Nonlinear spin-Seebeck diode in $f$-wave magnets, third-order spin-Nernst effects in $g$-wave magnets and spin-Nernst effects in $i$-wave altermagnets

A prominent feature of $d$-wave altermagnets is that spin current is generated by applying temperature gradient, which is known as the spin-Nernst effect. We show in $f$-wave magnets that spin current is generated proportional to the square of the temperature gradient, which we call the nonlinear spin-Seebeck current. It can be used as a spin current diode. In addition, we show in $g$-wave altermagnets that spin current is generated in the third order of the temperature gradient. We also show in $i$-wave altermagnets that spin current is generated perpendicular to the temperature gradient, which is the spin-Nernst current. We have derived analytic formulas for these spin currents. It is interesting that these phenomena occur in the absence of the spin-orbit interaction. On the other hand, we show in $p$-wave magnets that spin current is not generated by temperature gradient.