Algebraic methods for lattice models of statistical physics

统计物理晶格模型的代数方法

• 批准号: RGPIN-2019-05450
• 负责人: SaintAubin, Yvan
• 金额: $ 1.53万
• 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: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=764490
• 关键词: Algebraic; methods; lattice; models; statistical

Physics understands matter through its atomic or molecular structure. This description has been overwhelmingly successful to predict physical properties and discover new materials. It is particularly well-suited to systems with a finite, though not too large, number of atoms and molecules. Phenomena occurring only when a large number of atoms or molecules interact call for other techniques. Phase transition is one such phenomenon: it is known that no phase transitions may occur unless the number of interacting objects tends to infinity. The proposed project aims at understanding the passage from finite systems (say, those with a finite number of atoms) to their continuum limit (the theories that assume from the start an infinite number of interacting objects). In particular it wants to identify the properties of finite systems that reveal those of the infinite ones obtained by increasing the number of objects. The main mathematical tool of the project will be algebra. Several chapters of mathematics have played a role in the description of phase transitions. Analysis and algebra are probably the central ones. Algebra provides the tools to identify the symmetries of physical systems. Symmetries are operations that transforms a system into another one without changing its overall physical properties. These symmetries offer fundamental ways to approach physical systems and define them mathematically. The proposed project puts an emphasis on studying the algebraic structures arising in both the finite lattice models and their infinite continuum limits. The research will focus on two-dimensional lattice models of microscopic interactions. 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 and they are believed to go to (logarithmic) conformal field theories (a distinguished set of well-studied continuum models). Most importantly they rest upon an algebraic description that lends itself naturally to the study of the limit to large number of particles. Previous works in these directions, others' and mine, have contributed to both physics and mathematics. For example it is useful in physics to recognize emerging properties of a finite system, even though there are only partially realized. In mathematics, the study of algebraic structures of physical systems has suggested many new avenues of development or new ways of looking at existing results.

物理学通过其原子或分子结构理解至关重要。这种描述绝对成功地预测了物理特性并发现新材料。它特别适合具有有限的系统（尽管不是太大）原子和分子的系统。仅当大量原子或分子相互作用时，才会发生现象。相变是一种现象：众所周知，除非相互作用的对象的数量趋于无限，否则不会发生相变。拟议的项目旨在了解从有限系统（例如，原子数量有限的系统）到其连续限制的段落（从启动开始假设无限数量的相互作用对象的理论）。特别是要确定有限系统的属性，这些系统揭示了通过增加对象数量获得的无限系统的属性。该项目的主要数学工具将是代数。数学的几章在相变的描述中发挥了作用。分析和代数可能是核心。代数提供了识别物理系统对称性的工具。对称性是将系统转换为另一个系统而不改变其整体物理特性的操作。这些对称性提供了接近物理系统并数学定义它们的基本方法。拟议的项目重点是研究有限晶格模型及其无限连续体限制中产生的代数结构。该研究将重点放在微观相互作用的二维晶格模型上。这些模型被称为渗滤，伊辛模型，XXZ自旋链，密集和稀释的环模型等。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，并且据信它们转到（对数）保形场理论（一组杰出的经过良好研究的连续模型）。最重要的是，它们取决于代数描述，该描述自然而然地研究了大量粒子的极限。 以前在这些方向上的作品，他人和矿山都为物理和数学做出了贡献。例如，在物理学中识别有限系统的新兴特性很有用，即使只有部分实现。在数学中，对物理系统的代数结构的研究提出了许多新的发展途径或新的研究结果。