Constructions and Applications of Irreducible Representations of Spin-Space Groups

TL;DR

This paper systematically develops the theory of irreducible representations for spin-space groups, enabling detailed symmetry analysis of magnetic materials and their electronic and magnonic band structures.

Contribution

It introduces a comprehensive method for constructing irreps of SSGs, including projective irreps, and applies this to analyze band structures in magnetic materials like Mn3Sn.

Findings

Identified irreps for all SSGs and their application to band structure analysis.

Demonstrated degeneracies in magnetic materials due to SSG symmetries.

Provided computational tools and databases for SSG representations.

Abstract

Spin-space groups (SSGs), including the traditional space groups (SGs) and magnetic space groups (MSGs) as subsets, describe the complete symmetries of magnetic materials with weak spin-orbit coupling (SOC). In the present work, we systematically study the irreducible representations (irreps) of SSGs by focusing on the projective irreps of the little co-group $L(k)$ of any momentum point $\pmb k$. We analysis the factor systems of $L(k)$, and then reduce the projective regular representation of $L(k)$ into direct sum of irreps using the Hamiltonian approach. Especially, for collinear SSGs which contain continuous spin rotation operations, we adopt discrete subgroups to effectively capture their characteristics. Furthermore, we apply the representation theory of SSGs to study the band structure of electrons and magnons in magnetic materials. After identifying the SSG symmetry group, we extract relevant irreps and determine the $k\cdot p$ models. As an example, we illustrate how our approach works for the material \ch{Mn3Sn}. Degeneracies facilitated by SSG symmetry are observed, underscoring the effectiveness of application in material analysis. The SSG recognition and representation code is uploaded to GitHub, the information of irreps of all SSGs is also available in the online Database. Our work provides a practical toolkit for exploring the intricate symmetries of magnetic materials and paves the way for future advances in materials science.