---
title: Toroidal Susceptibility
url: https://www.emergentmind.com/topics/toroidal-susceptibility
type: topic
---

# Toroidal Susceptibility

Searching arXiv for recent and foundational papers on toroidal susceptibility across condensed matter, plasma physics, molecular magnetism, metamaterials, and toroidal-geometry susceptibilities.
Toroidal susceptibility denotes a family of response functions associated with toroidal order, toroidal multipoles, or toroidal geometry. The underlying toroidal moment is an independent object in the multipole expansion of electrodynamics and arises naturally from vortex-like or head-to-tail arrangements of spins, magnetic dipoles, or currents. The literature does not employ a single universal definition: in different settings it refers to the field derivative of a macroscopic toroidal moment, the antisymmetric part of a magnetic-susceptibility tensor, a second derivative of free energy with respect to the curl of a magnetic field, the sensitivity of an external magnetic field to the cross-section of a toroidal current channel, or an effective electromagnetic susceptibility extracted from a toroidal polarizability [2103.01360][2606.10077][2509.05424][2601.19401][1209.3400].

## 1. Definitions and formal scope

Three microscopic definitions recur across the literature. In a localized-spin trimer, the toroidal moment is
$$
\mathbf t \equiv \sum_{i=1}^3 \mathbf r_i \times \mathbf S_i .
$$
In a classical molecular picture of magnetic dipoles, it is
$$
\mathbf T = \tfrac12 \sum_i \mathbf r_i \times \mathbf m_i .
$$
For a time-harmonic current distribution, the volume toroidal dipole is
$$
\mathbf T=\frac{1}{10c}\int\!\Bigl[(\mathbf r\!\cdot\!\mathbf J)\,\mathbf r-2r^2\mathbf J\Bigr]\,d^3r .
$$
These constructions encode vortex-like or head-to-tail configurations that may carry a toroidal dipole even when ordinary electric and magnetic dipoles are suppressed or cancel. A distinct but related development appears in electric toroidal dipole order, where the local operator is written as
$$
\hat G_{zm}=(\hat{\mathbf l}_m\times \hat{\mathbf s}_m)^z=\hat l_m^x\hat s_m^y-\hat l_m^y\hat s_m^x .
$$
[2103.01360][2606.10077][1209.3400][2212.13018]

| Context | Quantity called susceptibility | Perturbation or conjugate variable |
|---|---|---|
| Chiral-lattice magnet BaCoSiO\(_4\) | $\chi_T \equiv \partial T_z/\partial H_z$ | Uniform magnetic field $H\parallel c$ |
| Dy\(_3\) single-molecule toroic | Antisymmetric part $\chi^{\rm A}_{ij}$ of $\chi_{ij}$ | Small applied field $H$ |
| Fe\(_{10}\)Dy\(_{10}\) ring | $\xi_{\alpha\beta}=-\partial^2F/\partial(\nabla\times\mathbf B)_\alpha\partial(\nabla\times\mathbf B)_\beta$ | Small non-vanishing curl of $\mathbf B$ |
| Toroidal current column | $\chi_t(\theta,p,\phi)=\partial B_{\rm ext}/\partial S$ | Cross-sectional area $S=\pi a_0^2$ |
| Planar ASRR metamaterial | $\chi_T(\omega)=N\alpha_T(\omega)/\varepsilon_0$ | Incident field through $T_x=\alpha_T E_x$ |
| ETD-ordered model | $\chi_{xyyy}$ | Cubic transverse response to magnetic field |

This diversity has a clear consequence: toroidal susceptibility is best understood as a class of response measures rather than a single invariant observable. The precise meaning is set by the microscopic toroidal variable and by the external perturbation to which it is conjugate.

## 2. Field derivative of toroidal order in chiral magnets

In the chiral triangular-lattice magnet BaCoSiO\(_4\), the toroidal susceptibility is defined operationally as
$$
\chi_T \equiv \frac{\partial T_z}{\partial H_z},
$$
with $T_z$ the macroscopic toroidal moment and $H\parallel c$. The microscopic Hamiltonian combines antiferromagnetic Heisenberg exchanges $J_t,J_t',J_t'',J_z,J_c$, easy-plane single-ion anisotropy $A>0$, Dzyaloshinskii-Moriya vectors with both $D_z$ and $D_{xy}$ components, and the Zeeman coupling. In zero field, the dominant frustrated $J_t+J_z$ network locks each trimer into a 120° in-plane vortex, generating $t_z\neq0$ on each trimer. The subleading $J_t',J_t'',J_c$ terms select a ferritoroidal arrangement in which two of the three sublattices carry $t_z=+t$ and one carries $t_z=-t$, giving net $T_z=\pm t$. An out-of-plane DM component further cants each spin slightly along $z$ in a sense locked to the vortex chirality, so that each sublattice also carries a small magnetization $M_z\propto t_z$ [2103.01360].

