inversion. Under time reversal, \(\mathbf{M}_i\to-\mathbf{M}_i\), hence \(\boldsymbol{\tau}\to-\boldsymbol{\tau}\); under inversion, \(\mathbf{r}_i\to-\mathbf{r}_i\), again implying \(\boldsymbol{\tau}\to-\boldsymbol{\tau}\). The microwave study describes \(\bm{\tau}\) as a polar vector of magnetic origin, with time-odd and space-odd character, while the neutron-diffraction study emphasizes that toroidal moments can stack ferro- or antiferrotoroidically in crystals and thereby enable magnetoelectric couplings and fourth-order ferroic phenomena [2504.08701].

Physically, a toroidal state is a head-to-tail loop of magnetic dipoles, or equivalently a vortex of spin and orbital currents, producing no net magnetic poles and typically a vanishing dipole moment. Its natural conjugate field is not a uniform magnetic field but a magnetic field with nonzero curl. In the Fe\(_{10}\)Dy\(_{10}\) formulation, the coupling is
\[
H_{\mathrm{Tor}}=\boldsymbol{\tau}\cdot(\nabla\times\mathbf{B}),
\]
so the degeneracy of counter-rotating toroidal partners is lifted by
\[
\Delta E_{\pm\boldsymbol{\tau}}=2\,\boldsymbol{\tau}\cdot(\nabla\times\mathbf{B}).
\]
Because \(\boldsymbol{\tau}\) is simultaneously odd under \(\mathcal{T}\) and \(\mathcal{P}\), toroidal polarization can mediate linear magnetoelectric coupling; in achiral systems this requires breaking \(\mathcal{T}\) with magnetic-field curls, whereas in chiral molecules broken \(\mathcal{P}\) can allow mixing so that uniform \(\mathbf{B}\) contributes as well [2509.05424].

## 2. Microscopic origin in lanthanide molecular clusters

The archetypal SMT is the Dy\(_3\) triangle. Its three Dy\(^{3+}\) ions carry strongly axial local moments whose easy axes are nearly tangential to the triangular rim, so the in-plane vector sum of dipoles vanishes while the vortex pattern generates a toroidal moment normal to the molecular plane. In the non-collinear Ising description used for Dy-based SMTs, pseudo-spins \(m_i=\pm 1\) define a toroidal ground doublet \(\ket{\pm\tau}=\ket{\pm1,\pm1,\pm1}\), consisting of time-reversal-related counter-rotating vortex textures with vanishing net magnetic moment [2504.08701].

The trimeric complex \([\mathrm{Dy}_3(\mathrm{OH})(\mathrm{teaH}_2)_3(\mathrm{paa})_3]\mathrm{Cl}(\mathrm{OMe})\cdot3\mathrm{MeOH}\cdot\mathrm{PhCN}\) provides a particularly explicit realization. Single-crystal neutron diffraction established trigonal \(P\bar{3}c1\) symmetry with a \(\mu_3\)-OH bridge, intramolecular Dy–Dy distance \(3.771(2)\,\text{\AA}\), and four molecules per unit cell. The Dy\(_3\) core has \(C_3\) symmetry through its center, and the three local easy axes lie approximately tangential to the triangle with a slight out-of-plane tilt, producing a head-to-tail arrangement in the \(ab\) plane and a toroidal moment along the crystallographic \(c\) axis. Within the \(P\bar{3}c1\) lattice, two molecular toroidal domains rotate clockwise and two counterclockwise per unit cell, yielding antiferrotoroidic stacking along \(c\) and zero net toroidization in the single crystal [2606.10077].

