相变和临界现象广泛存在于各种复杂系统中。无论是原子尺度的磁性物质还是行星尺度的气候过程，不同复杂系统在临界点处都会表现出一系列共同的行为特征。相变与临界现象的研究便是为了揭示不同复杂系统共同行为特征背后的物理原理，也是理解复杂系统的核心命题之一。

对于简单的物理模型，人们已经建立起了一套以哈密顿量、统计分布为出发点，基于系统序参量的平衡态统计物理研究框架，并在此基础上取得了一系列成果。然而在将传统相变与临界现象理论应用于各种复杂系统，尤其是与人类生存息息相关的地球系统时，却面临着严重的挑战。这类复杂系统往往具有序参量未知，哈密顿量形式不明确，非平衡的特点，这使得现有的统计物理学及相变临界现象理论无法提供充分的理论支持。发展新的描述复杂系统相变临界现象的理论研究手段，并将其应用于具体的复杂系统中，是目前临界现象研究最迫切的需求。

近些年发展起来的以本征微观态理论和气候网络为代表的复杂系统方法，在研究相变临界现象问题上展现出极大的潜能。此类方法直接从模拟和观测数据出发，通过分析数据内在的关联行为，给出了系统不同成分之间的关联模式和临界特征。本论文针对目前临界现象研究所面临的基本问题，开展了两方面的研究。首先，拓展了本征微观态理论研究框架，建立了基于本征微观态的新重整化群方法；提出了基于本征微观态理论的熵及其标度形式，扩充了相变临界现象研究的理论工具。其次，将本征微观态理论和气候网络方法分别应用到了各向异性系统和地球系统的临界现象研究中，取得了一系列突破传统认知的新发现。本论文的主要工作如下：

（1）为了解决重整化群方法无法直接应用于哈密顿量未知的复杂系统的问题，我们将重整化群的思想引入到本征微观态理论中，建立起了本征微观态重整化群理论框架。通过讨论经过Kadanoff块变化前后，系统本征微观态权重的不动点。给出了系统在临界点处，最大本征微观态权重sigma1满足的重整化群变换关系。并在一维、二维和三维具有外场的Ising模型中，对变换关系的正确性进行了检验。基于该理论，我们还提出了新的确定系统临界点和临界指数的方法。相比传统方法，新建立的重整化群理论不再需要对系统的哈密顿量作出特殊要求，可以直接从模拟或者观测数据出发研究系统的临界特征，为处理复杂系统相变临界现象问题提供了新的研究手段。

（2）对系统熵的概念进行了扩展。基于本征微观态理论，通过分析介观尺度上各本征模态的概率分布，提出了本征微观态的熵。通过理论研究和数值模拟发现，新定义的熵不仅可以作为系统混乱程度的量度，熵以及熵关于温度的变化量在临界点处将会表现出临界行为，并分别满足对应的有限尺度标度关系。具有临界特性的熵将为复杂系统临界点的预测提供新的理论工具。

（3）基于本征微观态方法系统地研究了各向异性相互作用对临界指数的影响。通过向二维Ising模型中引入次近邻相互作用，并考察次近邻相互作用从铁磁逐渐过渡到反铁磁的整个过程中系统临界指数的变化，我们发现：当次近邻为铁磁相互作用时，系统的普适类并不发生改变，但各向异性相互作用会使得系统的空间关联模态发生变化；当次近邻为反铁磁相互作用时，随着相互作用强度增加到一定比例之后，临界指数的比值beta/nu将会逐步变大，系统的普适类会在各向异性相互作用的影响下发生改变。该发现打破了短程相互作用细节不影响系统普适类的传统认知，首次建立了各向异性与系统临界指数改变的关系，为复杂物理系统相变临界现象的研究提供了新的研究模式。

（4）基于气候网络的方法，通过逐年构建含时间延迟的自相关温度关联网络，定量地研究了亚马逊雨林地区，这一重要的地球临界要素在过去四十年对全球的影响模式。首先，发现了亚马逊雨林对全球的影响受到ENSO的调控，存在明显的年际变化。其次，通过从气候网络中筛选出显著节点的方式，首次发现了亚马逊雨林地区与青藏高原地区和南极冰川在过去四十年存在着稳定的遥相关信号。通过构建网络最短路径的方法，揭示了亚马逊与青藏高原可能的遥相关传播路径及大气动力学原理，并基于气候模式数据发现该遥相关传播路径在全球变暖的影响下依旧鲁棒。进一步的研究还发现，亚马逊与青藏高原在极端事件上存在着显著的同步特征。除此之外，发现青藏高原的积雪覆盖呈现出临界慢化现象，其稳定性自2008年以来就逐步下降，第一次给出了青藏高原正在逐步接近气候临界点的预警信号。这进一步表明了，亚马逊雨林地区和青藏高原地区是气候临界要素之间的关联，为气候系统可能存在的级联失效效应提供了新的证据。

Phase transitions and critical phenomena are ubiquitous in various complex systems, spanning from atomic-scale magnetic materials to planetary-scale climate processes. At the critical point, different complex systems exhibit a set of common behavioral characteristics, which motivate investigations into the underlying physical principles governing these systems. Understanding the nature of phase transitions and critical phenomena is thus one of the central propositions in comprehending complex systems.

