量子力学的基本原理指出，希尔伯特空间中的厄米算符描述了微观系统的物理量，它的期望值是一个实数。系统在演化过程中遵循薛定谔方程，几率守恒，能量、粒子数和信息等物理量不与外界发生交换。然而，在实际应用中发现，所要研究的系统不可避免地会与周围环境或测量器件发生相互作用，导致它的能量、粒子数和信息与外界发生交换，进而形成了开放量子系统。在研究开放量子系统的过程中发现，一些可控的、与外界环境具有特殊耦合形式的开放量子系统可以通过后选择测量将量子跳跃项投影出去，进而等效成一个非厄米量子系统。这个非厄米量子系统的哈密顿量是非厄米的，它的本征值存在复数的情况，本征态不满足传统的正交归一性。这些现象与传统的量子力学的理论截然不同，因而吸引了大量研究者们的关注。

        随着理论方法和实验技术的进步，非厄米物理逐渐从开放量子系统中分化出来，形成了一门独立的学科。目前，非厄米物理的研究主要体现在它对新奇量子态的调控方面。对于新奇量子态的调控有多种方式，例如：周期驱动、杂质和无序、拓扑、多体和约束等。其中，非厄米物理与拓扑相结合形成了非厄米拓扑物理的研究方向，它又与关联相结合形成了非厄米多体物理的研究方向。非厄米物理的发展迅速，已经取得了很多成果，包括宇称-时间反演对称性破缺、非厄米趋肤效应、体边对应的破缺等现象的发现，因而成为了当今物理学研究的热点之一。然而，上述研究大多集中在零温的情况，对于非厄米量子系统在有限温下的统计规律和物理性质尚不完全清楚。另外，耗散对系统的调控也还没有研究彻底。

        本文的主旨在于探究非厄米量子系统在有限温下的物理性质和耗散对非厄米量子系统的调控作用。基于非厄米权重的特点，我们建立了非厄米量子系统在有限温下的统计理论，即量子刘维统计理论。根据该理论，我们发现非厄米量子系统在有限温下表现出一系列不同于厄米系统的性质，例如：刘维-玻尔兹曼分布、刘维权重和高温非热化等。随后，我们利用量子刘维理论分别探究了宇称-时间反演对称的非厄米量子系统和一维非厄米伊辛模型在有限温下的物理性质。利用一个可控的、与两个分离的环境相耦合的开放量子系统，模拟了有限温下的宇称-时间反演对称的非厄米量子系统。借助玻恩-马尔可夫近似下的 Gorini-Kossakowski-Sudarshan-Lindblad 形式的量子主方程，推导出它的密度矩阵的表达式。根据这个表达式，我们发现玻尔兹曼分布定律对于非厄米量子系统是
不适用的，而我们给出的量子刘维统计理论对非厄米量子系统是适用且正确的。进一步地，我们发现该系统在奇异点附近存在着一个“连续的”热力学相变，并且这个相变在零温时表现出反常的特点。在有限温下的一维非厄米伊辛模型中，我们发现了一个介于“拓扑相”和“非拓扑相”之间的“赝相变”，并伴随着刘维能隙的关闭，而不是一般的能隙关闭。另外，我们还研究了耗散对非厄米量子系统的影响。结果发现，系统的能带呈现出一个虚的线能隙，且它的本征态被束缚在某个特定的区域里。为了能够更好地描述这些新奇的现象，我们提出了“非厄米撕裂”的概念。在非厄米撕裂的过程中，系统的体态和边界态出现了宇称-时间反演相变。为了能够定量地描述本征态的撕裂程度，我们定义了撕裂度这一物理量，并发现它在奇异点处存在一个二级连续相变。我们将系统在实空间N×N 维的哈密顿量约化成一个k空间 2×2维的有效哈密顿量，以此来理解非厄米撕裂。此外，我们还探究了一维Su-Schrieffer-Heeger模型和Qi-Wu-Zhang模型的非厄米撕裂。

        本论文共六章。主要围绕非厄米量子系统在有限温下的物理性质和耗散对系统的调控作用而展开的，重点探究了宇称-时间反演对称的非厄米量子系统和一维非厄米伊辛系统在有限温下的物理性质、耗散对于一维 Su-Schrieffer-Heeger模型和Qi-Wu-Zhang模型的体态和边界态的影响。本论文的主要内容及结构如下：

        第一章主要介绍了开放量子系统的动力学过程，详细推导了Lindblad 主方程的一般形式，并由此给出非厄米物理的由来。随后，详细介绍了非厄米物理在理论和实验方面取得的成果，包括宇称-时间反演对称系统、赝厄米物理和体边不对应等。

