• BEC-BCS crossover
• boson-fermion local-field theory of Fermi gases

The interest in the BCS-BEC crossover has blossomed with the realisation of ultracold quantum gases as a versatile and combined theoretical-experimental platform where to play microscopic mechanisms under extremely controlled conditions. In ultra-cold Fermi gases the whole BEC-BCS crossover can be spanned under controlled conditions by means of a microscopic mechanism, hinging on the Fano-Feshbach (FF) resonance concept, which is due to the coupling between a free scattering state of the two atoms (so-called open channel) with a bound state (closed channel). The resonance width is related to the strength of the open-to-closed channel coupling and classifies Fano-Feshbach resonances into narrow and broad resonances. The intermediate regime bridging from narrow to broad FF resonance has been poorly investigated due to a lack of theoretical methods amenable to affordable numerics. The two cores of my master thesis work are the numerical implementation of a local field boson-fermion unifying theory and a first study of its physics consequences: in particular, I focus on the exchange and correlation effects in the critical region of the BCS-BEC crossover for the appearence of superfluidity with tunable interaction strength and effective range for the pairing potential. In order to do so, a renormalization procedure is to be first adopted for the contact-potentials interaction between Cooper pairs entering the equations. Then, the numerical solution of the BCS-BEC crossover equations is addressed by means of a C code built from scratch, for the critical temperature and the chemical potential.