Numerical Calculations of the Pairing Phenomenon in a Fermi Atomic Gas in the Strongly-Correlated Regime

The primary work presented in this thesis focuses on the calculation of the density-density dynamical correlations in a two dimensional Fermi gas of ultra-cold atoms in the strongly correlated regime, known in literature as the BEC-BCS crossover, using the state-of-the-art dynamical BCS theory, and improving the ability of dynamical BCS to describe the atom pairing behaviour at the crossover via renormalization. While dynamical BCS theory provides a good description of the BEC-BCS crossover, renormalizing the interaction strength between cold atoms and a parameter η, associated with energy resolution and tuned to the system size, improves the ability of dynamical BCS to describe local motions and pair structure. The validations of our density-density correlation calculations for the two dimensional system were done by using recently available exact, imaginary time Quantum Monte Carlo results for the two dimensional atomic gas, provided by Auxiliary Field Monte Carlo methodology (AFQMC). We deduce the feasibility of dynamical BCS theory as a description of Fermi atomic gases at the crossover after a renormalization procedure is applied. Our results suggest that dynamical BCS is able to provide quantitatively accurate results, and allows for the possibility of studying very large systems, something that QMC struggles with due to the complexity of algorithms involved.