Electric Toroidal Monopoles

• Electric toroidal monopoles are pseudoscalar, ℓ=0 entities that are inversion-odd and time-reversal-even, capturing the essence of chirality in various systems.
• They emerge as O(k²) corrections in the exact multipole expansion, clarifying the distinction between non-radiative formal terms and microscopic symmetry classifiers.
• Their quantification through tight-binding and many-body models provides a practical measure of chirality in materials such as twisted methane, elemental tellurium, and complex hidden-order phases.

Searching arXiv for the specified papers and closely related work on electric toroidal monopoles. arXiv search query: "electric toroidal monopole (Inda et al., 2024, Kusunose et al., 2024, Fernandez-Corbaton et al., 2015, Kuniyoshi et al., 11 Mar 2026)" Electric toroidal monopoles denote parity-odd, time-reversal-even $\ell=0$ objects whose precise meaning depends on the formalism in which they are introduced. In symmetry-adapted multipole theory for electronic states, the electric-toroidal monopole $G_0$ is the unique pseudoscalar microscopic variable used to characterize structural or electronic chirality in nonmagnetic and magnetic materials (Kusunose et al., 2024, Inda et al., 2024). In the exact multipole expansion of a localized time-harmonic current, a formally analogous “toroidal monopole” appears as the $O(k^2)$ term in the electric-parity family, but it does not define an independent radiative or coupling channel (Fernandez-Corbaton et al., 2015). Recent many-body work extends the same symmetry logic to interacting systems, where a two-body electric toroidal monopole can become active in spinless orbitals even though the corresponding one-body operator is absent (Kuniyoshi et al., 11 Mar 2026).

1. Conceptual scope and defining symmetries

The phrase “electric toroidal monopole” is used in two distinct literatures. One concerns the exact electromagnetic multipole expansion of localized currents; the other concerns symmetry-adapted multipoles of electronic degrees of freedom in molecules and solids. The common element is the appearance of an $\ell=0$ object with pseudoscalar character, but the physical interpretation is different.

Context	Representative quantity	Stated physical status
Exact current multipole expansion	$$T_{00}=\frac{1}{6\pi}\int d^3r\, r^2\,\mathbf J_\omega(\mathbf r)$$	Formal $O(k^2)$ term in the electric-parity expansion
Electronic symmetry-adapted multipoles	$G_0$	Unique $\mathcal T$-even, $\mathcal P$-odd pseudoscalar
Many-body spinless operator space	$G_0^{(2)}$	Two-body $G_0$0-odd, $G_0$1-even scalar

In the electronic setting, $G_0$2 is one of the symmetry-adapted multipole “charges” introduced to complete the classification of microscopic charge and current configurations. It is the unique $G_0$3 pseudoscalar that is $G_0$4-even and $G_0$5-odd, and it therefore matches “true chirality” in the symmetry sense (Kusunose et al., 2024, Inda et al., 2024). In the continuum language summarized for solids, $G_0$6 is axial, scalar under proper rotations, odd under inversion, and even under time reversal.

A common source of confusion is the use of similar terminology for the dynamical electromagnetic expansion. There, the toroidal monopole is not a separate family of multipoles alongside electric and magnetic ones. Rather, it is a higher-order term extracted from the electric-parity coefficients by a small-$G_0$7 expansion, and its separate isolation introduces non-radiative pieces that cancel only in the exact sum (Fernandez-Corbaton et al., 2015). This distinction is central: the electronic $G_0$8 is a symmetry classifier and possible order parameter, whereas the dynamical toroidal monopole is a formal subleading contribution within the electric multipole family.

2. Exact electromagnetic origin: higher-order electric-parity term

For a time-harmonic current $G_0$9 confined to $O(k^2)$0 with $O(k^2)$1, the exact transverse multipolar coefficients of electric parity are denoted $O(k^2)$2. Their coordinate-space expression involves spherical Bessel functions of orders $O(k^2)$3 and $O(k^2)$4, and each radial integral admits a power series in $O(k^2)$5 (Fernandez-Corbaton et al., 2015). Collecting terms by powers of $O(k^2)$6 yields the general structure

$O(k^2)$7

where $O(k^2)$8 is the lowest-order electric multipole contribution and $O(k^2)$9 is the next-order term customarily called the toroidal multipole.

