Parity-driven RKKY decoupling and anomalous $1/R$ Dzyaloshinskii-Moriya interaction in $p$-wave magnets

Abstract

Unconventional $p$-wave 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 a two-dimensional $p$-wave magnet 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 $p$-wave 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 $p$-wave momentum shift, while the in-plane chiral components are strictly relativistic. Furthermore, the competition between the $p$-wave 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 $1/R^2$ asymptote. These findings establish $p$-wave magnets as promising platforms for engineering robust, directionally tunable non-collinear spin textures.