Odd-Parity Multipole Order in Quantum Materials

• Odd-parity multipole order is a spontaneous phase where local parity mixing in noncentrosymmetric sites gives rise to inversion-odd magnetic and electric multipoles.
• This order converts sublattice-dependent magnetic and orbital textures into ferroic multipoles, leading to asymmetric band deformations and novel electronic responses.
• The phenomenon underpins magnetoelectric effects, nonlinear transport, and unconventional superconductivity, offering a framework for exploring correlated quantum materials.

Searching arXiv for recent and foundational papers on odd-parity multipole order to ground the article. Odd-parity multipole order is a spontaneous electronically or magnetically driven phase in which the order parameter is odd under spatial inversion. In condensed-matter systems, it is most commonly realized in crystals that are locally noncentrosymmetric but globally centrosymmetric in the paramagnetic state, so that staggered spin, orbital, or quadrupolar textures on inversion-partner sublattices become ferroic odd-parity multipoles at the unit-cell level. Depending on the time-reversal character, the active objects include magnetic toroidal dipoles and magnetic quadrupoles, which are time-odd and inversion-odd, as well as electric dipole- and electric toroidal quadrupole-type orders, which are inversion-odd and time-even (Hayami et al., 2015, Watanabe et al., 2018).

1. Symmetry definition and multipole taxonomy

The parity of a multipole follows from its tensor rank. For electric multipoles, inversion gives $\eta_{\mathrm{electric}} = (-1)^l$, whereas for magnetic multipoles inversion gives $\eta_{\mathrm{magnetic}} = (-1)^{l+1}$. An odd-parity multipole therefore changes sign under inversion, $P O P^{-1} = -O$. Electric dipoles and octupoles are odd parity, while magnetic quadrupoles and magnetic hexadecapoles are odd parity; magnetic toroidal moments are also odd under inversion and odd under time reversal (Ishizuka et al., 2018, Watanabe et al., 2018).

In crystals, the central distinction is between global and local inversion symmetry. A globally noncentrosymmetric crystal lacks an inversion center at the space-group level, whereas a locally noncentrosymmetric crystal can preserve global inversion while each crystallographic site lacks local inversion symmetry. The latter case generates sublattice-dependent antisymmetric spin-orbit coupling or, more generally, sublattice-dependent parity mixing. When an antiferromagnetic or antiferroic pattern develops without enlarging the primitive cell, it can break spatial inversion $P$ and time reversal $T$ separately while preserving the combined symmetry $PT$. This is the canonical setting for odd-parity magnetic multipole order in solids (Watanabe et al., 2020, Ishizuka et al., 2018).

A useful feature of the solid-state classification is the real-space versus momentum-space duality. For odd-parity electric multipoles, the allowed momentum-space bases are spin-dependent and encode spin-momentum locking. For odd-parity magnetic multipoles, the allowed momentum-space bases are spin-independent odd functions of $\boldsymbol{k}$, so the ordered state naturally produces asymmetric band deformations of the form $E(\boldsymbol{k}) \neq E(-\boldsymbol{k})$ while remaining compatible with the symmetry constraints of the magnetic space group (Watanabe et al., 2018, Watanabe et al., 2020).

2. Local parity mixing and microscopic origin

A microscopic route to odd-parity multipole order begins from local parity mixing between orbitals of opposite parity in a lattice that lacks inversion at each site but may retain inversion centers between sites. In the zig-zag-lattice formulation, the starting point is an extended periodic Anderson model

$H = H_0 + H_1,$

with

$$H_0 = - \sum_{i,j,\sigma} (t_{ij} c^\dagger_{i\sigma} c_{j\sigma} + \mathrm{H.c.}) + U \sum_i n^f_{i\uparrow} n^f_{i\downarrow} + E_0 \sum_{i,\sigma} n^f_{i\sigma},$$

and

$\eta_{\mathrm{magnetic}} = (-1)^{l+1}$0

For the quasi-one-dimensional zig-zag lattice, the sublattice-dependent antisymmetric hybridization takes the form

$\eta_{\mathrm{magnetic}} = (-1)^{l+1}$1

which follows from quasi-1D motion $\eta_{\mathrm{magnetic}} = (-1)^{l+1}$2, an odd-parity crystal field $\eta_{\mathrm{magnetic}} = (-1)^{l+1}$3, and $\eta_{\mathrm{magnetic}} = (-1)^{l+1}$4 (Hayami et al., 2015).