        第二章建立了非厄米量子系统在有限温下的统计理论，即量子刘维统计理论，并将其与厄米系统的量子统计理论进行比较。

        第三章中，利用量子刘维统计理论，探究了宇称-时间反演对称的非厄米量子系统在有限温下的热力学性质。首先，根据该系统的特点，分别给出它在刘维-玻尔兹曼分布定律和玻尔兹曼分布定律中所满足的密度矩阵和统计分布的具体形式。随后，模拟了有限温下的宇称-时间反演对称的非厄米量子系统，并利用量子主方程，推导出它的密度矩阵和统计分布的表达式。将这一结果与上述两种分布定律的结果进行比较，验证了我们提出的刘维-玻尔兹曼分布定律的正确性。最后，计算了该系统在有限温下的刘维哈密顿量和自旋算符的期望值及其导数，发现了位于奇异点附近的“连续的”热力学相变。

        第四章利用量子刘维统计理论探究了有限温下的一维非厄米伊辛模型的物理性质。我们计算了它的刘维哈密顿量和刘维能级，分析了它的“拓扑相图”，发现了一个“赝相变”。此外，还分别讨论了该模型在厄米情况、强非厄米情况、零温极限和高温极限下的热力学性质。

        第五章研究了耗散对非厄米量子系统的调控作用。以一维简单的均匀跳跃的紧束缚模型为例，分别讨论了它在实势和虚势下能谱、本征态、左右几率和撕裂度的性质，并给出了厄米撕裂和非厄米撕裂的概念。随后，探究了一维 Su-Schrieffer-Heeger 模型和Qi-Wu-Zhang 模型的非厄米撕裂。

        第六章是总结与展望。

        本文通过对非厄米量子系统在有限温下统计分布的研究，建立了量子刘维统计理论，为非厄米物理的研究开启了一个新的方向。同时，通过对耗散诱导的非厄米撕裂的探究，加深了人们对非厄米物理调控作用的认识，为研究复杂系统的非厄米撕裂提供了理论方法。

The fundamental principles of quantum mechanics point out that the physical quantity of a microscopic system is described by the Hermitian operator in Hilbert space, whose expected value is a real number. The system follows the Schrodinger equation, whose probability is conserved. Physical quantities, such as energy, particle number, and information, are not exchanged with the outside world. However, in practical applications, it is found that the system to be studied will interact inevitably with the  、surrounding environment or measurement devices, which results in the phenomenon that its energy, particle number and information are exchanged with the outside world and then forms an open quantum system. While studying open quantum systems, it is discovered that some controllable open quantum systems with particular coupling forms with the external environment can be equivalent to a non-Hermitian quantum system by postselection measurement which projects the quantum jump term. The Hamiltonian of this system is non-Hermitian, which its eigenvalues can be complex numbers and the eigenstates do not satisfy the traditional orthonormality and normalization. These phenomena are different from the traditional quantum mechanics, so they have attracted the attention of many researchers.

With the advancement of theoretical methods and experimental techniques, non-Hermitian physics gradually differentiates from open quantum systems and then forms an independent discipline. At present, the research on non-Hermitian physics mainly concentrates on controlling novel quantum states. There are many ways to control novel quantum states, such as periodic driving,
impurity and disorder, topology, many-body and constraining. Among them, the combination of non-Hermitian physics and topology forms the research direction of non-Hermitian topological physics, and the combination of non-Hermitian physics and correlation forms the research direction of non-Hermitian many-body physics. Therefore, non-Hermitian physics has developed rapidly and has made extensive achievements, including parity-time symmetry breaking, nonHermitian skin effect and breakdown of bulk-boundary correspondence. Thus, non-Hermitian has become one of the hot topics in current physics research. However, most of the above studies focus on the zero temperature case, and the statistical law and physical properties of non-Hermitian systems at finite temperature are inadequately reported. In addition, the control of dissipation on the system has not been thoroughly studied.

