A Unified Symmetry Classification of Magnetic Orders via Spin Space Groups: Prediction of Coplanar Even-Wave Phases

TL;DR

The paper develops a complete symmetry-based classification of magnetic orders using spin space groups (SSGs), highlighting that spin textures in real space S(\mathbf r) and reciprocal space S(\mathbf k) are constrained differently due to det(\hat U_s) effects in SSG operations. By enumerating all SSGs, the authors recover known phases such as FM, AFM, altermagnetism, and p-wave magnetism, and predict new textures, notably coplanar even-wave magnetism, which is coplanar in real space but collinear and even-wave in k-space. A minimal coplanar d-wave model illustrates the mechanism, including a non-quantized spin polarization and symmetry-enforced zero polarization on non-degenerate bands, alongside a proposed material realization in CoCrO4 validated by first-principles calculations. The framework extends to layered (2D) systems, enabling design principles for bilayer coplanar odd- and even-wave magnets and offering a roadmap for discovering unconventional magnetic materials with potentially novel transport and optical responses. The work establishes the completeness and predictive power of SSG-based classification for magnetic orders and sets the stage for experimental verification and broader exploration of coplanar and non-coplanar spin textures.

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.

Figures (3)

• Figure 1: Flowchart illustrating the SSG-based classification of magnetic orders. Starting from a given SSG $\mathcal{G}_s$, real space constraints (left) determine the local spin configuration and whether a net magnetization is allowed, while reciprocal space constraints (middle), formulated separately for bulk (3D) and layer (2D) systems, fix the allowed spin textures $S(\mathbf{k})$. Combining these operations yields the possible reciprocal space spin configurations (spin-polarized, coplanar, non-coplanar, or non-magnetic) and thus the corresponding magnetic phases.
• Figure 2: (a) Real-space lattice model for the coplanar $d$-wave magnet, fulfilling the required SSG symmetries. The dashed box indicates the magnetic unit cell. (b) Calculated band structure along high-symmetry lines and spin polarization ($\langle S_z \rangle$) projected onto the bands. (c) Spin polarization ($\langle S_z \rangle$) of four bands along the $\Gamma-M$ path. (d) Spin-polarized Fermi surface at $\mu = -2\,\text{eV}$, showing the characteristic $d$-wave anisotropy. (e) Magnetic structure of CoCrO4 exhibiting the coplanar magnetic configuration. (f) First-principles band structure of CoCrO4 and spin polarization ($\langle S_x \rangle$) projected onto the bands along the high-symmetry path.
• Figure S1: (a) Real-space lattice model for the coplanar $d$-wave magnet with $C_4$ symmetry, fulfilling the required SSG symmetries. The green and yellow planes show two layers of magnetic atoms related by the SSG symmetry $\{C_{2z}||M_z\}$. (b) Calculated band structure along high-symmetry lines and spin polarization $\langle S_z \rangle$ projected onto the bands for the $C_4$ model. (c) Spin-polarized Fermi surface for the $C_4$ model at $\mu = -2\,\text{eV}$, showing the characteristic $d$-wave anisotropy compatible with $C_4$ symmetry. (d, e, f) Schematics of various even-wave spin-polarized Fermi surface patterns: (d) a $d$-wave pattern (corresponding to the main text model), (e) a $g$-wave pattern, and (f) an $i$-wave pattern, generated by models detailed in the Supplementary Material.