Treating $\eta_{\mathrm{magnetic}} = (-1)^{l+1}$5 perturbatively about $\eta_{\mathrm{magnetic}} = (-1)^{l+1}$6 with a Schrieffer–Wolff transformation and assuming one $\eta_{\mathrm{magnetic}} = (-1)^{l+1}$7 electron per site, one obtains an extended Kondo lattice model with a standard on-site Kondo exchange and, crucially, a sublattice-dependent antisymmetric exchange between localized spins and conduction spins. In the quasi-1D reduction,

$\eta_{\mathrm{magnetic}} = (-1)^{l+1}$8

with

$\eta_{\mathrm{magnetic}} = (-1)^{l+1}$9

The third term is the central parity-mixing-induced coupling: it is odd under inversion in momentum and sublattice, and it converts sublattice-structured magnetic order into a ferroic odd-parity multipole texture over the unit cell (Hayami et al., 2015).

This mechanism is not identical to conventional Dzyaloshinskii–Moriya exchange or to a single-band Rashba term. Conventional DM interaction acts between localized spins on inversion-broken bonds, while the antisymmetric term here is between a localized spin and the conduction-electron spin at the same site, with a $P O P^{-1} = -O$0-odd form factor $P O P^{-1} = -O$1 and a staggered coefficient $P O P^{-1} = -O$2. Rashba-type spin-orbit coupling produces a uniform $P O P^{-1} = -O$3 in the unit cell, whereas local parity mixing produces a sublattice-dependent $P O P^{-1} = -O$4 tied to virtual $P O P^{-1} = -O$5–$P O P^{-1} = -O$6 processes and to the magnetic texture (Hayami et al., 2015).

3. Zig-zag-lattice realization: toroidal and quadrupolar order

On the quasi-1D zig-zag lattice in the $P O P^{-1} = -O$7 plane, the chain direction is $P O P^{-1} = -O$8 and the two sublattices are displaced along $P O P^{-1} = -O$9 within each $P$0 period. The odd-parity crystal field points along $P$1 on the two sublattices, and the ordered state stabilized at and near half filling is a Néel-type antiferromagnet with spins perpendicular to the zig-zag plane,

$P$2

with wave vector $P$3. This state is the $P$4-UD phase and is odd-parity multipolar: it carries a ferroic magnetic toroidal dipole along $P$5 and a magnetic quadrupole of $P$6 type (Hayami et al., 2015).

The toroidal moment and magnetic quadrupole are defined by

$P$7

and

$P$8

Because $P$9 changes sign on the two sublattices and $T$0 also alternates, the ferroic components are

$T$1

while the other components vanish by symmetry (Hayami et al., 2015).

Quantity	Component	Status in the $T$2-UD state
Toroidal dipole	$T$3	finite
Toroidal dipole	$T$4	vanish
Magnetic quadrupole	$T$5	finite
Magnetic quadrupole	other components	vanish

The ordered state breaks $T$6 and $T$7 but preserves $T$8, which guarantees a twofold band degeneracy without requiring inversion or time reversal separately. In the sublattice basis $T$9, the Bloch Hamiltonian for fixed spin $PT$0 yields the dispersion

$PT$1

The $PT$2 term shifts the band extrema away from high-symmetry points, producing the band-bottom shift characteristic of a toroidal-ordered state. For $PT$3, corresponding to the standard Kondo lattice, this $PT$4-odd term vanishes and the band-bottom shift disappears (Hayami et al., 2015).

Variational calculations, simulated annealing, and finite-temperature Monte Carlo support the stability of this multipolar phase. In the representative parameter set $PT$5, $PT$6, $PT$7, $PT$8, $PT$9, the $\boldsymbol{k}$0-UD phase appears near half filling, while the finite-temperature phase diagram shows a dome centered at $\boldsymbol{k}$1. Estimated critical temperatures from the inflection point of the order parameter are $\boldsymbol{k}$2 for $\boldsymbol{k}$3, $\boldsymbol{k}$4, and $\boldsymbol{k}$5 for $\boldsymbol{k}$6, $\boldsymbol{k}$7 (Hayami et al., 2015).

