Odd-Parity Magnetism

Updated 3 September 2025

Odd-parity magnetism is defined by parity-violating electronic and magnetic orders that emerge when spatial inversion symmetry is broken.
Mechanisms such as local parity mixing, antisymmetric exchange, and Floquet engineering yield momentum-dependent spin splitting and novel band deformations.
This phenomenon underpins distinctive magnetoelectric effects, topological phases, and promising applications in spintronics and multiferroics.

Odd-parity magnetism encompasses a class of magnetic and multipolar electronic states characterized by parity-violating order parameters—properties that change sign under spatial inversion. In condensed matter systems, odd-parity magnetism manifests when either local or global inversion symmetry is broken, frequently in combination with strong spin–orbit coupling or intricate multi-orbital interactions. These states give rise to a wide range of emergent phenomena, including unusual band deformations, hidden multipolar orders (such as magnetic toroidal or quadrupole moments), magnetoelectric effects, and non-standard transport responses (such as odd-parity magnetoresistance or nonlinear Hall effects). Odd-parity magnetism is central to the paper of unconventional topological matter, cross-correlated responses, and is increasingly relevant for the development of spintronic and multiferroic devices.

1. Microscopic Mechanisms and Symmetry Principles

Odd-parity magnetism arises due to either explicit breaking of spatial inversion symmetry at the atomic, sublattice, or structural level, or via symmetry-protected mechanisms in centrosymmetric but locally noncentrosymmetric or nonsymmorphic crystals. Common routes to odd-parity magnetism include:

Local Parity Mixing: Even when a crystal is globally inversion symmetric, the breaking of inversion at the site or sublattice level allows orbitals with different parity (e.g., d and f, or p and d) to hybridize. The resulting antisymmetric hybridization produces antisymmetric spin–orbit coupling of the form $H_\mathrm{ASOC}(k) = \mathbf{g}(k) \cdot \boldsymbol{\sigma} \simeq (\mathbf{k} \times \nabla V_\mathrm{pot}) \cdot \boldsymbol{\sigma}$ , leading to momentum-dependent spin-splitting and parity-odd responses (Hayami et al., 2015).
Antisymmetric Exchange via Multipole Couplings: In extended Kondo lattice and related models, sublattice-dependent antisymmetric exchange interactions arise, stabilizing antiferromagnetic (Néel-type) orders with odd-parity multipolar character (e.g., magnetic toroidal or quadrupole moments).
Hidden Symmetry in Nonsymmorphic Lattices: In antiferromagnets or collinear compensated magnets of nonsymmorphic symmetry, the action of inversion may interchange inequivalent sublattices or lead to mixed-parity irreducible representations. Vector or scalar products of local moments (e.g., $\mathbf{S}_1 \times \mathbf{S}_2$ or $|\mathbf{S}_1|^2 - |\mathbf{S}_2|^2$ ) can transform as parity-odd quantities, producing momentum-odd spin textures and associated nontrivial Berry curvatures (Yu et al., 3 Jan 2025, Lee et al., 8 Aug 2025).
Floquet Engineering: Periodic driving (e.g., circularly polarized light) is a universal strategy to induce odd-parity magnetism dynamically in collinear antiferromagnets. The light field breaks time-reversal or spatial symmetries, leading to effective Floquet Hamiltonians with odd-in-momentum spin splitting and realization of p-wave or f-wave odd-parity altermagnetic states (Huang et al., 28 Jul 2025, Zhu et al., 4 Aug 2025, Liu et al., 25 Aug 2025).

The generic feature unifying these mechanisms is the emergence of parity-odd, often higher-rank, magnetic or electronic order parameters that are odd under spatial inversion, yet frequently maintain time-reversal or hidden effective time-reversal symmetries.

