Quantum Phase Transitions Of Fermionic Atoms Trapped In A One-dimensional Optical Lattice

In this paper, firstly, we introduce some basic concepts about the cold atomic gases in condensed-matter systems. Then, we give a brief introduction of the ALPS(Algorithms and Libraries for Physics Simulations) and DMRG (Density Matrix Renormalization Group). And we study the ground-state properties of fermionic gas confined in a one-dimensional optical lattice by using the DMRG method. The results proved that our method is feasible and reliable.Secondly, we investigate the ground-state properties of fermions with attractive interaction and repulsive interaction trapped in a one-dimensional optical superlattice with two-site periodicity by using the DMRG method.When we study the ground-state properties of fermionic atoms with repulsive interaction confined in a one-dimensional optical superlattice, we find that in contrast to the ordinary Hubbard model in the optical lattice, new insulating regions appear with niuc=0.5and niuc=1.5caused by the superlattice potentials. On the other hand, we summarize that the region with nicu=1.0 is either in the band-insulating state, the BCDW state, or the Mott-insulating state, depending on the strength of on-site repulsion. These states are smoothly connected to each other with the increase in U. We also find that SOC effects enhance the metal phase. In the absence of SOC effects a large Zeeman field enhances the insulator. When the superlattice potentials exist, SOC effects enhance the metal phase and suppress the insulator of the system in the presence of both the SOC and a Zeeman field.We study the ground-state properties of fermionic atoms with attractive interaction loaded on a one-dimensional optical superlattice by using the DMRG method. We find that the local potential enhances the BCS superfluid phase while suppresses the FFLO phase. The phenomenon is most obvious at the filling factor n=1. We also find that, with the increasing of local potential, one must increase the intensity of Zeeman field to destroy degeneracy and generate to polarization in order to reach the FFLO phase. Finally, we make a conclusion and outlook.