From finite lattice models to continuum field theories

从有限晶格模型到连续介质场论

• 批准号: RGPIN-2014-05102
• 负责人: SaintAubin, Yvan
• 金额: $ 1.02万
• 依托单位: Université de Montréal
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2014
• 资助国家: 加拿大
• 起止时间: 2014-01-01 至 2015-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=566952
• 关键词: finite; lattice; models; continuum; field

Phase transitions are part of everyday experience: liquid water turning into ice upon freezing or into gas upon boiling, sparkling water or champagne releasing gas when the bottle is open, etc. Their study has been part of physics and mathematics since about a century. A complete theory should explain how microscopic interactions, known from classical or quantum physics, give rise to macroscopic phenomena when the number of particles involved is large. The difficulty of the field lies in this passage from the description of a few interacting particles to that of an infinite number of them. In physics many-body problems are tackled within statistical physics where one accepts to discard details of each particle involved and concentrates instead on global properties of the system. For example one might ask whether or not a piece of iron behaves as a magnet instead of trying to describe how the spins of iron atoms are aligned, even though it is this alignment that causes magnetisation. In mathematics these problems fall in probability theory: each state of the system, e.g. a particular alignment of spins, is given a probability and macroscopic behavior is described from the set of all these probabilities. Phase transitions are therefore part of statistical physics and probability theory. Properties of phase transitions depend on several external parameters. The behavior of water depends on temperature, obviously, but also on pressure. Water boils at a lower temperature at high altitude. Many scientists have concentrated their efforts to critical values of these parameters. (Water has a single critical point among all the pairs (temperature, pressure).) The reason for this is that it is believed (and has been proved in a few cases) that physical behavior at these critical points displays a large family of symmetries, that is, some deformations, known as conformal transformations, leave the physics unchanged. These symmetries help in the description of phase transitions. The research supported by this grant will focus on two-dimensional lattice models of microscopic interactions. In the community these models are known as percolation, the Ising model, the XXZ spin chain, dense and dilute loop models, etc. They offer a natural laboratory to probe physical properties and prove them rigorously. They have a finite number of “particles”, they can be probed on the computer, they are believed to go to (logarithmic) conformal field theories (a distinguished set of continuum models) and rest upon an algebraic description that lends itself naturally to the study of the limit to large number of particles. Because of the latter property, these models can be studied using algebra and representation theory. In two physical dimensions these properties are remarkably powerful and the research will concentrate on two-dimensional models. The goal of this research is to describe how the conformal field theories can be understood from finite lattice models and how the large family of symmetries at critical points arises through a mathematically sound limit from finite to infinite number of particles.

相过渡是每天经验的一部分：液体水在冰冻时变成冰，煮沸时会变成冰，泡沫或香槟在开放瓶时释放气体等。他们的研究一直是大约一个世纪以来物理和数学的一部分。一个完整的理论应解释当涉及的粒子数量较大时，从经典物理或量子物理学中知道的微观相互作用如何产生宏观现象。该领域的困难在于从描述一些相互作用的粒子到无限数量的段落。在物理学中，多体问题在统计物理学中解决了，其中人们接受丢弃每个粒子的细节，而将其集中在系统的全球特性上。例如，人们可能会问一块铁是否表现为磁铁，而不是试图描述铁原子的旋转是如何对齐的，即使是这种比对引起磁化的。在数学中，这些问题属于概率理论：系统的每个状态，例如给出了特定的旋转比对，并从所有这些可能性的集合中描述了宏观行为。因此，相变是统计物理和概率理论的一部分。相变的属性取决于几个外部参数。水的行为显然取决于温度，但也取决于压力。水在高海拔的温度下沸腾。许多科学家将努力集中在这些参数的关键价值上。 （水在所有对（温度，压力）之间具有一个临界点。）原因是，人们认为它（在少数情况下已经证明），这些临界点处的身体行为显示出很大的对称性家族，也就是说，某些变形（称为构型转换）使物理学保持不变。这些对称性有助于描述相变。该赠款支持的研究将重点放在微观相互作用的二维晶格模型上。在社区中，这些模型被称为渗透，ISING模型，XXZ自旋链，密集和稀释环模型等。它们提供了自然实验室来探测物理特性并严格证明它们。它们具有有限数量的“颗粒”，可以在计算机上进行探测，据信它们转到（对数）保形场理论（一组杰出的连续元模型），并基于代数描述，该代数描述自然而然地将其自然地用于研究极限的大量粒子。由于后一种属性，可以使用代数和表示理论研究这些模型。在两个物理方面，这些特性非常强大，研究将集中在二维模型上。这项研究的目的是描述如何从有限的晶格模型中理解保形场理论，以及在临界点处的大型对称性如何通过从有限到无限数量的粒子数的数学声音限制而产生。