4. Electronic responses and superconducting consequences

Odd-parity multipole order allows a linear magnetoelectric coupling because the tensor $\boldsymbol{k}$8 is odd under inversion. In the zig-zag model, the staggered $\boldsymbol{k}$9-magnetization induced by a uniform current along $E(\boldsymbol{k}) \neq E(-\boldsymbol{k})$0 is evaluated by the Kubo formula

$E(\boldsymbol{k}) \neq E(-\boldsymbol{k})$1

The correlation $E(\boldsymbol{k}) \neq E(-\boldsymbol{k})$2 is nonzero, which means that an electric current along $E(\boldsymbol{k}) \neq E(-\boldsymbol{k})$3 induces a staggered magnetization along $E(\boldsymbol{k}) \neq E(-\boldsymbol{k})$4. The response grows with increasing $E(\boldsymbol{k}) \neq E(-\boldsymbol{k})$5 and with stronger antisymmetric coupling $E(\boldsymbol{k}) \neq E(-\boldsymbol{k})$6, consistent with its microscopic origin in the staggered $E(\boldsymbol{k}) \neq E(-\boldsymbol{k})$7 term (Hayami et al., 2015).

In $E(\boldsymbol{k}) \neq E(-\boldsymbol{k})$8-symmetric odd-parity magnetic multipole metals, nonlinear transport has a distinctive hierarchy. The second-order conductivity decomposes as $E(\boldsymbol{k}) \neq E(-\boldsymbol{k})$9, but at zero magnetic field $H = H_0 + H_1,$0 symmetry enforces $H = H_0 + H_1,$1, $H = H_0 + H_1,$2, and therefore $H = H_0 + H_1,$3. The leading nonlinear response is instead the Drude term, $H = H_0 + H_1,$4, which depends on asymmetric band dispersion. In the BaMn$H = H_0 + H_1,$5As$H = H_0 + H_1,$6 model, the only nonzero independent tensor elements at $H = H_0 + H_1,$7 are

$H = H_0 + H_1,$8

so a transverse nonlinear Hall current survives even though the anomalous linear Hall effect is forbidden. In the strong-AF, low-density limit,

$H = H_0 + H_1,$9

and the sign change with the antiferromagnetic domain allows nonlinear-transport domain readout (Watanabe et al., 2020).

Odd-parity multipole fluctuations also supply unconventional pairing glue. In a two-dimensional two-sublattice Hubbard model motivated by Sr$$H_0 = - \sum_{i,j,\sigma} (t_{ij} c^\dagger_{i\sigma} c_{j\sigma} + \mathrm{H.c.}) + U \sum_i n^f_{i\uparrow} n^f_{i\downarrow} + E_0 \sum_{i,\sigma} n^f_{i\sigma},$$0IrO$$H_0 = - \sum_{i,j,\sigma} (t_{ij} c^\dagger_{i\sigma} c_{j\sigma} + \mathrm{H.c.}) + U \sum_i n^f_{i\uparrow} n^f_{i\downarrow} + E_0 \sum_{i,\sigma} n^f_{i\sigma},$$1, staggered ASOC suppresses intersublattice even-parity singlet pairing and enhances odd-parity magnetic multipole channels. Near the magnetic critical point, two odd-parity spin-triplet states become dominant: a $$H_0 = - \sum_{i,j,\sigma} (t_{ij} c^\dagger_{i\sigma} c_{j\sigma} + \mathrm{H.c.}) + U \sum_i n^f_{i\uparrow} n^f_{i\downarrow} + E_0 \sum_{i,\sigma} n^f_{i\sigma},$$2 state and an $$H_0 = - \sum_{i,j,\sigma} (t_{ij} c^\dagger_{i\sigma} c_{j\sigma} + \mathrm{H.c.}) + U \sum_i n^f_{i\uparrow} n^f_{i\downarrow} + E_0 \sum_{i,\sigma} n^f_{i\sigma},$$3 state. Both are two-dimensional class-DIII $$H_0 = - \sum_{i,j,\sigma} (t_{ij} c^\dagger_{i\sigma} c_{j\sigma} + \mathrm{H.c.}) + U \sum_i n^f_{i\uparrow} n^f_{i\downarrow} + E_0 \sum_{i,\sigma} n^f_{i\sigma},$$4 topological superconductors, and the intersublattice pairing components show gapped or nodal structures protected by nonsymmorphic glide symmetry (Ishizuka et al., 2018).

