From finite lattice models to continuum field theories

相变是日常经验的一部分：液态水在冻结时变成冰，或者在沸腾时变成气体，汽水或香槟在瓶子打开时释放气体，等等。大约一个世纪以来，他们的研究一直是物理和数学的一部分。一个完整的理论应该解释，当所涉及的粒子数量很大时，从经典或量子物理中已知的微观相互作用如何产生宏观现象。这一领域的困难在于这一段从几个相互作用的粒子的描述到无限多个相互作用的粒子的描述。在物理学中，多体问题是在统计物理学中处理的，其中一个人接受放弃所涉及的每个粒子的细节，而专注于系统的整体性质。例如，有人可能会问一块铁是否像磁铁一样，而不是试图描述铁原子的自旋是如何排列的，尽管正是这种排列导致了磁化。在数学中，这些问题属于概率论：系统的每种状态，例如特定的自旋排列，都被赋予一个概率，并从所有这些概率的集合中描述宏观行为。因此，相变是统计物理和概率理论的一部分。相变的性质取决于几个外部参数。显然，水的行为不仅取决于温度，还取决于压力。在高海拔地区，水在较低温度下沸腾。许多科学家集中精力研究这些参数的临界值。(水在所有对中只有一个临界点(温度、压力)。)这样做的原因是，人们相信(并在少数情况下得到了证明)，这些临界点的物理行为表现出一大族对称性，即一些变形，即所谓的保角变换，使物理保持不变。这些对称性有助于描述相变。这项拨款支持的研究将集中在微观相互作用的二维晶格模型上。在社会上，这些模型被称为渗流模型、伊辛模型、XXZ自旋链、密集和稀疏回路模型等。它们提供了一个天然的实验室来探索物理性质并严格证明它们。它们具有有限数量的“粒子”，它们可以在计算机上探测，它们被认为是(对数)共形场理论(一组杰出的连续介质模型)，并依赖于一种代数描述，这种描述自然地有助于研究大量粒子的极限。由于后者的性质，这些模型可以用代数和表示论来研究。在两个物理维度上，这些性质是非常强大的，研究将集中在二维模型上。本研究的目标是描述如何从有限晶格模型理解共形场理论，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上的大对称族。

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.