Spatially inhomogeneous condensate in asymmetric nuclear matter

We study the isospin singlet pairing in asymmetric nuclear matter with nonzero total momentum of the condensate Cooper pairs. The quasiparticle excitation spectrum is fourfold split compared to the usual BCS spectrum of the symmetric, homogeneous matter. A twofold splitting of the spectrum into separate branches is due to the finite momentum of the condensate, the isospin asymmetry, or the finite quasiparticle lifetime. The coupling of the isospin singlet and tripler paired states leads to further twofold splitting of each of these branches. The gap equation is solved numerically in the isospin singlet channel in the case where the pairing in the isospin tripler channel and the renormalization of the single particle energies are neglected. We find nontrivial solutions with finite total momentum of Cooper pairs. The corresponding condensate has a periodic spatial structure which carries a isospin density wave at constant total number of particles. The phase transition from the BCS to the inhomogeneous superconducting phase is found to be first order and occurs when the density asymmetry alpha = (rho (n) - rho (p))/(rho (n) + rho (p)) (defined in terms of partial densities of neutrons rho (n) and protons rho (p)) is increased above 0.25. The transition from the inhomogeneous superconducting to the unpaired normal state is second order. The maximal values of the critical total momentum (in units of the Fermi momentum) and the critical density asymmetry at which condensate disappears are P-c. /p(F) = 0.3 and alpha (c) = 0.41 (assuming free single particle spectrum). The possible spatial forms of the ground state of the inhomogeneous superconducting phase are briefly discussed.