Spin Symmetry Criteria for Odd-parity Magnets

Abstract: Inspired by the discovery of altermagnets, which exhibit even-parity nonrelativistic spin splitting, odd-parity magnets (OPMs) have been proposed and emerged as a novel research frontier. In this study, we perform a comprehensive spin group symmetry analysis to establish symmetry criteria for the emergence of OPMs. We identify eight distinct symmetry-driven cases that support OPMs, enabling their realization in collinear, coplanar, and noncoplanar magnetic orders. These OPMs are categorized into three types based on their spin textures for Bloch states: collinear (type-I), coplanar (type-II), and noncoplanar (type-III). For type-I OPMs, we further delineate additional symmetry requirements for $p$-wave and $f$-wave spin splitting. We identify 48 candidate materials in the Magndata database that satisfy these symmetry criteria. Additionally, we construct two theoretical models to validate the effectiveness of the established symmetry criteria. Finally, we show that OPMs can exhibit an intrinsic $\mathbb{Z}_2$ topology and construct a theoretical model to realize this phase.