对相和相变的认知构成了我们理解物质世界的基础。从朗道范式到拓扑相变，相变理论的发展树立了科学进步之路上的一座座丰碑。从演生论的角度看，拓扑激发之间的相互作用导致了丰富的物理现象。一个著名的例子是BKT相变，由没有自发破缺$U(1)$对称性下成对涡旋的解耦合所致。最新发展的张量网络方法为相变研究注入新的活力，利用统计模型转移矩阵对应的一维量子模型的纠缠信息，可以在热力学极限下精确研究各种相变。借此我们对二维经典XY模型相关的物理进行了系统深入的研究，揭示了由拓扑激发驱动的各种新奇的相和相变。我们研究了包含整数涡旋和半整数涡旋的拓展的XY模型。从纠缠熵的奇异性，我们精准地确定了相图。整数涡旋配对相和半整数涡旋配对相与无序相之间通过BKT相变分开。在两个不同的涡旋配对相之间存在一个连续相变，表现为比热的对数发散和自旋关联长度的指数发散。由此我们发现了一个混合了BKT和Ising相变特征的新的相变普适类。我们通过纠缠谱进一步表明，三条相变线在一个多临界点汇合，从该点延伸出一条解耦合交叉线进入无序相。对称性和拓扑激发的相互影响可以在耦合的双层XY系统中诱发新现象。我们发现，二阶约瑟夫森耦合效应可以在双层超导体中诱发一种奇异的超导相。上下超导层不同时，出现一个准长程有序的中间相，整数涡旋配对发生在层内耦合较大的层中，而另一层只存在由拓扑弦相连的半整数涡旋对，对应一种奇特的4电子超导性。我们还在弱耦合hexatic-nematic XY模型中发现一种Potts液相。当向列自由度的耦合较强时，相对Potts长程序经历两步融化，与$Z_3$变量中畴壁和涡旋的分次激发有关，由此揭示了由多种拓扑激发驱动的相变的隐秘结构。为研究阻挫XY系统，我们发展了一套张量网络表示方法。对于正方晶格，核心思想是将阻挫引起的基态局部规则编码到配分函数的局域张量里。我们在 $T_{c1} \approx 0.4459$ 和$T_{c2} \approx 0.4532$确定了两个紧挨的相变，分别对应BKT和Ising相变，一举澄清长久以来的相图争议。对于阻挫效应更强的笼目晶格反铁磁XY模型，我们通过对偶变换避免了对玻尔兹曼权重的有限截断。证实了系统中不存在整数涡旋配对，唯一的$T_c\approx0.075$处的BKT相变是由$1/3$涡旋对的解耦合所致。一言以蔽之，我们的工作不仅揭示了拓扑激发引起的各种新奇演生现象，也为今后研究具有连续对称性的二维系统提供了可行路线。
The fundamental concepts of phases and phase transitions constitute the cornerstone of our understanding of the physical universe. The historical development of the phase transition theory, from Landau‘s spontaneous symmetry breaking paradigm to modern topological phase transition theories, has represented a major milestone in the evolution of numerous scientific disciplines. From the perspective of emergent philosophy, the interplay of topological excitations leads to enriched physical phenomena. One prominent prototype is the Berezinskii-Kosterlitz-Thouless (BKT) phase transition where unbinding of integer vortices occurs in the absence of spontaneous breaking of continuous $U(1)$ symmetry. Leveraging state-of-the-art tensor network methods, we present a systematic investigation of a broad range of XY-related systems in two dimensions (2D), revealing a diverse array of exotic phases and phase transitions driven by topological excitations.We study the generalized 2D XY model after a thorough review of the classical XY model. From the singularities of the entanglement entropy, we accurately determine the complete phase diagram of the generalized model. Both the integer vortex-antivortex binding and half-integer vortex-antivortex binding phases are separated from the disordered phase by the usual BKT transitions, while a continuous topological phase transition exists between two different vortex binding phases, exhibiting a logarithmic divergence of the specific heat and exponential divergence of the spin correlation length. A hybrid BKT and Ising universality class of topological phase transition is thus established. We further prove that three phase transition lines meet at a multicritical point from which a deconfinement crossover line extends into the disordered phase by analysis of entanglement spectrum.Interplay of symmetries and topological excitations can induce new phenomena in coupled XY systems. We find that the second-order Josephson coupling can induce an exotic superconducting phase in a bilayer superconductor. When two layers coupled inequivalently, there emerges an intermediate quasi-long-range order phase, where the vortex-antivortex bindings occur in the layer with the larger intralayer coupling, but only half-vortex pairs with topological strings exist in the other layer, corresponding to the phase coherence of pairs of Cooper pairs. Such results provide a promising way to realize the charge-4e superconductivity in a bilayer system. We also find an intermediate Potts liquid phase in the study of a weakly coupled two-dimensional hexatic-nematic XY model. When the coupling of nematic fields is larger than that of hexatic fields, the inter-component Potts long-range order undergoes a two-stage melting process. As increasing the temperature, there emerges an intermediate Potts liquid phase with an algebraic correlation with the formation of charge-neutral pairs of hexatic and nematic vortices. These two-stage phase transitions are associated with the proliferation of the domain walls and vortices of the relative $Z_3$ Potts variable, respectively. Our results thus reveal the hidden structure of the phase transitions driven by multiple topological defects.A general framework is proposed to solve the 2D frustrated XY systems. For square lattices, the essential idea is to encode the ground-state local rules induced by frustrations in the local tensors of the partition function. The singularity of the entanglement entropy for the 1D quantum analog provides a stringent criterion to distinguish various phase transitions without identifying \textit{a priori} order parameter. Two very close phase transitions are determined at $T_{c1} \approx 0.4459$ and $T_{c2} \approx 0.4532$, respectively. The former corresponds to a BKT phase transition describing the phase coherence of XY spins, and the latter is an Ising-like continuous phase transition below which a chirality order with spontaneously broken $Z_2$ symmetry is established. For the kagome lattices with stronger frustrations, we apply the duality transformation to avoid the finite cut of Boltzmann weights. The phase structure of a single BKT transition is confirmed which is driven by the unbinding of $1/3$ vortex-antivortex pairs at $T_c\approx0.075$, in the absence of phase coherence between integer vortices at all temperatures.In summary, our works provide new insights into the emergent phenomenon driven by topological excitations, and shed a new light on future studies of 2D systems with continuous symmetries.
