由於拓樸材料具有拓樸保護性質，因為有保持不受環境干擾破壞的量子態，在量子電腦的應用上有發展的潛力。Su-Schrieffer-Heeger(SSH)模型是一個簡單的一維拓樸絕緣體模型，描述一維交錯排列分子上電子躍遷的模型。另外，在光學系統中，增益和損耗的非厄米特哈米頓是討論宇稱時間對稱性很好的例子。本論文結合一維拓樸絕緣體及增益損耗系統，從一維簡單的拓樸絕緣體模型，延伸至增益和損耗的非厄米特哈米頓系統。在此，一維非厄米特SSH模型，當能量為實數時，沒有破壞宇稱時間對稱性之下，我們可以得到貝里相位為0或是π。但是在能量為虛數時，破壞宇稱時間對稱性，在計算貝里相位積分內的函數不是連續的，要使用其他方法討論貝里相位。論文研究的結果，期望推廣到非厄米特光子系統的應用，同時拓展對拓樸絕緣體模型、非厄米特量子系統與光子系統領域的認識。

The Su-Schrieffer-Heeger (SSH) model perhaps is the simplest and the standard model for topological insulators. Parity–time symmetry is extending quantum theories to non-Hermitian Hamiltonians. This thesis is dedicated to discussing the topological property of the non-Hermitian SSH model. Using optical gain and loss is a way to prepare non-Hermitian eigenstates and hence in this non-Hermitian SSH model with additional gain and loss potential. There is a method of studying topological invariant in the non-Hermitian SSH model by use of a biorthogonal basis. We find that the Berry phase as a topological invariant in this non-Hermitian SSH model with parity-time symmetry. We suggest finding another method to get the Berry phase in this model with broken parity-time symmetry. We justify our analysis numerically and discuss relevant applications.

致謝 II
摘要 III
ABSTRACT IV
目錄 V
圖目錄 VI
1. 簡介 1
1.1. 引言 1
1.2. 拓樸不變量(topological invariant) 3
1.3. 拓樸絕緣體(topological insulator) 5
1.4. 動機 7
2. 模型 9
2.1. Su-Schrieffer-Heeger(SSH)模型 9
2.2. SSH模型的哈米頓算符(Hamiltonian) 11
2.3. SSH模型塊體(bulk)的哈米頓算符 13
3. 研究方法 17
3.1. 貝里相位(Berry phase) 17
3.2. SSH模型的色散關係和貝里相位 21
4. 研究結果 26
4.1. Non-Hermitian SSH模型與拓樸不變量 26
4.2. Non-Hermitian SSH模型與dilated Hamiltonian 32
5. 結論 35
6. 參考資料 37

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