---
title: Odd-Parity Multipole Order in Quantum Materials
url: https://www.emergentmind.com/topics/odd-parity-multipole-order
type: topic
---

# Odd-Parity Multipole Order in 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 [1502.00057, 1805.10828].

## 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 [1807.05707, 1805.10828].

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 [2010.08480, 1807.05707].

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 [1805.10828, 2010.08480].

## 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
$$
H_1 = \sum_{\ell,k,\sigma} V_\ell(k) (c^\dagger_{\ell k \sigma} f_{\ell k \sigma} + \mathrm{H.c.}) + \sum_{\ell,k} \boldsymbol{g}^{cf}_\ell(k)\cdot \boldsymbol{s}^{cf}_\ell(k).
$$
For the quasi-one-dimensional zig-zag lattice, the sublattice-dependent antisymmetric hybridization takes the form
$$
\boldsymbol{g}^{cf}_\ell(k) = (-1)^\ell g \hat{z} \sin k_x,
$$
which follows from quasi-1D motion $k \parallel x$, an odd-parity crystal field $\nabla V_{\mathrm{pot}} \parallel y$, and $\boldsymbol{g} \sim (\boldsymbol{k} \times \nabla V_{\mathrm{pot}})$ [1502.00057].

Treating $H_1$ perturbatively about $H_0$ with a Schrieffer–Wolff transformation and assuming one $f$ 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,
$$
J^\ell_{k'k} \to J>0,\qquad
D^\mu_{\ell k'k} \to D_\ell \delta_{\mu z}\sin k_x,\qquad
G^{\mu\nu}_{\ell k'k} \to G \delta_{\mu z}\delta_{\nu z}\sin k_x \sin k'_x,
$$
with
$$
D_\ell = (-1)^\ell \sqrt{JG}.
$$
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 [1502.00057].

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 $k$-odd form factor $\sin k_x$ and a staggered coefficient $D_\ell = (-1)^\ell \sqrt{JG}$. Rashba-type spin-orbit coupling produces a uniform $\boldsymbol{g}(\boldsymbol{k})\cdot \boldsymbol{\sigma}$ in the unit cell, whereas local parity mixing produces a sublattice-dependent $\boldsymbol{g}^{cf}_\ell(\boldsymbol{k})$ tied to virtual $c$–$f$ processes and to the magnetic texture [1502.00057].

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

On the quasi-1D zig-zag lattice in the $xy$ plane, the chain direction is $x$ and the two sublattices are displaced along $\pm y$ within each $x$ period. The odd-parity crystal field points along $\pm y$ 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,
$$
\boldsymbol{S}_{p\ell} = (-1)^\ell \frac{\hat{z}}{2},
$$
with wave vector $\boldsymbol{Q} = (\pi,0,0)$. This state is the $z$-UD phase and is odd-parity multipolar: it carries a ferroic magnetic toroidal dipole along $x$ and a magnetic quadrupole of $yz$ type [1502.00057].

The toroidal moment and magnetic quadrupole are defined by
$$
\boldsymbol{T} = \sum_i \boldsymbol{r}_i \times \boldsymbol{m}_i,
$$
and
$$
Q_{\mu\nu} = \sum_i \frac{1}{2}(r_{i\mu} m_{i\nu} + r_{i\nu} m_{i\mu}) - \frac{1}{3}\delta_{\mu\nu}\sum_i \boldsymbol{r}_i \cdot \boldsymbol{m}_i.
$$
Because $r_{iy}$ changes sign on the two sublattices and $m_{iz} \propto S_{iz}$ also alternates, the ferroic components are
$$
T_x \neq 0,\qquad Q_{yz} = Q_{zy} \neq 0,
$$
while the other components vanish by symmetry [1502.00057].

| Quantity | Component | Status in the $z$-UD state |
|---|---:|---|
| Toroidal dipole | $T_x$ | finite |
| Toroidal dipole | $T_y, T_z$ | vanish |
| Magnetic quadrupole | $Q_{yz}=Q_{zy}$ | finite |
| Magnetic quadrupole | other components | vanish |

The ordered state breaks $P$ and $T$ but preserves $PT$, which guarantees a twofold band degeneracy without requiring inversion or time reversal separately. In the sublattice basis $(A,B)$, the Bloch Hamiltonian for fixed spin $\sigma=\pm1$ yields the dispersion
$$
E(\boldsymbol{k}) = \varepsilon_2 + \frac{G}{4}\sin^2 k_x \pm \sqrt{\varepsilon_1^2 + \frac{1}{4}\left(\frac{J}{2} + \sqrt{JG}\sin k_x\right)^2 }.
$$
The $\sin k_x$ term shifts the band extrema away from high-symmetry points, producing the band-bottom shift characteristic of a toroidal-ordered state. For $G \to 0$, corresponding to the standard Kondo lattice, this $k$-odd term vanishes and the band-bottom shift disappears [1502.00057].

