The Studies On The Quantum Phases And Disorder Effects In One-dimensional Spin-orbit Coupled Atomic Fermi Gases

We investigate the properties of one-dimensional spin-orbit coupled atomic Fermi gases and the effect of disorder on a one-dimensional p-wave superconduc-tor. There are five chapters in the thesis.In chapter one, we introduce the background of the topological superconduc-tor and Majorana fermion in condensed matter physics. We also introduce the progress of spin-orbit coupling in ultracold atomic experiments and theoretical research of the spin-orbit coupled ultracold-atom systems.In chapter two, we introduce the origin of spin-orbit coupling, and derive the Bogoliubov-de Gennes equation from the model Hamiltonian in details. We introduce how to solve this equation self-consistently as well. Furthermore, we discuss the dispersion relations of quasiparticles and obtain the critical Zeeman field when the topological phase transition happens.In chapter three, by solving the Bogoliubov-de Gennes equation self-consiste-ntly, we find that the system supports the topological superfluid state and the Fulde-Ferrell-Larkin-Ovchinnikov superfluid state respectively when the system under the different strength of Zeeman field and filling factors. When the system turns into topological superfluid state, a pair of zero-energy Majorana fermions are found. Finally, the phase diagrams are given.In chapter four, we investigate the effect of disorder on a one-dimensional p-wave superconductor. By using the method of multifractal analysis, we find that the system not only supports extended phase and localized phase, but also supports a critical phase. We investigate the influence of disorder on Majorana edge modes under open boundary conditions as well.Conclusions and outlook are given in chapter five.