From finite lattice models to continuum field theories

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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.