Because the DM interaction locks $M_z$ to $t_z$, a field $H\parallel z$ favors trimers with $t_z>0$. At a critical field $H_C\approx1.2\,{\rm T}$, one trimer-sublattice flips its chirality and the system undergoes a ferri-to-ferrotoroidal transition, with net $T_z$ jumping from $+t$ to $+3t$. Experimentally, both $M(H)$ and the neutron-refined $T_z(H)=\sum_i\mathbf r_i\times\mathbf S_i$ exhibit a multi-step evolution, with kinks at the weak ferromagnetic-domain step ($\lesssim0.15\,{\rm T}$), the ferri-to-ferro toroidal step ($1.2\,{\rm T}$), and further high-field transitions near $10\,{\rm T}$ and $40\,{\rm T}$. Correspondingly, $\chi_T(H)=dT_z/dH$ shows sharp peaks at each metamagnetic transition. In this formulation, toroidal susceptibility is not an independent probe disconnected from conventional magnetometry; it is measured indirectly via $\partial M_z/\partial H$ and validated by neutron-diffraction reconstruction of the spin texture.

## 3. Tensorial and thermodynamic formulations in molecular toroics

In single-molecule toroics, toroidal susceptibility is often encoded in the magnetic-susceptibility tensor rather than in a direct derivative $dT/dH$. For the trimeric Dy(III) single-molecule toroic \([{\rm Dy}_3({\rm OH})({\rm teaH}_2)_3({\rm paa})_3]{\rm Cl}({\rm OMe})\), the induced magnetization is written as
$$
M_i=\sum_j \chi_{ij}H_j,
$$
with decomposition
$$
\chi_{ij}=\chi^{\rm S}_{ij}+\chi^{\rm A}_{ij},\qquad
\chi^{\rm A}_{ij}=\tfrac12(\chi_{ij}-\chi_{ji}) .
$$
The antisymmetric part generates a toroidal response through
$$
T_k \propto \sum_{i,j}\epsilon_{kij}\chi^{\rm A}_{ij}H .
$$
Polarized neutron diffraction at $T=5\,{\rm K}$ and $H=0.6\,{\rm T}$ in the $ab$ plane was used to refine the nine components of the Dy\(^{3+}\) susceptibility tensor; its antisymmetric part gives direct evidence of a toroidal response in the $ab$ plane, with $\mathbf T$ along the crystallographic $c$ axis. Variable-field neutron diffraction further showed field-induced magnetization along $c$ with toroidal moments anti-parallelly stacked, and successive layers along $c$ were found to be antiferrotoroidically stacked, explaining the absence of net toroidal moment in zero field [2606.10077].

A distinct thermodynamic formulation was introduced for the Fe\(_{10}\)Dy\(_{10}\) ring through the molar toroidal susceptibility tensor
$$
\xi_{\alpha\beta}=
-\,\frac{\partial^2 F}
{\partial(\nabla\times\mathbf B)_\alpha\,\partial(\nabla\times\mathbf B)_\beta}
\Bigg|_{\nabla\times B=0},
$$
which measures the linear response of the induced average toroidal moment per mole to a magnetic field with a small non-vanishing curl. The microscopic Hamiltonian contains a toroidal coupling
$$
H_{\rm Tor}=\sum_{i=1}^{10}(\mathbf r_i\times\mathbf M_i)\cdot(\nabla\times\mathbf B)
=\boldsymbol\tau\cdot(\nabla\times\mathbf B),
$$
and the partition function is evaluated through a product of ten $24\times24$ transfer matrices. The resulting Van Vleck-type expression for $\xi_{\alpha\beta}$ yields a low-temperature sum rule,
$$
\lim_{T\to0}\xi_{\alpha\alpha}T=\bigl|\langle0|\tau_\alpha|0\rangle\bigr|^2.
$$
For Fe\(_{10}\)Dy\(_{10}\), $\xi_{zz}T\approx(1.5\times10^3\;\mu_B\!\cdot\!\text{\AA})^2$ as $T\to0$, implying $\langle0|\tau_z|0\rangle\approx1.5\times10^3\;\mu_B\text{\AA}$; above $T\sim1\,{\rm K}$, $\xi_{zz}$ decays roughly like $1/T$ but remains sizable up to $T\approx10\,{\rm K}$, and a modest uniform field $\lesssim1\,{\rm T}$ slightly enhances the susceptibility by mixing excited toroidal states [2509.05424].