For simple physical models, a research framework has been established based on the system's Hamiltonian and statistical distribution, starting from the system's order parameters and focusing on equilibrium state physics. This framework has led to a wealth of theoretical achievements. However, applying traditional critical phenomena theory to complex systems, particularly systems that are closely linked to real life, such as the Earth system, poses significant challenges. Those complex systems often have unknown order parameters, unclear Hamiltonian forms, and complex statistical distributions, rendering existing critical phenomena theories inadequate for providing sufficient theoretical support. Developing new theoretical research methods to describe the critical phenomena of complex systems and applying them to solve problems in specific complex systems is one of the most urgent need in current critical phenomena research.

In recent years, complex systems methods, such as the Eigen Microstate Approach (EMA) and Climate Networks (CNs), have shown great potential in studying critical phenomena. These data-driven methods analyze the inherent correlation behavior within the data, and provide the correlation patterns and critical characteristics among the different components of the system. This paper investigates the fundamental issues facing critical phenomena research and conducts two studies. Firstly, we expand the EMA framework by establishing a new renormalization group framework. We propose a new entropy and its scaling form based on EMA, thereby expanding the theoretical tools for studying critical phenomena. Secondly, we apply the EMA and CNs to the critical phenomena research of the anisotropic system and the climate tipping points, respectively, achieving a series of new findings that go beyond traditional understanding. The main work of this paper includes the following:

(1) In order to address the problem that Hamiltonian-dependent renormalization group (RG) methods cannot be directly applied to complex systems with unknown Hamiltonians, we introduce the idea of RG into the framework of EMA and establish the renormalization group theory of eigen microstates. By discussing the fixed-point behavior of the largest eigenvalue, sigma1, under Kadanoff block transformation, we give the RG transformation relation satisfied by the $\sigma_1$ at the critical point. The correctness of the transformation relation is verified in one-dimensional, two-dimensional, and three-dimensional Ising models with the external field. Based on this theory, we propose new methods for determining the critical point and critical exponents of the system. Compared with the traditional method, our RG theory no longer requires special requirements on the system's Hamiltonian and can directly study the critical characteristics of the system from simulation or observation data, providing a new research framework for handling complex system phase transition and critical phenomenon problems.

(2) The concept of entropy was extended in this study. Based on the EMA, the entropy of eigen microstates was proposed by analyzing the probability distribution of each eigen mode at the mesoscale. Theoretical studies and numerical simulations have shown that the newly defined entropy can not only measure the degree of chaos in the system but also exhibit critical behavior at the critical point, and the entropy and its temperature-dependent variation will satisfy certain finite-scale scaling relations. This entropy with critical characteristics will provide a novel theoretical tool for exploring the critical behavior of complex systems.

(3) The impact of anisotropic interactions on the critical exponents was systematically studied using the EMA. By introducing next-nearest-neighbor interactions to the two-dimensional Ising model, and examining the change in the critical exponents of the system as it changes from ferromagnetic to antiferromagnetic interactions, we found that when the next-nearest-neighbor interaction is ferromagnetic, the universality class of the system remains unchanged, but the eigen correlation modes of the system are affected by anisotropic interactions. When the next-nearest-neighbor interaction is antiferromagnetic, the ratio of the critical exponents beta/nu increases obviously when the strength of the interaction is above a certain degree, and the universality class of the system is altered by antiferromagnetic anisotropic interactions. This conclusion challenges the traditional view that the details of short-range interactions do not affect the universality class of the system.

(4) We propose a climate network approach to quantitatively analyze the global impacts of a well-known tipping element, the Amazon Rainforest Area (ARA). We find that regions, such as, the Tibetan Plateau and West Antarctic Ice Sheet, are characterized by higher network weighted links and exhibit strong correlations with the ARA. A teleconnection propagation path is identified between the ARA and the Tibetan Plateau. We found this path is robust under climate change as simulated by various climate models of CMIP5 and CMIP6. Furthermore, we detect early warning signals for a critical transition in the snow cover extent on the Tibetan Plateau by applying critical slowing down indicators, lag-1 autocorrelation and detrended fluctuation analysis. We find that the snow cover of the Tibetan Plateau has been losing stability since 2008, revealing that the Tibetan Plateau is operating like a tipping element and approaching a potential tipping point. We further uncover that various climate extremes between the ARA and the Tibetan Plateau are significantly synchronized under climate change. Our framework provides new insights into how tipping elements are linked to each other and into the potential predictability of cascading tipping dynamics.