Effective theory for the Goldstone field in the BCS-BEC crossover at T=0

Abstract

We perform a detailed study of the effective Lagrangian for the Goldstone mode of a superfluid Fermi gas at zero temperature in the whole BCS-BEC crossover. By using a derivative expansion of the response functions, we derive the most general form of this Lagrangian at the next to leading order in the momentum expansion in terms of four coefficient functions. This involves the elimination of all the higher order time derivatives by careful use of the leading order field equations. In the infinite scattering length limit where conformal invariance is realized, we show that the effective Lagrangian must contain an unnoticed invariant combination of higher spatial gradients of the Goldstone mode, while explicit couplings to spatial gradients of the trapping potential are absent. Across the whole crossover, we determine all the coefficient functions at the one-loop level, taking into account the dependence of the gap parameter on the chemical potential in the mean-field approximation. These results are analytically expressed in terms of elliptic integrals of the first and second kind. We discuss the form of these coefficients in the extreme BCS and BEC regimes and around the unitary limit, and compare with recent work by other authors.

arXiv comment27 pages. 4 references added, typos corrected, expanded Section III B

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• Validating Simple Dynamical Simulations of the Unitary Fermi Gas
• Three-phonon and four-phonon interaction processes in a pair-condensed Fermi gas
• Effective field theory and dispersion law of the phonons of a non-relativistic superfluid
• Phononic collective excitations in superfluid Fermi gases at nonzero temperatures
• Finite temperature damping of collective modes of a BCS-BEC crossover superfluid
• Derivation of the effective action of a dilute Fermi gas in the unitary limit of the BCS-BEC crossover
• Phonon contribution to the shear viscosity of a superfluid Fermi gas in the unitarity limit
• Collective modes in the anisotropic unitary Fermi gas and the inclusion of a backflow term
• Numerical study of unitary fermions in one spatial dimension
• Transition from supersonic to subsonic waves in superfluid Fermi gases
• Hamiltonian formulation of the effective kinetic theory for superfluid Fermi liquids
• Exact evaluation of large-charge correlation functions in non-relativistic conformal field theory