Algebraic methods for lattice models of statistical physics

点击翻译按钮获取中文摘要

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.**