臺灣博碩士論文加值系統

穩定的邊界態是拓樸絕緣體眾所周知的性質。我們用SSH model 來說明零能量邊界態在任意擾動下的穩定性。此外，已知極化和札克相位成正比。藉由加總各個能帶的札克相位，我們發現即便不存在手徵對稱性，系統的札克相位依然總是2 pi 的整數倍。我們據此建立了札克相位和繞圈數的明確聯繫，並且演示了在沒有手徵對稱性的系統中札克相位和邊界態數量之間的對應關係。

A well known property of topological insulators is the existence of robust edge states. We use the SSH model to illustrate the robustness of the zero energy edge state under arbitrary perturbations. Also, it is known that the polarization is proportional to the Zak phase. By summing over the Zak phase of all the energy bands, the total Zak phase is shown to be always an integer multiple of 2pi even if there is no chiral symmetry. Therefore, we may relate the total Zak phase and the winding number, and we may generalize the bulk-edge correspondence to systems without chiral symmetry.

口試委員會審定書 i
致謝 ii
中文摘要 iii
Abstract iv
Contents v
List of Figures vii
1 Edge state of the SSH model 1
1.1 Topological invariant of SSH model . . . . . . . . . . . . . . . . . . . . 1
1.2 Bulk-edge correspondence . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 SSH model with even number of particles . . . . . . . . . . . . . . . . . 10
1.4 SSH model with odd number of particles . . . . . . . . . . . . . . . . . . 13
1.5 Robustness of the edge state . . . . . . . . . . . . . . . . . . . . . . . . 16
2 Edge states in the extended SSH model 25
2.1 Bulk-edge correspondence . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2 Extended SSH model with even number of particles . . . . . . . . . . . . 36
2.3 Extended SSH model with odd number of particles . . . . . . . . . . . . 39
3 Polarization 45
3.1 Wannier state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.2 Polarization and the Zak phase . . . . . . . . . . . . . . . . . . . . . . . 50
3.3 Zak phase and winding number . . . . . . . . . . . . . . . . . . . . . . . 51
3.4 Polarization of the extended SSH model . . . . . . . . . . . . . . . . . . 53
Conclusion 61
A The algebra of the eigenvalue problem in section 1.5 63
Bibliography 67

Alexei Kitaev. Periodic table for topological insulators and superconductors. 2009.
W. P. Su, J. R. Schrieffer, and A. J. Heeger. Solitons in polyacetylene. Phys. Rev. Lett., 42:1698–1701, Jun 1979.
János K. Asbóth, László Oroszlány, and András Pályi. A Short Course on Topological Insulators: Band-structure topology and edge states in one and two dimensions. arXiv e-prints, page arXiv:1509.02295, Sep 2015.
Andrei Bernevig and Titus Neupert. Topological Superconductors and Category Theory. arXiv e-prints, page arXiv:1506.05805, Jun 2015.
Callum W. Duncan, Patrik Öhberg, and Manuel Valiente. Exact edge, bulk, and bound states of finite topological systems. Phys. Rev. B, 97:195439, May 2018.
Bo-Hung Chen and Dah-Wei Chiou. A rigorous proof of bulk-boundary correspondence in the generalized Su-Schrieffer-Heeger model. arXiv e-prints, page arXiv: 1705.06913, May 2017.
Wladimir A. Benalcazar, B. Andrei Bernevig, and Taylor L. Hughes. Quantized electric multipole insulators. Science, 357(6346):61–66, Jul 2017.
Gregory H. Wannier. The structure of electronic excitation levels in insulating crystals. Phys. Rev., 52:191–197, Aug 1937.
Nicola Marzari, Arash A. Mostofi, Jonathan R. Yates, Ivo Souza, and David Vanderbilt. Maximally localized Wannier functions: Theory and applications. Reviews of Modern Physics, 84(4):1419–1475, Oct 2012.
R. D. King-Smith and David Vanderbilt. Theory of polarization of crystalline solids. Phys. Rev. B, 47:1651–1654, Jan 1993.
A. S. Sergeev. Geometry of projected connections, zak phase, and electric polarization. Phys. Rev. B, 98:161101, Oct 2018.
Marcel Franz and Laurens Molenkamp. Topological insulators. Contemporary concepts of condensed matter science. Elsevier Science, Amsterdam, 2013.
Qian Niu, Ming-Che Chang, Biao Wu, Di Xiao, and Ran Cheng. Physical Effects of Geometric Phases. WORLD SCIENTIFIC, 2017.
J. Zak. Berry’s phase for energy bands in solids. Phys. Rev. Lett., 62:2747–2750, Jun 1989.
Roger S. K. Mong and Vasudha Shivamoggi. Edge states and the bulk-boundary correspondence in dirac hamiltonians. Phys. Rev. B, 83:125109, Mar 2011.