 |
|
|
|
以作者查詢圖書館館藏 、以作者查詢臺灣博碩士 、以作者查詢全國書目 、勘誤回報 、線上人數：9 、訪客IP：3.142.197.212
姓名 張立宏(Li-Hong Chang) 查詢紙本館藏 畢業系所 光電科學與工程學系 論文名稱 四方八角晶格非厄米特拓樸電路系統研究
(The Research of Non-Hermitian Topological Circuit System of Square Octagonal Lattices)相關論文 檔案 [Endnote RIS 格式]
[相關文章]
[文章引用]
[完整記錄]
[館藏目錄]
[檢視]
- 本電子論文使用權限為同意立即開放。
- 已達開放權限電子全文僅授權使用者為學術研究之目的，進行個人非營利性質之檢索、閱讀、列印。
- 請遵守中華民國著作權法之相關規定，切勿任意重製、散佈、改作、轉貼、播送，以免觸法。
摘要(中) 本文主要研究四方八角晶格 (square-octagon lattice) 之緊束縛模型 (tight binding
model) 的拓樸性質。首先研究量子版本的晶格模型的能帶結構，探討有限晶格結構的角
態與邊緣態，以及它們與根據布洛赫波函數 (Bloch wave function) 計算札克相 (Zak
phases) 所得的拓樸不變量之間的對應關係。對於此晶格系統的電路模型，在每兩個節
點連線上除了布置有電感之外，還引入並聯電阻使系統成為非厄米特 (Non-Hermitian)
系統。對非厄米特系統的計算是藉著將節點電壓與其時間導數都納入波函數中，而將系
統演化方程式寫成非厄米薛丁格方程式的形式。對於此電路模型的角態與邊緣態的模擬，
採用雙層區域結構：外層為拓樸平凡區域，內核為拓樸非平凡區域。我們對此系統分析
其角態與邊緣態隨邊界條件改變的變化，並探討它們與拓樸不變量的對應關係。摘要(英) This thesis focuses on the studies of the topological properties of the tight binding models
on square-octagon lattice. First, the band structure of the quantum version of the lattice model
is studied, investigating the corner states and edge states of finite lattice structures, as well as
their correspondence with the topological invariants related to the Zak phases of the Bloch wave
functions. For the circuit model of this lattice system, in addition to placing inductors on each
edge of two connecting nodes, parallel resistors are also introduced to make the system nonHermitian. The calculations for the non-Hermitian system involve incorporating both the node
voltages and their time derivatives into the wave function, and expressing the system′s evolution
equation in the form of a non-Hermitian Schrödinger equation. For the simulation of corner
states and edge states in the circuit model, a bi-region structure is adopted, with the outer layer
being topologically trivial and the inner core being topologically non-trivial. We analyze the
variations of corner states and edge states in this system as the boundary conditions change, and
explore their correspondence with the topological invariants.關鍵字(中) ★ 四方八角晶格
★ 拓撲
★ 非厄米特
★ 邊緣態
★ 角態關鍵字(英) ★ Square Octagonal Lattices
★ Topological
★ Non-hermitian
★ Edge state
★ Corner state論文目次 摘要....................................................................................................................................................... iv
Abstract .................................................................................................................................................. v
致謝....................................................................................................................................................... vi
目錄...................................................................................................................................................... vii
圖目錄................................................................................................................................................. viii
第一章 緒論............................................................................................................................................1
1-1 拓樸簡介 ...........................................................................................................................................1
1-2 貝瑞相 ...............................................................................................................................................2
1-3 章節安排 ...........................................................................................................................................5
第二章 研究理論....................................................................................................................................6
2-1 二維 Su-Schrieffer-Heeger 模型......................................................................................................6
2-2 2D SSH 模型拓樸不變量之計算 ...................................................................................................10
2-3 以 LC 電路實現 SSH 模型............................................................................................................12
第三章 非厄米特系統..........................................................................................................................19
3-1 非厄米特系統..................................................................................................................................19
3-2 非厄米特 2D SSH 模型 ..................................................................................................................20
3-3 非厄米特 2D SSH 模型模擬結果 ..................................................................................................23
第四章 研究模擬與討論......................................................................................................................26
4-1 四方八角晶格..................................................................................................................................26
4-2 四方八角晶格之邊緣態與角態......................................................................................................28
4-3 LC 電路實現四方八角晶格............................................................................................................43
4-4 非厄米特四方八角模型..................................................................................................................47
第五章 結論與未來展望......................................................................................................................50
5-1 結論 .................................................................................................................................................50
5-2 未來展望 .........................................................................................................................................50
參考文獻................................................................................................................................................51參考文獻 [1] 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).
[2] DAI Xi, Topological phases and transitions in condensed matter systems,
PHYSICS, 45, 757 (2016).
[3] Klitzing, K.v., G. Dorda, and M. Pepper, New Method for High-Accuracy
Determination of the Fine-Structure Constant Based on Quantized Hall Resistance,
Phys. Rev. Lett. 45, 494 (1980).
[4] Thouless, D.J., et al., Quantized Hall Conductance in a Two-Dimensional
Periodic Potential, Phys. Rev. Lett. 49, 405 (1982).
