Dynamic properties of superconductors: Anderson-Bogoliubov mode and Berry phase in BCS and BEC regimes

We analyze the evolution of the dynamics of a neutral s-wave superconductor between BCS and BEC regimes. We consider 2d case, when BCS-BEC crossover occurs already at weak coupling as a function of the ratio of the two scales -- the Fermi energy $E_F$ and the bound state energy for two fermions in a vacuum, $E_0$. BCS and BEC limits correspond to $E_F \gg E_0$ and $E_F \ll E_0$, respectively. The chemical potential $\mu = E_F-E_0$ changes the sign between the two regimes. We develop an approach to derive the leading terms in the expansion of the effective action in the spatial and time derivative of the slowly varying superconducting order parameter $\Delta (r, \tau)$, and express the action in terms of derivatives of the phase $\phi (r,\tau)$ of $\Delta (r, \tau) = \Delta e^{i\phi (r, \tau)}$. In the long wavelength limit the second gradients $(\nabla \phi)^2$ and ${\dot \phi}^2$, describe Bogoliubov-Anderson mode, which, as we find, does not change through BCS-BEC crossover. The effective action also contains first order gradient term, $i \pi A {\dot \phi}$, which is meaningful only if $\phi$ contains topological defects, such as vortices. We apply our approach to evaluate coefficient $A$ for a moving vortex. We find that $A_{vort}$ has two contributions, $A_{vort}= A_{vort,1} - A_{vort,2}$. One comes from the states away from the vortex core and has $A_{vort,1} = n/2$, where $n$ is the fermion density. The other comes from electronic states inside the vortex core and has $A_{vort,2} = -n_0/2$, where $n_0$ is the fermion density at the vortex core. We interpret this term as the reaction force of normal electrons to the vortex displacement in the limit when the spacing between energy levels is set to zero. The difference $(n-n_0)/2$ changes through the BEC-BCS crossover as $n_0$ nearly compensates $n$ in the BCS regime, but vanishes in the BEC regime.