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 formation of a finite-density polaronic state is analyzed in the context of the Holstein model using the dynamical mean-field theory. The spinless and spinful fermion cases are compared to disentangle the polaron crossover from the bipolaron formation. The exact solution of dynamical mean-field theory is compared with weak-coupling perturbation theory, noncrossing (Migdal), and vertex correction approximations. We show that polaron formation is not associated with a metal-insulator transition, which is instead due to bipolaron formation.