Abstract
In this thesis, I will discuss the effect of spin-orbit coupling in different systems: spinor Bose-Einstein Condensate (BEC) system, bilayer Fermionic system, and 2D topological insulator. In the spinor BEC system, I focus on the hybrid effect of spin-orbit coupling and two-body interaction. By using adiabatic approximation, I systematically investigate the ground state, elementary excitations and related effects of a BEC within a synthetic vector potential. In the bilayer Fermionic system, I consider the effect of spin-layer coupling and superfluidity. By mapping to an effective model, I demonstrate that at zero temperature the critical value of the magnetic field for pairing can be significantly increased by including a spin-flip tunnelling between layers. In the 2D topological insulator, I focus on the Kane-Mele (KM) Hubbard model and consider the effect of a single spin-flip impurity at the Zigzag edge. I analytically obtain the spectra and wavefunction of the KM model and then discuss its electronic transport property. Furthermore, I develop a low energy effective 1D model to describe the system.