2. Multipole Orders and Quantum Geometry

A central focus of odd-parity magnetism is the formation of higher-rank multipolar states:

Magnetic Toroidal and Quadrupole Moments: Local parity mixing and sublattice antisymmetry can stabilize magnetic toroidal moments ( $\bm{T} \propto \sum_i (\mathbf{r}_i \times \mathbf{S}_i)$ ) and quadrupole moments as primary or induced order parameters. Such moments underlie unique cross-coupling between electric and magnetic fields, acting as the source of unconventional magnetoelectric effects (Hayami et al., 2015, Yatsushiro et al., 2019, Hayami et al., 2021).
Spin–Orbital Momentum Locking: Odd-parity magnetic quadrupole (MQ) order results in the locking of composite spin–orbital operators (e.g., $l_z \sigma_x$ , $l_z \sigma_y$ ) to crystal momentum. This produces momentum-dependent antisymmetric spin–orbital polarization, which modulates the band structure and leads to “hidden” spin textures that are only revealed under symmetry-breaking perturbations (Hayami et al., 2021).
Quantum-Geometric Multipole Magnetism: The quantum geometry of Bloch functions—encompassing the quantum metric and momentum-space overlap—directly governs ferroic multipole fluctuations, and strongly enhances the odd-parity channel susceptibility. Interaction effects (e.g., Hubbard U) can condense quantum-geometric multipole fluctuations into true ordered states, producing complex intertwined in-plane and interlayer correlations that are experimentally accessible (Kudo et al., 27 May 2025).

These multipolar phenomena fundamentally exceed simple dipole magnetism and are at the heart of many of the exotic band structure effects and cross-correlated responses associated with odd-parity magnets.

3. Band Structure Deformations and Topological Effects

Odd-parity magnetism induces characteristic deformations and topological restructuring of electronic bands:

Momentum-Odd Spin Splitting: In odd-parity antiferromagnets or altermagnets, the energy spectrum satisfies $E(k, s) = E(-k, -s)$ , with spin splitting that is an odd function of momentum (typical of p-wave, f-wave, or h-wave symmetry), even when the net magnetization is zero (Yu et al., 3 Jan 2025, Lin, 12 Mar 2025, Zhu et al., 4 Aug 2025). For example, in Fe-based superconductors, h-wave spin splitting of the form $\Delta_E \sim (k_x^2 - k_y^2)k_xk_y \sin(k_z)$ emerges in coplanar AFM states (Dsouza et al., 29 Aug 2025).
Fermi Surface Reconstruction: The modulation of spin and orbital textures leads to “hidden” Fermi surface topology changes, which may underlie unconventional superconductivity, anomalous transport, and nematicity-assisted phenomena.
Topological Phases: Odd-parity multipole magnets and altermagnets can stabilize insulating or semimetallic topological phases (e.g., Chern insulators, Weyl semimetals, quantum spin Hall states) as a result of parity-violating band inversions near Dirac points or the light-induced acquisition of nontrivial Chern numbers (Zhu et al., 4 Aug 2025, Liu et al., 25 Aug 2025, Zhuang et al., 25 Aug 2025). Topologically protected edge states with helical or chiral character may coexist with magnetic compensation, providing robust platforms for dissipationless transport or Majorana excitations.

4. Transport and Cross-Correlated Responses

Odd-parity magnetism leads to nonstandard electrical and cross-correlated phenomena, frequently characterized by unconventional symmetry properties:

Odd-Parity Magnetoresistance (OMR) and Planar Hall Effect: Magnetoresistance and Hall effects that are odd in the magnetic field (and sometimes controlled by domain structure or external gating) can arise in systems with broken time-reversal symmetry (e.g., ferromagnets, domain walls in AFMs) or simultaneous breaking of TRS and SIS (e.g., magnetically proximitized InAs quantum wells) (Fujita et al., 2015, Wang et al., 2019, Takiguchi et al., 2020).
Giant Nonlinear Hall and Magnetoelectric Effects: In $\mathcal{PT}$ -symmetric odd-parity magnetic multipole systems (e.g., Mn-based compounds), the equilibrium Berry curvature is odd in momentum and globally vanishes, suppressing conventional Berry curvature dipole effects. Nonlinear transport (such as a giant nonlinear Hall response) is instead dominated by higher-order “Drude-like” terms, scaling as $\tau^2$ , signaling sensitivity to band-structure anharmonicity and accessible in clean samples (Watanabe et al., 2020).
Current-Induced Distortion and Magnetoelectric Coupling: The complex interplay between MQ order, spin–orbital momentum locking, and electric fields leads to current-induced lattice distortions and linear magnetoelectric responses, activated only by the specific symmetry and momentum locking in the underlying multipolar state (Hayami et al., 2021, Yatsushiro et al., 2019).
Non-Relativistic Edelstein Effect: Odd-parity AFM states without SOC can host a non-relativistic Edelstein response, enabling charge–to–spin conversion via antiferromagnetic exchange alone (Yu et al., 3 Jan 2025).