For $\ell=0$0, this procedure reproduces the familiar electric dipole at leading order and the standard toroidal dipole moment at $\ell=0$1. The same logic may be pushed formally to $\ell=0$2. In that case,

$\ell=0$3

with

$\ell=0$4

This object may formally be called the toroidal monopole moment (Fernandez-Corbaton et al., 2015).

The paper’s key result is that this formal splitting does not endow the toroidal contribution with independent physical meaning. The exact coefficients $\ell=0$5 contain only on-shell Fourier components $\ell=0$6 with $\ell=0$7, which are the only components that genuinely couple to real electromagnetic waves outside the source region. By contrast, each truncated term obtained from the Taylor expansion of the spherical Bessel functions fails to enforce the restriction $\ell=0$8, so the separate pieces $\ell=0$9, $$T_{00}=\frac{1}{6\pi}\int d^3r\, r^2\,\mathbf J_\omega(\mathbf r)$$0, and higher orders acquire out-of-shell, non-radiative Fourier components. These spurious components cancel only when the full series is recombined (Fernandez-Corbaton et al., 2015).

The immediate consequence is that there is no toroidal radiation or independent toroidal electromagnetic coupling. Any on-shell measurement, whether near-field or far-field, sees only the exact coefficients $$T_{00}=\frac{1}{6\pi}\int d^3r\, r^2\,\mathbf J_\omega(\mathbf r)$$1. Conversely, attempts to detect $$T_{00}=\frac{1}{6\pi}\int d^3r\, r^2\,\mathbf J_\omega(\mathbf r)$$2 alone would require access to longitudinal or static Fourier components of the current, and those are canceled in the total field by the charge-density contribution. In particular, the toroidal monopole $$T_{00}=\frac{1}{6\pi}\int d^3r\, r^2\,\mathbf J_\omega(\mathbf r)$$3 has no transverse on-shell component by itself and therefore does not appear in any multipolar radiation pattern and does not obey any independent selection rule or coupling coefficient (Fernandez-Corbaton et al., 2015).

3. Microscopic electronic definition of the electric-toroidal monopole

In the electronic multipole framework, the electric-toroidal monopole $$T_{00}=\frac{1}{6\pi}\int d^3r\, r^2\,\mathbf J_\omega(\mathbf r)$$4 is defined as the unique scalar that is odd under spatial inversion and even under time reversal: $$T_{00}=\frac{1}{6\pi}\int d^3r\, r^2\,\mathbf J_\omega(\mathbf r)$$5 This identifies $$T_{00}=\frac{1}{6\pi}\int d^3r\, r^2\,\mathbf J_\omega(\mathbf r)$$6 as a pseudoscalar and makes it the natural microscopic descriptor of chirality in a quantum-mechanical setting (Inda et al., 2024).

Several operator realizations are used. In an atomic-scale spin-orbital description,

$$T_{00}=\frac{1}{6\pi}\int d^3r\, r^2\,\mathbf J_\omega(\mathbf r)$$7

where $$T_{00}=\frac{1}{6\pi}\int d^3r\, r^2\,\mathbf J_\omega(\mathbf r)$$8 is the orbital angular momentum and $$T_{00}=\frac{1}{6\pi}\int d^3r\, r^2\,\mathbf J_\omega(\mathbf r)$$9 is the magnetic-toroidal dipole operator with $O(k^2)$0 (Kusunose et al., 2024). In the alternative formulation summarized for twisted methane, one may construct an on-site electric-toroidal monopole as

$O(k^2)$1

with $O(k^2)$2 the magnetic toroidal dipole and $O(k^2)$3 or $O(k^2)$4 the spin or orbital magnetic dipole (Inda et al., 2024).

Cluster constructions are equally important. If a cluster of sites $O(k^2)$5 at positions $O(k^2)$6 carries electric-toroidal dipoles $O(k^2)$7, then the cluster-multipole electric-toroidal monopole is

$O(k^2)$8

This form makes explicit that the monopole is built from lower-rank toroidal moments arranged in a spatially chiral pattern (Kusunose et al., 2024).

