我們研究Haldane系統和Rashba自旋軌道耦合作用於二維電子系統的陳數數值 (Chern number) 相變，陳數相變對應於系統結構的邊緣狀態變化。這些系統可以建立在單層石墨烯 (Single-layer Graphene)，並且在光學晶格 (optical lattice) 中通過使用超冷原子 (Ultracold atom) 來實現。因此，我們研究了Haldane的Rashba系統中的量子反常霍爾效應。我們首先描述如何建構緊束縛系統的哈密頓並獲得在費米能量 (Fermi energy) 附近能帶的交點，也稱作狄拉克點 (Dirac point) 。然後，我們展示了鋸齒狀邊緣的石墨烯帶 (Graphene ribbon with zigzag edge) 的電子能帶結構和相應波函數電子機率密度分布。並且通過引入Haldane系統的相變邊界關係，我們得到塊材 (Bulk) 與鋸齒狀邊緣相對應的Haldane-Rashba系統。最後，我們討論Haldane-Rashba系統的陳數數值和相變邊界值關係變化。

We study the possible Chern insulator phase transitions in the Haldane-Rashba system. These systems could be established on single-layer graphene and realized by using the ultracold atoms. Therefore, we investigate the quantum anomalous Hall effect in the Haldane-Rashba systm. We first describe how to construct tight-binding model Hamiltonian and obtain the Dirac points. We then show the electronic band structure and the corresponding wave functions of electrons. By using the boundary of phase transformation we study the change in Chern number in Haldane-Rashba system. Finally, we discuss the variation of Chern number values and its corresponding phase transitions.

論文審定書 i
中文摘要 ii
英文摘要 iii
致謝 iv
目錄 v
圖次 vi
第一章 緒論 1
1.1 量子霍爾效應 (Quantum Hall effect) 1
1.2 塊材與邊緣對應 (Bulk-edge correspondence) 1
1.3 反常霍爾效應 (Anomalous Hall effect) 2
1.4 量子反常霍爾效應 (Quantum Anomalous Hall effect) 2
第二章 理論與方法 5
2.1 單層石墨烯的緊束縛系統 (Tight-binding model) 5
2.2 Haldane系統 12
2.3 第三次鄰近跳躍耦合 22
2.4 Rashba自旋軌道耦合 26
第三章 結果與討論 31
3.1 Haldane-N3系統 31
3.2 Haldane-N3-N3'系統 37
3.3 Haldane-Rashba系統 45
3.4 陳數數值計算 51
第四章 結論 56
參考文獻 57

[1] Klaus von Klitzing, et al, “New Method for High-Accuracy Determination of the Fine-Structure Constant Based on Quantized Hall Resistance”, Phys. Rev. Lett. 45, 494 (1980).
[2] Landau, L. D. and E. M. Lifshitz, “Quantum Mechanics: Nonrelativistic Theory”. (1977).
[3] D. R. Yennie, “Integral quantum Hall effect for nonspecialists”, Rev. Mod. Phys. 59, 781.
[4] Anushya Chandran, et al., “Bulk-edge correspondence in entanglement spectra”, Phys. Rev. B. 84, 205136 (2011).
[5] Gian Michele Graf and Marcello Porta, “Bulk-Edge Correspondence for Two-Dimensional Topological Insulators”, Commun. Math. Phys. 324, 851–895 (2013).
[6] Roger S. K. Mong and Vasudha Shivamoggi, “Edge states and the bulk-boundary correspondence in Dirac Hamiltonians”, Phys. Rev. B 83, 125109 (2011).
[7] Mark S. Rudner, et al., “Anomalous Edge States and the Bulk-Edge Correspondence for Periodically Driven Two-Dimensional Systems”, Phys. Rev. X. 3, 031005 (2013).
[8] Naoto Nagaosa, et al., “Anomalous Hall effect”, Rev. Mod. Phys. 82, 1539.
[9] M. Gradhand, et al., “Skew Scattering Mechanism by an Ab Initio Approach:extrinsic spin Hall effect in noble metals”, Solid State Phenomena Vols. 168-169 (2011).
[10] Lindemann FA (1910). “The calculation of molecular vibration frequencies". Physik. Z. 11: 609–612.
[11] K. S. Novoselov et al., “ Electric Field Effect in Atomically Thin Carbon Films”, Science 306, 666 (2004).
[12] K. S. Novoselov et al., “Two-dimensional gas of massless Dirac fermions in graphene”, Nature 438, 197-200.
[13] P. R. Wallace,” The Band Theory of Graphite”, Phys. Rev. 71, 622.
[14] Yuanbo Zhang et al.,” Experimental observation of the quantum Hall effect and Berry’s phase in graphene”, Nature 438, 201-204 (2005).
[15] Changgu Lee et al., “Measurement of the Elastic Properties and Intrinsic Strength of Monolayer Graphene”, Science 321, 385 (2008).
[16] A. H. Castro Neto et al., “The electronic properties of graphene”, Rev. Mod. Phys. 81, 109.
[17] Cristina Bena1 and Gilles Montambaux, “Remarks on the tight-binding model of graphene”, New J. Phys. 11 095003
[18] F. D. M. Haldane, “Model for a Quantum Hall Effect without Landau Levels Condensed-Matter Realization of the Parity Anomaly”, Phys. Rev. Lett. 61, 2015.
[19] C. L. Kane and E. J. Mele, “Quantum Spin Hall Effect in Graphene”, Phys. Rev. Lett. 95, 226801.
[20] C. L. Kane and E. J. Mele, “Z_2 Topological Order and the Quantum Spin Hall Effect”, Phys. Rev. Lett. 95, 146802.
[21] Shuichi Murakami, “Quantum Spin Hall Phases”, Progress of Theoretical Physics Supplement No. 176, 2008.
[22] Zhigang Wang et al., “Topological winding properties of spin edge states in the Kane-Mele graphene model”, Phys. Rev. B 80, 115420
[23] L. B. Shao et al., “Realizing and Detecting the Quantum Hall Effect without Landau Levels by Using Ultracold Atoms”, Phys. Rev. Lett. 101, 246810.
[24] Xiong-Jun Liu et al., “Detecting Topological Phases in Cold Atoms”, Phys. Rev. Lett. 111, 120402.
[25] Julen Ibanez-Azpiroz et al., “Tight-binding models for ultracold atoms in honeycomb optical lattices”, Phys. Rev. A 87, 011602(R).
[26] T. Thonhauser and David Vanderbilt, “Insulator/Chern-insulator transition in the Haldane model”, Phys. Rev. B 74, 235111 .
[27] Cristina Bena and Laurent Simon, “Dirac point metamorphosis from third-neighbor couplings in graphene and related materials”, Phys. Rev. B 83, 115404.
[28] Doru Sticlet et al., “Geometrical engineering of a two-band Chern insulator in two dimensions with arbitrary topological index”, Phys. Rev. B 85, 165456.
[29] Doru Sticlet et al., “Distant-neighbor hopping in graphene and Haldane models”, Phys. Rev. B 87, 115402.
[30] Mahdi Zarea and Nancy Sandler, “Rashba spin-orbit interaction in graphene and zigzag nanoribbons”, Phys. Rev. B 79, 165442.
[31] Zhenhua Qiao et al., “Quantum anomalous Hall effect in graphene from Rashba and exchange effects”, Phys. Rev. B 82, 161414(R).
[32] Tsung-Wei Chen et al., “High Chern number quantum anomalous Hall phases in single-layer grapheme with Haldane orbital coupling”, Phys. Rev. B 84, 165453.
[33] Charles Kittel, “Introduction to Solid State Physics”, New York, 1996.