Variational calculations, simulated annealing, and finite-temperature Monte Carlo support the stability of this multipolar phase. In the representative parameter set $t_1=1$, $t_2=0.1$, $t_3=t_4=0.2$, $J=4$, $G=0.5$, the $z$-UD phase appears near half filling, while the finite-temperature phase diagram shows a dome centered at $n_e \approx 1.0$. Estimated critical temperatures from the inflection point of the order parameter are $T_c \approx 0.09$ for $\mu=-0.1$, $n_e \approx 0.96$, and $T_c \approx 0.075$ for $\mu=-0.5$, $n_e \approx 0.85$ [1502.00057].

## 4. Electronic responses and superconducting consequences

Odd-parity multipole order allows a linear magnetoelectric coupling because the tensor $\alpha$ is odd under inversion. In the zig-zag model, the staggered $z$-magnetization induced by a uniform current along $x$ is evaluated by the Kubo formula
$$
K^{(s)}_{zx} = \frac{g\mu_B}{2}\frac{1}{iV_s}
\sum_{m,n,\boldsymbol{k}}
\frac{f(\varepsilon_{n\boldsymbol{k}})-f(\varepsilon_{m\boldsymbol{k}})}{\varepsilon_{n\boldsymbol{k}}-\varepsilon_{m\boldsymbol{k}}}
\frac{\sigma^{z,\boldsymbol{k}}_{nm} J^{x,\boldsymbol{k}}_{mn}}
{\varepsilon_{n\boldsymbol{k}}-\varepsilon_{m\boldsymbol{k}}+i\eta}.
$$
The correlation $K^{(s)}_{zx}$ is nonzero, which means that an electric current along $x$ induces a staggered magnetization along $z$. The response grows with increasing $t_1/t_2$ and with stronger antisymmetric coupling $G$, consistent with its microscopic origin in the staggered $D_\ell \sin k_x$ term [1502.00057].

In $PT$-symmetric odd-parity magnetic multipole metals, nonlinear transport has a distinctive hierarchy. The second-order conductivity decomposes as $\sigma=\sigma^D+\sigma^{\mathrm{BCD}}+\sigma^{\mathrm{int}}$, but at zero magnetic field $PT$ symmetry enforces $\Omega(\boldsymbol{k})=0$, $D_{\mu\nu}=0$, and therefore $\sigma^{\mathrm{BCD}}=0$. The leading nonlinear response is instead the Drude term, $\sigma^D \sim O(\tau^2)$, which depends on asymmetric band dispersion. In the BaMn$_2$As$_2$ model, the only nonzero independent tensor elements at $H=0$ are
$$
\sigma^D_{z;xy}=\sigma^D_{x;yz}=\sigma^D_{y;zx},
$$
so a transverse nonlinear Hall current survives even though the anomalous linear Hall effect is forbidden. In the strong-AF, low-density limit,
$$
\sigma^D_{z;xy}=\sigma^D_{x;yz}=\sigma^D_{y;zx}
=\frac{e^3 \alpha_\parallel n}{4\gamma^2}\,\mathrm{sgn}(h_{\mathrm{AF}}),
$$
and the sign change with the antiferromagnetic domain allows nonlinear-transport domain readout [2010.08480].

Odd-parity multipole fluctuations also supply unconventional pairing glue. In a two-dimensional two-sublattice Hubbard model motivated by Sr$_2$IrO$_4$, 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 $B_{1u}$ state and an $A_{1u}$ state. Both are two-dimensional class-DIII $\mathbb{Z}_2$ topological superconductors, and the intersublattice pairing components show gapped or nodal structures protected by nonsymmorphic glide symmetry [1807.05707].

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
$$
\mathrm{Im}\,d^{\mathrm{AB}}_z(\boldsymbol{k})\, s_z i s_y \otimes \sigma_y
$$
activates an odd-parity $B_u$ pairing channel. For larger inter-sublattice hybridization, the Eliashberg eigenvalue in the $B_u$ channel increases with field, and the resulting $H$–$T$ phase diagrams show re-entrant or field-enhanced odd-parity superconductivity [2312.07053].

## 5. Material realizations and experimental identification

The odd-parity multipole framework now spans magnetic metals, dilute semiconductors, locally noncentrosymmetric $f$-electron systems, and spin-orbit-coupled metals. In CeCoSi, staggered $x$-type antiferromagnetism induces a $y$-type magnetic toroidal dipole, staggered $x^2-y^2$ antiferroquadrupole order induces an $xy$-type electric toroidal quadrupole, and staggered $3z^2-r^2$ antiferroquadrupole order induces a $z$-type electric dipole. The resulting ^59Co NQR and NMR spectra are sublattice dependent: only the $xy$-type electric toroidal quadrupole exhibits zero-field NQR splitting, while the three candidate orders produce distinct field-direction dependences in NMR [1912.02422, 2011.06212].