Ab initio calculations on this Dy\(_3\) system show strongly axial ground Kramers doublets at each Dy site with \(g_{zz}\approx19.69\)–\(19.72\), \(g_{xx}\approx0.036\)–\(0.037\), \(g_{yy}\approx0.049\)–\(0.050\), and near-pure \(|m_J|=15/2\) character \(\sim0.98\). The principal magnetic axes have out-of-plane tilt \(\theta\approx17^\circ\) and in-plane canting \(\phi\approx38^\circ\)–\(41^\circ\), consistent with the experimentally inferred \(\sim35^\circ\) canting. The lowest eight Kramers doublets span \(\approx0\)–\(538\,\mathrm{cm}^{-1}\), with the first excited doublet at \(\sim112\,\mathrm{cm}^{-1}\), yielding a computed Orbach barrier \(U_{\mathrm{cal}}\approx112\,\mathrm{cm}^{-1}\) [2606.10077].

In coupled double triangles MDy\(_6\), two Dy\(_3\) toroidal moieties are stacked and linked by a central ion. The Dy sites retain approximately tangential anisotropy axes with weak out-of-plane canting angles \(\eta_t\) and \(\eta_b\), and the low-energy configurations can be ferrotoroidic (FT, con-rotating) or antiferrotoroidic (AFT, counter-rotating) depending on geometry, exchange, and dipolar interactions. The microwave-control study emphasizes that weak inversion-symmetry breaking, expressed as \(\eta_t-\eta_b\neq0\), becomes a crucial spectroscopic handle because perfect inversion symmetry prevents a static field from distinguishing inversion-related toroidal configurations [2504.08701].

## 3. Scaling from Dy\(_3\) to Fe\(_{10}\)Dy\(_{10}\)

A central development in SMT research is the move from small triangles to larger rings in order to amplify the toroidal response. Since \(\boldsymbol{\tau}\propto\sum_i\mathbf{r}_i\times\mathbf{M}_i\), both larger molecular radii and larger local moments increase \(|\boldsymbol{\tau}|\). The Fe\(_{10}\)Dy\(_{10}\) icosanuclear wheel alternates Fe and Dy ions around an elliptical ring of average radius \(\sim10\,\text{\AA}\), compared with \(\sim2\,\text{\AA}\) for Dy\(_3\). Using \(m_{\mathrm{Dy}}\approx10\,\mu_B\) and \(m_{\mathrm{Fe}}\approx5\,\mu_B\), the estimated toroidal moment scale rises from
\[
|\boldsymbol{\tau}|_{\mathrm{Dy}_3}\approx60\,\mu_B\,\text{\AA}
\]
to
\[
|\boldsymbol{\tau}|_{\mathrm{Fe}_{10}\mathrm{Dy}_{10}}\approx1500\,\mu_B\,\text{\AA},
\]
which the Fe\(_{10}\)Dy\(_{10}\) study presents as an approximate \(25\times\) enhancement [2509.05424].

The compound \([\mathrm{Fe}_{10}\mathrm{Dy}_{10}(\mathrm{Me\mbox{-}tea})_{10}(\mathrm{Me\mbox{-}teaH})_{10}(\mathrm{NO}_3)_{10}]\) contains alternating \(3d\)–\(4f\) centers around an elliptical wheel. Ab initio CAHF/CASCI-SO calculations show that the Dy\(^{3+}\) sites possess well-isolated ground Kramers doublets composed of nearly pure \(\ket{m_J=\pm15/2}\) states with axial \(g\)-tensors,
\[
g_Z\approx18.85\text{–}19.64,\qquad g_X,g_Y\lesssim0.68,
\]
and first excited Kramers doublets at \(60\)–\(150\,\mathrm{cm}^{-1}\). Their principal axes are arranged tangentially around the ring, with planar projections following a vortex-like pattern. Fe\(^{3+}\) ions contribute orbitally nondegenerate \(S=5/2\) ground multiplets with negligible zero-field splitting and \(g\approx2\). Broken-symmetry DFT yields asymmetric nearest-neighbor ferromagnetic Fe–Dy couplings \(J_A\approx0.13\)–\(0.58\,\mathrm{cm}^{-1}\) and \(J_B\approx0.86\)–\(1.13\,\mathrm{cm}^{-1}\), together with weak antiferromagnetic Fe–Fe couplings \(J_C\approx-0.03\) to \(-0.26\,\mathrm{cm}^{-1}\) [2509.05424].