The distinctive selection rules for these responses, often forbidden in conventional dipole magnets or nonmagnetic crystals, highlight the experimental signatures of odd-parity multipolar order.

5. Phenomenology in Real Materials and Controlled Tuning

Odd-parity magnetic phenomena have material realizations and can be widely tuned:

Representative Material Systems: Predicted and observed candidates include quasi-1D and multi-orbital lattices (zig-zag chains, honeycomb, diamond), pyrochlore iridates (Eu₂Ir₂O₇ with all-in–all-out order), strongly correlated oxides (BaMn₂As₂, Sr₂IrO₄), Fe-based superconductors (FeSe, LaFeAsO), and type-II multiferroics (NiI₂) (Hayami et al., 2015, Fujita et al., 2015, Dsouza et al., 29 Aug 2025, Song et al., 29 Apr 2025). Bilayer heterostructures, heavy-fermion superlattices (CeCoIn₅/YbCoIn₅), and oxide interfaces (SrTiO₃/LaAlO₃) are platforms for odd-parity superconductivity (Watanabe et al., 2015).
Field, Gating, and Light Tuning: The magnitude and symmetry of odd-parity order parameters and their responses can be manipulated by electrical gating (modulating magnetic proximity and SIS breaking in thin heterostructures (Takiguchi et al., 2020)), by direct voltage switching (reversing ferroelectric polarization and spiral chirality in multiferroics (Song et al., 29 Apr 2025)), or via photonic Floquet engineering (CPL or other light fields to induce and reconfigure p-/f-wave odd-parity splitting (Huang et al., 28 Jul 2025, Zhu et al., 4 Aug 2025, Liu et al., 25 Aug 2025)).
Topological and Superconducting Applications: Odd-parity altermagnetic and multipolar states can host quantum spin Hall or Chern insulating phases, Weyl semimetals, or even promote odd-parity (staggered) superconductivity with nontrivial $Z_2$ topology, protected edge modes, and potential for Majorana fermion realization (Watanabe et al., 2015, Ishizuka et al., 2018).

6. Stability, Incommensuration, and Phase Transitions

The stability of odd-parity magnetic orders is sensitive to symmetry-allowed gradient terms and band structure details:

Incommensuration: The Lifshitz invariant, permitted by the symmetry of mixed-parity irreps in nonsymmorphic crystals, produces linear-in-gradient terms in the Ginzburg–Landau free energy that preclude continuous phase transitions into commensurate unit-cell-doubling odd-parity AFM states. As a result, odd-parity orders often arise from an incommensurate precursor phase or via a first-order transition. Type-II van Hove singularities and weak spin–orbit coupling further enhance the propensity to incommensuration (Lee et al., 8 Aug 2025).
Competing Orders: Phenomenological models show that AFM systems in nonsymmorphic lattices can support competing odd-parity vector, scalar, and nematic orders, each characterized by a distinct symmetry, response, and topological signature (Yu et al., 3 Jan 2025). The competition between these channels may be resolved by details of interactions, lattice anisotropy, or external perturbations.

7. Outlook and Experimental Verification

The theoretical and computational advances reviewed here establish odd-parity magnetism as a unifying concept bridging multipolar magnetic order, momentum-odd spin splitting, topological physics, and unconventional cross-correlated phenomena. Table 1 (below) summarizes key features and experimental handles:

Mechanism/Order	Symmetry Condition	Experimental Consequence
Local Parity Mixing	Broken local inversion	Band deformation, magnetoelectricity
Multipole (Toroidal/Quadrupole)	Higher-rank, odd parity	Linear ME effect, spin–orbital locking
Antisymmetric Exchange in AFM	Nonsymmorphic, 2D rep.	Momentum-odd spin splitting
Floquet Light-Induced	TRS/SIS broken by light	Dynamically tuned odd-parity ALM
Competing Orders (Scalar/Nematic)	Mixed-parity irreps	Nonlinear Hall, Berry curvature dipole

Experimental probes include neutron scattering (for magnetic correlations and multipole signatures), inelastic light scattering (for multipole order), angle-resolved photoemission (for band deformation and spin splitting), nonlinear transport, CPGE, and voltage/polarization control to establish the presence and utility of odd-parity order.

Odd-parity magnetism offers a robust framework for designing quantum materials and devices with tailored, symmetry-driven responses and topological characteristics, with prospects for future developments in spintronics, multiferroics, and quantum information applications.