以複數布洛赫波向量法探索一維拓撲系統邊界條件與邊界態對應關係

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：36

、訪客IP：3.137.223.153

姓名

李昱賢(Yu-Xian Li) 查詢紙本館藏

畢業系所

光電科學與工程學系

論文名稱

以複數布洛赫波向量法探索一維拓撲系統邊界條件與邊界態對應關係
(Exploring the correspondence between boundary conditions and Edge States in One-dimensional Topological Systems using the complex Bloch vector)

相關論文

★ 平坦化陣列波導光柵分析和一維光子晶體研究	★ 光子晶體波導與藕合共振波導之研究
★ 光子晶體異常折射之研究	★ 光子晶體傳導帶與介電質柱波導之研究
★ 平面波展開法在光子晶體之應用	★ 偏平面光子晶體能帶之研究
★ 通道選擇濾波器之探討	★ 廣義光子晶體元件之研究與分析
★ 新式光子晶體波導濾波器之研究	★ 廣義非均向性介質的光傳播研究
★ 光子晶體耦合濾波器之研究	★ 聲子晶體傳導帶與週期性彈性柱波導之研究
★ 對稱與非對稱波導光柵之特性研究	★ 雙曲透鏡之研究
★ 電磁波與聲波隱形斗篷之研究	★ 一維光子晶體等效非均向介值之研究

檔案

[Endnote RIS 格式]

[Bibtex 格式]

[相關文章]

[文章引用]

[完整記錄]

[館藏目錄]

[檢視]

[下載]

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)

本論文探討 Su-Schrieffer-Heeger (SSH) 模型以及其相關的推廣模型的邊緣態。當引入實數布洛赫波向量時，根據週期性邊界條件所導出的哈密頓算符僅能得到系統之拓樸與體態能譜，得不到邊緣態能譜與邊緣態解。一般而言，後者必須藉由開放性邊界條件在開鍊情況下才能解出。引入了複數布洛赫波向量後，就可以藉由解析週期性哈密頓算符而解出對應於開放性邊界條件中的邊緣態。本篇探討的系統，除了原始的 SSH 模型外，還有具有 PT 對稱之 SSH 模型以及利用左右躍遷不對稱之非厄米特 SSH 模型。引入複數布洛赫波向量能使週期性邊界條件與開放性邊界條件的物理有對應關係。

摘要(英)

This thesis investigates the edge states of the Su-Schrieffer-Heeger (SSH) model and its related extended models. When real Bloch wave vectors are introduced, the
Hamiltonian derived from periodic boundary conditions can only obtain the system′s topology and bulk energy spectrum, but not the edge state energy spectrum and edge state solutions. Generally, the latter must be solved using open boundary conditions in an open chain scenario. However, by introducing complex Bloch wave vectors, the edge states corresponding to those in open boundary conditions can be obtained through analytic continuation of the periodic Hamiltonian. This study examines not only the original SSH model but also the PT-symmetric SSH model and the non-Hermitian SSH model with nonreciprocal left and right hopping coefficients. The introduction of complex Bloch wave vectors enables a correspondence between the physics of periodic boundary conditions and open boundary conditions.

關鍵字(中)

★ 拓撲
★ 布洛赫波

關鍵字(英)

論文目次

目錄

第一章緒論 1
1-1 拓樸材料 (Topological material) 1
1-2 布洛赫定理 (Bloch Theorem) 3
1-3 非厄米特系統 (non-Hermitian system) 與PT對稱性 (Parity-time symmetry, PT-symmetry) 4
1-4 章節編排 7
第二章拓樸模型理論 8
2-1 Su-Schrieffer-Heeger模型 8
2-2 改變邊界條件之SSH模型 14
2-3 左右躍遷不對稱之非厄米特SSH模型 24
第三章研究方法 30
3-1 衰減因子與複數布洛赫波向量 30
3-2 一維厄米特SSH模型以遞迴關係表現其開鍊能譜之數學推導 31
3-3 在邊界上加入位勢之SSH模型在以遞迴關係推得其邊界條件及能譜關係式之推導 34
3-4 左右躍遷不對稱之非厄米特SSH模型以遞迴關係推得其能譜關係式之推導 37
第四章模擬結果與討論 40
4-1 一維厄米特SSH模型能譜模擬 40
4-2 在邊界上加入反對稱位勢之SSH模型之邊緣態及邊界條件模擬 45
4-3 左右躍遷不對稱之非厄米特SSH模型模擬結果 54
第五章結論與未來展望 60
5-1 結論 60
5-2 未來展望 60
參考文獻 61

