Collective mode across the BCS-BEC crossover in Holstein model

We investigate the emergence of the collective mode in the phonon spectra of the superconducting state within the Holstein model by varying the electron-phonon coupling. Using dynamical mean field theory combined with the numerical renormalization group technique, we calculate the phonon spectra. In the superconducting state with a pairing gap (Delta(P)), the peak position of the collective mode (omega(col)) evolves from the Bardeen-Cooper-Schrieffer (BCS) regime, manifesting near 2 Delta(P) and increasing with coupling, to the Bose-Einstein condensation (BEC) regime, where omega(col) decreases with increasing coupling. The decrease of omega(col) matches well with the reduction of superfluid stiffness, which originates from the increasing phase fluctuations of local pairs with coupling strength. In the crossover regime with intermediate coupling, omega(col) aligns with the soft phonon mode (omega(s)) of the normal state and decreases with increasing coupling when omega(s) < 2 Delta(P). Additionally, comparing the collective mode weight to Delta(P )suggests that the collective mode predominantly stems from U(1) gauge symmetry breaking across all coupling strengths.