Phonon-Induced Renormalization and Interaction: An Improvement on Frohlich Transformation

Starting from the Flohlich electron-phonon model, which has a history of 50 years, a new unitary transformation is proposed to implement the perturbation treatment. Our main results are: (1)The phonon-induced interaction shows a crossover from the BCS-like potential when the phonon frequency ${\omega }_p$ is much smaller than the Fermi energy $E_F$ to that of the small polarons when ${\omega }_p\mathrm{/}{E}_F\ge 1$. (2)The jump of momentum distribution of electron number $n_{\mathbf{k}}$ at the Fermi surface goes to zero when the dimensionless coupling constant $\lambda$ increases to the critical value ${\lambda }_c\ge 1$, which means a possible broken down of the Fermi-liquid description.