Toroidal Susceptibility

• Toroidal susceptibility is a class of response functions that quantify how toroidal moments—arising from vortex-like or head-to-tail configurations—react to varying external fields.
• Definitions differ by context, ranging from field derivatives of macroscopic toroidal moments in chiral magnets to antisymmetric tensor components in single-molecule systems.
• Experimental and numerical methods, including neutron diffraction and FDTD simulations, validate these responses across condensed matter, plasma physics, and metamaterials.

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 (Ding et al., 2021, Jenkins et al., 8 Jun 2026, Soncini et al., 5 Sep 2025, Aich et al., 27 Jan 2026, Fan et al., 2012).

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

$$\hat G_{zm}=(\hat{\mathbf l}_m\times \hat{\mathbf s}_m)^z=\hat l_m^x\hat s_m^y-\hat l_m^y\hat s_m^x .$$

(Ding et al., 2021, Jenkins et al., 8 Jun 2026, Fan et al., 2012, Inda et al., 2022)

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 $$\mathbf T = \tfrac12 \sum_i \mathbf r_i \times \mathbf m_i .$$0
Fe$$\mathbf T = \tfrac12 \sum_i \mathbf r_i \times \mathbf m_i .$$1Dy$$\mathbf T = \tfrac12 \sum_i \mathbf r_i \times \mathbf m_i .$$2 ring	$$\mathbf T = \tfrac12 \sum_i \mathbf r_i \times \mathbf m_i .$$3	Small non-vanishing curl of $$\mathbf T = \tfrac12 \sum_i \mathbf r_i \times \mathbf m_i .$$4
Toroidal current column	$$\mathbf T = \tfrac12 \sum_i \mathbf r_i \times \mathbf m_i .$$5	Cross-sectional area $$\mathbf T = \tfrac12 \sum_i \mathbf r_i \times \mathbf m_i .$$6
Planar ASRR metamaterial	$$\mathbf T = \tfrac12 \sum_i \mathbf r_i \times \mathbf m_i .$$7	Incident field through $$\mathbf T = \tfrac12 \sum_i \mathbf r_i \times \mathbf m_i .$$8
ETD-ordered model	$$\mathbf T = \tfrac12 \sum_i \mathbf r_i \times \mathbf m_i .$$9	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$$\mathbf T=\frac{1}{10c}\int\!\Bigl[(\mathbf r\!\cdot\!\mathbf J)\,\mathbf r-2r^2\mathbf J\Bigr]\,d^3r .$$0, the toroidal susceptibility is defined operationally as

$$\mathbf T=\frac{1}{10c}\int\!\Bigl[(\mathbf r\!\cdot\!\mathbf J)\,\mathbf r-2r^2\mathbf J\Bigr]\,d^3r .$$1

