Euclidean path integral for superfluid fermions in the one dimensional BCS-BEC crossover

In the first part of the thesis the student will learn the Euclidean path integral formulation for non interacting, repulsive and attractive fermions in one dimension at zero temperature with some extension also at finite temperature. In the second part of the thesis, the student will focus on the repulsive case analyzing some properties of the system emerging in the mean field approximation and comparing it with the exactly solvable Gaudin-Yang model. In the third part of the thesis, the student will face the BCS-BEC crossover. By tuning the interaction strength (related to the s-wave scattering length), the fermionic system goes from the BCS state of weakly attractive fermions with Cooper pairs to a BEC state of strongly attractive fermions, where Cooper pairs form a bosonic dimer. In the fourth part of the thesis, the student will investigate the 1D BCS-BEC crossover, where the fermionic system is confined in quasi-one-dimensional spatial configuration which can be experimentally realized with neutral atoms under optical confinement. The student will compare a zero-temperature exact solvable model of the 1D BCS-BEC proposed by Fuchs, Recati and Zwerger in 2004 with a Euclidean path integral approach based on the mean-field solution. In the last part of the thesis, the student will analyze the role of the superfluid density and the critical temperature of the 1D superfluid-to-normal phase transition in the BCS-BEC crossover.