Spin Group Symmetry Criteria for Odd-parity Magnets

Odd-parity magnets (OPMs) have recently emerged as a new magnetic class, but their general symmetry criteria remain elusive. In this Letter, we establish these criteria through a comprehensive spin group symmetry analysis. Concretely, we identify eight distinct symmetry-driven cases that support OPMs with collinear, coplanar, or noncoplanar magnetic order. These are classified into three classes based on their spin textures: collinear (type-I), coplanar (type-II), and noncoplanar (type-III). From the Magndata database, we identify 33 candidate OPM materials and diagnose their spin-splitting character ($p$- or $f$-wave) by analyzing the representation of spin textures within an emergent Laue group derived from the spin space group, which reveals a variety of novel spin textures. To validate the symmetry criteria, we construct and analyze two theoretical models. Furthermore, we demonstrate that OPMs can host an intrinsic $\mathbb{Z}_2$ topology and propose a model for their realization. Our work provides a foundational framework for the future exploration of OPMs.

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