A Unified Symmetry Classification of Magnetic Orders via Spin Space Groups: Prediction of Coplanar Even-Wave Phases
Authors
Ziyin Song
Ziyue Qi
Chen Fang
Zhong Fang
Hongming Weng
Abstract
Spin space groups (SSGs) impose fundamentally different constraints on magnetic configurations in real and reciprocal spaces. As a consequence, the correspondence between real-space and momentum-space spin arrangements is far richer than traditionally assumed. Building on the complete enumeration of SSGs, we develop a systematic, symmetry-based framework that classifies all possible spin arrangements allowed by these groups. This unified approach naturally incorporates conventional magnetic orders, altermagnetism, and p-wave magnetism as distinct symmetry classes. Crucially, our classification predicts a variety of novel magnetic phases, highlighted by the discovery of the coplanar even-wave magnet: a state that is non-collinear in real space but hosts a collinear even-wave spin polarization in k-space. Analysis of a minimal model reveals that this phase is characterized by non-quantized spin polarization and exhibits a novel mechanism for symmetry-enforced zero polarization on non-degenerate bands. Extending the framework from bulk crystals to layer SSGs appropriate for two-dimensional systems, we further predict layered counterparts and provide symmetry guidelines for designing bilayer coplanar p-wave and even-wave magnets. We further validate this finding through first-principles calculations and propose CoCrO4 as a promising candidate for its experimental realization, thereby demonstrating the completeness and predictive power of the SSG-based classification of magnetic orders.