with $$\mathbf T=\frac{1}{10c}\int\!\Bigl[(\mathbf r\!\cdot\!\mathbf J)\,\mathbf r-2r^2\mathbf J\Bigr]\,d^3r .$$2 the macroscopic toroidal moment and $$\mathbf T=\frac{1}{10c}\int\!\Bigl[(\mathbf r\!\cdot\!\mathbf J)\,\mathbf r-2r^2\mathbf J\Bigr]\,d^3r .$$3. The microscopic Hamiltonian combines antiferromagnetic Heisenberg exchanges $$\mathbf T=\frac{1}{10c}\int\!\Bigl[(\mathbf r\!\cdot\!\mathbf J)\,\mathbf r-2r^2\mathbf J\Bigr]\,d^3r .$$4, easy-plane single-ion anisotropy $$\mathbf T=\frac{1}{10c}\int\!\Bigl[(\mathbf r\!\cdot\!\mathbf J)\,\mathbf r-2r^2\mathbf J\Bigr]\,d^3r .$$5, Dzyaloshinskii-Moriya vectors with both $$\mathbf T=\frac{1}{10c}\int\!\Bigl[(\mathbf r\!\cdot\!\mathbf J)\,\mathbf r-2r^2\mathbf J\Bigr]\,d^3r .$$6 and $$\mathbf T=\frac{1}{10c}\int\!\Bigl[(\mathbf r\!\cdot\!\mathbf J)\,\mathbf r-2r^2\mathbf J\Bigr]\,d^3r .$$7 components, and the Zeeman coupling. In zero field, the dominant frustrated $$\mathbf T=\frac{1}{10c}\int\!\Bigl[(\mathbf r\!\cdot\!\mathbf J)\,\mathbf r-2r^2\mathbf J\Bigr]\,d^3r .$$8 network locks each trimer into a 120° in-plane vortex, generating $$\mathbf T=\frac{1}{10c}\int\!\Bigl[(\mathbf r\!\cdot\!\mathbf J)\,\mathbf r-2r^2\mathbf J\Bigr]\,d^3r .$$9 on each trimer. The subleading $$\hat G_{zm}=(\hat{\mathbf l}_m\times \hat{\mathbf s}_m)^z=\hat l_m^x\hat s_m^y-\hat l_m^y\hat s_m^x .$$0 terms select a ferritoroidal arrangement in which two of the three sublattices carry $$\hat G_{zm}=(\hat{\mathbf l}_m\times \hat{\mathbf s}_m)^z=\hat l_m^x\hat s_m^y-\hat l_m^y\hat s_m^x .$$1 and one carries $$\hat G_{zm}=(\hat{\mathbf l}_m\times \hat{\mathbf s}_m)^z=\hat l_m^x\hat s_m^y-\hat l_m^y\hat s_m^x .$$2, giving net $$\hat G_{zm}=(\hat{\mathbf l}_m\times \hat{\mathbf s}_m)^z=\hat l_m^x\hat s_m^y-\hat l_m^y\hat s_m^x .$$3. An out-of-plane DM component further cants each spin slightly along $$\hat G_{zm}=(\hat{\mathbf l}_m\times \hat{\mathbf s}_m)^z=\hat l_m^x\hat s_m^y-\hat l_m^y\hat s_m^x .$$4 in a sense locked to the vortex chirality, so that each sublattice also carries a small magnetization $$\hat G_{zm}=(\hat{\mathbf l}_m\times \hat{\mathbf s}_m)^z=\hat l_m^x\hat s_m^y-\hat l_m^y\hat s_m^x .$$5 (Ding et al., 2021).

Because the DM interaction locks $$\hat G_{zm}=(\hat{\mathbf l}_m\times \hat{\mathbf s}_m)^z=\hat l_m^x\hat s_m^y-\hat l_m^y\hat s_m^x .$$6 to $$\hat G_{zm}=(\hat{\mathbf l}_m\times \hat{\mathbf s}_m)^z=\hat l_m^x\hat s_m^y-\hat l_m^y\hat s_m^x .$$7, a field $$\hat G_{zm}=(\hat{\mathbf l}_m\times \hat{\mathbf s}_m)^z=\hat l_m^x\hat s_m^y-\hat l_m^y\hat s_m^x .$$8 favors trimers with $$\hat G_{zm}=(\hat{\mathbf l}_m\times \hat{\mathbf s}_m)^z=\hat l_m^x\hat s_m^y-\hat l_m^y\hat s_m^x .$$9. At a critical field $_4$0, one trimer-sublattice flips its chirality and the system undergoes a ferri-to-ferrotoroidal transition, with net $_4$1 jumping from $_4$2 to $_4$3. Experimentally, both $_4$4 and the neutron-refined $_4$5 exhibit a multi-step evolution, with kinks at the weak ferromagnetic-domain step ($_4$6), the ferri-to-ferro toroidal step ($_4$7), and further high-field transitions near $_4$8 and $_4$9. Correspondingly, $\chi_T \equiv \partial T_z/\partial H_z$0 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 $\chi_T \equiv \partial T_z/\partial H_z$1 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 $\chi_T \equiv \partial T_z/\partial H_z$2. For the trimeric Dy(III) single-molecule toroic $\chi_T \equiv \partial T_z/\partial H_z$3, the induced magnetization is written as

