本篇論文主要探究的是對於非厄米哈密頓量滿足宇稱與時間反演對稱下，一維柘樸系統中的孤立子解對於環境造成的能量增益及耗損其本身的穩定性。首先我會簡易介紹宇稱與時間反演對稱性在非厄米性統中的性質，我們注意到獨特點是在非厄米系統中獨有的性質，我們可以發現此點可以用來刻畫全實本徵值到本徵值具有複數值中過渡出現的轉變點。在二章中，我則會介紹如何從開放系統及量子資訊理論中密度矩陣演化兩種不同的的觀點去推導出非厄米哈密頓量。接下來，我們將介紹一維SSH 模型中的孤立子並考慮孤立子處在具有能量增益及耗散的系統中，當系統具備宇稱及時間返演對稱時，隨著增加系統中非厄米強度，我們發現到孤立子不受其影響並保持本身具有的穩定性。最後，我們簡單探討如何利用柘樸不變量保真度去描述拓樸系統中的平庸相及拓樸相並拓廣至非厄米系統情況。

For the non-Hermitian system to satisfy the parity and time reversal symmetry, this thesis aims to study the stability of the soliton state in the one-dimensional topological system with gain and loss caused by the environment. First, I will briefly introduce the properties of parity and time reversal symmetry in the non-Hermitian system. We notice that the exceptional point has a special property that can characterize the transition point for the eigenvalues with all real to the eigenvalues with complex values. In the second chapter, I will introduce how to derive the non-Hermitian Hamiltonian from two different perspectives, one is an open quantum system and another is the evolution of the density matrix in the quantum information theory. Then, I will introduce the soliton state in the one-dimensional SSH model and consider the soliton state in the system with energy gain and loss. When the system has the PT -symmetry, as increasing strength of non-Hermitian, we found that the soliton state would not be affected and still keeps the stability itself. Finally, we briefly discuss how to describe the trivial phase and topological phase using the topological invariant and fidelity and extend it into the non-Hermitian case.

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