  Since the concept of spin superconductor was proposed, all the related
studies concentrate on spin-polarized case. Here, we generalize the study to
spin-non-polarized case. The free energy of non-polarized spin superconductor
is obtained, and the Ginzburg-Landau-type equations are derived by using the
variational method. These Ginzburg-Landau-type equations can be reduced to the
spin-polarized case when the spin direction is fixed. Moreover, the expressions
of super linear and angular spin currents inside the superconductor are
derived. We demonstrate that the electric field induced by super spin current
is equal to the one induced by equivalent charge obtained from the second
Ginzburg-Landau-type equation, which shows self-consistency of our theory. By
applying these Ginzburg-Landau-type equations, the effect of electric field on
the superconductor is also studied. These results will help us get a better
understanding of the spin superconductor and the related topics such as
Bose-Einstein condensate of magnons and spin superfluidity.
