We present a self-consistent theory for the thermodynamics of the BCS-BEC crossover in the normal and superfluid phase which is both conserving and gapless. It is based on the variational many-body formalism developed by Luttinger and Ward and by DeDominicis and Martin. Truncating the exact functional for the entropy to that obtained within a ladder approximation, the resulting self-consistent integral equations for the normal and anomalous Green functions are solved numerically for arbitrary coupling. The critical temperature, the equation of state and the entropy are determined as a function of the dimensionless parameter 1/kF a, which controls the crossover from the BCS-regime of extended pairs to the BEC-regime of tightly bound molecules. The tightly bound pairs turn out to be described by a Popov-type approximation for a dilute, repulsive Bose gas. Even though our approximation does not capture the critical behaviour near the continuous superfluid transition, our results provide a consistent picture for the complete crossover thermodynamics which compare well with recent numerical and field-theoretic approaches at the unitarity point.

We compute the shear viscosity of the unitary Fermi gas above the superfluid transition temperature, using a diagrammatic technique that starts from the exact Kubo formula. The formalism obeys a Ward identity associated with scale invariance which guarantees that the bulk viscosity vanishes identically. For the shear viscosity, vertex corrections and the associated Aslamazov-Larkin contributions are shown to be crucial to reproduce the full Boltzmann equation result in the high-temperature, low fugacity limit. The frequency dependent shear viscosity η(ω) exhibits a Drude-like transport peak and a power-law tail at large frequencies which is proportional to the Tan contact. The weight in the transport peak is given by the equilibrium pressure, in agreement with a sum rule due to Taylor and Randeria. Near the superfluid transition the peak width is of the order of 0.5 T F , thus invalidating a quasiparticle description. The ratio η/s between the static shear viscosity and the entropy density exhibits a minimum near the superfluid transition temperature whose value is larger than the string theory bound /(4πk B ) by a factor of about seven.

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We present a calculation of the spectral functions and the associated rf response of ultracold fermionic atoms near a Feshbach resonance. The single particle spectra are peaked at energies that can be modeled by a modified BCS dispersion. However, even at very low temperatures their width is comparable to their energy, except for a small region around the dispersion minimum. The structure of the excitation spectrum of the unitary gas at infinite scattering length agrees with recent momentum-resolved rf spectra near the critical temperature. A detailed comparison is made with momentum integrated, locally resolved rf spectra of the unitary gas at arbitrary temperatures and shows very good agreement between theory and experiment. The pair size defined from the width of these spectra is found to coincide with that obtained from the leading gradient corrections to the effective field theory of the superfluid.