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Quasiclassical description of a superconductor with a spin density wave

We derive equations for the quasiclassical Green's functions $\check{g}$ within a simple model of a two-band superconductor with a spin-density-wave (SDW). The elements of the matrix $\check{g}$ are the retarded, advanced, and Keldysh functions each of which is an $8\times 8$ matrix in the Gor'kov-Nambu, the spin and the band space. In equilibrium, these equations are a generalization of the Eilenberger equation. On the basis of the derived equations we analyze the Knight shift, the proximity and the dc Josephson effects in the superconductors under consideration. The Knight shift is shown to depend on the orientation of the external magnetic field with respect to the direction of the vector of the magnetization of the SDW. The proximity effect is analyzed for an interface between a superconductor with the SDW and a normal metal. The function describing both superconducting and magnetic correlations is shown to penetrate the normal metal or a metal with the SDW due to the proximity effect. The dc Josephson current in an $S_{SDW}/N/S_{SDW}$ junction is also calculated as a function of the phase difference $ϕ$. It is shown that in our model the Josephson current does not depend on the mutual orientation of the magnetic moments in the superconductors $S_{SDW}$ and is proportional to $ \sin ϕ$. The dissipationless spin current $j_{sp}$ depends on the angle $α$ between the magnetization vectors in the same way ($j_{sp} \sim \sin α$) and is not zero above the superconducting transition temperature.