A complementary mechanism appears in a two-sublattice Hubbard model with staggered Rashba-type spin-orbit coupling and Zeeman field. There, odd-parity multipole fluctuations are enhanced by inter-sublattice hybridization and Fermi-surface nesting, and a field-induced inter-sublattice, antisymmetric, spin-triplet component

$$H_0 = - \sum_{i,j,\sigma} (t_{ij} c^\dagger_{i\sigma} c_{j\sigma} + \mathrm{H.c.}) + U \sum_i n^f_{i\uparrow} n^f_{i\downarrow} + E_0 \sum_{i,\sigma} n^f_{i\sigma},$$5

activates an odd-parity $$H_0 = - \sum_{i,j,\sigma} (t_{ij} c^\dagger_{i\sigma} c_{j\sigma} + \mathrm{H.c.}) + U \sum_i n^f_{i\uparrow} n^f_{i\downarrow} + E_0 \sum_{i,\sigma} n^f_{i\sigma},$$6 pairing channel. For larger inter-sublattice hybridization, the Eliashberg eigenvalue in the $$H_0 = - \sum_{i,j,\sigma} (t_{ij} c^\dagger_{i\sigma} c_{j\sigma} + \mathrm{H.c.}) + U \sum_i n^f_{i\uparrow} n^f_{i\downarrow} + E_0 \sum_{i,\sigma} n^f_{i\sigma},$$7 channel increases with field, and the resulting $$H_0 = - \sum_{i,j,\sigma} (t_{ij} c^\dagger_{i\sigma} c_{j\sigma} + \mathrm{H.c.}) + U \sum_i n^f_{i\uparrow} n^f_{i\downarrow} + E_0 \sum_{i,\sigma} n^f_{i\sigma},$$8–$$H_0 = - \sum_{i,j,\sigma} (t_{ij} c^\dagger_{i\sigma} c_{j\sigma} + \mathrm{H.c.}) + U \sum_i n^f_{i\uparrow} n^f_{i\downarrow} + E_0 \sum_{i,\sigma} n^f_{i\sigma},$$9 phase diagrams show re-entrant or field-enhanced odd-parity superconductivity (Nogaki et al., 2023).

5. Material realizations and experimental identification

The odd-parity multipole framework now spans magnetic metals, dilute semiconductors, locally noncentrosymmetric $\eta_{\mathrm{magnetic}} = (-1)^{l+1}$00-electron systems, and spin-orbit-coupled metals. In CeCoSi, staggered $\eta_{\mathrm{magnetic}} = (-1)^{l+1}$01-type antiferromagnetism induces a $\eta_{\mathrm{magnetic}} = (-1)^{l+1}$02-type magnetic toroidal dipole, staggered $\eta_{\mathrm{magnetic}} = (-1)^{l+1}$03 antiferroquadrupole order induces an $\eta_{\mathrm{magnetic}} = (-1)^{l+1}$04-type electric toroidal quadrupole, and staggered $\eta_{\mathrm{magnetic}} = (-1)^{l+1}$05 antiferroquadrupole order induces a $\eta_{\mathrm{magnetic}} = (-1)^{l+1}$06-type electric dipole. The resulting ^59Co NQR and NMR spectra are sublattice dependent: only the $\eta_{\mathrm{magnetic}} = (-1)^{l+1}$07-type electric toroidal quadrupole exhibits zero-field NQR splitting, while the three candidate orders produce distinct field-direction dependences in NMR (Yatsushiro et al., 2019, Yatsushiro et al., 2020).

CeRh$\eta_{\mathrm{magnetic}} = (-1)^{l+1}$08As$\eta_{\mathrm{magnetic}} = (-1)^{l+1}$09 provides an example inside a superconducting phase. NQR detects antiferromagnetic order at $\eta_{\mathrm{magnetic}} = (-1)^{l+1}$10 K below the superconducting transition $\eta_{\mathrm{magnetic}} = (-1)^{l+1}$11 K. The site-selective broadening at the two inequivalent As sites implies a magnetic configuration that breaks global inversion, so that odd-parity magnetic multipoles become active in the superconducting ground state (Kibune et al., 2021).

In Mn-based magnets such as BaMn$\eta_{\mathrm{magnetic}} = (-1)^{l+1}$12As$\eta_{\mathrm{magnetic}} = (-1)^{l+1}$13, the antiferromagnetic phase is a $\eta_{\mathrm{magnetic}} = (-1)^{l+1}$14-symmetric odd-parity magnetic multipole state with staggered exchange field $\eta_{\mathrm{magnetic}} = (-1)^{l+1}$15 and staggered ASOC $\eta_{\mathrm{magnetic}} = (-1)^{l+1}$16. The antisymmetric cubic distortion $\eta_{\mathrm{magnetic}} = (-1)^{l+1}$17 near time-reversal-invariant momenta controls the allowed nonlinear conductivity components, and the field-free nonlinear Hall effect, nematicity-assisted dichroism, and field-induced Berry-curvature-dipole response are proposed as symmetry diagnostics (Watanabe et al., 2020).

