Multicolor groups for molecules and solids

TL;DR

The paper addresses the limitations of conventional magnetic space groups, especially their exclusion of SOC and difficulty accommodating altermagnets and $p$-wave magnets. It introduces multicolor groups, a unified framework where local moments are colored and a color-exchange symmetry $\mathcal{P}$ of order $m$ organizes both spin and lattice degrees of freedom, with separate treatments for without SOC ($G^m=H+\mathcal{P}H+\cdots=H\\times\\mathbb{Z}_m(\\mathcal{P})$) and with SOC, plus complex $n\\times m$-color extensions $G^{n\\times m}=G^m+\mathcal{A}G^m+\\cdots+\mathcal{A}^{n-1}G^m$. The framework yields a comprehensive classification of paramagnets, ferromagnets, antiferromagnets, noncollinear magnets, altermagnets, and $p$-wave magnets, including altermagnetic topological matter, and resolves long-standing confusions in conventional MSG descriptions. This unified approach offers a powerful language for predicting band topology, optical responses, and spintronic phenomena across magnetic and nonmagnetic materials. Overall, multicolor groups advance symmetry classification toward a broader, topology-aware description of matter beyond traditional magnets.

Abstract

The local magnetic moments of atoms in a molecule or solid can be designated by different colors. Magnetic groups, or 2-color groups, or black-and-white groups have been applied in crystallography to classify different magnets. Despite its successes in the past decades, the recent advents of altermagnets and p-wave magnets raise new challenges to this long-standing framework, which urges for a new and unified one. Here we develop a multicolor group classification framework to classify all kinds of molecules and solids, including nonmagnetic materials and magnets with collinear or non-collinear magnetism, and with or without spin-orbit couplings (SOC). This new scheme can unify the classifications of matters into a single framework, including the recently identified altermagnets and p-wave magnets. Especially, altermagnetic topological matters and p-wave magnets with SOC, can also be diagnosed with multicolor groups, a task which can not be accomplished by magnetic space groups and spin space groups. Moreover, insufficiencies and misconceptions of conventional magnetic group classification can be supplemented through this new framework. Multicolor group will serve as a new stage in the symmetry classification of matters.

Figures (3)

• Figure 1: Different partitions of multicolor groups (MCG). The magnetic space groups (MSG) are defined for collinear magnets without including of SOC, which belong to a subpatition of the spin space groups (SSG). Multicolor color groups are applicable for collinear or noncollinear magnets with or without SOC.
• Figure 2: Illustrations of the confusion brought by conventional magnetic groups. (a) An antiferromagnetic lattice, with symmetry operations [$E, C_2$], which belongs to Type 1 magnetic groups. (b) A ferromagnetic lattice, with symmetry operations [$E, C_2\Theta$], which belongs to magnetic group Type 3.
• Figure 3: Illustrations of the $m$-color groups. (a) A simple 6-color group, generated by $C_{6z}\Theta$ or $S_{6z}$. (b) A complex 2$\times$2-color group, composed of i$\Theta\times C_{2x}$.