Topological correspondence between magnetic space group representations and subdimensions

The past years have seen rapid progress in the classification of topological band structures using symmetry eigenvalue indicated methods. Given their importance in condensed matter systems, these ideas are increasingly getting explored in the pertinent context of magnetic structures. We here adopt this viewpoint to address the physical implications of extending space groups to magnetic variants. In particular, we introduce a simple model as a generic example of magnetic fragile topology. Most interestingly, we find that this antiferromagnetic-compatible model can be tuned via Zeeman terms to a ferro/ferrimagnetic (FM) counterpart in the same space-group family. This correspondence manifests itself by ensuring that the fragile topology produces bands of finite Chern number in the FM phase. In addition, we discuss how the system can be tuned into a stable topological semimetallic phase, characterized by a simple expression for the $\mathbf{Z}_2$ symmetry indicator that results from the combination of $C_4$ symmetry and $C_2T$-protected Euler class topology. This scenario features a similar correspondence that can even relate to higher Chern numbers. Pointing out the generality of such relations for a variety of space group families, we believe our results pave the way for new pursuits in magnetic topologies.