Pseudogap in Eliashberg approach based on electron-phonon and electron-electron-phonon interaction

The properties of the superconducting and the anomalous normal state have been described by using the Eliashberg method. The pairing mechanism has been reproduced with help of the Hamiltonian, which models the electron-phonon and electron-electron-phonon interaction (EEPh). The set of the Eliashberg equations, which determines the order parameter function ($\varphi$), the wave function renormalization factor ($Z$), and the energy shift function ($\chi$) has been derived. It has been proven that for the sufficiently large values of EEPh potential, the doping dependence of order parameter ($\varphi/Z$) has the analogous course to that observed experimentally in cuprates. The energy gap in the electron density of states is induced by $Z$ and $\chi$ - the contribution from $\varphi$ is negligible. The electron density of states possesses the characteristic asymmetric form and the pseudogap is observed above the critical temperature.