彈簧力學系統中的谷拓樸邊緣態探討

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姓名

田皓升(Hao-Sheng Tian) 查詢紙本館藏

畢業系所

光電科學與工程學系

論文名稱

彈簧力學系統中的谷拓樸邊緣態探討
(Exploring Valley Topological Edge States in Spring-Mechanical Systems)

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摘要(中)

本文首先討論彈簧力學蜂窩晶格 (honeycomb lattice) 以及正方晶格於不同邊界條件下的邊界態，並解釋其存在原因。而後加入科氏力變量，破壞時間反演對稱性 (Time Reversal symmetry) 產生手徵邊緣態 (chiral edge state)，探討相關的能帶以及拓樸相變。最後改變球與彈簧的質量，破壞 PT 對稱，藉此打開帶隙產生谷拓樸邊緣態 (valley edge state)，討論能帶與拓樸邊界態的變化以並藉由計算貝里曲率 (berry curvature) 觀察整個布里渊區中的局部陳數是否符合預期。最後發現引入旋轉以後對邊緣態有較強的干預作用，除此之外，外部邊界的選擇也會導致內部所產生的谷拓樸邊緣態產生不同程度的影響。

摘要(英)

This thesis first explores the boundary states of spring-mechanical honeycomb and square lattices under various boundary conditions, explaining the reasons for their ex- istence. Next, the inclusion of the Coriolis force term breaks time-reversal symmetry (TRS), leading to the emergence of chiral edge states. The study examines the associated band structures and topological phase transitions.
Finally, by altering the masses of the balls and springs to break PT symmetry, a band gap is introduced, resulting in the formation of valley topological edge states. The paper discusses the changes in band structures and topological edge states and calculates the Berry curvature to observe whether the local Chern number across the entire Brillouin zone aligns with expectations.
This thesis concludes that the introduction of rotation significantly interferes with the edge states. Furthermore, the selection of external boundaries also affects the topological valley edge states generated within the system, leading to varying degrees of impact.

關鍵字(中)

★ 力學系統
★ 谷拓樸
★ 邊緣態

關鍵字(英)

論文目次

摘要 ix Abstract xi
目錄圖目錄
xiii xv
一、緒論 1
1.1 量子霍爾效應 ................................................................. 1
1.2 拓樸絕緣體.................................................................... 2
1.3 拓樸能帶理論 ................................................................. 2
1.4 時間反演對稱性............................................................... 3
1.5 空間對稱性.................................................................... 3
1.6 谷拓樸邊緣態 ................................................................. 3
二、理論與計算 5
2.1 貝里相位和貝里曲率.......................................................... 5
2.2 陳數計算 ...................................................................... 8
2.3 一維 Su-Schrieffer-Heeger(SSH) 模型......................................... 9
三、彈簧力學模型計算 13 3.1 正方晶格運動方程式.......................................................... 13 3.1.1 二維能帶結構.......................................................... 16 3.1.2 沿同一方向傳播之邊界態 ............................................. 18 3.2 破壞 PT 對稱性之正方晶格運動方程式...................................... 21 3.2.1 正方晶格運動方程式 .................................................. 21
3.3 蜂窩晶格運動方程式.......................................................... 23 3.3.1 二維能帶結構.......................................................... 25 3.3.2 沿同一方向傳播之邊界態 ............................................. 26
3.4 貝瑞曲率與陳數計算.......................................................... 27
3.5 威爾森迴圈.................................................................... 29
四、數值計算結果與討論 31 4.1 蜂窩晶格之能帶結構以及邊緣態 ............................................. 31
4.1.1 無旋轉及谷拓樸現象之蜂窩晶格邊緣態分析 ......................... 35
4.1.2 谷拓樸蜂窩晶格邊緣態分析........................................... 38
4.1.3 引入旋轉時的谷拓樸蜂窩晶格邊緣態分析............................ 42
五、結論與未來展望 49
5.1 結論 ........................................................................... 49
5.2 未來展望 ...................................................................... 49
參考文獻
51

