非厄米特二階拓樸電路之研究

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姓名

鍾佺翰(Chuan-Han Chung) 查詢紙本館藏

畢業系所

光電科學與工程學系

論文名稱

非厄米特二階拓樸電路之研究
(The Research of Non-Hermitian Second-Order Topological Electric Circuits)

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摘要(中)

本論文主要探討二維二階非厄米特拓樸絕緣體 (second-order topological insulator) 的物理特性。我們分別考慮方形晶格與可果美晶格兩種晶格型態，然後以電路的方式實現此拓樸絕緣體，並觀察零維的角態 (corner states) 與一維的非無能隙邊緣態 (non-gapless edge states) 如何形成。此電路系統可利用克希荷夫電路定律 (Kirchhoff′s circuit laws) 分別針對完全週期性的晶格 (periodic lattice) 結構與具有開放邊界 (open boundaries) 的有限週期結構兩種情況推導出電路拉普拉斯算符 (Circuit Laplacian) 與哈密頓矩陣 (Hamiltonian matrix)，並解出後者的本徵值與本徵向量。其中本徵向量給出此系統的本徵振盪模態 (modes)，而本徵值就是模態的 (複數) 振動頻率。分別考慮方形晶格與可果美晶格的拓樸不變量 (topological invariant)，就可以在這兩種晶格結構中區分不同的二階拓樸相，以確認角態的存在。

摘要(英)

In the thesis, we mainly discuss the physical properties of two-dimensional second-order non-Hermitian topological insulators. Structures of square lattice and Kagome lattice are considered, and the topological insulators are realized by means of appropriately defined electric circuits with the corresponding lattice structures. The study focuses on the zero-dimensional corner states instead of the one-dimensional non-gapless edge states.
　　We use Kirchhoff′s circuit laws to derive the circuit Laplacian and Hamiltonian of the circuits for both the periodic lattice structure without boundary and the finite periodic structure with open boundaries. The circuit Hamiltonian matrix is used to solve for the eigenmodes/eigenstates of the system and their corresponding (complex valued) eigenfrequencies.
　　Different topological phases can be distinguished by the topological invariants defined according to the band structures of the system, and the existence of the corner states can be confirmed by checking the topological invariants.

關鍵字(中)

★ 非厄米特
★ 二階拓樸
★ 角態

關鍵字(英)

論文目次

摘要 I
Abstract II
謝誌 III
目錄 IV
圖目錄 VI
第一章緒論 1
1-1 拓樸絕緣體 1
1-2 非厄米特系統 3
1-3 布洛赫定理、貝瑞相與陳數 5
第二章研究理論 8
2-1 Su-Schrieffer-Heeger 模型 8
2-2 非厄米特 Su-Schrieffer-Heeger 模型 11
2-3 廣義布洛赫能帶理論 14
2-4 線性電路理論與電路拉普拉斯算符 17
第三章電路模型與研究方法 19
3-1 非厄米特系統電路架構 19
3-2 導納譜和阻抗峰值 20
3-3 二階非厄米特方形拓樸電路 21
3-4 二階厄米特可果美拓樸電路 24
3-5 二階非厄米特可果美拓樸電路 29
3-5-1 並聯結構 29
3-5-2 串聯結構 31
第四章研究模擬與討論 39
4-1 一維 Su-Schrieffer-Heeger 模型 39
4-2 一維非厄米特 Su-Schrieffer-Heeger 模型 40
4-3 非厄米特系統電路模型與局域態密度 44
4-4 二階非厄米特方形拓樸電路與角態 52
4-5 二階厄米特可果美拓樸電路與角態 57
4-6 二階非厄米特可果美拓樸電路與角態 63
4-6-1 並聯結構 63
第五章結論與未來展望 70
5-1 結論 70
5-2 未來展望 70
參考文獻 71

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指導教授

欒丕綱

審核日期

2022-9-27

推文