相变是多体系统中粒子间相互作用导致的宏观现象，核心在于系统中的协同效应。XY自旋模型是相变的典型研究对象。二维XY模型中存在BKT相变，相变点左侧存在正反涡旋对，涡旋对密度受温度影响，并在升温至相变温度时发生解缚。三维及更高维度的XY模型中存在从铁磁相到顺磁相的转变，其中自发磁化强度在相变温度以上归零。在XY模型的研究中，一个普适的序参量的缺乏和介观尺度分析的不足是当前存在的主要问题，目前其序参量的具体形式和计算方法往往取决于模型的类型以及相互作用的特性。

本征微观态理论以系统全局系综信息中本征微观态的凝聚作为相变的标志，用介观尺度的本征微观态分析复杂系统的相变临界现象。本文通过分析三维与二维XY模型的本征微观态及其权重随温度的变化，探索了两个分属不同相变普适类的模型在本征微观态视角下的相变临界现象，得到了具有系统特征意义的本征微观态，并扩展了本征微观态理论的应用范畴。由于以往本征微观态方法在物理系统中主要应用于连续相变，本文在XY模型中的研究工作先基于具有连续相变的三维XY模型展开，并进一步拓展到具有特殊拓扑类型的BKT相变中。

在三维XY模型中，本文将本征微观态方法在连续相变情景下的应用范畴从一维自旋扩展到XY模型的二维自旋。基于平均磁化强度在本征微观态下的展开，本文提出了三维XY模型中本征微观态权重的有限尺寸标度形式，并基于连续相变中本征微观态权重的有限尺寸标度与重整化群估计了三维XY模型的相变点与本征微观态权重的临界指数。本文同时讨论了三维XY模型的本征微观态，发现其分不同类别，同一类别中本征微观态权重大小相等、构型相近，且存在两两相互垂直的构型对。第一类本征微观态为均匀构型，对应于系统的平均磁化强度。第二、三、四类本征微观态中存在不同数目的具有平行自旋的团，相邻团方向相反，团的交界位置存在涡旋区域。同一类别的不同本征微观态之间旋转对称，使得类别整体的平均磁化强度表现为0。

在二维XY模型中，本文将本征微观态方法的应用范畴由连续相变扩展到BKT拓扑相变。由于涡旋对破坏了系统的长程关联，通过连续相变中本征微观态的有限尺寸标度和重整化群确定系统相变点的方式不再适用。本文发现不同本征微观态权重之比存在不光滑的尖峰，意味着不同本征微观态的大小顺序位置发生了变化，代表系统发生了相变。本文基于尖峰产生的温度依照有限尺寸标度方法确定了系统的相变点，进而基于平均磁化强度在本征微观态下的展开形式提出了相变点左、右侧的本征微观态权重的有限尺寸标度形式。对于相变点附近的对数修正形式，本文发现第一类本征微观态权重的修正项指数与其他类别不同，并推得第一类本征微观态对关联函数不构成影响，其他类别本征微观态权重的修正项指数才能被用于估计关联函数中的对数修正项指数。本文同时讨论了二维XY模型的本征微观态，发现其分不同类别，同一类别中本征微观态权重大小相等、构型相近，且存在两两相互垂直的构型对。除第一类为前两大本征微观态外，第二、三、四类均包含八个本征微观态。第一类本征微观态为均匀构型，对应于系统的平均磁化强度。第二、三、四类本征微观态中存在不同数目的具有平行自旋的团，相邻团方向相反，团的交界位置存在正反涡旋对，本文认为这一特征对应二维XY模型的正反涡旋对构型。

Phase transitions are macroscopic phenomena resulting from the interactions among particles in many-body systems, with the core being the cooperative effects within the system. The XY spin model is a typical object of research for phase transitions. In the two-dimensional XY model, there exists a BKT phase transition, where pairs of vortices and antivortices exist to the left of the transition point, and the density of these vortex pairs is influenced by temperature, becoming unbound upon heating to the transition temperature. In three-dimensional and higher-dimensional XY models, there is a transition from a ferromagnetic phase to a paramagnetic phase, where the spontaneous magnetization vanishes above the transition temperature. In the research of the XY model, the lack of a universal order parameter and the insufficiency of mesoscale analysis are the main issues currently present, and the specific form and calculation method of the order parameter often depend on the type of model and the characteristics of the interactions.

