热力学相变、平均场理论与重整化（半）群：教学式导论
Thermodynamical Phase transitions, the mean-field theories, and the renormalization (semi)group: A pedagogical introduction
AI总结：
本文以教学视角梳理热力学相变研究脉络，从序参量与朗道平均场理论出发，分析其在临界点的失效问题，介绍威尔逊重整化方法对大关联长度难题的突破，并探讨相变非解析性、普适性起源等核心问题。
摘要：
在分析二阶热力学相变时，著名俄罗斯物理学家列夫·朗道提出了一个极为关键的概念——“序参量”。这一概念不仅整合了此前关于相变的零散理论认识（一系列平均场理论），还提出了重要的“自发对称性破缺”理论。如今，序参量概念已成为凝聚态物理与高能物理领域的范式，而朗道理论更是所有平均场理论的巅峰之作。平均场理论是对相变行为的优质定性描述工具，但所有平均场理论（包括朗道理论）在临界点处都会失效（即大关联长度问题）。量子多体系统中的大关联长度问题是理论物理学（涵盖凝聚态与粒子物理领域）已知的最棘手难题。正是肯·威尔逊的物理洞见与强大的数学能力，为解决这类难题开辟了道路。\n本手稿是对上述议题的综述性探讨。我们从冰/水、金刚石/石墨等简单相变实例入手，探讨以下重要问题：当热力学函数（如自由能等）是性质良好的解析函数时，非解析性（突变式相变）为何会出现？朗道的理论框架如何统一此前所有的平均场理论？为何所有平均场理论在临界点附近都会失效？威尔逊的理论框架如何超越所有平均场理论？涌现性与普适性的起源是什么？
英文摘要：
While analyzing second order thermodynamical phase transitions, Lev Landau (the famous Russian physicist) introduced a very vital concept, the concept of an "order parameter". This not only amalgamated the previous fragmentary theoretical understanding of phase transitions (an arsenal of mean-field theories) but also it put forward the important theory of "spontaneous symmetry breaking". Today, order parameter concept is a paradigm both in condensed matter physics and in high energy physics, and Landau theory is a pinnacle of all mean-field theories. Mean field theories are good qualitative descriptors of the phase transition behavior. But all mean-field theories (including Landau's theory) fail at the critical point (the problem of large correlation length). The problems with large correlation length in quantum many-body systems are the hardest problems known in theoretical physics (both in condensed matter and in particle physics). It was Ken Wilson's physical insights and his powerful mathematical skills that opened a way to the solution of such hard problems. This manuscript is a perspective on these issues. Starting with simple examples of phase transitions (like ice/water; diamond/graphite etc.) we address the following important questions: Why does non-analyticity (sharp phase transitions) arise when thermodynamical functions (i.e., free energies etc) are good analytic functions? How does Landau's program unify all the previous mean-field theories? Why do all the mean-field theories fail near the critical point? How does Wilson's program go beyond all the mean-field theories? What is the origin emergence and universality?
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