Kagome-lattice magnets RMn6Sn6 recently emerged as a new platform to exploit the interplay between magnetism and topological electronic states 1. Some of the most exciting features of this family are the dramatic dependence of the easy magnetization direction on the rare-earth specie, despite other magnetic and electronic properties being essentially unchanged, and the Kagome geometry of the Mn planes that in principle can generate flat bands and Dirac points; gapping of the Dirac points by spin-orbit coupling has been suggested recently to be responsible for the observed anomalous Hall response in the member TbMn6Sn6. In this work 2, we address both issues with density functional calculations and are able to explain, with full quantitative agreement, the evolution of magnetic anisotropy, including a complete reversal upon adding an $f$-electron with zero magnetic orbital quantum number when going from Ho to Er. We also show the microscopic origin of this computational result using a simple and physically transparent analytical model. We analyze in detail the topological properties of Mn-dominated bands and demonstrate how they emerge from the multiorbital planar Kagome model. We further show that, despite this fact, most of the topological features at the Brillouin zone corner K are strongly 3D, and therefore cannot explain the observed quasi-2D AHE, while those few that show a quasi-2D dispersion are too far removed from the Fermi level. We conclude that, contrary to previous claims, Kagome-derived topological band features bear little relevance to transport in RMn6Sn6, albeit they may possibly be brought to focus by electron or hole doping.
References:
1 J.-X. Yin, S. S. Zhang, H. Li, K. Jiang, G. Chang, B. Zhang, B. Lian, C. Xiang, I. Belopolski, H. Zheng, T. A. Cochran, S.-Y. Xu, G. Bian, K. Liu, T.-R. Chang, H. Lin, Z.-Y. Lu, Z. Wang, S. Jia, W. Wang, and M. Z. Hasan, Giant and anisotropic many-body spin-orbit tunability in a strongly correlated kagome magnet, Nature 562, 91 (2018)
2 Y. Lee, R. Skomski, X. Wang, P. P. Orth, A. K. Pathak, B. N. Harmon, R. J. McQueeney, I. I. Mazin, and L. Ke, Interplay between magnetism and band topology in Kagome magnets RMn6Sn6, arXiv 2201, 11265 (2022), https://arxiv.org/abs/2201.11265

Fig. 1 Variation of magnetic energy (in meV/f.u.) as a function of spin-axis rotation in RMn6Sn6, with R = Gd, Tb, Dy, Ho, and Er, calculated (a) with and (b) without R-4f contributions. $\theta$ is the angle between the spin direction and the out-of-plane direction. The experimental easy directions for each compound are denoted by arrows in panel (a). The lines are just guides for the eye.

Fig. 2 Band structures projected on surface BZ calculated without (blue) and with (red) SOC in TbMn6Sn6. The k-dependent DOS are integrated along kz and are calculated in DFT.