CeRh$_2$As$_2$ provides an example inside a superconducting phase. NQR detects antiferromagnetic order at $T_{\mathrm N}\approx 0.25$ K below the superconducting transition $T_{\mathrm{SC}}\sim 0.37$ 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 [2112.07081].

In Mn-based magnets such as BaMn$_2$As$_2$, the antiferromagnetic phase is a $PT$-symmetric odd-parity magnetic multipole state with staggered exchange field $h_{\mathrm{AF}}$ and staggered ASOC $g_0(\boldsymbol{k})\cdot \boldsymbol{\sigma}\,\tau_z$. The antisymmetric cubic distortion $k_x k_y k_z$ 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 [2010.08480].

In the dilute-carrier semiconductor family $\mathit{Ln}$Cd$_3$P$_3$ ($\mathit{Ln}=\mathrm{La},\mathrm{Ce},\mathrm{Pr},\mathrm{Nd}$), optical second harmonic generation reveals a bulk inversion-breaking transition with an in-plane polar axis and three domain variants related by $120^\circ$ rotations. The high-temperature point group $6/mmm$ lowers to $mm2$, and the order parameter transforms as the odd-parity irrep $E_u$. The ordered phase appears only in lightly self-hole-doped compounds and is absent in insulating SmCd$_3$P$_3$, which indicates an essential role for itinerant carriers near the semiconductor limit [2606.26338].

In the pyrochlore oxide Pb$_2$Re$_2$O$_{7-\delta}$, x-ray diffraction and SHG identify a continuous inversion-symmetry-breaking transition at $T_s \approx 300$ K from cubic $Fd\bar{3}m$ to tetragonal $I4_122$. By comparison with Cd$_2$Re$_2$O$_7$, this low-temperature phase is assigned to an odd-parity electric toroidal quadrupole of $G_u$ type, originating from a $k=0$ $E_u$ electronic instability [2405.04018].

| System | Odd-parity order discussed | Experimental or theoretical identifier |
|---|---|---|
| Zig-zag Kondo lattice | $T_x$ and $Q_{yz}$ | band-bottom shift and linear magnetoelectric response |
| CeCoSi | MTD, ETQ, ED candidates | sublattice-dependent NQR/NMR splittings |
| CeRh$_2$As$_2$ | odd-parity magnetic multipoles in SC | inequivalent As-site NQR broadening |
| BaMn$_2$As$_2$-type magnets | $PT$-symmetric odd-parity magnetic multipoles | field-free nonlinear Hall response |
| $\mathit{Ln}$Cd$_3$P$_3$ | $E_u$ odd-parity electronic order | bulk SHG onset and three domain variants |
| Pb$_2$Re$_2$O$_{7-\delta}$ | ETQ in $I4_122$ | XRD extinction rules and SHG |

More broadly, the candidate-material lists emphasize $f$-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$_2$, URhGe, UCoGe, $\mathrm{Ln}M_2\mathrm{Al}_{10}$, YbAlB$_4$, and $RT_2X_{20}$ cages, as well as Mn-based and nonsymmorphic antiferromagnets cataloged by symmetry analysis [1502.00057, 2501.02057].

## 6. Limits of current models and emerging directions

The zig-zag-lattice formulation makes several controlled approximations: large-$U$ localized $f$ moments with $n_f=1$, Schrieffer–Wolff transformation to second order, neglected pair-hopping and $f$-hopping terms, omission of conduction-electron atomic SOC in the $c$–$c$ channel, classical localized spins with $|\boldsymbol{S}_i|=1/2$, and a quasi-1D reduction in which only the essential $k$-odd form factor $\sin k_x$ is retained. Variational and Monte Carlo studies also involve finite-size effects and restricted unit cells. The stability of the $z$-UD multipole phase therefore rests on a controlled but not fully general hierarchy of approximations [1502.00057].

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 $p$-, $f$-, or $h$-wave, and scalar odd-parity order gives a non-zero Berry curvature dipole without SOC [2501.02057].

Quantum geometry provides a second extension. In a bilayer Lieb lattice without SOC, the odd-parity spin multipole
$$
M_O = \sum_j (S_{j,1}-S_{j,2})
$$
has a susceptibility
$$
\chi_0^{O}(\boldsymbol{q}) = 2\int \frac{d^2k}{(2\pi)^2}\sum_{n,m,p}
L^{p,-p}_{nm}(\boldsymbol{k},\boldsymbol{q})\,[1-D_{nm}(\boldsymbol{k},\boldsymbol{q})^2],
$$
and the quantum-metric terms in the curvature of $\chi_0^{O}(\boldsymbol{q})$ are always negative, favoring ferroic $q=0$ fluctuations. With Hubbard interaction, the generalized Stoner condition
$$
1-U\chi_0^{O}(q=0,\omega=0)=0
$$
signals condensation of the odd-parity multipole order [2505.20907].

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 [1502.00057, 1805.10828, 2505.20907].

Source: https://www.emergentmind.com/topics/odd-parity-multipole-order