Unbalanced superconductivity induced by the constant electron-phonon coupling on a square lattice

In the present paper, we analyze the properties of the unbalanced superconducting state on a square lattice with the constant value of the electron-phonon coupling function. We conduct our analysis in the framework of the Eliashberg formalism, explicitly considering {\bf k}-dependence of the electron and phonon dispersion relations. It is found that the balanced superconducting state does not induce itself in the system due to the high value of the electron effective mass. However, in the unbalanced case the thermodynamic properties of the superconducting condensate can be distinctly different from the predictions of the Bardeen-Cooper-Schieffer theory; when the coupling constant value in the diagonal channel of the self-energy is diminished comparing to the non-diagonal channel. This is due to the reduced dimensionality of the tested system and the strong-coupling effects included in the Eliashberg formalism.