[5] Haldane, F.D.M., Model for a quantum Hall effect without Landau levels:
Condensed-matter realization of the" parity anomaly", Phys. Rev. Lett. 61, 2015
(1988).
[6] F. D. M. Haldane and S. Raghu, Possible Realization of Directional Optical
Waveguides in Photonic Crystals with Broken Time-Reversal Symmetry, Phys. Rev.
Lett. 100, 013904 (2008).
[7] Yang, Y., Jiang, H., & Hang, Z. H, Topological valley transport in twodimensional honeycomb photonic crystals, Scientific reports, 8, 1588 (2018).
[8] C. L. Kane and E. J. Mele, Z2 Topological Order and the Quantum Spin Hall
Effect, Phys. Rev. Lett. 95, 146802 (2005)
[9] C. L. Kane, and E.J. Mele, Quantum Spin Hall Effect in Graphene, Phys. Rev.
Lett. 95, 226801 (2005).
[10] Imhof, Stefan, et al., Topolectrical-circuit realization of topological corner
modes, Nat. Phys. 14, 925 (2018).
[11] I. Rotter, A non-Hermitian Hamilton operator and the physics of open quantum
systems, J. Phys. A 42, 153001 (2009).
[12] R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S. Rotter, and
D. N. Christodoulides, Non-Hermitian physics and PT symmetry, Nat. Phys. 14, 11
(2018).
[13] Y. Xiong, Why does bulk boundary correspondence fail in some non-Hermitian
topological models, J. Phys. Commun. 2, 035043 (2018)
[14] Bittner, S. et al., PT symmetry and spontaneous symmetry breaking in a
microwave billiard. Phys. Rev. Lett. 108, 024101 (2012).
[15] Schindler, J., Li, A., Zheng, M. C., Ellis, F. M. & Kottos, T., Experimental study
of active LRC circuits with PT symmetries. Phys. Rev. A 84, 040101 (2011).
[16] Hang, C., Huang, G. & Konotop, V. V., PT symmetry with a system of threelevel atoms. Phys. Rev. Lett. 110, 083604 (2013).
[17] Fleury, R., Sounas, D. & Alù, A., An invisible acoustic sensor based on paritytime symmetry. Nat. Comm. 6, 5905 (2015).
[18] Berry, Michael, Anticipations of the geometric phase, Physics Today 43, 34
(1990).
[19] Su, W-P, J. R. Schrieffer, and A. J. Heeger, Soliton excitations in polyacetylene,
Phys. Rev. B 22, 2009 (1980).
[20] Xun-Wei Xu, Yu-Zeng Li, Zheng-Fang Liu, and Ai-Xi Che, General bounded
corner states in the two-dimensional Su-Schrieffer-Heeger model with intracellular
next-nearest-neighbor hopping, Phys. Rev. A 101, 063839 (2020).
[21] Obana, D., Liu, F., and Wakabayashi, K. Topological edge states in the SuSchrieffer-Heeger model. Phys. Rev. B, 100, 075437 (2019).
[22] Imhof, S., Berger, C., Bayer, F., Brehm, J., Molenkamp, L. W., Kiessling, T., and
Thomale, R., Topolectrical-circuit realization of topological corner modes. Nat. Phys.
14, 925 (2018).
[23] Wang, H. X., Liang, C., Poo, Y., Luan, P. G., & Guo, G. Y, The topological edge
modes and Tamm modes in Su–Schrieffer–Heeger LC-resonator circuits, J. Phys. D:
Applied Physics, 54, 435301 (2021).
[24] Zhang, Z. Q., Wu, B. L., Song, J., and Jiang, H., Topological Anderson insulator
in electric circuits, Phys. Rev. B 100, 184202 (2019).
[25] Zeng, J., Chen, C., Ren, Y., Liu, Z., Ren, W., & Qiao, Z., Topological corner
states in graphene by bulk and edge engineering, Phys. Rev. B, 106, L201407 (2022).
[26] Ezawa, Motohiko, Non-Hermitian higher-order topological states in
nonreciprocal and reciprocal systems with their electric-circuit realization, Phys.
Rev. B 99, 201411 (2019).
[27] Wu, Jien, et al., Non-Hermitian second-order topology induced by resistances in
electric circuits, Phys. Rev. B 105,.195127 (2022).
[28] Kohmoto, Mahito, and Yasumasa Hasegawa, Zero modes and edge states of the
honeycomb lattice, Phys. Rev. B 76, 205402 (2007).
[29] Olekhno, N. A., Rozenblit, A. D., Kachin, V. I., Dmitriev, A. A., Burmistrov, O.
I., Seregin, P. S.,and Gorlach, M. A., Experimental realization of topological corner
states in long-range-coupled electrical circuits, Phys. Rev. B, 105, L081107 (2022).
[30] Zhao, E., Topological circuits of inductors and capacitors, Annals of Physics
399, 289 (2018).指導教授 欒丕綱(Pi-Gang Luan) 審核日期 2023-8-21 推文 plurk
funp
live
udn
HD
myshare
netvibes
friend
youpush
delicious
baidu
網路書籤 Google bookmarks
del.icio.us
hemidemi
facebook
twitter
google
reddit