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研究生： 陳柏宏
Chen, Bo-Hung
論文名稱： Two-Dimensional Extended Su-Schrieffer-Heeger Model
Two-Dimensional Extended Su-Schrieffer-Heeger Model
指導教授： 高賢忠
Kao, Hsien-Chung
學位類別： 碩士
Master
系所名稱： 物理學系
Department of Physics
論文出版年： 2018
畢業學年度： 106
語文別： 英文
論文頁數： 63
中文關鍵詞： Su-Schrieffer-Heeger model 、Topoglogy 、Topological insulator 、Topological semimetal 、SSH model 、Winding number
英文關鍵詞： Su-Schrieffer-Heeger model, Topoglogy, Topological insulator, Topological semimetal, SSH model, Winding number
DOI URL： http://doi.org/10.6345/THE.NTNU.DP.008.2018.B04
論文種類： 學術論文
相關次數： 點閱：215  下載：32 
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    • 無中文摘要。

    The edge state is known to be a characteristic of a topological material. In two-dimensional topological systems, one can use the \emph{Chern number} to describe the topological property of the systems. However, the Chern number fails to discern the topology for two special cases of two-band systems: (a) when the parameter space is restricted to a plane, and (b) when the system is a semimetal. One should find another way instead to characterize the nontrivial topology.

    In this thesis, the SSH model is extended from one dimension to two dimensions by four different ways. None of them can be described by the Chern number. However, by applying the dimensional reduction, the systems are reduced to one dimension and are equivalent to the generalized SSH model, whose topological nontriviality is characterized by the \emph{winding number}. Since the open boundary conditions are preserved under the dimensional reduction, the edge effect should be described by the reduced Hamiltonian. Therefore, we find the quasi-bulk-boundary correspondence to connect the edge states of the two-dimensional systems and the winding number of the reduced Hamiltonian. Moreover, if the edges of SSH chains are preserved under the extension in the thesis, the edge states are also preserved.

    1 Introduction to Topology 3 1.1 Adiabatic deformation 4 1.2 Berry Phase 4 1.3 Z Invariant in the Two-Band System 6 2 Su-Schrieffer-Heeger Model 10 2.1 Bulk Hamiltonian 10 2.2 Chiral Symmetry 11 2.3 Winding Number 13 2.4 Edge State 13 2.5 Bulk-Boundary Correspondence 15 3 Generalized SSH Model 17 3.1 Bulk Hamiltonian 17 3.2 Winding Number 18 3.3 Zero-Energy Edge Modes 19 4 Two-Dimensional Extended SSH Model 21 4.1 Type I Extension 21 4.1.1 Single Inter-Chain Hopping 22 4.1.2 Staggered Inter-Chain Hopping 25 4.2 Type II Extension 30 4.2.1 Single Inter-Chain Hopping 30 4.2.2 Staggered Inter-Chain Hopping 40 5 Discussion and Conclusion 48 5.1 Discussion about the weak topological insulator 48 5.2 Conclusion 48 Appendices 50 A Exact Calculation of Edge Modes 50 A.1 SSH Model 50 A.1.1 zero-energy edge state 51 A.2 Generalized SSH Model 53 A.2.1 zero-energy edge modes 55 B Carbon Nanotubes & Graphene nanoribbons 56 B.1 Zigzag and Beard type 57 B.2 Armchair type 61 Bibliography 63

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    [2] J. K. Asbóth, L. Oroszlány, A. P´ alyi, A Short Course on Topological Insulators: Band-structure topology and edge states in one and two dimensions, (Springer, Switzerland, 2016).
    [3] M. Nakahara. Geometry, Topology and Physics, 2nd Edition, (CRC Press, 2003)
    [4] D. J. Griffiths, Introduction to Quantum Mechanics, (Pearson Education Limited, Harlow, 2014)
    [5] A. Kitaev, “Periodic table for topological insulators and superconduc tors”, AIP. Conf. Proc. 1134, 22 (2009).
    [6] P. K. Nayak Recent Advances in Graphene Research (Intechopen, 2016)
    [7] B.-H. Chen and D.-W. Chiou, “An elementary proof of bulk-boundary correspondence in the generalized Su-Schrieffer-Heeger model”, arXiv:1705.06913 [cond-mat.mes-hall].
    [8] D. Varjas, F. de Juan, and Y.-M. Lu, “Space group constraints on weak indices in topological insulators”, Phys. Rev. B 96, (2017)
    [9] E. Burstein, A. H. MacDonald and P. J. Stiles, Topological Insulators Volume 6, (Elsevier, 2013).

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