The main purpose of this paper is to explore the physical properties of non-Hermitian quantum systems at finite temperature and the control of dissipation on non-Hermitian quantum systems. Firstly, based on the non-Hermitian weight, we develop a statistical theory of non-Hermitian quantum systems at finite temperature, quantum Liouvillian statistical theory. According to this theory, we find that non-Hermitian quantum systems at finite temperature show a series of abnormal properties which are different from the Hermitian systems at finite temperature, such
as Liouvillian-Boltzmann distribution, Liouvillian weight and nonthermalization in high temperature. Subsequently, we investigate the physical properties of the parity-time-symmetric nonHermitian quantum system and the one-dimensional non-Hermitian Ising model at finite temperature by quantum Liouvillian statistical theory, respectively. By a controllable open quantum system coupling to two separate environments, we simulate the parity-time-symmetric non-Hermitian quantum system at finite temperature. With the help of the quantum master equation in the GoriniKossakowski-Sudarshan-Lindblad form under the Born-Markov approximation, the expression of the system’s density matrix at finite temperature is derived. According to this expression, we find that the Boltzmann distribution law is not applicable for non-Hermitian quantum systems, while
the quantum Liouvillian statistical theory we gave is applicable and correct to non-Hermitian quantum systems. Furthermore, a “continuous” thermodynamic phase transition occurs at the exceptional point where a zero-temperature anomaly exists. In the one-dimensional non-Hermitian Ising model at finite temperature, “pseudo-phase transition” is explored between the “topological
phase” and the “non-topological phase”, at which the Liouvillian energy gap is closed rather than the usual energy gap. Finally, we study the effect of dissipation on non-Hermitian quantum systems. The results show that the energy band shows an imaginary line gap and energy eigenstates are bound to a specific region. To describe these novel phenomena, we propose the concept of“non-Hermitian tearing”. In the process of non-Hermitian tearing, bulk states and boundary states show the parity-time transition. To characterize the degree to which an eigenstate is torn, we define the tearability, which it exhibits a second-order and continuous phase transition at the exceptional point. We give the effective 2 × 2 Hamiltonian in the k space by reducing the N × N Hamiltonian of the system in the real space to better understand the non-Hermitian tearing. Besides, we also explore the non-Hermitian tearing in the one-dimensional Su-Schrieffer-Heeger model and the Qi-Wu-Zhang model.

There are six chapters in this paper, which mainly study the physical properties of nonHermitian quantum systems at finite temperature and the control of dissipation on the system. We focus on the physical properties of the parity-time-symmetric non-Hermitian quantum system and the one-dimensional non-Hermitian Ising model at finite temperature, and the effect of dissipation on the bulk states and the boundary states of the one-dimensional Su-Schrieffer-Heeger model and the Qi-Wu-Zhang model. The main content and structure of this paper are as follows:

In the first chapter, we mainly introduce the dynamics of the open quantum system, deduce the general form of the Lindblad equation in detail. From this, we give the origin of non-Hermitian
physics. Subsequently, the theoretical and experimental achievements of non-Hermitian physics are introduced in detail, including parity-time-symmetric system, pseudo-Hermitian physics, and breakdown of bulk-boundary correspondence.

In the second chapter, we develop the statistical theory for non-Hermitian systems at finite temperature, the quantum Liouvillian statistical theory, and then compare it with the quantum statistical theory for the Hermitian system.

In the third chapter, we explore the thermodynamic properties for parity-time-symmetric nonHermitian quantum systems at finite temperature by the quantum Liouvillian statistical theory. Firstly, based on the system’s characteristics, the specific forms of the density matrix and statistical distribution satisfying the Liouvillian-Boltzmann distribution law and the Boltzmann distribution law are given, respectively. Secondly, we simulate the parity-time-symmetric non-Hermitian quantum system at finite temperature, and derive its density matrix and statistical distribution using the
quantum master equation. This result is compared with the results of the above two distribution laws, verifying the correctness of the proposed Liouvillian-Boltzmann distribution law. Thirdly,
the Liouvillian Hamiltonian, and the expected values of the spin operators and its derivatives for the system at finite temperature are calculated and a “continuous” thermodynamic phase transition at the exceptional point is found.

In the fourth chapter, according to the quantum Liouvillian statistical theory, we study the one-dimensional non-Hermitian Ising model at finite temperature. We calculate the Liouvillian
Hamiltonian and the Liouvillian energy level, analyze the “topological phase diagram” and find the“pseudo-phase transition”. In addition, we discuss its thermodynamic properties in Hermitian and strong non-Hermitian cases, and in the zero-temperature and high-temperature limits, respectively.

In the fifth chapter, we explore the control of dissipation on non-Hermitian quantum systems. The properties of energy spectrum, energy eigenstates, left and right probabilities and tearability are discussed by a simple one-dimensional tight binding model with uniform hopping under the real and imaginary potential, respectively. Then we propose the concepts of Hermitian tearing and non-Hermitian tearing. Subsequently, we explore the non-Hermitian tearing in the onedimensional Su-Schrieffer-Heeger model and the Qi-Wu-Zhang model.

In the sixth chapter, we give the conclusion and prospect of this paper.

In this paper, the quantum Liouvillian statistical theory is established through the study of the statistical distribution of non-Hermitian quantum systems at finite temperature, which opens a new direction for non-Hermitian physics. At the same time, exploring the non-Hermitian tearing by dissipation deepens people’s understanding of the control of non-Hermitian physics, and provides a theoretical method for studying non-Hermitian tearing of complex systems.