These interactions generate a dense manifold of toroidal excitations. Even when the description is restricted to Dy ground Kramers doublets and Fe \(S=5/2\) manifolds, the product space contains
\[
2^{10}\times6^{10}\simeq6.2\times10^{10}
\]
states, described in the paper as a “\(\sim62\) billion dimensional toroidal space.” The spectrum is organized by an Ising-band structure with \(s\)-, \(p\)-, \(d\)-, \(f\)-, \(g\)-, and \(h\)-wave distortions of the vortex pattern. A magnetic state with moment \(\sim83\,\mu_B\) lies only \(0.2\,\mathrm{cm}^{-1}\) above the nonmagnetic toroidal ground doublet, so a uniform field of \(\sim5\,\mathrm{mT}\) along an easy axis is sufficient to interchange toroidal and magnetic ground states; the same model reproduces powder magnetization at \(2\,\mathrm{K}\) and specific-heat data up to \(3\,\mathrm{T}\) [2509.05424].

## 4. Theoretical descriptions and toroidal response functions

SMT modeling typically starts from a non-collinear Ising or anisotropic spin Hamiltonian in which strong crystal-field anisotropy projects the lanthanide sites onto local Kramers-doublet pseudospins. For the Dy\(_3\) neutron study the spin-Hamiltonian context is written
\[
\mathcal{H}=\sum_i \mathbf{J}_i\cdot\mathbf{D}_i\cdot\mathbf{J}_i+\sum_{i<j}J_{ij}\,\mathbf{J}_i\cdot\mathbf{J}_j-\mu_B\sum_i \mathbf{J}_i\cdot\mathbf{g}_i\cdot\mathbf{B},
\]
with local response expressed through
\[
\mathbf{M}_i=\boldsymbol{\chi}_i\cdot\mathbf{B},
\]
so that polarized neutron diffraction can refine the site-resolved susceptibility tensors \(\boldsymbol{\chi}_i\) directly [2606.10077].

For Fe\(_{10}\)Dy\(_{10}\), the effective Hamiltonian is decomposed as
\[
H=H_{\mathrm{ex}}+H_{\mathrm{dip}}+H_{\mathrm{Zee}}+H_{\mathrm{Tor}},
\]
where the Dy sites are treated as Ising spins \(\mathbf{S}_i^{\mathrm{Dy}}=\tfrac52\sigma_i\,\mathbf{u}_i\) with \(\sigma_i=\pm1\), the Fe ions remain explicit quantum \(S=5/2\) spins, \(H_{\mathrm{dip}}\) contains intrafragment dipolar interactions, \(H_{\mathrm{Zee}}=\mathbf{M}\cdot\mathbf{B}\), and \(H_{\mathrm{Tor}}=\boldsymbol{\tau}\cdot(\nabla\times\mathbf{B})\). Because the weak Fe–Fe exchange is treated perturbatively to first order, the zeroth-order Hamiltonian factorizes into local Fe-site Hamiltonians \(h_i(\sigma_i,\sigma_{i+1})\), and the partition function can be compressed into traces of products of \(24\times24\) transfer matrices,
\[
Z=\mathrm{Tr}\,[A^{1,2}A^{2,3}\cdots A^{10,1}],\qquad F=-\frac1\beta\ln Z.
\]
This reduction turns the \(6.2\times10^{10}\)-state Hilbert space into a computationally manageable thermodynamic problem while retaining excellent agreement with experiment [2509.05424].