Taken together, these works show two non-equivalent but complementary molecular usages. One identifies toroidal response with the antisymmetric sector of $\chi_{ij}$ at the single-ion level; the other defines a bona fide thermodynamic susceptibility conjugate to $\nabla\times\mathbf B$.

## 4. Nonlinear magnetic susceptibility induced by electric toroidal dipole order

Electric toroidal dipole ordering generates a different susceptibility problem. In the five-\(d\)-orbital model under a tetragonal crystalline electric field, the ETD order parameter is time-reversal even and inversion even, so it does not produce an ordinary linear antisymmetric magnetic susceptibility. Instead, the leading transverse response is third order:
$$
M_\eta=\chi_{\eta\mu}H_\mu+\chi_{\eta\mu\nu\kappa}H_\mu H_\nu H_\kappa+\cdots,
$$
with
$$
M_x=\chi_{xyyy}H_y^3,\qquad
M_y=-\chi_{xyyy}H_x^3 .
$$
The nonlinear coefficient $\chi_{xyyy}$ is obtained from a static, uniform nonlinear Kubo formula and becomes symmetry-allowed because ETD order lowers $D_{4h}$ to $C_{4h}$, thereby permitting $\chi_{xyyy}=-\chi_{yxxx}\neq0$ [2212.13018].

The microscopic ingredients identified as important are a low-lying first excited crystal-field doublet and strong spin-orbital entanglement. The dominant contribution comes from processes involving the transition between the ground-state Kramers pair and the first excited pair, with enhancement that scales roughly as $1/\Delta^2$ when the CEF gap $\Delta$ is small. A decomposition into orbital, spin, and mixed pieces shows that the mixed contribution $\chi_{LS}$ dominates, consistent with the ETD operator itself entangling $\ell$ and $s$. Numerically, for a typical spin-orbit coupling $\lambda=0.2J$ and a CEF gap of order unity, $\chi_{xyyy}\sim O(10^2)$ at $T\sim0.1$ in units where $J=1$, giving $M_x\sim10^{-7}$ for $H_y=10^{-3}$; larger fields or a smaller $\Delta$ can raise $M_x/M_y$ into the percent range. This establishes a nonlinear route by which toroidal order controls magnetic susceptibility even when the linear antisymmetric channel is forbidden.

## 5. Sensitivity of external magnetic fields to toroidal current cross-section

In tokamak plasma physics, toroidal susceptibility is defined neither from a toroidal moment nor from a multipole tensor. For a toroidal current channel of circular cross-section and total current $I$, with cross-sectional area $S=\pi a_0^2$, the susceptibility is
$$
\chi_t(\theta,p,\phi)=\frac{\partial B_{\rm ext}(\theta,p,\phi)}{\partial S},
$$
where $B_{\rm ext}$ is the magnitude of the poloidal component of the external magnetic field at an external point. The toroidal current is modeled by $N$ co-axial filaments with current-density profile
$$
J(r)=J_0\,[1-(r/a_0)^2]^\gamma,
$$
and finite-cross-section effects are obtained by Biot-Savart superposition. Since $\Delta S=2\pi a_0\Delta a_0$ for small $\Delta a_0$,
$$
\frac{\partial B_{\rm ext}}{\partial S}
=\frac{1}{2\pi a_0}\frac{\partial B_{\rm ext}}{\partial a_0}.
$$
A sign change in $\partial B_{\rm ext}/\partial a_0$ separates sensitive and insensitive regimes, and the insensitive point is defined by
$$
\frac{\partial B_{\rm ext}(\theta_{\rm inv},p)}{\partial S}=0.
$$
Numerically, a single angle of invariance is found on the circular arc of radius $p$, obeying the empirical relation
$$
\theta_{\rm inv}=|1.297-0.571\,(p/R_0)|
$$
in radians [2601.19401].

Experimental validation was carried out on Aditya Upgrade using a 16-channel Mirnov-probe garland in a single poloidal plane, with probes at equal angular steps $\Delta\theta=22.5^\circ$ and radius $p=27.5\,{\rm cm}$. Probe signals were corrected by subtracting vacuum-shot pickups, low-pass filtered below $30\,{\rm Hz}$, and calibrated individually. To mimic a change in $a_0$ at fixed $R_0$ and geometric center, the fields from probe pairs at $\pm\theta$ were averaged:
$$
\langle B\rangle(\theta)=\frac{B(+\theta)+B(-\theta)}{2}.
$$
Numerical tests showed that this reproduces the exact $B_{\rm ext}$ for a shifted-in-$a_0$ plasma with maximum uncertainty $\lesssim0.9\%$. Analysis of approximately 50 discharges found that for inboard angles $(\theta<\theta_{\rm inv})$ the field decreases with increasing $a_0$, whereas for outboard angles $(\theta>\theta_{\rm inv})$ it increases; no measurable change was detected near $\theta\simeq62.3^\circ$, confirming the predicted insensitive point. In this usage, toroidal susceptibility is a diagnostic sensitivity measure for equilibrium reconstruction and plasma-shape monitoring.