In the dilute-carrier semiconductor family $\eta_{\mathrm{magnetic}} = (-1)^{l+1}$18Cd$\eta_{\mathrm{magnetic}} = (-1)^{l+1}$19P$\eta_{\mathrm{magnetic}} = (-1)^{l+1}$20 ($\eta_{\mathrm{magnetic}} = (-1)^{l+1}$21), optical second harmonic generation reveals a bulk inversion-breaking transition with an in-plane polar axis and three domain variants related by $\eta_{\mathrm{magnetic}} = (-1)^{l+1}$22 rotations. The high-temperature point group $\eta_{\mathrm{magnetic}} = (-1)^{l+1}$23 lowers to $\eta_{\mathrm{magnetic}} = (-1)^{l+1}$24, and the order parameter transforms as the odd-parity irrep $\eta_{\mathrm{magnetic}} = (-1)^{l+1}$25. The ordered phase appears only in lightly self-hole-doped compounds and is absent in insulating SmCd$\eta_{\mathrm{magnetic}} = (-1)^{l+1}$26P$\eta_{\mathrm{magnetic}} = (-1)^{l+1}$27, which indicates an essential role for itinerant carriers near the semiconductor limit (Tregidga et al., 24 Jun 2026).

In the pyrochlore oxide Pb$\eta_{\mathrm{magnetic}} = (-1)^{l+1}$28Re$\eta_{\mathrm{magnetic}} = (-1)^{l+1}$29O$\eta_{\mathrm{magnetic}} = (-1)^{l+1}$30, x-ray diffraction and SHG identify a continuous inversion-symmetry-breaking transition at $\eta_{\mathrm{magnetic}} = (-1)^{l+1}$31 K from cubic $\eta_{\mathrm{magnetic}} = (-1)^{l+1}$32 to tetragonal $\eta_{\mathrm{magnetic}} = (-1)^{l+1}$33. By comparison with Cd$\eta_{\mathrm{magnetic}} = (-1)^{l+1}$34Re$\eta_{\mathrm{magnetic}} = (-1)^{l+1}$35O$\eta_{\mathrm{magnetic}} = (-1)^{l+1}$36, this low-temperature phase is assigned to an odd-parity electric toroidal quadrupole of $\eta_{\mathrm{magnetic}} = (-1)^{l+1}$37 type, originating from a $\eta_{\mathrm{magnetic}} = (-1)^{l+1}$38 $\eta_{\mathrm{magnetic}} = (-1)^{l+1}$39 electronic instability (Nakayama et al., 2024).

System	Odd-parity order discussed	Experimental or theoretical identifier
Zig-zag Kondo lattice	$\eta_{\mathrm{magnetic}} = (-1)^{l+1}$40 and $\eta_{\mathrm{magnetic}} = (-1)^{l+1}$41	band-bottom shift and linear magnetoelectric response
CeCoSi	MTD, ETQ, ED candidates	sublattice-dependent NQR/NMR splittings
CeRh$\eta_{\mathrm{magnetic}} = (-1)^{l+1}$42As$\eta_{\mathrm{magnetic}} = (-1)^{l+1}$43	odd-parity magnetic multipoles in SC	inequivalent As-site NQR broadening
BaMn$\eta_{\mathrm{magnetic}} = (-1)^{l+1}$44As$\eta_{\mathrm{magnetic}} = (-1)^{l+1}$45-type magnets	$\eta_{\mathrm{magnetic}} = (-1)^{l+1}$46-symmetric odd-parity magnetic multipoles	field-free nonlinear Hall response
$\eta_{\mathrm{magnetic}} = (-1)^{l+1}$47Cd$\eta_{\mathrm{magnetic}} = (-1)^{l+1}$48P$\eta_{\mathrm{magnetic}} = (-1)^{l+1}$49	$\eta_{\mathrm{magnetic}} = (-1)^{l+1}$50 odd-parity electronic order	bulk SHG onset and three domain variants
Pb$\eta_{\mathrm{magnetic}} = (-1)^{l+1}$51Re$\eta_{\mathrm{magnetic}} = (-1)^{l+1}$52O$\eta_{\mathrm{magnetic}} = (-1)^{l+1}$53	ETQ in $\eta_{\mathrm{magnetic}} = (-1)^{l+1}$54	XRD extinction rules and SHG

