From finite lattice models to continuum field theories

相变是日常生活的一部分：液态水在冻结时变成冰，在沸腾时变成气体，气泡水或香槟在打开瓶子时释放气体，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，当涉及的粒子数量很大时，从经典或量子物理学中已知的微观相互作用是如何产生宏观现象的。这个领域的困难在于从描述几个相互作用的粒子过渡到描述无数个相互作用的粒子。**在物理学中，多体问题是在统计物理学中解决的，在统计物理学中，人们接受抛弃所涉及的每个粒子的细节，而专注于系统的整体特性。例如，人们可能会问一块铁是否具有磁铁的行为，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：系统的每个状态，例如自旋的特定排列，都有一个概率，宏观行为是由所有这些概率的集合描述的。因此，相变是统计物理和概率论的一部分。**相变的性质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。水在高海拔地区沸腾的温度较低。许多科学家都把精力集中在这些参数的临界值上。（水在所有对（温度、压力）中只有一个临界点。）这样做的原因是，人们相信（并在少数情况下得到了证明），在这些临界点处的物理行为显示了大量的对称性，也就是说，一些变形，称为保形变换，使物理不变。这些对称性有助于描述相变。**本基金支持的研究将集中于微观相互作用的二维晶格模型。在学界，这些模型被称为渗流模型、Ising模型、XXZ自旋链模型、密集环模型和稀释环模型等。它们提供了一个天然的实验室来探索物理特性并严格地证明它们。它们有有限数量的“粒子”，它们可以在计算机上进行探测，它们被认为是（对数）共形场论（一组杰出的连续体模型），并依赖于一种代数描述，这种描述自然地适合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理论来研究。在两个物理维度中，这些特性非常强大，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上的大量对称性。

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.