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

We analyze the evolution of the dynamics of a neutral $s$-wave superconductor between the BCS and BEC regimes. We consider 2d case when a 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, ${E}_{0}$. The BCS and BEC limits correspond to ${E}_{F}\ensuremath{\gg}{E}_{0}$ and ${E}_{F}\ensuremath{\ll}{E}_{0}$, respectively. The chemical potential $\ensuremath{\mu}={E}_{F}\ensuremath{-}{E}_{0}$ changes the sign between the two regimes. We use the effective action approach, derive the leading terms in the expansion of the effective action in the spatial and time derivative of the slowly varying superconducting order parameter $\mathrm{\ensuremath{\Delta}}(r,\ensuremath{\tau})$, and express the action in terms of a derivatives of the phase $\ensuremath{\phi}(r,\ensuremath{\tau})$ of $\mathrm{\ensuremath{\Delta}}(r,\ensuremath{\tau})=\mathrm{\ensuremath{\Delta}}{e}^{i\ensuremath{\phi}(r,\ensuremath{\tau})}$. The action contains ${(\ensuremath{\nabla}\ensuremath{\phi})}^{2}$ and ${\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\phi}}}^{2}$ terms, which determine the dispersion of collective phase fluctuations, and $i\ensuremath{\pi}A\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\phi}}$ term. For continuous $\ensuremath{\phi}(r,\ensuremath{\tau})$, the latter reduces to the contribution from the boundary and does not affect the dynamics. We show that this long-wavelength action does not change through the BCS-BEC crossover. We apply our approach to a moving vortex, for which $\ensuremath{\phi}$ is singular at the center of the vortex core, and $i\ensuremath{\pi}{A}_{\mathrm{vort}}\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\phi}}$ term affects vortex dynamics. We find that this term has two contributions. One comes from the states away from the vortex core and has ${A}_{\mathrm{vort},1}=n/2$, where $n$ is the fermion density. The other comes from electronic states inside the vortex core and has ${A}_{\mathrm{vort},2}=\ensuremath{-}{n}_{0}/2$, where ${n}_{0}$ is the fermion density at the vortex core. This last term comes from the continuous part of the electronic spectrum and has no contribution from discrete levels inside the core; it also does not change if we add impurities. We interpret this term as the contribution to vortex dynamics in the continuum limit, when the spacing between energy levels $\ensuremath{\omega}$ is set to zero, while fermionic lifetime $\ensuremath{\tau}$ can be arbitrary. The total ${A}_{\mathrm{vort}}=(n\ensuremath{-}{n}_{0})/2$ determines the transversal force acting on the vortex core, $\ensuremath{\pi}{A}_{\mathrm{vort}}\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\mathbf{R}}\ifmmode\times\else\texttimes\fi{}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{z}$, where $\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\mathbf{R}}$ is the velocity of the vortex core and $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{z}$ a unit vector perpendicular to the 2d sample. The difference $(n\ensuremath{-}{n}_{0})/2$ changes through the BEC-BCS crossover as ${n}_{0}$ nearly compensates $n$ in the BCS regime, but vanishes in the BEC regime.