Collective mode across the BCS-BEC crossover in the Holstein model

We present the collective mode emerging in bosonic spectra for the Holstein model with a superconducting state in terms of the variation of the electron-phonon coupling. Phonon spectra and density-density correlation functions are analyzed using dynamical mean field theory (DMFT) calculation combined with the numerical renormalization group (NRG) technique. In the superconducting state with a pairing gap (∆_P), the peak position of the collective mode (ω_col) undergoes an evolution from the Bardeen-Cooper-Schrieffer (BCS) regime to the Bose-Einstein condensation (BEC) regime with an intermediate crossover regime. In the BCS regime, ω_col manifests near 2∆_P in bosonic spectra, increasing with electron-phonon coupling. Conversely, in the crossover regime, ω_col aligns with the soft phonon mode (ω_s), decreasing when ω_s < 2∆_P. In the BEC regime, ω_col decreases with increasing coupling, indicative of the superfluid stiffness originating from phase fluctuations of local pairs. Additionally, our analysis reveals a monotonic increase in phase fluctuation with coupling strength. Comparing the collective mode weight to ∆_P suggests that the collective mode predominantly stems from U(1) gauge symmetry breaking across all coupling strengths.