參考文獻

[1] Stormer, Horst L, Nobel lecture: the fractional quantum Hall effect, Rev. Mod. Phys. 71, 875 (1999).
[2] 蔡雅雯、吳杰倫、欒丕綱, 從量子霍爾效應到拓樸光子學與拓樸聲子學, 科儀新知, 211 期, 68 (2017).
[3] Bernevig, B. Andrei, Topological insulators and topological superconductors, Princeton uni- versity press (2013).
[4] H. Xue, Y. Yang, and B. Zhang, Topological valley photonics: physics and device applica- tions, Advanced Photonics Research 2, 2100013 (2021).
[5] J. Noh, S. Huang, K. P. Chen, M. C. Rechtsman, Observation of photonic topological valley Hall edge states, Phys. Rev. Lett. 120. 063902 (2018)
[6] B.A. Bernevig, S.-C. Zhang, Quantum spin Hall effect, Phys. Rev. Lett. 96. 106802 (2006).
[7] D.J Thouless, et al, Quantized Hall conductance in a two-dimensional periodic potential, Phys. Rev. Lett. 49, 405 (1982).
[8] Klitzing, K. V., Gerhard Dorda, and Michael Pepper, New method for high-accuracy de- termination of the fine-structure constant based on quantized Hall resistance, Phys. Rev. Lett 45, 494 (1980).
[9] Tsui, D. C., Stormer, H. L. and Gossard, A. C., Two-dimensional magnetotransport in the extreme quantum limit, Phys. Rev. Lett. 48, 1559 (1982).
[10] Kane, C. L. and Mele, E. J., Quantum spin Hall effect in graphene, Phys. Rev. Lett. 95, 226801 (2005).
[11] Hasan, M. Z. and Kane, C. L., Colloquium: topological insulators, Rev. Mod. Phys. 82, 3045-3067 (2010).
[12] Young, A. F., et al., Tunable symmetry breaking and helical edge transport in a graphene quantum spin Hall state, Nature 505, 528-532 (2014).
[13] Kane, C. L. and Mele, E. J., Z2 topological order and the quantum spin Hall effect, Phys. Rev. Lett. 95, 146802 (2005).
[14] J.-W. Dong, et al., Valley photonic crystals for control of spin and topology, Nature Mate- rials 16, 298-302 (2017).
[15] Davis, R., et al., Photonic topological insulators: A beginner’s introduction [electromag- netic perspectives, IEEE Antennas and Propagation Magazine 63, 112-124 (2021).
[16] Samuel, J, and Bhandari, R., General setting for Berry’s phase, Phys. Rev. Lett. 60, 2339 (1988).
[17] Xiao, D., Chang, M.-C. and Niu, Q., Berry phase effects on electronic properties, Rev. Mod. Phys. 82, 1959-2007 (2010).
[18] Simon, B., Holonomy, the quantum adiabatic theorem, and Berry’s phase. Phys, Rev. Lett. 51, 2167 (1983).
[19] Vanderbilt, D., Berry phases in electronic structure theory: electric polarization, orbital magnetization and topological insulators. Cambridge University Press, (2018).
[20] Hatsugai, Y., Chern number and edge states in the integer quantum Hall effect, Phys. Rev. Lett. 71, 3697 (1993).
[21] Fukui, T., Yasuhiro Hatsugai, and Hiroshi Suzuki., Chern numbers in discretized Brillouin zone: eﬀicient method of computing (spin) Hall conductances, Journal of the Physical Society of Japan 74, 1674-1677 (2005).
[22] Goudarzi, K., H Maragheh, H.G. and Moonjoo L., Calculation of the Berry curvature and Chern number of topological photonic crystals, Journal of the Korean Physical Society 81, 386-390 (2022).
[23] Asbóth, J.K., Oroszlány, L. and Pályi, A., A short course on topological insulators, Lecture notes in physics 919.1 (2016).
[24] Kohmoto, M, and Hasegawa, Y., Zero modes and edge states of the honeycomb lattice, Phys. Rev. B, Condensed Matter and Materials Physics 76, 205402 (2007).
[25] Borges-Silva, D., Costa, C.H.O. and Bezerra, C.G., Robust Topological Edge States in C6 Photonic Crystals, Photonics. Vol. 10 961 (2023).
[26] Carpentier, D., Topology of bands in solids: From insulators to dirac matter. Dirac Matter. Cham: Springer International Publishing, 95-129 (2017).
[27] Niu,Q.,Berryphaseeffectsonelectronicproperties,APSMarchMeetingAbstracts.(2007).
[28] Landau, L.D., Lifshits, E.M. and Lifshits, E.M., Mechanics, Vol. 1. CUP Archive, (1960).
[29] Marion, J.B., Classical dynamics of particles and systems, Academic Press, (2013).
[30] Liu, Y., et al., Model for topological phononics and phonon diode, Phys. Rev. B, 96, 064106 (2017).
[31] Benalcazar, W.A., Bernevig, B.A. and Hughes, T.L. Electric multipole moments, topologi- cal multipole moment pumping, and chiral hinge states in crystalline insulators, Phys. Rev. B, 96, 245115 (2017).
[32] Schindler, F., Higher-order topological insulators. APS March Meeting Abstracts, Vol.68 (2023).

指導教授

欒丕綱

審核日期

2024-8-15

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