We investigate the dynamical effects of pairing interaction on superconductivity in BCS-BEC crossover by studying the Holstein model at half filling where the electron-phonon coupling
g controls the crossover. The dynamical mean-field theory was employed in combination with the numerical renormalization group technique to study the frequency dependence of phonon mediated interaction between electrons in a superconductive state. The frequency dependence induces distinct features such as the absence of the dispersion back bending of Bogoliubov quasiparticles, nonmonotonous coupling dependence of the pairing gap, and the critical temperature
Tc being maximum at
gc1, the lower critical coupling of the normal state insulator to metal transition. Some of these features have intriguing similarities with the recent observations in the ${\mathrm{FeSe}}_{1\ensuremath{-}x}{\mathrm{S}}_{x}$ and ${\mathrm{Fe}}_{1+y}{\mathrm{Se}}_{x}{\mathrm{Te}}_{1\ensuremath{-}x}$ iron chalcogenides in the BCS-BEC crossover.
The renormalized mode frequencies are obtained with the help of quantum dynamical approach of many body phonon Green’s function technique via a general Hamiltonian (excluding BCS Hamiltonian) including the effects of phonons and electrons, anharmonicities and electron-phonon interactions. The numerical estimates have been carried out to study the renormalized mode frequency of high temperature cuprate superconductor (HTS) YBa2Cu3O7-δ using modified Born-Mayer-Huggins interaction potential (MBMHP) best applicable to study the dynamical properties of all HTS.