拓扑物态是最近四十年来凝聚态物理的前沿研究方向，它最初是在二维量子霍尔体系中发现的。随着实验技术的进步，拓扑在物理学各个领域中的作用变得越来越重要。
然而，在最近二十年里，由于“Ten-fold way”对拓扑物态的分类取得的重大成功，使得拓扑的概念也被推广到了非平衡系统中。
非平衡系统一般指的是和环境存在耦合的开放系统，由于其和实验密切相关，且存在新奇的拓扑物态以及独特的动力学效应，因此受到了广泛的关注与研究。

本文研究了若干非平衡自由费米子系统中的物理性质，包括其中的拓扑结构和相应的动力学效应。主要涉及到三种非平衡系统：开放量子系统、受到周期驱动的Floquet（弗洛凯）系统以及非厄米系统。
主要工作包括：系统地阐述了研究自由费米子开放系统的方法；基于“38-fold way”，对自由费米子开放系统中的边界暗态进行了拓扑分类；分析了陈绝缘体在周期脉冲驱动下的拓扑性质，以及其中的拓扑结构在相应开放系统中的动力学效应；同时还研究了具有复费米波速的非厄米手性边界态的独特性质。

第一章为绪论，我们首先详细介绍了对厄米系统中的拓扑物态进行分类的“Ten-fold way”，以及对非厄米系统中的拓扑物态进行分类的“38-fold way”。
然后，我们引入了开放系统的Lindblad主方程，并简单介绍了由这个方程所描述的系统中的拓扑结构。
最后，我们还介绍了Floquet系统，对此类系统中的拓扑物态以及Floquet开放系统的性质进行了简单的综述。

第二章介绍了研究自由费米子开放系统的一般方法。此类系统的动力学演化由一个非厄米矩阵（阻尼矩阵）完备地描述，其本征值谱对应于这个系统的刘维尔谱。
结合自由费米子系统的密度矩阵为高斯二次型这一性质，建立了该系统的单粒子关联函数和密度矩阵之间的一一对应关系。
利用单粒子关联函数，我们能够得到系统中的粒子数分布、粒子流以及von-Neumann熵等物理量，进而探究系统中的动力学过程。
然后，我们考虑了阻尼矩阵中的非厄米效应以及拓扑结构对这些物理量的影响，包括$\mathcal{PT}$对称性自发破缺导致的退耦现象、非厄米趋肤效应导致的手性阻尼振荡现象、边界暗态导致的阻尼振荡的手性中断以及阻尼矩阵中的拓扑结构与非平衡稳态之间的关系。

在第三章中，我们提出了双重阻尼矩阵的概念，这个非厄米矩阵中存在定义良好的虚能隙。
然后，基于“38-fold way”，根据双重阻尼矩阵的对称性对自由费米子开放系统中的边界暗态进行了拓扑分类。
利用这个方案，我们对拓扑绝缘体开放系统和拓扑超导体开放系统中的边界暗态进行了拓扑分类，给出了相应的拓扑周期表并分析了边界暗态的鲁棒性。

第四章研究了周期脉冲驱动的陈绝缘体中的拓扑性质，其中周期脉冲驱动力为其Dirac哈密顿量中的质量项。以周期脉冲驱动的Qi-Wu-Zhang（QWZ）模型为例，其拓扑性质由Floquet算符描述。通过分析其所对应的Bloch矢量空间中的拓扑结构，我们讨论了这个模型的拓扑性质并给出了它的拓扑相图，其中我们用Floquet陈数$C_F$来表征该模型的拓扑性质。
该模型一共存在六个不等价的拓扑相，分别对应于$C_F=\{-1_0,-2,-1_\pi,1_\pi,2,1_0\}$。
然后，我们还考虑了当Floquet系统耦合了马尔可夫环境之后的物理，此时Lindblad主方程变成含时的。据此我们给出了一个Sylvester（西尔维斯特）方程，它可以用来求解一个受到周期脉冲驱动的自由费米子开放系统中的Floquet非平衡稳态。
结合这个方程，我们研究了受到周期脉冲驱动的QWZ模型开放系统中的Floquet稳态，然后从稳态的von-Neumann熵中得到了该模型拓扑相变的阶数，同时还分析了周期驱动力对稳态粒子流的调控行为。

第五章考虑了在二维拓扑物态的边界上引入不可逆跃迁之后的现象，此时系统的边界上存在着具有复费米波速的非厄米手性边界态。
通过求解该非厄米手性边界态的薛定谔方程，我们发现此时存在着由手性决定的局域化行为，且其局域化强度是尺寸依赖的。
相关的结论在二维陈绝缘体和二维$p$-波拓扑超导体中得到了验证。

第六章中，我们总结了本文的主要工作，对可能的实验现象以及未来的研究方向进行了讨论。    

About forty years ago, the topological state of matter is discovered in the two dimensional quantum hall system,  since then, the topological states become the frontier of condense matter physics, and it is still true up till now.  From then on, the topology becomes more and more important in the field of physics as the development of experimental techniques.  Furthermore, due to the great success of "Ten-fold way" that achieved in the classification of topological states, the concept of topology has extended into non-equilibrium systems in the recent twenty years. Generally speaking, the non-equilibrium systems are correspond to the open systems which are coupled to the environment, these systems are closely related to the real experiment, and it has attracts lots of attention because there exist extraordinary topological states and have astonishing dynamical effects.
    
