Molar Toroidal Susceptibility: Theory & Applications

• Molar toroidal susceptibility is defined as the linear-response coefficient quantifying the toroidal polarization per mole induced by a conjugate field derived from Helmholtz free energy.
• The topic covers multiple microscopic models—from molecular rings to plaquette systems—and employs symmetry-allowed free-energy couplings and magnetoelectric tensors to infer susceptibility.
• Its analysis reveals critical behavior and giant susceptibility anomalies in tailored materials, emphasizing both thermodynamic scaling and cross-correlated experimental proxies.

Searching arXiv for the cited literature and topic scope. arXiv search: molar toroidal susceptibility Molar toroidal susceptibility is the thermodynamic linear-response coefficient that quantifies how a bulk ensemble develops a toroidal polarization per mole under a field conjugate to a toroidal moment. In the strictest formulation presently available in the cited literature, it is introduced for the Fe$_{10}$Dy$_{10}$ molecular ring as a response to a magnetic field with a small non-vanishing curl, in direct analogy with the Van Vleck magnetic susceptibility (Soncini et al., 5 Sep 2025). Across the broader toroidal-multipole literature, the same quantity is often implicit rather than explicitly named: it is reconstructed from free-energy couplings, symmetry-allowed magnetoelectric tensors, nonlinear transport, or microscopic toroidal operators in localized-spin, plaquette, cluster, and atomic-scale models (Harris, 2010, Hayami et al., 2022, Yatsushiro et al., 2019, Kusunose et al., 2024).

1. Definition, conjugate fields, and thermodynamic status

In the Fe$_{10}$Dy$_{10}$ work, the toroidal susceptibility tensor is defined through the Helmholtz free energy $F$ as

$$\xi_{\alpha \beta} = - \left. \frac{\partial^2 F}{\partial \left( \nabla \times \mathbf{B} \right)_{\alpha} \partial \left( \nabla \times \mathbf{B} \right)_{\beta} } \right|_{\nabla \times \mathbf{B} = 0},$$

with $\nabla\times\mathbf B$ treated as the external variable conjugate to the toroidal moment operator $\boldsymbol{\tau}$ (Soncini et al., 5 Sep 2025). The corresponding toroidal polarization satisfies

$$P_{\tau,\alpha} = -\frac{\partial F}{\partial (\nabla\times\mathbf B)_\alpha},$$

so that, in linear response, $$P_{\tau,\alpha} = \sum_\beta \xi_{\alpha\beta}(\nabla\times\mathbf B)_\beta$$ (Soncini et al., 5 Sep 2025).

A more general symmetry framework distinguishes electric, magnetic, electric-toroidal, and magnetic-toroidal multipoles. Within that classification, the magnetic toroidal dipole $_{10}$0 is time-odd and parity-odd, while the electric toroidal dipole $_{10}$1 is time-even and parity-even; their conjugated fields differ accordingly. For rank-1 multipoles, $_{10}$2 is conjugated to $_{10}$3 or $_{10}$4, whereas $_{10}$5 is conjugated to $_{10}$6 or $_{10}$7 (Kusunose et al., 2024). This point is fundamental: a toroidal susceptibility is not, in general, the response to a uniform $_{10}$8 or $_{10}$9, even though effective low-field proxies can emerge through symmetry-allowed couplings.

A closely related construction appears in the two-dimensional plaquette model of toroidal order, where the magnetoelectric free energy is written as

$_{10}$0

and the effective conjugate field is therefore

$_{10}$1

In that formulation, the toroidal susceptibility is

$_{10}$2

This establishes a second, equally legitimate thermodynamic definition: the conjugate field may be a symmetry-allowed bilinear combination of $_{10}$3 and $_{10}$4, rather than a bare curl field (Harris, 2010).

2. Microscopic realizations of toroidal moments

The microscopic object whose molar response is being measured is model dependent. In cluster and spin-texture language, toroidal dipoles are vortex-like moments of dipole distributions, schematically $_{10}$5 for magnetic dipoles and $_{10}$6 for electric dipoles (Kusunose et al., 2024). In a localized-spin cluster, this reduces to the familiar discrete spin form used for trimers and molecular rings.

