Parity-driven RKKY decoupling and anomalous 1/R Dzyaloshinskii-Moriya interaction in odd-parity magnets

The gist: The study theoretically investigates how parity-driven symmetry in p-wave magnets causes a fundamental decoupling of magnetic responses and leads to an anomalous 1/R spatial decay for in-plane Dzyaloshinskii-Moriya interactions along nodal lines.

How it works

1. The model utilizes an effective two-band continuum Hamiltonian for a two-dimensional (2D) unconventional p-wave magnet in the presence of Rashba spin-orbit coupling (RSOC) where the Hamiltonian is described by Hˆ = Pk ψ†kHˆ (k)ψ; this Hamiltonian is given by Hˆ = αkσ0 − Mp k σz + λ(kyσx − kxσy). The unconventional magnetic order is captured by the momentum-dependent exchange field Mp k, which has an odd-parity harmonic defined as Mp k = JkF (kx cos β + ky sin β); this odd-parity inversion symmetry is the defining feature of p-wave magnets.

2. The spin-dependent momentum shift in the p-wave order rigidly shifts the Fermi surfaces, and when RSOC is introduced, it lifts nodal degeneracies, leading to band hybridization and a rich, momentum-dependent spin texture. The real-space Green’s function formalism is used to evaluate the RKKY interaction mediated by itinerant electrons where the Green’s function can be expressed as G(k, E) = G0(k, E)σ0 + G(k, E) · σ.

Parity-Driven Decoupling

1. The odd-parity symmetry of the p-wave order fundamentally alters the RKKY exchange tensor compared to the d-wave case. Because the p-wave magnetic order rigidly shifts the Fermi surfaces, a profound decoupling occurs where the out-of-plane Ising interaction does not exhibit macroscopic beating and oscillates exclusively at the shifted Fermi-surface radial wave vector because of the odd-parity exchange field, we uncover a profound decoupling in the spatial magnetic responses.

2. The analysis reveals a parity-driven role reversal when compared to even-parity dwave altermagnets. In conventional ferromagnets or evenparity dwave altermagnets, the momentum-dependent exchange field is an even function in momentum space (M−k = Mk), but because the p-wave exchange field is strictly odd in momentum space (Mp−k = −Mp k), the resulting real-space spin propagator inherits an odd parity, Gz(−R) = −Gz(R).

Anomalous Spatial Crossovers

1. The competition between the p-wave nodal geometry and the Rashba gap drives an anomalous, dimension-reducing crossover where inplane chiral components follow a 1D-like 1/R spatial decay along the nodal lines over an extended intermediate-distance window before ultimately recovering the conventional 2D 1/R2 asymptote.

2. Along the nodal direction where the p-wave shift vanishes (kM = 0), the retention of the RSOC-induced gap fundamentally prevents mathematical divergences and instead drives an anomalous, dimension-reducing 1/R spatial scaling. This crossover is rigorously defined within a finite intermediate-distance window, 1/Q ≪ R ≲ R∗ ∼ 1/(2kR).

Chiral Interaction Hierarchy

1. The antisymmetric DM vector JDM,z is driven primarily by the macroscopic p-wave momentum shift and emerges as a massive, nonrelativistic out-of-plane chiral component (JDM,z). This component is present even at λ = 0, where the p-wave exchange shifts the two spin sectors by opposite momenta.

2. The in-plane DM components JDM,x and JDM,y are strictly relativistic Rashba features that scale proportionally to kR/κ. In contrast, the out-of-plane component JDM,z is built exclusively from the scalar and pwave channels alone, meaning it is present even at λ = 0.

3. The in-plane DM components exhibit a striking spatial consequence where they crossover from conventional 2D 1/R2 scaling into a massively enhanced, 1D-like 1/R power-law decay over a macroscopic intermediate-distance window. This anomalous enhancement is observed along the nodal line (kM = 0), yielding Jnodal DM,x ≈ A(R) sin ϕ sin(2kRR) sin(2kF R).

Conclusion

The findings establish p-wave magnets as promising platforms for engineering robust, directionally tunable non-collinear spin textures. By leveraging this finite 1D-like crossover window, one could theoretically stabilize extended, onedimensional chains of non-collinear spin spirals or magnetic skyrmions along specific crystallographic axes of unconventional p-wave hosts. The highly anisotropic and parity-decoupled RKKY tensor, combined with the anomalously enhanced, directionally filtered nodal DM interaction, identifies p-wave magnets as exceptionally promising platforms. These materials offer tunable, purely nonrelativistic and RSOC-regularized mechanisms to stabilize low-dimensional non-collinear spin textures. The study provides a physical and mathematical blueprint for two-body interactions in odd-parity magnetic ground states.

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Morteza Salehi, Department of Physics, Bu-Ali Sina University, Hamadan, Iran; Tohid Farajollahpour, Department of Physics, Norwegian University of Science and Technology (NTNU), NO-7491 Trondheim, Norway (Dated: July 14, 2026). Unconventional p-wave magnets offer a fundamentally distinct paradigm for non-collinear spintronics.

