  Superconducting spintronics focuses on the interplay between
superconductivity and magnetism and has sparked significant interest in
nonunitary superconductors as a platform for novel magneto-superconducting
phenomena. However, a direct test for nonunitary superconductors is currently
absent, and their identification is challenging. In this paper, we demonstrate
that spin current driven by the thermal gradient sensitively probes the nature
of the condensate in nonunitary superconductors. We should note that the spin
polarization of the condensate in the momentum space induces the spin-Seebeck
effect, and the spin-dependent chirality (spin-chirality) of the condensate
induces the spin-Nernst effect in nonunitary superconductors. Notably, the
nonvanishing spin-Seebeck effect provides a smoking gun evidence of nonunitary
superconductivity in materials because it reflects the spin polarization of the
Cooper pairs in the momentum space, irrespective of whether the net pair spin
magnetization vanishes. Our results position the spin caloritronics phenomena
as a definitive probe of nonunitary superconductors.
