Spin Group Symmetry Criteria For Unconventional Magnetism

Abstract

Unconventional magnetism has typically been classified into two fundamental classes: even-parity magnets (EPMs) and odd-parity magnets (OPMs). These two classes exhibit identical and opposite spin splittings, respectively, under momentum inversion, while both maintain symmetry-compensated magnetization. In this Letter, we present a unified spin space group-based framework that establishes comprehensive symmetry criteria for both classes. Our framework not only yields a complete classification of EPMs and OPMs but also uncovers a wealth of new symmetry-driven mechanisms for them. Specifically, we classify both classes into three types based on their spin textures: collinear (type-I), coplanar (type-II), and noncoplanar (type-III), and we demonstrate that both classes can be realized across collinear, coplanar, and noncoplanar magnetic orders. We identify eight distinct symmetry-driven mechanisms for OPMs and seven for EPMs, among which some paradigms of unconventional magnetism, for instance, altermagnets naturally emerge as one specific mechanism of EPMs. Using these established criteria, we identify numerous candidate materials from the Magndata database, realizing some new symmetry mechanisms for OPMs and EPMs. Our work establishes a foundational symmetry framework for understanding, predicting, and designing unconventional magnetic materials.

Figures (4)

• Figure 1: Schematic illustration of the unified SSG framework for unconventional magnetism. From a SSG $G$, symmetries preserving momentum ($g_i\bm{k}=\bm{k}$) constrain the spin texture dimensionality, yielding three types: collinear (type-I), coplanar (type-II), and noncoplanar (type-III). Symmetries flipping momentum ($g_i\bm{k}=-\bm{k}$) determine the parity of spin splitting, yielding odd-parity and even-parity classes. For even-parity NSS, additional operations satisfying $g_i\bm{k}\neq\pm\bm{k}$ is required to enforce zero magnetization. The resulting spin textures form representations of emergent Laue group $\tilde{G}$ with odd-parity or even-parity.
• Figure 2: (a) Schematic illustration of $120^{\circ}$ antiferromagnetic order on a triangle lattice. (b) The isoenergy-surface characterized by $s_z(\bm k)$ for the model shown in (a). (c) Magnetic unit cell for a 2D lattice model on a square lattice. The vectors $(x,y,z)$ denotes the magnetic moment directions. (d) The isoenergy-surface characterized by the vector $(s_x(\bm k,s_y(\bm k))$, shown by the blue arrow, for the lattice model in (c). We take $t=J=1$, $\mu=-2.5$ for (b), and $\mu=-4.3$ for (d).
• Figure 3: (a) Schematic illustration of a bilayer kagome breathing lattice. (b) The wannier center $W(k_y)$ for the lowest energies two bands. (c) The energy spectrum for the system with a nanowire geometry along the $x$ direction. The red bands denote the edge states. (d) The spin polarization Fermi contour. The color encodes the value $s_z(\bm k)$. The pink arrows denote the direction of the vector $(s_x(\bm k), s_y(\bm k))$ at $\bm k$. See SM supp for model parameters.
• Figure S1: (a) Schematic illustration of a 1D OPM. (b) The energy bands for the illustrated model in (a). (c) Schematic illustration of Haldane model with antiferromagnet order. (d) The energy bands of the illustrated model in (c). We take $t=J=1$ and $\lambda=0.2$ in (d).