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] Imhof, Stefan, et al., Topolectrical-circuit realization of topological corner modes,
Nat. Phys. 14, 925 (2018)
[3] 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)
[4] Fleury, R., Sounas, D. & Alù, A., An invisible acoustic sensor based on paritytime symmetry, Nat. Comm. 6, 5905 (2015)
[5] Bittner, S. et al., PT symmetry and spontaneous symmetry breaking in a
microwave billiard, Phys. Rev. Lett. 108, 024101 (2012)
[6] Hang, C., Huang, G. & Konotop, V. V., PT symmetry with a system of threelevel atoms, Phys. Rev. Lett. 110, 083604 (2013)
[7] DAI Xi, Topological phases and transitions in condensed matter systems,
Physics, 45, 757 (2016)
[8] K. v. Klitzing, G. Dorda, and M. Pepper, New Method for High-Accuracy
62
Determination of the Fine-Structure Constant Based on Quantized Hall Resistance,
Phys. Rev. Lett. 45, 494 (1980)
[9] Thouless, D.J., et al., Quantized Hall Conductance in a Two-Dimensional Periodic
Potential, Phys. Rev. Lett. 49, 405 (1982)
[10] Chern, Shiing-Shen, Characteristic classes of Hermitian manifolds, The Annals of
Mathematics. 2nd Ser., Vol. 47, No. 1 (Jan., 1946), 85 - 121
[11] Chern, Shiing-Shen, Complex Manifolds Without Potential Theory: With an
Appendix on the Geometry of Characteristic Classes (Universitext), Springer, 2nd Ed.
(1979)
[12] 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)
[13] 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)
[14] Yang, Y., Jiang, H., & Hang, Z. H, Topological valley transport in two
dimensional honeycomb photonic crystals, Scientific reports, 8, 1588 (2018)
63
[15] C. L. Kane, and E.J. Mele, Quantum Spin Hall Effect in Graphene, Phys.
Rev.Lett. 95, 226801 (2005)
[16] C. L. Kane and E. J. Mele, Z2 Topological Order and the Quantum Spin Hall
Effect, Phys. Rev. Lett. 95, 146802 (2005)
[17] Charles Kittel, Introduction to Solid State Physics, Wiley, 8th ed (2004)
[18] Shunyu Yao and Zhong Wang, Edge States and Topological Invariants of NonHermitian Systems, Phys. Rev. Lett. 121, 086803 (2018)
[19] Nobuyuki Okuma, Kohei Kawabata, Ken Shiozaki, and Masatoshi Sato,
Topological Origin of Non-Hermitian Skin Effects, Phys. Rev. Lett. 124, 086801
(2020)
[20] Hu Yu-Min, Song Fei and Wang Zhong, Generalized Brillouin zone and nonHermitian band theory, Acta Phys. Sin., 70, 230307 (2021)
[21] Ramy El-Ganainy, Konstantinos G. Makris, Mercedeh Khajavikhan, Ziad
H.Musslimani, Stefan Rotter & Demetrios N. Christodoulides, Non-Hermitian physics
and PT symmetry, Nat. Phys. 14, 11 (2018)
64
[22] Youngsun Choi , Choloong Hahn, Jae Woong Yoon & Seok Ho Song,
Observation of an anti-PT-symmetric exceptional point and energy-difference
conserving dynamics in electrical circuit resonators, Nat. Commun. 9, 2182 (2018)
[23] Yi-Cheng Wang, Jhih-Shih You and H. H. Jen, A non-Hermitian optical atomic
mirror, Nature Communications 13, 4598 (2022)
[24] 游至仕、王奕誠, 能帶拓樸：從厄米走向非厄米, 物理雙月刊, 6, 37
(2022)
[25] János K. Asbóth, László Oroszlány, András Pályi, A Short Course on
Topological Insulators Band Structure and Edge States in One and Two Dimensions,
Springer International Publishing Switzerland (2016)
[26] Baogang Zhu, Rong L ̈u and Shu Chen, PT -Symmetry in Non-Hermitian SuSchrieffer-Heeger model with complex boundary potentials, Phys. Rev. A 98, 013628
(2018)
[27] Shunyu Yao and Zhong Wang, Edge States and Topological Invariants of NonHermitian Systems, Phys. Rev. Lett. 121, 086803 (2018)
[28] Kazuki Yokomizo and Shuichi Murakami, Non-Bloch Band Theory of NonHermitian Systems, Phys. Rev. Lett. 123, 066404 (2019)
65
[29] Mohammad-Ali Miri, Andrea Alù, Exceptional points in optics and photonics,
Science 363, 42 (2019)
[30] Fleury, R., Sounas, D. & Alù, A. An invisible acoustic sensor based on paritytime symmetry. Nat. Commun. 6, 5905 (2015)
[31] Youngsun Choi , Choloong Hahn, Jae Woong Yoon & Seok Ho Song,
Observation of an anti-PT-symmetric exceptional point and energy-difference
conserving dynamics in electrical circuit resonators, Nat. Commun. 9, 2182 (2018)
[32] 欒丕綱、陳啟昌,光子晶體：從蝴蝶翅膀到奈米光子學 (第二版), 五南出
版社 (2010)

指導教授

欒丕綱

審核日期

2024-8-16

推文