The same formulation yields standard magnetic observables,
\[
M_\alpha=-\frac{\partial F}{\partial B_\alpha},\qquad
\chi_{\alpha\beta}=-\frac{\partial^2F}{\partial B_\alpha\,\partial B_\beta},
\]
and also the toroidal expectation and response,
\[
\langle\tau_\alpha\rangle=-\frac{\partial F}{\partial (\nabla\times B)_\alpha},\qquad
\xi_{\alpha\beta}=-\left.\frac{\partial^2F}{\partial (\nabla\times B)_\alpha\,\partial (\nabla\times B)_\beta}\right|_{\nabla\times\mathbf{B}=0}.
\]
The Fe\(_{10}\)Dy\(_{10}\) paper introduces \(\boldsymbol{\xi}\) as the toroidal susceptibility tensor, and its molar form \(\xi^{(\mathrm{mol})}_{\alpha\beta}=N_A\xi_{\alpha\beta}\), as a thermodynamic linear-response function that measures finite-temperature toroidal polarization induced by a weak magnetic-field curl. In analogy with Van Vleck susceptibility, \(\xi\) contains a diagonal fluctuation term proportional to \(\beta\langle\tau_\alpha\tau_\beta\rangle\) and an off-diagonal virtual-transition term. In the \(T\to0\) limit,
\[
\xi_{\alpha\alpha}T\propto|\langle0|\tau_\alpha|0\rangle|^2,
\]
so \(\xi\) directly encodes the ground-state toroidal moment [2509.05424].

Direct evaluation of \(\xi_{\alpha\beta}\) for Fe\(_{10}\)Dy\(_{10}\) predicts a sizable ground-state toroidal moment and an enhanced finite-temperature toroidal polarization up to \(\sim10\,\mathrm{K}\) under modest uniform fields, attributed to field-induced spreading of toroidal levels across the spectrum. This suggests that toroidal thermodynamics in larger wheels is not limited to a single isolated doublet but reflects the structure of a broad, densely packed vortex manifold [2509.05424].

## 5. Experimental identification of toroidal order

Historically, SMT assignments relied largely on magnetometry and ab initio calculations. The Dy\(_3\) neutron-diffraction study identifies this as a limitation, because bulk magnetization suppression and simulated local axes cannot directly resolve site-specific non-collinearity inside a crystal. Its central methodological advance is the combined use of polarized neutron diffraction (PND), variable-field single-crystal neutron diffraction, ab initio calculations, and magnetometry as a quantitative framework for probing molecular toroidal order [2606.10077].

In PND, flipping ratios
\[
R(\mathbf{k})=\frac{I_+}{I_-}
\]
are measured at Bragg reflections and fitted to obtain local susceptibility tensors \(\boldsymbol{\chi}_i\) through \(\mathbf{M}_i=\boldsymbol{\chi}_i\cdot\mathbf{B}\). For Dy\(_3\)-2, the measurements were performed on DEMAND at \(\lambda=1.542\,\text{\AA}\), neutron polarization \(76\%\), \(H\parallel ab=0.6\,\mathrm{T}\), and \(T=5\,\mathrm{K}\), using \(27\) good-quality flipping ratios and yielding a goodness of fit \(\chi^2/n=0.48\). The refined Dy-site susceptibility ellipsoids are strongly anisotropic, nearly tangential to the triangle, and contain nonzero off-diagonal terms that directly evidence non-collinearity. This constitutes an atomic-scale visualization of the head-to-tail arrangement responsible for toroidicity [2606.10077].

Variable-field neutron diffraction complements PND by resolving field-induced order. With \(H\parallel c\) at \(T=1.5\,\mathrm{K}\), the \((2\,0\,0)\) Bragg reflection shows the onset of field-induced magnetic order at \(\approx0.2\,\mathrm{T}\), peaking near \(\approx2\,\mathrm{T}\). At \(4\,\mathrm{T}\), magnetic scattering appears below \(\approx40\,\mathrm{K}\). Magnetic symmetry refinement gives the best description in \(P\bar{3}c'1\), with an in-plane ordered moment \(4.6(3)\,\mu_B\) per Dy and a fixed \(c\)-axis component \(2.25\,\mu_B\) chosen to match bulk data. The refined pattern combines antiferromagnetic alignment of toroidal domains along \(c\) with head-to-tail in-plane alignment, thereby establishing antiferrotoroidic stacking together with a field-orderable \(c\)-axis dipolar component [2606.10077].