$\chi_T \equiv \partial T_z/\partial H_z$4

with decomposition

$\chi_T \equiv \partial T_z/\partial H_z$5

The antisymmetric part generates a toroidal response through

$\chi_T \equiv \partial T_z/\partial H_z$6

Polarized neutron diffraction at $\chi_T \equiv \partial T_z/\partial H_z$7 and $\chi_T \equiv \partial T_z/\partial H_z$8 in the $\chi_T \equiv \partial T_z/\partial H_z$9 plane was used to refine the nine components of the Dy$H\parallel c$0 susceptibility tensor; its antisymmetric part gives direct evidence of a toroidal response in the $H\parallel c$1 plane, with $H\parallel c$2 along the crystallographic $H\parallel c$3 axis. Variable-field neutron diffraction further showed field-induced magnetization along $H\parallel c$4 with toroidal moments anti-parallelly stacked, and successive layers along $H\parallel c$5 were found to be antiferrotoroidically stacked, explaining the absence of net toroidal moment in zero field (Jenkins et al., 8 Jun 2026).

A distinct thermodynamic formulation was introduced for the Fe$H\parallel c$6Dy$H\parallel c$7 ring through the molar toroidal susceptibility tensor

$H\parallel c$8

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\parallel c$9

and the partition function is evaluated through a product of ten $_3$0 transfer matrices. The resulting Van Vleck-type expression for $_3$1 yields a low-temperature sum rule,

$_3$2

For Fe$_3$3Dy$_3$4, $_3$5 as $_3$6, implying $_3$7; above $_3$8, $_3$9 decays roughly like $\chi^{\rm A}_{ij}$0 but remains sizable up to $\chi^{\rm A}_{ij}$1, and a modest uniform field $\chi^{\rm A}_{ij}$2 slightly enhances the susceptibility by mixing excited toroidal states (Soncini et al., 5 Sep 2025).

Taken together, these works show two non-equivalent but complementary molecular usages. One identifies toroidal response with the antisymmetric sector of $\chi^{\rm A}_{ij}$3 at the single-ion level; the other defines a bona fide thermodynamic susceptibility conjugate to $\chi^{\rm A}_{ij}$4.

4. Nonlinear magnetic susceptibility induced by electric toroidal dipole order

Electric toroidal dipole ordering generates a different susceptibility problem. In the five-$\chi^{\rm A}_{ij}$5-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:

$\chi^{\rm A}_{ij}$6

with

$\chi^{\rm A}_{ij}$7

The nonlinear coefficient $\chi^{\rm A}_{ij}$8 is obtained from a static, uniform nonlinear Kubo formula and becomes symmetry-allowed because ETD order lowers $\chi^{\rm A}_{ij}$9 to $\chi_{ij}$0, thereby permitting $\chi_{ij}$1 (Inda et al., 2022).

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 $\chi_{ij}$2 when the CEF gap $\chi_{ij}$3 is small. A decomposition into orbital, spin, and mixed pieces shows that the mixed contribution $\chi_{ij}$4 dominates, consistent with the ETD operator itself entangling $\chi_{ij}$5 and $\chi_{ij}$6. Numerically, for a typical spin-orbit coupling $\chi_{ij}$7 and a CEF gap of order unity, $\chi_{ij}$8 at $\chi_{ij}$9 in units where $$\mathbf T = \tfrac12 \sum_i \mathbf r_i \times \mathbf m_i .$$00, giving $$\mathbf T = \tfrac12 \sum_i \mathbf r_i \times \mathbf m_i .$$01 for $$\mathbf T = \tfrac12 \sum_i \mathbf r_i \times \mathbf m_i .$$02; larger fields or a smaller $$\mathbf T = \tfrac12 \sum_i \mathbf r_i \times \mathbf m_i .$$03 can raise $$\mathbf T = \tfrac12 \sum_i \mathbf r_i \times \mathbf m_i .$$04 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 $$\mathbf T = \tfrac12 \sum_i \mathbf r_i \times \mathbf m_i .$$05, with cross-sectional area $$\mathbf T = \tfrac12 \sum_i \mathbf r_i \times \mathbf m_i .$$06, the susceptibility is

$$\mathbf T = \tfrac12 \sum_i \mathbf r_i \times \mathbf m_i .$$07

where $$\mathbf T = \tfrac12 \sum_i \mathbf r_i \times \mathbf m_i .$$08 is the magnitude of the poloidal component of the external magnetic field at an external point. The toroidal current is modeled by $$\mathbf T = \tfrac12 \sum_i \mathbf r_i \times \mathbf m_i .$$09 co-axial filaments with current-density profile

