  Nematicity and magnetism are two key features in Fe-based superconductors,
and their interplay is one of the most important unsolved problems. In FeSe,
the magnetic order is absent below the structural transition temperature
$T_{str}=90$K, in stark contrast that the magnetism emerges slightly below
$T_{str}$ in other families. To understand such amazing material dependence, we
investigate the spin-fluctuation-mediated orbital order ($n_{xz}\neq n_{yz}$)
by focusing on the orbital-spin interplay driven by the strong-coupling effect,
called the vertex correction. This orbital-spin interplay is very strong in
FeSe because of the small ratio between the Hund's and Coulomb interactions
($\bar{J}/\bar{U}$) and large $d_{xz},d_{yz}$-orbitals weight at the Fermi
level. For this reason, in the FeSe model, the orbital order is established
irrespective that the spin fluctuations are very weak, so the magnetism is
absent below $T_{str}$. In contrast, in the LaFeAsO model, the magnetic order
appears just below $T_{str}$ both experimentally and theoretically. Thus, the
orbital-spin interplay due to the vertex correction is the key ingredient in
understanding the rich phase diagram with nematicity and magnetism in Fe-based
superconductors in a unified way.
