Unified magnetoelectric mechanism for spin splitting in magnets

We identify a magnetoelectric correction that completes the theoretical description of spin splitting (SS) in magnetic systems. Derived from the Dirac equation, this term couples local magnetic moments to the scalar electric potential, providing a third fundamental mechanism---alongside Zeeman and spin-orbit coupling (SOC)---that governs SS in ferromagnets, antiferromagnets, and altermagnets. In compensated magnets, the proposed relativistic correction depends on the difference in electric potential between symmetry-inequivalent motifs, ${\mathcal{H}}_{\text{ME}}=\ensuremath{-}{\ensuremath{\mu}}_{\text{B}}{\ensuremath{\eta}}_{0}({\mathcal{V}}_{1}\ensuremath{-}{\mathcal{V}}_{2})\mathbit{\ensuremath{\sigma}}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbit{m}$, which explains how finite SS emerges in the absence of SOC and enables a complete classification of momentum dependence and motif connectivity across all 32 point groups. Through illustrative examples, we show that distinct SS behaviors---quadratic ($d$-wave altermagnets), linear ($p$-wave altermagnets or spin Zeeman effect), and $k$ independent (SS at $\mathrm{\ensuremath{\Gamma}}$ or fully compensated ferrimagnets)---are specific manifestations of the proposed magnetoelectric relativistic mechanism, each governed by electric quadrupoles, dipoles, or monopoles, respectively. The formalism naturally extends to higher-order multipoles and more complex symmetries. This work establishes a unified framework for SS in magnets and provides a predictive tool for analyzing symmetry-allowed SS in magnetic materials.