本論文參考經典的 Su-Schrieffer-Heeger (SSH) 模型，探討在一維與二維緊束縛模型 (tight binding model) 系統以及光子晶體 (photonic crystal) 中，藉由改變晶胞內的跳躍振幅 (intracell hopping amplitude ) 與相鄰晶胞間的躍遷振幅 (intercell hopping amplitude) 之比例而導致的拓樸相變 (topological phase transition)。此外，我們也分析此拓樸相變中的拓樸不變量 (topological invariant) 與對應的拓樸邊緣態 (topological edge state)。
第一章主要是介紹石墨烯蜂窩晶格 (honeycomb lattice) 的能帶特色，知道可以利用K跟K’有不同方向的假自旋 (pseudospin) 產生量子能谷霍爾效應 (quantum valley Hall effect)。第二章則把一維和二維的 SSH 模型應用在電路系統上，計算出它們的哈密頓量 (Hamiltonian) 以便求出能帶結構 (band structure)。第三章是說明COMSOL Multiphysics 商用模擬軟體如何計算能帶結構，並介紹用來判斷拓樸性質 (topological properties) 的拓樸不變量 (topological invariants)。第四章通過數值計算出電路系統的能帶結構與捲繞數 (winding number) 和用超晶胞法 (supercell method) 模擬出可果美晶格 (Kagome lattice) 光子晶體 (photonic crystals) 的拓樸邊緣態 (topological edge states)。之後，改變SSH 模型的邊界位能，會出現不同於拓樸邊緣態的邊緣態，這是一種稱為 "Tamm mode" 的缺陷態 (defect states)。
在電路系統的 SSH 模型得知，增加兩邊邊界的當地位能 (on-site potential)，會把非平庸拓樸 (nontrivial topology) 變成平庸拓樸 (trivial topology) 的拓樸性質，等於我們只藉著增加邊界的位能，就能使其發生拓樸相變，最後在光子晶體系統中，也能通過改變邊界位能，造成邊緣態的改變。

In this thesis, we treat the Su-Schrieffer-Heeger (SSH) model as a porotype, and study the topological phase transition occurring in some one-dimensional (1D) and two-dimensional (2D) tight binding models and photonic crystals. The topological phase transition is due to changing the ratio between the intracell and intercell hopping amplitude in the models. We also study the topological invariants in the topological phase transition and analyze the properties of the topologically protected edge states.
In the first chapter we mainly study the characteristics of the band structure of the honeycomb lattice system, and discuss how to use the two different pseudospins at the K and K’ point of the Brillion zone to generate the quantum valley Hall effect. In the second chapter we mimic the 1D and 2D SSH models to define the corresponding circuit systems, formulate their Hamiltonians, and then calculate the band structures. The third chapter explains how to use the commercial simulation software COMSOL Multiphysics to calculate the band structures, and introduce the topological invariants used to analyze the topological properties. In chapter four, we first numerically calculate the band structures and winding numbers of the circuit systems. We then use the supercell method to simulate the topological edge states of the photonic crystal defining by the periodically arranged dielectric rods located on the Kagome lattice. We also study the influence of the boundary potentials in the SSH model. We find that by adding the boundary potentials, new edge state emerges, which is different from the topological edge state. This might be the defect state named " Tamm mode".
In the SSH model of circuit systems, just by adding the on-site potentials on the two boundaries, the topological property of the system turns from nontrivial into trivial. This indicates that we can have a topological phase transition only by increasing the on-site potential at the boundary. Corresponding to this, in the photonic crystal systems, the edge states can also be changed by changing the “on-site potential” at the boundaries.

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