$$\mathbf T = \tfrac12 \sum_i \mathbf r_i \times \mathbf m_i .$$10

and finite-cross-section effects are obtained by Biot-Savart superposition. Since $$\mathbf T = \tfrac12 \sum_i \mathbf r_i \times \mathbf m_i .$$11 for small $$\mathbf T = \tfrac12 \sum_i \mathbf r_i \times \mathbf m_i .$$12,

$$\mathbf T = \tfrac12 \sum_i \mathbf r_i \times \mathbf m_i .$$13

A sign change in $$\mathbf T = \tfrac12 \sum_i \mathbf r_i \times \mathbf m_i .$$14 separates sensitive and insensitive regimes, and the insensitive point is defined by

$$\mathbf T = \tfrac12 \sum_i \mathbf r_i \times \mathbf m_i .$$15

Numerically, a single angle of invariance is found on the circular arc of radius $$\mathbf T = \tfrac12 \sum_i \mathbf r_i \times \mathbf m_i .$$16, obeying the empirical relation

$$\mathbf T = \tfrac12 \sum_i \mathbf r_i \times \mathbf m_i .$$17

in radians (Aich et al., 27 Jan 2026).

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 $$\mathbf T = \tfrac12 \sum_i \mathbf r_i \times \mathbf m_i .$$18 and radius $$\mathbf T = \tfrac12 \sum_i \mathbf r_i \times \mathbf m_i .$$19. Probe signals were corrected by subtracting vacuum-shot pickups, low-pass filtered below $$\mathbf T = \tfrac12 \sum_i \mathbf r_i \times \mathbf m_i .$$20, and calibrated individually. To mimic a change in $$\mathbf T = \tfrac12 \sum_i \mathbf r_i \times \mathbf m_i .$$21 at fixed $$\mathbf T = \tfrac12 \sum_i \mathbf r_i \times \mathbf m_i .$$22 and geometric center, the fields from probe pairs at $$\mathbf T = \tfrac12 \sum_i \mathbf r_i \times \mathbf m_i .$$23 were averaged:

$$\mathbf T = \tfrac12 \sum_i \mathbf r_i \times \mathbf m_i .$$24

Numerical tests showed that this reproduces the exact $$\mathbf T = \tfrac12 \sum_i \mathbf r_i \times \mathbf m_i .$$25 for a shifted-in-$$\mathbf T = \tfrac12 \sum_i \mathbf r_i \times \mathbf m_i .$$26 plasma with maximum uncertainty $$\mathbf T = \tfrac12 \sum_i \mathbf r_i \times \mathbf m_i .$$27. Analysis of approximately 50 discharges found that for inboard angles $$\mathbf T = \tfrac12 \sum_i \mathbf r_i \times \mathbf m_i .$$28 the field decreases with increasing $$\mathbf T = \tfrac12 \sum_i \mathbf r_i \times \mathbf m_i .$$29, whereas for outboard angles $$\mathbf T = \tfrac12 \sum_i \mathbf r_i \times \mathbf m_i .$$30 it increases; no measurable change was detected near $$\mathbf T = \tfrac12 \sum_i \mathbf r_i \times \mathbf m_i .$$31, 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,

$$\mathbf T = \tfrac12 \sum_i \mathbf r_i \times \mathbf m_i .$$32

and the effective susceptibility is then

$$\mathbf T = \tfrac12 \sum_i \mathbf r_i \times \mathbf m_i .$$33

Here $$\mathbf T = \tfrac12 \sum_i \mathbf r_i \times \mathbf m_i .$$34 is the unit-cell density. The same current distribution enters the standard multipole expansion, alongside the electric dipole $$\mathbf T = \tfrac12 \sum_i \mathbf r_i \times \mathbf m_i .$$35 and magnetic dipole $$\mathbf T = \tfrac12 \sum_i \mathbf r_i \times \mathbf m_i .$$36, so that the toroidal term can be isolated by direct numerical evaluation of the multipole integrals (Fan et al., 2012).