## 6. Effective toroidal susceptibility in metamaterials

In planar metamaterials, toroidal susceptibility is formulated as an effective electromagnetic response derived from a toroidal polarizability. For the asymmetric split-ring-resonator (ASRR) structure, the toroidal moment induced in one unit cell is taken to be linearly related to the incident field,
$$
T_x(\omega)=\alpha_T(\omega)\,E_x(\omega),
$$
and the effective susceptibility is then
$$
\chi_T(\omega)=\frac{N\,\alpha_T(\omega)}{\varepsilon_0}
=\frac{\alpha_T(\omega)}{\varepsilon_0 V_{\rm cell}} .
$$
Here $N=1/V_{\rm cell}$ is the unit-cell density. The same current distribution enters the standard multipole expansion, alongside the electric dipole $\mathbf p$ and magnetic dipole $\mathbf m$, so that the toroidal term can be isolated by direct numerical evaluation of the multipole integrals [1209.3400].

The ASRR metamaterial is designed so that horizontal mirror symmetry and vertical $C_2$ symmetry suppress the net electric and magnetic dipoles at the lower resonance while enabling in-phase vertical coupling of the two layer currents to form the head-to-tail magnetic vortex of $\mathbf T$. FDTD calculations show that the radiated toroidal power $P_T$ peaks three orders of magnitude above $P_p$ at the toroidal resonance. Field maps at $6.378\,{\rm GHz}$ display a closed loop of magnetic flux threading the four ASRRs, with deep-subwavelength confinement. Experimentally, a PCB slab of size $540\,{\rm mm}\times440\,{\rm mm}$ was measured in an anechoic chamber using a vector network analyzer, and the narrow transmission feature at $6.378\,{\rm GHz}$ agreed with the calculated toroidal response. In this setting, toroidal susceptibility is an effective-medium quantity characterizing how a structured current distribution converts incident electromagnetic drive into a toroidal multipole.

## 7. Terminological boundaries: toroidal topology versus toroidal moment

A recurrent source of ambiguity is that some susceptibilities are studied on a torus without being toroidal susceptibilities in the multipolar sense. In lattice QCD, the magnetic susceptibility is defined from the free-energy density by
$$
\chi=-\frac{\partial^2 f}{\partial B^2}\Big|_{B=0},
$$
but the adjective “toroidal” enters through the four-dimensional torus on which the theory is formulated. The uniform magnetic flux through an $x$–$y$ slice is quantized,
$$
B=\frac{2\pi b}{qL_xL_y},\qquad b\in\mathbb Z,
$$
and the free-energy shift is reconstructed by analytically continuing $b$ to real values and integrating $M(b)=\partial f/\partial b$. After $T=0$ subtraction, the renormalized shift is fitted quadratically to extract $\chi$. The result is that the susceptibility is compatible with zero for $T\lesssim150\,{\rm MeV}$, rises sharply across the deconfinement crossover at $T_c\approx160$–$170\,{\rm MeV}$, reaches $\chi\approx(2$–$6)\times10^{-2}$ in SI units for $T\gtrsim200\,{\rm MeV}$, and remains positive, so strongly interacting matter is paramagnetic on the torus [1312.5070].

An analogous distinction appears in the two-flavor four-fermion interaction model in toroidal topology, where the response quantity is the chiral susceptibility
$$
\chi_m=-\frac{\partial^2\Omega}{\partial m^2}=\sum_{i=u,d}\frac{\partial\phi_i}{\partial m}.
$$
Finite temperature, finite box size, chemical potential, and magnetic field are implemented through generalized Matsubara sums and proper-time integrals with Jacobi theta functions. The dependence on boundary conditions is decisive: with antiperiodic spatial boundary conditions, infrared modes are suppressed, the condensate and $\chi_m$ decrease as $L$ decreases, and below a critical size $L_c\approx0.5\,{\rm fm}$ the crossover peak disappears; with periodic boundary conditions, a zero mode survives, the condensate grows as $L\to0$, and no finite-volume chiral restoration occurs [2004.11237].

These cases do not define toroidal susceptibility as a response of a toroidal moment. They instead show that toroidal geometry or topology can alter how conventional susceptibilities are defined, renormalized, and interpreted. A plausible implication is that the phrase “toroidal susceptibility” should always be read together with its conjugate variable—$H$, $\nabla\times\mathbf B$, $S$, or topological flux—before any comparison across subfields is attempted.

Source: https://www.emergentmind.com/topics/toroidal-susceptibility