Magnetometry is consistent with this toroidal ground state. For Dy\(_3\)-2, Curie–Weiss temperatures are \(\theta_{\mathrm{CW}}\approx+25.45\,\mathrm{K}\) for \(H\parallel ab\), \(\theta_{\mathrm{CW}}\approx-151\,\mathrm{K}\) for \(H\parallel c\), and \(-15.8\,\mathrm{K}\) for powder; \(\chi T\) at \(300\,\mathrm{K}\) is \(50.6\) and \(24.4\,\mathrm{cm}^3\,\mathrm{K}\,\mathrm{mol}^{-1}\) for \(H\parallel ab\) and \(H\parallel c\), respectively; and at \(5\,\mathrm{K}\) the isothermal magnetization is \(17.4\,\mu_B\) per trimer for \(H\parallel a\) and \(8.4\,\mu_B\) for \(H\parallel c\). Powder samples show hysteresis at \(1.9\,\mathrm{K}\), whereas single crystals do not, consistent with cancellation between clockwise and counterclockwise toroidal domains within the unit cell [2606.10077].

The Fe\(_{10}\)Dy\(_{10}\) study addresses a different experimental bottleneck: direct detection of the toroidal degree of freedom itself. Its calculated splittings under realistic magnetic-field curls and its sizable finite-temperature toroidal susceptibility are presented as signatures amenable to direct observation, in contrast with Dy\(_3\), where the smaller molecular radius strongly limits detectability [2509.05424].

## 6. Preparation, manipulation, and design principles

A major obstacle in SMT research has been the selective preparation of one member of a degenerate toroidal doublet. The microwave-control study proposes a realistic pulsed-EPR protocol that avoids nanometer-scale static field gradients. In Dy\(_3\), an in-plane static field
\[
\mathbf{B}=|B_\parallel|(0,-1,0),\qquad |B_\parallel|=1\,\mathrm{T},
\]
moves the toroidal doublet \(\ket{\pm\tau}\) above a singly degenerate magnetic ground state \(\ket{-1,+1,+1}\). Because linearly polarized microwave radiation perpendicular to the triangle plane flips one Dy pseudospin through the small transverse components \(g_x\approx g_y\approx0.05\), the transition \(\ket{-1,+1,+1}\to\ket{+\tau}\) is resonant while the transition to \(\ket{-\tau}\) requires two flips and is off-resonant. A single resonant \(\pi\) pulse at \(\omega=45\,\mathrm{GHz}\) therefore selectively prepares one toroidal state, with pulse duration
\[
t_p=\hbar\pi\big/(g_x\mu_B|B_\perp|).
\]
This is a genuinely coherent preparation mechanism, not a thermally biased one [2504.08701].

In AlDy\(_6\), preparation of a ferrotoroidic state requires three Dy flips. With
\[
\mathbf{B}=|B_\parallel|(0.1,0,1),\qquad |B_\parallel|=1\,\mathrm{T},
\]
and a weak canting asymmetry \(\eta_t-\eta_b=1^\circ\), three sequential resonant pulses at
\[
\omega_1=115\,\mathrm{GHz},\qquad \omega_2=43\,\mathrm{GHz},\qquad \omega_3=101\,\mathrm{GHz}
\]
selectively transfer population from \(\ket{-\tau,+\tau}\) to \(\ket{+\tau,+\tau}\). The toroidal polarization is monitored through
\[
\langle\tau_z\rangle=\bra{\psi(t)}\tau_z\ket{\psi(t)}.
\]
A simultaneous-pulse variant, using three pulses over \(35\,\mathrm{ns}\) with \(|B_\perp^{(2)}|=75\,\mathrm{mT}\) and \(|B_\perp^{(1)}|/|B_\perp^{(2)}|=|B_\perp^{(3)}|/|B_\perp^{(2)}|=0.95\), yields \(100\%\) population in \(\ket{+\tau,+\tau}\) in the simulations. Dissipative dynamics treated with a Redfield-type master equation show robustness over the literature-estimated relaxation range \(\Gamma\sim10^{-2}\)–\(10^2\,\mathrm{s}^{-1}\,\mathrm{cm}^3\), while very large relaxation rates drive population back toward ground or metastable states [2504.08701].