The ASRR metamaterial is designed so that horizontal mirror symmetry and vertical $$\mathbf T = \tfrac12 \sum_i \mathbf r_i \times \mathbf m_i .$$37 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 $$\mathbf T = \tfrac12 \sum_i \mathbf r_i \times \mathbf m_i .$$38. FDTD calculations show that the radiated toroidal power $$\mathbf T = \tfrac12 \sum_i \mathbf r_i \times \mathbf m_i .$$39 peaks three orders of magnitude above $$\mathbf T = \tfrac12 \sum_i \mathbf r_i \times \mathbf m_i .$$40 at the toroidal resonance. Field maps at $$\mathbf T = \tfrac12 \sum_i \mathbf r_i \times \mathbf m_i .$$41 display a closed loop of magnetic flux threading the four ASRRs, with deep-subwavelength confinement. Experimentally, a PCB slab of size $$\mathbf T = \tfrac12 \sum_i \mathbf r_i \times \mathbf m_i .$$42 was measured in an anechoic chamber using a vector network analyzer, and the narrow transmission feature at $$\mathbf T = \tfrac12 \sum_i \mathbf r_i \times \mathbf m_i .$$43 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

$$\mathbf T = \tfrac12 \sum_i \mathbf r_i \times \mathbf m_i .$$44

but the adjective “toroidal” enters through the four-dimensional torus on which the theory is formulated. The uniform magnetic flux through an $$\mathbf T = \tfrac12 \sum_i \mathbf r_i \times \mathbf m_i .$$45–$$\mathbf T = \tfrac12 \sum_i \mathbf r_i \times \mathbf m_i .$$46 slice is quantized,

$$\mathbf T = \tfrac12 \sum_i \mathbf r_i \times \mathbf m_i .$$47

and the free-energy shift is reconstructed by analytically continuing $$\mathbf T = \tfrac12 \sum_i \mathbf r_i \times \mathbf m_i .$$48 to real values and integrating $$\mathbf T = \tfrac12 \sum_i \mathbf r_i \times \mathbf m_i .$$49. After $$\mathbf T = \tfrac12 \sum_i \mathbf r_i \times \mathbf m_i .$$50 subtraction, the renormalized shift is fitted quadratically to extract $$\mathbf T = \tfrac12 \sum_i \mathbf r_i \times \mathbf m_i .$$51. The result is that the susceptibility is compatible with zero for $$\mathbf T = \tfrac12 \sum_i \mathbf r_i \times \mathbf m_i .$$52, rises sharply across the deconfinement crossover at $$\mathbf T = \tfrac12 \sum_i \mathbf r_i \times \mathbf m_i .$$53–$$\mathbf T = \tfrac12 \sum_i \mathbf r_i \times \mathbf m_i .$$54, reaches $$\mathbf T = \tfrac12 \sum_i \mathbf r_i \times \mathbf m_i .$$55–$$\mathbf T = \tfrac12 \sum_i \mathbf r_i \times \mathbf m_i .$$56 in SI units for $$\mathbf T = \tfrac12 \sum_i \mathbf r_i \times \mathbf m_i .$$57, and remains positive, so strongly interacting matter is paramagnetic on the torus (Bonati et al., 2013).

An analogous distinction appears in the two-flavor four-fermion interaction model in toroidal topology, where the response quantity is the chiral susceptibility

$$\mathbf T = \tfrac12 \sum_i \mathbf r_i \times \mathbf m_i .$$58

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 $$\mathbf T = \tfrac12 \sum_i \mathbf r_i \times \mathbf m_i .$$59 decrease as $$\mathbf T = \tfrac12 \sum_i \mathbf r_i \times \mathbf m_i .$$60 decreases, and below a critical size $$\mathbf T = \tfrac12 \sum_i \mathbf r_i \times \mathbf m_i .$$61 the crossover peak disappears; with periodic boundary conditions, a zero mode survives, the condensate grows as $$\mathbf T = \tfrac12 \sum_i \mathbf r_i \times \mathbf m_i .$$62, and no finite-volume chiral restoration occurs (Abreu et al., 2020).

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—$$\mathbf T = \tfrac12 \sum_i \mathbf r_i \times \mathbf m_i .$$63, $$\mathbf T = \tfrac12 \sum_i \mathbf r_i \times \mathbf m_i .$$64, $$\mathbf T = \tfrac12 \sum_i \mathbf r_i \times \mathbf m_i .$$65, or topological flux—before any comparison across subfields is attempted.