More broadly, the candidate-material lists emphasize $\eta_{\mathrm{magnetic}} = (-1)^{l+1}$55-electron systems with local inversion breaking and strong spin-orbit coupling, zig-zag or honeycomb geometries, and inversion-related bilayers. Specific examples named in the literature include UGe$\eta_{\mathrm{magnetic}} = (-1)^{l+1}$56, URhGe, UCoGe, $\eta_{\mathrm{magnetic}} = (-1)^{l+1}$57, YbAlB$\eta_{\mathrm{magnetic}} = (-1)^{l+1}$58, and $\eta_{\mathrm{magnetic}} = (-1)^{l+1}$59 cages, as well as Mn-based and nonsymmorphic antiferromagnets cataloged by symmetry analysis (Hayami et al., 2015, Yu et al., 3 Jan 2025).

6. Limits of current models and emerging directions

The zig-zag-lattice formulation makes several controlled approximations: large-$\eta_{\mathrm{magnetic}} = (-1)^{l+1}$60 localized $\eta_{\mathrm{magnetic}} = (-1)^{l+1}$61 moments with $\eta_{\mathrm{magnetic}} = (-1)^{l+1}$62, Schrieffer–Wolff transformation to second order, neglected pair-hopping and $\eta_{\mathrm{magnetic}} = (-1)^{l+1}$63-hopping terms, omission of conduction-electron atomic SOC in the $\eta_{\mathrm{magnetic}} = (-1)^{l+1}$64–$\eta_{\mathrm{magnetic}} = (-1)^{l+1}$65 channel, classical localized spins with $\eta_{\mathrm{magnetic}} = (-1)^{l+1}$66, and a quasi-1D reduction in which only the essential $\eta_{\mathrm{magnetic}} = (-1)^{l+1}$67-odd form factor $\eta_{\mathrm{magnetic}} = (-1)^{l+1}$68 is retained. Variational and Monte Carlo studies also involve finite-size effects and restricted unit cells. The stability of the $\eta_{\mathrm{magnetic}} = (-1)^{l+1}$69-UD multipole phase therefore rests on a controlled but not fully general hierarchy of approximations (Hayami et al., 2015).

Recent work has broadened the microscopic landscape beyond local parity mixing plus SOC. In nonsymmorphic antiferromagnets, coplanar period-doubling AFM exchange can induce odd-parity spin-vector, nematic, or scalar odd-parity ground states even without SOC. The group-theory construction spans 421 conventional period-doubling AFM systems in nonsymmorphic space groups, minimal microscopic models were constructed for 119 of them, and 67 materials were identified in the Magndata database. In that setting, odd-parity spin splitting can be $\eta_{\mathrm{magnetic}} = (-1)^{l+1}$70-, $\eta_{\mathrm{magnetic}} = (-1)^{l+1}$71-, or $\eta_{\mathrm{magnetic}} = (-1)^{l+1}$72-wave, and scalar odd-parity order gives a non-zero Berry curvature dipole without SOC (Yu et al., 3 Jan 2025).

Quantum geometry provides a second extension. In a bilayer Lieb lattice without SOC, the odd-parity spin multipole

$\eta_{\mathrm{magnetic}} = (-1)^{l+1}$73

has a susceptibility

$\eta_{\mathrm{magnetic}} = (-1)^{l+1}$74

and the quantum-metric terms in the curvature of $\eta_{\mathrm{magnetic}} = (-1)^{l+1}$75 are always negative, favoring ferroic $\eta_{\mathrm{magnetic}} = (-1)^{l+1}$76 fluctuations. With Hubbard interaction, the generalized Stoner condition

$\eta_{\mathrm{magnetic}} = (-1)^{l+1}$77

signals condensation of the odd-parity multipole order (Kudo et al., 27 May 2025).

A plausible implication is that the most favorable materials platform combines three ingredients that recur across the literature: a multi-sublattice or bilayer structure that makes inversion act by site exchange, low-energy states connected by local parity mixing or by strong quantum geometry, and either strong spin-orbit coupling or a nonsymmorphic constraint that converts conventional AFM or orbital order into inversion-odd cluster multipoles. Under those conditions, odd-parity multipole order is not an isolated lattice-specific anomaly but a generic symmetry-entangled instability of correlated Bloch electrons (Hayami et al., 2015, Watanabe et al., 2018, Kudo et al., 27 May 2025).