In the 3-spin toroidal plaquette model, the plaquette toroidicity is defined so that the leading magnetostatic interaction between two plaquettes is proportional to $_{10}$7. The explicit scalar toroidal strength is

$_{10}$8

where the sign is an Ising-like variable set by chirality (Harris, 2010). That definition is deliberately adapted to the plaquette interaction and is not universal, but it provides a concrete microscopic toroidal variable whose susceptibility can be defined thermodynamically.

In Ce$_{10}$9TiBi$_{10}$0, the relevant object is a magnetic toroidal dipole generated by staggered antiferromagnetic order on locally noncentrosymmetric zigzag chains. The local toroidal component is written as

$_{10}$1

with staggered local crystal fields $_{10}$2 and staggered moments $_{10}$3, producing a ferroic in-plane toroidal dipole. At the unit-cell level, these chain toroidal moments form cluster multipoles $_{10}$4, $_{10}$5, and $_{10}$6, and the partial-disordered toroidal phase is identified with an in-plane cluster toroidal dipole (Hayami et al., 2022).

At the atomic scale, the two-orbital $_{10}$7-$_{10}$8 model with odd-parity hybridization yields toroidal operators directly in orbital-spin space. The in-plane and out-of-plane magnetic toroidal dipoles are

$_{10}$9

with $F$0 and $F$1 spin dependent and $F$2 purely orbital (Yatsushiro et al., 2019). In that setting, toroidal susceptibility refers to the field response of expectation values $F$3, not to a geometric spin texture alone.

3. From microscopic response to a molar quantity

The transition from a microscopic toroidal response to a molar susceptibility is conceptually straightforward but convention dependent. The core step is to identify the toroidal moment per microscopic unit—plaquette, site, unit cell, or molecule—and then multiply the corresponding single-unit response by the number of such units per mole.

For the plaquette model, the molar toroidal susceptibility is written as

$F$4

where $F$5 is the number of toroidal plaquettes per formula unit and $F$6 is Avogadro’s number (Harris, 2010). The same paper notes that the precise units depend on how $F$7 and the conjugate field $F$8 are defined.

For Ce$F$9TiBi$$\xi_{\alpha \beta} = - \left. \frac{\partial^2 F}{\partial \left( \nabla \times \mathbf{B} \right)_{\alpha} \partial \left( \nabla \times \mathbf{B} \right)_{\beta} } \right|_{\nabla \times \mathbf{B} = 0},$$0, where the order parameter is a cluster toroidal dipole $$\xi_{\alpha \beta} = - \left. \frac{\partial^2 F}{\partial \left( \nabla \times \mathbf{B} \right)_{\alpha} \partial \left( \nabla \times \mathbf{B} \right)_{\beta} } \right|_{\nabla \times \mathbf{B} = 0},$$1, the corresponding molar quantity is

$$\xi_{\alpha \beta} = - \left. \frac{\partial^2 F}{\partial \left( \nabla \times \mathbf{B} \right)_{\alpha} \partial \left( \nabla \times \mathbf{B} \right)_{\beta} } \right|_{\nabla \times \mathbf{B} = 0},$$2

with $$\xi_{\alpha \beta} = - \left. \frac{\partial^2 F}{\partial \left( \nabla \times \mathbf{B} \right)_{\alpha} \partial \left( \nabla \times \mathbf{B} \right)_{\beta} } \right|_{\nabla \times \mathbf{B} = 0},$$3 Ce ions per formula unit if the susceptibility is normalized per mole of Ce, or $$\xi_{\alpha \beta} = - \left. \frac{\partial^2 F}{\partial \left( \nabla \times \mathbf{B} \right)_{\alpha} \partial \left( \nabla \times \mathbf{B} \right)_{\beta} } \right|_{\nabla \times \mathbf{B} = 0},$$4 if it is normalized per mole of formula units (Hayami et al., 2022). This makes explicit that “molar toroidal susceptibility” is not a unique number until the normalization convention is stated.

