We investigate the competition of various exotic superfluid states in a chain of spin-polarized ultracold fermionic atoms with hyperfine spin $F = 3/2$ and s-wave contact interactions. We show that the ground state is an exotic inhomogeneous mixture
in which two distinct superfluid phases --- spin-carrying pairs and singlet quartets --- form alternating domains in an extended region of the parameter space.
We have studied the Metal-Insulator like Transition (MIT) in lithium and beryllium ring-shaped clusters through ab initio Density Matrix Renormalization Group (DMRG) method. Performing accurate calculations for different interatomic distances and usi
ng Quantum Information Theory (QIT) we investigated the changes occurring in the wavefunction between a metallic-like state and an insulating state built from free atoms. We also discuss entanglement and relevant excitations among the molecular orbitals in the Li and Be rings and show that the transition bond length can be detected using orbital entropy functions. Also, the effect of different orbital basis on the effectiveness of the DMRG procedure is analyzed comparing the convergence behavior.
The system of ultracold atoms with hyperfine spin $F=3/2$ might be unstable against the formation of quintet pairs if the interaction is attractive in the quintet channel. We have investigated the behavior of correlation functions in a model includin
g only s-wave interactions at quarter filling by large-scale density-matrix renormalization-group simulations. We show that the correlations of quintet pairs become quasi-long-ranged, when the system is partially polarized, leading to the emergence of various mixed superfluid phases in which BCS-like pairs carrying different magnetic moment coexist.
