Spin Currents at the Interface and Spin Hall Torque

We compare two different approaches to compute the spin Hall torque at the interface between a spin Hall metallic layer and a ferromagnet. In one approach, one attributes a spin mixing conductance
$g_{\uparrow \downarrow }$
to the interface, while the other employs the thermodynamic theory for the magnetic moment currents. The main difference between the outcomes of the two approaches is the field-like torque term that, in the first approach, is due to the imaginary part of the spin mixing conductance
$\text {Im}[g_{\uparrow \downarrow }]$
and, in the second one, is proportional to the effective damping
$\alpha ^{\prime }$
of the ferromagnet.