A complementary control route is direct coupling to \(\nabla\times\mathbf{B}\). For Fe\(_{10}\)Dy\(_{10}\), Maxwell–Ampère,
\[
\nabla\times\mathbf{B}(\mathbf{r},t)=\mu_0\mathbf{J}(\mathbf{r},t)+\mu_0\epsilon_0\frac{\partial\mathbf{E}(\mathbf{r},t)}{\partial t},
\]
suggests two experimental realizations: local current injection perpendicular to the molecular ring, for example by STM tips, and displacement-current curls generated by spatially focused, shaped femtosecond laser pulses. For a Ti:sapphire pulse at \(800\,\mathrm{nm}\), \(10\,\mathrm{fs}\), \(200\,\mu\mathrm{m}\) spot, and peak power \(\sim1\,\mathrm{GW}\), the estimated peak field is \(E_{\max}\approx5\times10^9\,\mathrm{V/m}\), giving
\[
|\nabla\times\mathbf{B}|\approx5\times10^{-4}\,\mathrm{T}\,\text{\AA}^{-1}.
\]
The corresponding toroidal splittings are then
\[
\Delta E_{\pm\boldsymbol{\tau}^{\mathrm{Dy}_3}}\approx0.06\,\mathrm{cm}^{-1},\qquad
\Delta E_{\pm\boldsymbol{\tau}^{\mathrm{Fe}_{10}\mathrm{Dy}_{10}}}\approx1.5\,\mathrm{cm}^{-1},
\]
with the larger wheel predicted to be readily observable at liquid-helium temperatures. Proposed signatures include pump–probe changes in magnetization, shifts or enhancements in low-temperature specific heat, and AC susceptibility under modulated \(\nabla\times\mathbf{B}\) [2509.05424].

Several design rules recur across these studies. Larger rings amplify \(|\mathbf{r}_i|\) and hence \(|\boldsymbol{\tau}|\); high-moment ions such as Dy\(^{3+}\) maximize \(|\mathbf{M}_i|\); and tangential alignment of Dy easy axes around a ring or triangle stabilizes the vortex texture. Mixed \(3d\)–\(4f\) architectures can use Fe\(^{3+}\) spins to shape the spectrum while keeping Fe–Fe antiferromagnetism weak enough not to suppress toroidicity. Large Kramers-doublet gaps, such as the \(60\)–\(150\,\mathrm{cm}^{-1}\) range in Fe\(_{10}\)Dy\(_{10}\), help justify Ising projections at liquid-helium temperatures. In Dy\(_3\)-2, the \(\mu_3\)-OH proton is chemically decisive: removing it and replacing \(\mu_3\)-OH by \(\mu_3\)-O\(^{2-}\) reorients the anisotropy to \(\theta\approx15.3^\circ\), \(\phi\approx70.6^\circ\), destroying toroidal alignment [2606.10077].

These properties motivate qubit-oriented interpretations. Toroidal doublets have zero total magnetic dipole moment and are first-order insensitive to uniform magnetic fields, so they are less sensitive to stray-field noise than conventional dipolar states. The microwave study presents coherent \(\pi\)-pulse-like transfers between toroidal configurations, while the Fe\(_{10}\)Dy\(_{10}\) analysis argues that protection against uniform \(\mathbf{B}\) and direct addressability by field curls make larger SMTs attractive for robust spin states. A plausible implication is that future progress will depend on combining chemically engineered toroidicity, direct toroidal probes, and coherent control protocols in the same molecular platform [2504.08701].

Source: https://www.emergentmind.com/topics/single-molecule-toroics-smts