In second quantization, the twisted-CH$O(k^2)$9 model uses a basis

$G_0$0

and each electric-toroidal monopole takes the form

$G_0$1

where the $G_0$2 blocks $G_0$3 are purely imaginary, spin-diagonal matrices proportional to $G_0$4 (Inda et al., 2024). One representative operator is the on-site C(s)–C(p) electric-toroidal monopole, while additional bond-cluster electric-toroidal monopoles live on C–H(s), p–H(s), and H–H bonds.

These constructions emphasize that the electronic $G_0$5 is not introduced through radiative fields but through the irreducible decomposition of microscopic electronic variables under rotational, inversion, and time-reversal symmetries.

4. Tight-binding realization and quantification of chirality

The twisted-methane model provides a concrete computational realization of the electric-toroidal monopole as a quantitative measure of chirality (Inda et al., 2024). The full tight-binding Hamiltonian is

$G_0$6

with

$G_0$7

$G_0$8

$G_0$9

The real hoppings are the usual overlaps, whereas the imaginary hoppings are spin-dependent, purely imaginary, preserve $\mathcal T$0, and break parity locally (Inda et al., 2024).

In perfectly tetrahedral CH$\mathcal T$1 with point group $\mathcal T$2, all $\mathcal T$3’s vanish by symmetry because $\mathcal T$4 transforms as the $\mathcal T$5 irreducible representation of $\mathcal T$6. To activate the monopole, the model breaks all mirror planes while keeping the three $\mathcal T$7 axes, thereby lowering the symmetry to point group $\mathcal T$8. This is implemented by bond modulations

$\mathcal T$9

with the signs chosen across the bonds so as to produce a right- or left-handed twist (Inda et al., 2024).

Once the Hamiltonian is constructed, one diagonalizes it to obtain eigenstates $\mathcal P$0 and eigenvalues $\mathcal P$1, and then evaluates

$\mathcal P$2

Within this model,

$\mathcal P$3

and

$\mathcal P$4

Thus the sign of $\mathcal P$5 tracks handedness, while its magnitude supplies a quantitative measure of chirality (Inda et al., 2024).

The same study also isolates the dominant microscopic ingredient. The fitted on-site spin-orbit coupling $\mathcal P$6 is extremely small, and explicit variation of $\mathcal P$7 produces no appreciable change in any $\mathcal P$8. By contrast, the imaginary H–H hopping modulation $\mathcal P$9 is identified as the only essential parameter for generating $G_0^{(2)}$0 (Inda et al., 2024). A general computational recipe is then stated: all-electron or DFT calculation, extraction of local orbitals by symmetrized Wannierization using SymClosestWannier + MultiPie, construction of the tight-binding Hamiltonian, enumeration of the SAMB multipole operators, imposition of a small chiral distortion, diagonalization, and evaluation of $G_0^{(2)}$1.

5. Symmetry classification, materials, and response phenomena

Because $G_0^{(2)}$2 is $G_0^{(2)}$3-even, $G_0^{(2)}$4-odd, and scalar under proper rotations, it can be active only in point groups that lack inversion, lack any mirror or roto-inversion symmetry, and preserve time reversal. In the 32 crystallographic point groups, precisely 11 noncentrosymmetric “pure-rotation” chiral groups admit an electric-toroidal monopole as a primary order parameter. In magnetic point groups, $G_0^{(2)}$5 remains active only in Type II and some Type III groups that break inversion but preserve $G_0^{(2)}$6 or $G_0^{(2)}$7 (Kusunose et al., 2024). The compact selection rule given is that $G_0^{(2)}$8 belongs to $G_0^{(2)}$9 groups with no mirror or $G_0$00, and is forbidden if any mirror $G_0$01, inversion $G_0$02, or $G_0$03 axes remain.

