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: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=588213
• 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.

相变是日常经验的一部分：液态水在冻结时变成冰或在沸腾时变成气体，当瓶子打开时，气泡水或香槟释放气体等。 一个完整的理论应该解释，当涉及的粒子数量很大时，从经典物理或量子物理中已知的微观相互作用如何引起宏观现象。场的困难在于从描述几个相互作用的粒子到描述无限多个相互作用的粒子这一过程。 在物理学中，多体问题是在统计物理学中解决的，其中一个接受放弃每个粒子的细节，而专注于系统的全局属性。例如，人们可能会问一块铁是否具有磁铁的性质，而不是试图描述铁原子的自旋是如何排列的，即使正是这种排列导致了磁化。在数学中，这些问题属于概率论：系统的每个状态，例如自旋的特定排列，都被赋予一个概率，宏观行为从所有这些概率的集合中描述。因此，相变是统计物理学和概率论的一部分。 相变的性质取决于几个外部参数。很明显，水的行为取决于温度，但也取决于压力。水在高海拔地区沸腾的温度较低。许多科学家都集中精力，这些参数的临界值。（水在所有对（温度、压力）中只有一个临界点。）这是因为人们相信（并在少数情况下得到了证明），在这些临界点的物理行为显示出一大群对称性，也就是说，一些变形，称为保角变换，使物理学保持不变。这些对称性有助于描述相变。 这项资助支持的研究将集中在微观相互作用的二维晶格模型上。在社区中，这些模型被称为渗流，伊辛模型，XXZ自旋链，稠密和稀环模型等，它们提供了一个天然的实验室来探测物理性质并严格证明它们。它们有有限数量的“粒子”，它们可以在计算机上探测，它们被认为是（对数）共形场论（一组杰出的连续模型），并依赖于代数描述，这自然有助于研究大量粒子的极限。由于后者的性质，这些模型可以使用代数和表示论来研究。在两个物理维度上，这些属性非常强大，研究将集中在二维模型上。 本研究的目的是描述如何从有限格点模型中理解共形场论，以及如何通过从有限到无限数量的粒子的数学合理限制来产生临界点处的大家族对称性。