本文主要研究四方八角晶格 (square-octagon lattice) 之緊束縛模型 (tight binding model) 的拓樸性質。首先研究量子版本的晶格模型的能帶結構，探討有限晶格結構的角 態與邊緣態，以及它們與根據布洛赫波函數 (Bloch wave function) 計算札克相 (Zak phases) 所得的拓樸不變量之間的對應關係。對於此晶格系統的電路模型，在每兩個節 點連線上除了布置有電感之外，還引入並聯電阻使系統成為非厄米特 (Non-Hermitian) 系統。對非厄米特系統的計算是藉著將節點電壓與其時間導數都納入波函數中，而將系 統演化方程式寫成非厄米薛丁格方程式的形式。對於此電路模型的角態與邊緣態的模擬， 採用雙層區域結構：外層為拓樸平凡區域，內核為拓樸非平凡區域。我們對此系統分析 其角態與邊緣態隨邊界條件改變的變化，並探討它們與拓樸不變量的對應關係。 ;This thesis focuses on the studies of the topological properties of the tight binding models on square-octagon lattice. First, the band structure of the quantum version of the lattice model is studied, investigating the corner states and edge states of finite lattice structures, as well as their correspondence with the topological invariants related to the Zak phases of the Bloch wave functions. For the circuit model of this lattice system, in addition to placing inductors on each edge of two connecting nodes, parallel resistors are also introduced to make the system non Hermitian. The calculations for the non-Hermitian system involve incorporating both the node voltages and their time derivatives into the wave function, and expressing the system′s evolution equation in the form of a non-Hermitian Schrödinger equation. For the simulation of corner states and edge states in the circuit model, a bi-region structure is adopted, with the outer layer being topologically trivial and the inner core being topologically non-trivial. We analyze the variations of corner states and edge states in this system as the boundary conditions change, and explore their correspondence with the topological invariants.