Eigen microstate theory takes the condensation of eigen microstates in the global ensemble information of the system as the sign of a phase transition, using mesoscale analysis of eigen microstates to study the critical phenomena and phase transitions in complex systems. This paper explores the critical phenomena of phase transitions from the perspective of eigen microstates for two models belonging to different universality classes, through analyzing the eigen microstates and their weights as a function of temperature in three-dimensional and two-dimensional XY models. It identifies eigen microstates with significant meaning for the system characteristics and expands the application scope of eigen microstate theory. Since the application of the eigen microstate method in physical systems has been mainly based on continuous phase transitions, this paper's research on the XY model first focuses on the three-dimensional XY model with continuous phase transitions and further extends to BKT topological transitions with special topological types.

In the three-dimensional XY model, this paper extends the application scope of the eigen microstate method from one-dimensional spins to two-dimensional spins in the XY model under continuous phase transition scenarios. Based on the expansion of the average magnetization under eigen microstates, this paper proposes a finite-size scaling form of the eigen microstate weights in the three-dimensional XY model and estimates the transition point as well as the critical exponents of the eigen microstate weights using the finite-size scaling and renormalization group of eigen microstate weights in continuous phase transitions. The paper also discusses the eigen microstates of the three-dimensional XY model, finding that they are divided into different categories, with equal weights and similar configurations within the same category, and there exist pairs of configurations that are perpendicular to each other. The first category of eigen microstates is a uniform configuration, corresponding to the system's average magnetization. In the second, third, and fourth categories of eigen microstates, there are clusters with parallel spins of different numbers, where adjacent clusters have opposite directions, and there are vortex regions at the boundaries between the clusters. Different eigen microstates within the same category exhibit rotational symmetry, which results in the overall average magnetization of the category being zero.

In the two-dimensional XY model, this paper extends the application scope of the eigen microstate method from continuous phase transitions to BKT topological phase transitions. Since the long-range correlation of the system is destroyed by vortex pairs, the method of determining the system's transition point through the finite-size scaling of eigen microstates in continuous phase transitions is no longer applicable. This paper finds that the ratio of weights of different eigen microstates has an irregular sharp peak, indicating a change in the order of the sizes of different eigen microstates, representing a phase transition of the system. Based on the temperature at which the peak occurs, the paper estimates the system's transition point using the finite-size scaling method and proposes a finite-size scaling form for the eigen microstate weights on the left and right sides of the transition point based on the expansion form of the average magnetization under eigen microstates. For the logarithmic correction form near the transition point, this paper finds that the exponent of the correction term for the weight of the first category of eigen microstates is different from other categories, and it is deduced that the first category of eigen microstates does not affect the correlation function, while the exponent of the correction term for the weights of other categories of eigen microstates can be used to estimate the logarithmic correction term exponent in the correlation function. The paper also discusses the eigen microstates of the two-dimensional XY model, finding that they are divided into different categories, with equal weights and similar configurations within the same category, and there exist pairs of configurations that are perpendicular to each other. Except for the first category being the top two eigen microstates, the second, third, and fourth categories each contain eight eigen microstates. The first category of eigen microstates is a uniform configuration, corresponding to the system's average magnetization. The second, third, and fourth categories of eigen microstates contain clusters with a varying number of parallel spins, where adjacent clusters have opposite directions, and the interfaces between the clusters feature configurations of vortex-antivortex pairs. This paper posits that this characteristic corresponds to the vortex-antivortex pair configurations of the two-dimensional XY model.