Collective mode across the BCS-BEC crossover in Holstein model

Abstract

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 (ΔP), the peak position of the collective mode (ωcol) evolves from the Bardeen-Cooper-Schrieffer (BCS) regime, manifesting near 2ΔP and increasing with coupling, to the Bose-Einstein condensation (BEC) regime, where ωcol decreases with increasing coupling. The decrease of ωcol 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, ωcol aligns with the soft phonon mode (ωs) of the normal state and decreases with increasing coupling when ωs<2ΔP. Additionally, comparing the collective mode weight to ΔP suggests that the collective mode predominantly stems from U(1) gauge symmetry breaking across all coupling strengths. Published by the American Physical Society 2024