A complementary route starts from ordinary magnetic susceptibility tensors of localized centers. For a cluster of spins at positions $$\xi_{\alpha \beta} = - \left. \frac{\partial^2 F}{\partial \left( \nabla \times \mathbf{B} \right)_{\alpha} \partial \left( \nabla \times \mathbf{B} \right)_{\beta} } \right|_{\nabla \times \mathbf{B} = 0},$$5 with local linear responses $$\xi_{\alpha \beta} = - \left. \frac{\partial^2 F}{\partial \left( \nabla \times \mathbf{B} \right)_{\alpha} \partial \left( \nabla \times \mathbf{B} \right)_{\beta} } \right|_{\nabla \times \mathbf{B} = 0},$$6, the toroidal moment is

$$\xi_{\alpha \beta} = - \left. \frac{\partial^2 F}{\partial \left( \nabla \times \mathbf{B} \right)_{\alpha} \partial \left( \nabla \times \mathbf{B} \right)_{\beta} } \right|_{\nabla \times \mathbf{B} = 0},$$7

and the toroidal susceptibility tensor becomes

$$\xi_{\alpha \beta} = - \left. \frac{\partial^2 F}{\partial \left( \nabla \times \mathbf{B} \right)_{\alpha} \partial \left( \nabla \times \mathbf{B} \right)_{\beta} } \right|_{\nabla \times \mathbf{B} = 0},$$8

Once the unit-cell geometry is known, this tensor can be converted to a molar toroidal susceptibility by the usual scaling from unit-cell to per-mole quantities (Pelka, 2013).

The Fe$$\xi_{\alpha \beta} = - \left. \frac{\partial^2 F}{\partial \left( \nabla \times \mathbf{B} \right)_{\alpha} \partial \left( \nabla \times \mathbf{B} \right)_{\beta} } \right|_{\nabla \times \mathbf{B} = 0},$$9Dy$\nabla\times\mathbf B$0 paper uses the same logic at the molecular level. It computes $\nabla\times\mathbf B$1 for a single ring and then interprets it as the molar toroidal susceptibility by the usual scaling with Avogadro’s number when estimating experimental signals (Soncini et al., 5 Sep 2025).

4. Magnetoelectric, transport, and optical proxies

In much of the literature, molar toroidal susceptibility is not measured directly; it is inferred from symmetry-related response functions. The most common proxy is the linear magnetoelectric tensor. In the ferrotoroidal phase of the plaquette model, the allowed linear magnetoelectric components follow directly from the free energy $\nabla\times\mathbf B$2, and they vanish when the uniform toroidal moment $\nabla\times\mathbf B$3 vanishes (Harris, 2010). This makes the slope of toroidal-order-dependent magnetoelectric coefficients a practical surrogate for toroidal susceptibility.

Ce$\nabla\times\mathbf B$4TiBi$\nabla\times\mathbf B$5 provides a metallic realization of this logic. The paper does not introduce an explicit tensor named toroidal susceptibility, but it identifies toroidal-origin response functions in the partial-disordered toroidal phase. The linear magnetoelectric effect is written as $\nabla\times\mathbf B$6, with symmetry-allowed components in the PD-MT state given by $\nabla\times\mathbf B$7 and $\nabla\times\mathbf B$8. Nonlinear transport coefficients,

$\nabla\times\mathbf B$9

also become finite only in phases with a net toroidal dipole. The paper therefore treats toroidal susceptibility as something that must be inferred from magnetoelectric and nonlinear transport responses, not read literally from the Hamiltonian (Hayami et al., 2022).

The atomic-scale $\boldsymbol{\tau}$0-$\boldsymbol{\tau}$1 theory makes this relation even more explicit. The symmetry form of the field-driven magnetoelectric tensor is

$\boldsymbol{\tau}$2

so the antisymmetric in-plane part tracks $\boldsymbol{\tau}$3, while the $\boldsymbol{\tau}$4 and $\boldsymbol{\tau}$5 components track in-plane toroidal dipoles $\boldsymbol{\tau}$6 and $\boldsymbol{\tau}$7 (Yatsushiro et al., 2019). Within that model, odd-parity hybridization enhances the magnetoelectric effect for in-plane magnetic toroidal dipoles and suppresses it for the out-of-plane one. A plausible implication is that the associated molar toroidal susceptibilities inherit the same enhancement and suppression trends.

