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

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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)

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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

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指導教授

欒丕綱

審核日期

2024-8-16

推文