Unified Magnetoelectric Mechanism for Spin Splitting in Magnets

Abstract

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, ${\mathcal{H}}_{\text{ME}}=-{\mu }_{\text{B}}{\eta }_0({\mathcal{V}}_1-{\mathcal{V}}_2)\mathbf{\sigma }\cdot \mathbf{m}$, 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 $\Gamma$ 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.

8 pages and 2 figure

References in corpus (18)

• Altermagnetism: spin-momentum locked phase protected by non-relativistic symmetries
• Prediction of unconventional magnetism in doped FeSb2
• Spin-polarization coupling in multiferroic transition-metal oxides
• Collinear-to-Spiral Spin Transformation without Changing Modulation Wavelength upon Ferroelectric Transition in Tb1-xDyxMnO3
• Antiferroelectric Altermagnets: Antiferroelectricity Alters Magnets
• Ferroelectric switchable altermagnetism
• Magnetic octupoles as the order parameter for unconventional antiferromagnetism
• Minimal Models for Altermagnetism
• Multipole expansion for magnetic structures: A generation scheme for symmetry-adapted orthonormal basis set in crystallographic point group
• Spontaneous Formation of Altermagnetism from Orbital Ordering
• Two-dimensional fully-compensated Ferrimagnetism
• Non-relativistic spin splitting in compensated magnets that are not altermagnets
• Structure, control, and dynamics of altermagnetic textures
• The Rashba Scale: Emergence of Band Anti-Crossing as a Design Principle for Materials with Large Rashba coefficient
• Strong influence of non-magnetic ligands on the momentum dependent spin splitting in antiferromagnets
• Giant proximity exchange and flat Chern band in 2D magnet-semiconductor heterostructures
• Zeeman effect in centrosymmetric antiferromagnets controlled by an electric field
• Electric Quadrupolar Contributions in the Magnetic Phases of UNi${}_4$B