Optical activity provides a further, nonlocal proxy. In the metamaterial study of toroidal optical activity, the toroidal dipole moment

$\boldsymbol{\tau}$8

must be included to reproduce the observed circular dichroism at one resonance, and the analysis attributes the effect to toroidal dipole plus electric quadrupole contributions rather than to the usual electric and magnetic dipoles (Raybould et al., 2015). This suggests a frequency-dependent toroidal susceptibility in the optical channel, although the paper does not normalize it in molar form.

5. Material realizations and characteristic temperature or field dependence

The first explicit molar toroidal susceptibility in the cited corpus is the Fe$\boldsymbol{\tau}$9Dy$$P_{\tau,\alpha} = -\frac{\partial F}{\partial (\nabla\times\mathbf B)_\alpha},$$0 ring. There the molecular toroidal moment operator is

$$P_{\tau,\alpha} = -\frac{\partial F}{\partial (\nabla\times\mathbf B)_\alpha},$$1

and the Hamiltonian contains an explicit toroidal coupling

$$P_{\tau,\alpha} = -\frac{\partial F}{\partial (\nabla\times\mathbf B)_\alpha},$$2

The calculated $$P_{\tau,\alpha} = -\frac{\partial F}{\partial (\nabla\times\mathbf B)_\alpha},$$3 tends to a finite constant as $$P_{\tau,\alpha} = -\frac{\partial F}{\partial (\nabla\times\mathbf B)_\alpha},$$4, which the authors use to identify a non-zero toroidal moment in the ground state. The maximal toroidal moment is estimated as $$P_{\tau,\alpha} = -\frac{\partial F}{\partial (\nabla\times\mathbf B)_\alpha},$$5, compared with $$P_{\tau,\alpha} = -\frac{\partial F}{\partial (\nabla\times\mathbf B)_\alpha},$$6 for Dy$$P_{\tau,\alpha} = -\frac{\partial F}{\partial (\nabla\times\mathbf B)_\alpha},$$7, and for an estimated curl $$P_{\tau,\alpha} = -\frac{\partial F}{\partial (\nabla\times\mathbf B)_\alpha},$$8 the corresponding toroidal splitting is $$P_{\tau,\alpha} = -\frac{\partial F}{\partial (\nabla\times\mathbf B)_\alpha},$$9 rather than $$P_{\tau,\alpha} = \sum_\beta \xi_{\alpha\beta}(\nabla\times\mathbf B)_\beta$$0 (Soncini et al., 5 Sep 2025). In this literature, that scale difference is the most direct sense in which a “giant” molar toroidal susceptibility is claimed.

In the two-dimensional plaquette model, the relevant critical behavior is Ising-like. The gauge-toroid phase is characterized by ordering of the composite variable $$P_{\tau,\alpha} = \sum_\beta \xi_{\alpha\beta}(\nabla\times\mathbf B)_\beta$$1, whereas true ferrotoroidal order appears only when the gauge symmetry is broken at a lower temperature. The paper states that the toroidal susceptibility should diverge or show a strong peak at the ferrotoroidal transition; within Landau mean-field theory $$P_{\tau,\alpha} = \sum_\beta \xi_{\alpha\beta}(\nabla\times\mathbf B)_\beta$$2, while in the 2D Ising universality class one expects $$P_{\tau,\alpha} = \sum_\beta \xi_{\alpha\beta}(\nabla\times\mathbf B)_\beta$$3 with $$P_{\tau,\alpha} = \sum_\beta \xi_{\alpha\beta}(\nabla\times\mathbf B)_\beta$$4 (Harris, 2010).