The microscopic interpretation supplied in the same review is that $G_0$04 measures the “divergence of vorticity” of an electric-dipole distribution. A hedgehog arrangement of electric-toroidal dipoles radiating from a center provides an intuitive picture. In elemental tellurium, the $G_0$05-orbital tight-binding description near the H-point yields three contributions,

$G_0$06

$G_0$07

where the superscripts $G_0$08, $G_0$09, and $G_0$10 denote atomic, bond-cluster, and site-cluster bases (Kusunose et al., 2024). The same review states that $G_0$11 is a static divergence of the atomic electric-toroidal dipole $G_0$12, whereas $G_0$13 and $G_0$14 arise from spin-dependent complex hoppings along and transverse to the helical chains. An analogous spin-independent divergence of dipole currents realizes $G_0$15 for chiral phonons in $G_0$16-HgS and $G_0$17-Te.

Representative material and molecular examples include elemental Te, twisted methane, and the hidden-order phase of URu$G_0$18Si$G_0$19 (Kusunose et al., 2024). In elemental Te with space groups $G_0$20, three-fold helical chains realize a finite $G_0$21, and circularly polarized NMR ($G_0$22Te) together with current-induced magnetization directly probe the $G_0$23 and $G_0$24 components. Twisted methane is identified as a prototypical chiral molecule with finite $G_0$25, and the sign of $G_0$26 quantifies enantiomeric handedness. For URu$G_0$27Si$G_0$28, the review summarizes proposals that a small chiral charge $G_0$29 may be the hidden-order parameter.

Once $G_0$30, the free energy admits trilinear couplings such as

$G_0$31

together with the optical-chirality coupling

$G_0$32

The associated responses include current-induced magnetization, electric-field-induced structural rotation, magnetic-field-induced electric-toroidal dipole, and optical-chirality coupling (Kusunose et al., 2024). Experimental probes listed in the review are optical rotation under static $G_0$33 or $G_0$34, circular dichroism in transmission or reflection, current-induced magnetization measured by SQUID or NMR shift, second-harmonic generation and rotational SHG for domain imaging, and NMR/EPR line shifts.

6. Many-body extension and the status of “independence”

The recent many-body formulation generalizes multipole operators from the one-body space to the many-body operator space by treating fermionic creation and annihilation operators as spherical tensors and combining Clebsch–Gordan coupling with exterior algebra (Kuniyoshi et al., 11 Mar 2026). For spinless electrons, the four-fermion tensor is written as

$G_0$35

and the electric-toroidal monopole is the rank-$G_0$36 Hermitian combination with odd intermediate rank $G_0$37: $G_0$38 Under $G_0$39, this operator is a scalar, inversion odd, and time-reversal even; in group-theory notation its irreducible representation is $G_0$40 (Kuniyoshi et al., 11 Mar 2026).

The same work identifies a sharp structural distinction between one-body and two-body sectors. In the one-body sector, spinless single-centered hybrid orbitals do not admit an electric-toroidal operator of this type, because the required antisymmetric combination vanishes when the orbital labels coincide. In the two-body sector, however, different orbitals may be coupled in the creation and annihilation parts, so parity-odd, Hermitian, $G_0$41-even combinations survive. The paper therefore concludes that the electric toroidal monopole and the magnetic toroidal monopole become active in spinless interacting many-body systems although they are absent in the spinless one-body hybrid orbital space (Kuniyoshi et al., 11 Mar 2026).

This many-body extension sharpens the broader conceptual issue. In interacting condensed-matter systems, the electric-toroidal monopole is a legitimate symmetry-classified operator and, in suitable settings, a candidate order parameter or response channel. In the exact electrodynamics of a localized current source, by contrast, the toroidal monopole extracted from a truncated multipole expansion carries no independent on-shell information beyond the full electric-parity coefficients (Fernandez-Corbaton et al., 2015). The two statements are compatible because they concern different operator spaces and different notions of observability.

For this reason, the most persistent misconception is to treat all “toroidal monopoles” as independent radiative multipoles. The electrodynamic result rules that out for the current-expansion object $G_0$42, while the electronic and many-body results show that the symmetry-adapted $G_0$43 remains meaningful as a microscopic pseudoscalar characterizing chirality, order, and allowed cross-correlations in molecules and solids (Kusunose et al., 2024, Kuniyoshi et al., 11 Mar 2026).