本論文，我將討論不同系統中自旋軌道耦合的效應：旋量玻色-愛因斯坦凝聚系統，雙層費米系統，以及二維拓撲絕緣體系統。在旋量玻色-愛因斯坦凝聚系統中，我將主要討論自旋軌道耦合和兩體相互作用的混合效應。運用絕熱近似，我系統的研究了在合成規範場中旋量玻色-愛因斯坦凝聚的基態，激發態以及相關的效應。在雙層費米系統中，我主要考慮自旋-層耦合及超流的效應。通過映射到有效模型，我證明了在零溫下，加入層之間帶有自旋翻轉的隧穿項，可以大大提高超導配對的臨界磁場。在二維拓撲絕緣體中，我主要研究Kane-Mele模型，并考慮在鋸齒形邊界上單顆磁性雜質的效應。我得到Kane-Mele模型譜以及波函數的解析解并討論了它的電輸運性質。進一步構造了一維有效模型來描述該系統。

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.

1 Introduction 1
1.1 Two-level toy model . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Artificial gauge field . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.1 Adiabatic approximation . . . . . . . . . . . . . . . . . . . . . 7
1.2.2 Validity of adiabatic approximation . . . . . . . . . . . . . . . 7
1.2.3 Simple example . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3 Spin-orbit coupling vs. topological property . . . . . . . . . . . . . . 11
1.4 Structure of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2 Topological Condensate in an Interaction-induced Gauge Potential 14
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2 System Hamiltonian and its mean-field equation . . . . . . . . . . . . 17
2.2.1 Two-component GP equation . . . . . . . . . . . . . . . . . . 17
2.2.2 Interaction induced synthetic gauge field (Adiabatic approximation)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.3 Results in uniform space . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.3.1 Ferromagnetism in strong interaction limit . . . . . . . . . . . 22
2.3.2 Excitations and energetic instability of superfluid current (weak
interaction limit) . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.4 Results in a harmonic trap . . . . . . . . . . . . . . . . . . . . . . . . 27
2.4.1 Coreless vortex-antivortex pair in the 2D trap . . . . . . . . . 29
2.4.2 Coreless vortex ring in the 3D trap . . . . . . . . . . . . . . . 31
2.5 Experiment measurement . . . . . . . . . . . . . . . . . . . . . . . . 32
2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3 Superconductivity Enhanced by Spin-flip Tunnelling in The Presence
of A Magnetic Field 35
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.2 System and methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.2.1 System Hamiltonian in original basis . . . . . . . . . . . . . . 38
3.2.2 Equivalent description in a rotated basis . . . . . . . . . . . . 39
3.2.3 Single-particle spectrum . . . . . . . . . . . . . . . . . . . . . 40
3.2.4 General framework in meanfield theory . . . . . . . . . . . . . 42
3.3 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.3.1 Single layer limit . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.3.2 Zero Raman coupling limit . . . . . . . . . . . . . . . . . . . 46
3.3.3 Raman coupling with ϕ = 0 . . . . . . . . . . . . . . . . . . . 48
3.3.4 Raman coupling with ϕ = π/2 . . . . . . . . . . . . . . . . . . 50
3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4 Electric Transport Property for Kane-Mele Model with A Spin-flip
Impurity 56
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.2 Hamiltonian and boundary conditions . . . . . . . . . . . . . . . . . . 57
4.3 Spectra for edge states . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.4 Wavefunction for the Semi-infinite System . . . . . . . . . . . . . . . 67
4.4.1 Wavefunction for edge States . . . . . . . . . . . . . . . . . . 68
4.4.2 Wavefunction for Bulk States . . . . . . . . . . . . . . . . . . 69
4.5 Green function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.6 Transport properties with a single impurity . . . . . . . . . . . . . . . 74
4.7 Basis for constructing an effective model . . . . . . . . . . . . . . . . 77
4.8 Effective one dimensional model . . . . . . . . . . . . . . . . . . . . . 79
4.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5 Conclusions 85
6 Appendix-A 86
7 Appendix-B 93

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