本論文主要探討 Su-Schrieffer-Heeger (SSH) 模型在厄米特 (Hermitian) 與非厄米特 (Non-Hermitian) 情況下的行為，以及比較其不同的拓樸形態。以電路系統為主要模擬的對象，在非厄米特的拓樸能帶圖可以發現在厄米特系統中所沒有的新拓樸相。論文中分析非厄米特趨膚效應 (Non-Hermitian skin effect) 對拓樸邊緣模態 (topological edge modes) 與 Tamm 模態 (Tamm mode) 的影響。 第一章主要介紹拓樸材料的近代發展，先從介紹整數量子霍爾效開始，再 延伸至不同系統的拓樸相 (topological phase)，此外也介紹布洛赫能定理 (Bloch theorem)、貝瑞相(Berry phase)以及厄米特與非厄米特矩陣等與拓樸相有關的知識。第二章介紹一維和二維厄米特和非厄米特量子系統的矩陣表達式的相關知識並應用在電路系統。藉由使用運算放大器 (Operational Amplifier) 建構出非厄米特電路系統，導出其漢米頓量 (Hamiltonian) 並計算出能帶圖。第三章介紹柯希荷夫電流定律 (KCL) 以及運用節點分析法進行電路分析和計算札克相 (Zak phase)來建立區分拓樸性質的拓樸不變量 (topological invariant)。第四章為此論文主體，在其中我們主要比較 Tamm 模態 (Tamm mode) 、體模態 (bulk eigenmodes)， 以及拓樸零模 (topological zero mode) 之邊界局域化程度，以及探討三種模態在調整運算放大器電壓增幅下的變化趨勢。第五章對厄米特與非厄米特系統進行了總結比較，並探討了對未來的展望。 ;This thesis mainly studiesthe behaviors of the one-dimensional and two-dimensional Su-Schrieffer-Heeger (SSH) models in the Hermitian and Non-Hermitian cases, and compare their different topological properties. Circuit periodic systems consisting of capacitors, inductors, and amplifiers (for the non-hermitian cases) are chosen as the model systems. As the main simulation results, new topological phases are revealed in the non-Hermitian topological band diagrams, which are not found in the corresponding Hermitian systems. We study the influence of the non-Hermitian skin effect on the topological edge modes and Tamm mode numerically. The first chapter mainly introduces the development of topological physics in modern times. We start from the introduction of integer quantum Hall effect, and then extend the concepts extracted from topological insulators to other systems with topological phases. We also introduce the corresponding knowledge about Bloch theorem, Berry phase, and Hermitian and non-Hermitian matrices. In chapter two the knowledge of matrix expressions of typical one-dimensional and two-dimensional Hermitian and non-Hermitian systems is introduced and applied to the circuit systems. Operational Amplifiers are used to help construct the non-hermitian circuit systems. The Hamiltonian is derived, and the energy band diagrams are calculated. Chapter 3 introduces Kochhoff’s Laws (KCL) and the use of nodal analysis method for circuit analysis and the calculation of Zak phase to characterize the topological properties of the topological circuit system. In chapter 4 we simulate the topological circuits numerically. We compare the band diagrams of one-dimensional and two-dimensional non-Hermitian circuit systems, and find a new edge state: Tamm mode. The localization degrees of these modes were compared by tuning the amplification factor of the Operational Amplifiers. In chapter 5 we give the conclusive comparison of Hermitian and non-Hermitian systems, and discuss possible future works.
