  TbMn$_6$Sn$_6$ has attracted a lot of recent interest for a variety of
reasons, most importantly, because of the hypothesis that it may support
quantum-limit Chern topological magnetism, derived from the kagome geometry.
Besides, TbMn$_6$Sn$_6$ features a highly unusual magnetic reorientation
transition about 100 K below the Curie point, whereby all spins in the system,
remaining collinear, rotate by 90$^\circ$. In this work, we address both issues
combining experiment, mean-field theory and first-principle calculations. Both
magnetic reorientation and the unusual temperature dependence of the anomalous
Hall conductivity (AHC) find quantitative explanation in the fact that Mn and
Tb, by virtue of the Mermin-Wagner theorem, have very different spin dynamics,
with Tb spins experiencing much more rapid fluctuation. We were able to cleanly
extract the intrinsic AHC from our experiment, and calculated the same
microscopically, with good semiquantitative agreement. We have identified the
points in the band structure responsible for the AHC and showed that they are
not the kagome-derived Dirac points at the K-corner of the Brillouin zone, as
conjectured previously.
