  The spin anomalous Hall effect (SAHE) in ferromagnetic metals, which can
generate spin-orbit torque to rotate the magnetization of another ferromagnetic
layer through a non-magnetic spacer in magnetic junctions, has attracted much
attention. We theoretically investigated the spin anomalous Hall conductivity
(SAHC) of the L1$_0$-type alloys $X$Pt($X$=Fe,Co,Ni) on the basis of
first-principles density functional theory and linear response theory. We found
that the SAHC of FePt is much smaller than the anomalous Hall conductivity
(AHC), leading to very small polarization for the anomalous Hall effect
$\zeta$=SAHC/AHC of around 0.1. On the other hand, the SAHC increases with
increasing number of valence electrons($N_{\rm v}$), and CoPt and NiPt show
relatively large values of $|\zeta|$, greater than 1. The negative contribution
of the spin-down-down component of AHC is the origin of the large SAHC and
$\zeta$ in CoPt and NiPt, which is due to the anti-bonding states of Pt around
the Fermi level in the minority-spin states.