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FIG. 1. (a) Schematic illustration of 2D p-wave unconventional magnet in the presence of two magnetic impurities with spins S1 and S2, located at spatial positions R1 and R2. The localized impurity spins interact with each other via the indirect RKKY interaction mediated by the itinerant electrons. (b) to (e) Constant-energy contours in momentum space showing the spin-splitting effect. The color map corresponds to the out-of-plane spin expectation value ⟨σz⟩, highlighting the momentum-dependent spin polarization. The top row shows the unperturbed (b) px-wave (β = 0) and (c) pywave (β = π/2) unconventional magnets in the absence of RSOC, where the bands exhibit a uniform momentum shift with perfectly intersecting nodes. The bottom row displays the corresponding (d) px-wave and (e) py-wave magnets in the presence of RSOC (dimensionless Rashba splitting λkF /EF = 1/4). The introduction of RSOC lifts the nodal degeneracies, inducing band hybridization and a smooth mixing of the spin projections driven by the competition between the p-wave magnetic order and relativistic spin-momentum locking. channels is a cornerstone for engineering twisted, noncollinear spin textures such as spin spirals and magnetic skyrmions.

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A. Model Hamiltonian for 2D p-wave Magnets We characterize the electronic properties of a twodimensional (2D) unconventional p-wave magnet in the presence of RSOC by utilizing an effective two-band continuum Hamiltonian. In momentum space, the Hamiltonian Hˆ = Pk ψˆ† kHˆ (k)ψ is described by the 2 × 2 matrix Hˆ (k) = αkσ0 − Mp k σz + λ(kyσx − kxσy), where σ0 is the 2×2 identity matrix, and σ = (σx, σy, σz) is the vector of Pauli matrices acting on the electron spin subspace. The unconventional magnetic order is entirely captured by the momentum-dependent exchange field Mp k. Unlike conventional ferromagnetism (where the exchange field is a constant) or d-wave altermagnetism (where the exchange field has an even-parity symmetry such as k2x −k2y), the p-wave magnetic order is defined by an oddparity harmonic, Mp k = JkF (kx cos β + ky sin β); this odd-parity inversion symmetry is the defining feature of p-wave magnets and, as we will show, fundamentally dictates the resulting spatial symmetries of the RKKY interaction tensor. Setting β = 0 corresponds to a px-wave magnet, while β = π/2 corresponds to a py-wave magnet. It is crucial to note that Mp−k = −Mp k. This odd-parity inversion symmetry is the defining feature of p-wave magnets and, as we will show, fundamentally dictates the resulting spatial symmetries of the RKKY interaction tensor. Diagonalizing the Hamiltonian in Eq. (1) yields two spin-split energy branches E±(k) = αk ± qλ2k2 + (Mp k)2, where k2 = k2x + k2y.

1. Analysis of p-wave magnet interaction mapping: The AI can accurately predict which magnetic exchange components (Heisenberg, Ising, DM) will be dominant based on the orientation of the crystalline p-wave axis by analyzing how the projected momentum shift kM = Q cos(beta) tunes these amplitudes.

2. Prediction of spatial decay crossover regimes: The system can determine whether an impurity interaction will exhibit a conventional 1/R2 decay or an anomalous, extended 1D-like power-law decay based on the ratio of the Rashba coupling strength (kR) to the magnetic modulation wavevector (Q).

3. Engineering noncollinear spin textures: The AI can design materials or impurity configurations that stabilize specific noncollinear spin spirals or skyrmions by calculating the precise spatial distance R required to maximize the desired chiral twisting, such as finding distances where 2kR equals n pi.

4. Directional control of chiral coupling: The system can calculate the magnitude and sign of Dzyaloshinskii-Moriya interactions (JDM) in three dimensions, allowing for orientation-driven engineering of noncollinear spin textures by tuning the p-wave polarization angle beta.

5. Identification of relativistic hierarchy breaking points: The AI can identify specific material parameters where the strict hierarchy between Rashba and p-wave couplings breaks down, indicating regimes where all off-diagonal exchange components become comparable in magnitude, necessitating a full noncollinear response model rather than a simplified DM vector approximation.

Unconventional odd-parity magnets, characterized by an odd-parity momentum-dependent spin splitting, offer a fundamentally distinct paradigm for non-collinear spintronics. Here, we theoretically investigate the Ruderman-Kittel-Kasuya-Yosida indirect exchange in two-dimensional odd-parity magnets subjected to Rashba spin-orbit coupling. Using an analytical real-space Green's function formalism, we uncover a parity-driven spatial decoupling in the magnetic response. Because of the odd-parity exchange field, the out-of-plane Ising interaction is structurally insulated from the macroscopic magnetic modulation, oscillating isotropically at the shifted Fermi wavevector. Conversely, the in-plane Heisenberg components exhibit pronounced, directionally tunable spatial beating. Beyond collinear exchange, the hybridized bands generate a highly tunable, three-component Dzyaloshinskii-Moriya interaction alongside symmetric off-diagonal anisotropies. We reveal that the out-of-plane chiral twisting is driven by the massive, nonrelativistic momentum shift, while the in-plane chiral components are relativistic. Furthermore, the competition between the nodal geometry and the Rashba gap drives an anomalous, dimension-reducing crossover, in which the in-plane chiral components follow a 1D-like 1/R spatial decay along the nodal lines over an extended intermediate-distance window before ultimately recovering the conventional 2D 1/R squared asymptote. We demonstrate that these findings hold universally across the odd-parity class by explicitly generalizing our framework to f-wave magnets. Finally, numerical tight-binding calculations verify the striking robustness of this parity-driven decoupling against both scalar and random magnetic exchange disorder. These findings establish odd-parity magnets as promising platforms for engineering robust, directionally tunable non-collinear spin textures.