In the present thesis, we are researching on certain non-equilibrium free fermionic systems, in which the topological structures and the correlated dynamical effects are studied. We are mainly focusing on three kinds of non-equilibrium systems: the open quantum systems, the periodically driven Floquet systems, and the non-Hermitian systems. Main works of this thesis are as the follows: a systematic way to study the open free fermionic system is elaborated; the edge dark states in the open free fermionic  system are classified topologically by using the "38-fold way"; the topological phases of periodically kicked Chern insulator is obtained, as well as the dynamical effects of its topological structure in the corresponding open system;  furthermore, the remarkable properties of non-Hermitian chiral edge modes with complex fermi velocity is uncovered.
    
The first chapter is the introduction, the "Ten-fold way" that is used to classify the topological states of Hermitian systems is introduced in detail, as well as the "38-fold way" which classifying the topological states of non-Hermitian systems. Then, the Lindblad master equation that used to describing the time evolution of open system is introduced in detail, and the topological structure behind the Lindblad master equation is reviewed briefly. At last, we give a brief review of the Floquet systems, in which the topological states of Floquet systems and the properties of Floquet open systems are summarized.
    
In the chapter two, the general framework to study the open free fermionic system is provided, in which it's dynamical information is completely captured by a non-Hermitian matrix (the damping matrix), in which the eigenvalues of  damping matrix corresponds to the Liouvillian spectrum of system. Due to the Gaussian quadratic property of free fermionic systems, the one-to-one correspondence between the single-particle correlation function and the density matrix is established, that the time evolution of single-particle correlation function can reflects the time evolution of states. Despite this, the physical quantities such as the distribution of occupation number, the particle currents and the von-Neumann entropy of the system are encoded in it's single-particle correlation function. Then, how these quantities are related to the non-Hermitian effect and the topological structure of damping matrix is investigated correspondingly. Including the decoupling effect induced by the spontaneous breaking of $\mathcal{PT}$ symmetry; chiral damping effect induced by the non-Hermitian skin effect; chiral termination of damping that result from the edge dark states; as well as the relationship between the topological structure and the property of steady state.
    
In the chapter three, the concept of double damping matrix is proposed, in which the imaginary line gap is well defined in such non-Hermitian matrix. Then, based on the symmetry of double damping matrix, the edge dark states in the open free fermionic system is classified topologically by using the "38-fold way". Then, the symmetry classes of the edge dark states are given for the topological insulators open system and the topological superconductors open system, in which the corresponding periodical tables are provided, and the robustness of edge dark states are checked.
    
In the chapter four,  the topological property of Floquet Chern insulator which it's Dirac mass is periodically kicked is investigated, that the driven force corresponds to the mass of Dirac Hamiltonian. Take periodically kicked Qi-Wu-Zhang (QWZ) model as an example, that it's topological properties are completely captured by the Floquet operator. Then, the topological property and the corresponding topological phase diagram are revealed by studying the topological structure in the Bloch vector space of Floquet operator. We use the Floquet Chern number $C_F$ to characterize it's topological property,  that there are six different topological phases, i.e. $C_F=\{-1_0,-2,-1_\pi,1_\pi,2,1_0\}$. Furthermore, the physics when a Floquet system is coupled to a Markov system is studied, in which the Lindblad master equation is valid under the condition that the driven is strong enough and the coupling is comparable weak. Then, we derived a matrix equation, which belongs to the Sylvester equation and can be used to find the Floquet non-equilibrium steady states of periodically kicked open free ferminonic system. At last, the Floquet steady states of periodically kicked QWZ model is studied, that the order of topological phase transition is obtained by studying the von-Neumann entropy of steady state, and the amazing behavior of steady state currents which is altered by the strength of driven is demystified.
    
In the chapter five, the physics that there is non-reciprocal hopping at the edge of 2-dimensional (2D) topological materials are studied, that there are non-Hermitian chiral edge modes with complex fermi velocity at the edge of system. By solving the Schr\"{o}dinger equation of non-Hermitian chiral edge modes, we find that they become the localized states, in which the position where they are localized is completely determined by the chirality of edge modes, and it is the scale-relevant localization.Our findings are verified in the 2D Chern insulator and the 2D $p$-wave superconductor.
    
We conclude our works in chapter six, as well as some experimental implication and future directions are discussed.