In BaCoSiO$$P_{\tau,\alpha} = \sum_\beta \xi_{\alpha\beta}(\nabla\times\mathbf B)_\beta$$5, the paper does not define toroidal susceptibility explicitly, but it presents a field-tunable toroidal order parameter. The ferritoroidal state in zero field carries a net $$P_{\tau,\alpha} = \sum_\beta \xi_{\alpha\beta}(\nabla\times\mathbf B)_\beta$$6 or $$P_{\tau,\alpha} = \sum_\beta \xi_{\alpha\beta}(\nabla\times\mathbf B)_\beta$$7 per macroscopic domain, while a field $$P_{\tau,\alpha} = \sum_\beta \xi_{\alpha\beta}(\nabla\times\mathbf B)_\beta$$8 drives a transition at $$P_{\tau,\alpha} = \sum_\beta \xi_{\alpha\beta}(\nabla\times\mathbf B)_\beta$$9 to a ferrotoroidal $_{10}$00 or $_{10}$01 state (Ding et al., 2021). In the paper’s own reconstruction, one can naturally define

$_{10}$02

or, in molar form, $_{10}$03. Under that definition, the first-order toroidal transition yields a large anomaly in toroidal susceptibility (Ding et al., 2021).

Ce$_{10}$04TiBi$_{10}$05 contributes a different hallmark: an unusual anisotropic magnetic susceptibility in the partial-disordered toroidal regime. At $_{10}$06 and $_{10}$07, $_{10}$08 shows a cusp at $_{10}$09, whereas $_{10}$10 does not show a cusp and continues to increase below $_{10}$11, with $_{10}$12 across the full temperature range (Hayami et al., 2022). The paper explicitly interprets this as an indirect fingerprint of a soft toroidal order parameter, although it does not compute $_{10}$13 itself.

6. Conceptual limits, common confusions, and current directions

A first recurrent confusion is to identify toroidal susceptibility with ordinary magnetic susceptibility. The literature does not support that identification. In the strict thermodynamic construction, the magnetic toroidal dipole is conjugated to $_{10}$14 or $_{10}$15, or to symmetry-allowed field bilinears such as $_{10}$16; a uniform magnetic field becomes relevant only indirectly, through composite couplings or through specific lattice symmetries (Kusunose et al., 2024, Harris, 2010, Soncini et al., 5 Sep 2025).

A second confusion is to assume that toroidal order is equivalent to a net magnetic dipole or to a spontaneous polarization. The plaquette model explicitly states that no phase permits a spontaneous polarization under its assumed mirror symmetry, even in the ferrotoroidal phase (Harris, 2010). Conversely, Ce$_{10}$17TiBi$_{10}$18 realizes an in-plane toroidal dipole generated by staggered antiferromagnetic order in a globally centrosymmetric but locally noncentrosymmetric lattice, and its partial-disordered toroidal phase contains one nonmagnetic chain rather than a conventional uniform magnetization (Hayami et al., 2022).

A third issue is methodological. Many papers provide all ingredients needed to define molar toroidal susceptibility but do not actually tabulate it. The atomic-scale visualization of toroidal order in Dy$_{10}$19 shows how polarized neutron diffraction yields local Dy$_{10}$20 susceptibility tensors, variable-field neutron diffraction resolves field-induced toroidal ordering, and ab initio calculations recover the toroidal ground state, yet the paper states that toroidal susceptibility is not explicitly defined or numerically tabulated there (Jenkins et al., 8 Jun 2026). Similarly, the coherent-control work on Dy-based molecular toroics computes time-dependent $_{10}$21 under pulsed microwave fields and argues that the ensuing magneto-electric properties can be used as a read-out mechanism, but it does not introduce a closed-form molar susceptibility (Hymas et al., 11 Apr 2025).

The present state of the subject is therefore bifurcated. On one side, a fully thermodynamic molar toroidal susceptibility has been written down and evaluated for a specific molecular ring (Soncini et al., 5 Sep 2025). On the other, a larger body of work treats toroidal susceptibility as a derived or inferred quantity, reconstructed from toroidal order parameters, symmetry-allowed free-energy terms, local susceptibility tensors, or cross-correlated responses (Hayami et al., 2022, Yatsushiro et al., 2019, Harris, 2010, Kusunose et al., 2024). This suggests that, in current usage, “molar toroidal susceptibility” is best understood not as a single universally standardized observable, but as a family of response coefficients whose common structure is the derivative of a molar toroidal polarization with respect to the correctly specified conjugate field.