Collective mode across the BCS-BEC crossover in Holstein model

TL;DR

This study explores how a collective mode in phonon spectra evolves across the BCS-BEC crossover in the Holstein model, revealing its dependence on coupling strength and phase fluctuations.

Contribution

It provides a detailed analysis of the collective mode behavior across the BCS-BEC crossover using DMFT and NRG, highlighting the mode's relation to superfluid stiffness and phonon softening.

Findings

Collective mode peaks near 2Δ_P in BCS regime

Mode frequency decreases in BEC regime with increasing coupling

Mode primarily arises from U(1) gauge symmetry breaking

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 (DMFT) combined with the numerical renormalization group (NRG) 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.