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 EF and the bound state energy for two fermions in a vacuum, E0. The BCS and BEC limits correspond to EF≫E0 and EFE0, respectively. The chemical potential μ=EF-E0 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 Δ(r,τ), and express the action in terms of a derivatives of the phase φ(r,τ) of Δ(r,τ)=Δeiφ(r,τ). The action contains (φ)2 and φdot;2 terms, which determine the dispersion of collective phase fluctuations, and iπAφ term. For continuous φ(r,τ), 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 φ is singular at the center of the vortex core, and iπAvortφ term affects vortex dynamics. We find that this term has two contributions. One comes from the states away from the vortex core and has Avort,1=n/2, where n is the fermion density. The other comes from electronic states inside the vortex core and has Avort,2=-n0/2, where n0 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 ω is set to zero, while fermionic lifetime τ can be arbitrary. The total Avort=(n-n0)/2 determines the transversal force acting on the vortex core, πAvortṘ×ẑ, where Ṙ is the velocity of the vortex core and ẑ a unit vector perpendicular to the 2d sample. The difference (n-n0)/2 changes through the BEC-BCS crossover as n0 nearly compensates n in the BCS regime, but vanishes in the BEC regime.
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The authors thank I. Aleiner, P. Ao, J. Dziarmaga, I. Martin, J. Sauls, D. Solenov, and particularly G.E. Volovik, for valuable discussions. Our special thanks to M. Stone for in-depth discussions (and respectful disagreement), highly relevant comments and suggestions, and for pointing out an error in the earlier version of the MS. The work by DM was supported by the US Department of Energy through the Los Alamos National Laboratory. Los Alamos National Laboratory is operated by Triad National Security, LLC, for the National Nuclear Security Administration of the US Department of Energy (Contract No. 89233218NCA000001). The work by AVC was supported by the Office of Basic Energy Sciences US Department of Energy under Award DE-SC0014402. AVC is thankful to KITP at UCSB where part of the work has been done. KITP is supported by NSF Grant PHY-1125915.