The unbalanced phonon-induced superconducting state on a square lattice beyond the static boundary

The paper presents our verification of induction of the superconducting state on a square lattice by the linear electron-phonon interaction for values of the unbalance parameter ($\gamma=\lambda_{D}/\lambda_{ND}$) less than $\gamma_{C}=0.42$. Symbols $\lambda_{D}$ and $\lambda_{ND}$ denote the values of the coupling constant in the diagonal and the non-diagonal channel of the self-energy. Calculations were carried out using the Eliashberg equations, in which the order parameter ($\Delta_{\bf k}\left(i\omega_{n}\right)$) and the wave function renormalising factor ($Z_{\bf k}\left(i\omega_{n}\right)$) depend explicitly on the Matsubara frequency ($\omega_{n}$) and the wave vector (${\bf k}$). The value of $\gamma_{C}$ in the static boundary ($\Delta_{\bf k}\left(i\omega_{n}\right)\rightarrow \Delta_{\bf k}\left(i\omega_{n=1}\right)$), equal to ($0.93$), is significantly greater than the obtained limit value of $0.42$. Values of the thermodynamic functions of the superconducting state determined for our assumptions are significantly different from the values calculated in accordance with the BCS theory. The results were obtained for the electron-phonon interaction function explicitly dependent on the momentum transfer between electron states.