Microscopic origin of the spin-reorientation transition in the kagome topological magnet TbMn6Sn6

${\mathrm{TbMn}}_{6}{\mathrm{Sn}}_{6}$ is a correlated topological magnet with a Mn-based kagome lattice, in which a Chern gap opens at the Dirac point at low temperatures. The magnetic moment direction of the ferrimagnetic order changes from in the kagome plane to out-of-plane upon cooling, which is essential for generating the Chern gap, but the underlying mechanism for the spin reorientation remains elusive. Here, we investigate the spin-reorientation transition in ${\mathrm{TbMn}}_{6}{\mathrm{Sn}}_{6}$ using neutron scattering. We provide direct evidence for the spin-reorientation transition and unveil the coexistence of two Tb modes at 200 K. To account for these results, we put forward a model based on SU(N) spin-wave theory, in which there is a temperature evolution of the ground state Tb $4f$ orbitals, driven by the crystalline electric field, single-ion anisotropy, and exchange interactions between Tb and Mn ions. Our findings shed light on the complex magnetism of ${\mathrm{TbMn}}_{6}{\mathrm{Sn}}_{6}$, despite its relatively simple ground state magnetic structure, and provide insights into the mechanisms for tuning magnetic topological materials.