Integrable systems and statistical physics

We study mathematical questions arising in integrable systems and statistical physics. The questions that interest us come from several sources, such as

• Models for phase transition, e.g. the Ising- and Heisenberg models,
• Field theory, e.g. lattice gauge theories,
• Many-particle systems, e.g. the Boltzmann equation and Calogero-Moser systems,
• Integrable wave equations.

The study of these questions often involves a combination of algebra, analysis and probability theory.

Possible Master Projects

Quantum algebras in mathematical physics (Martin Hallnäs). In mathematical models of physical systems, symmetries have classically been described by groups, e.g. permutation- and rotation groups, and related (infinitesimal) objects such as Lie algebras. In more recent years, it has been realized that symmetries of many interesting and important physical systems are naturally described by `quantizations’ of such classical objects (groups, Lie algebras, etc.). One particularly important class of examples are the so called quantum groups, introduced in the 1980s independently by Drinfeld and Jimbo, and which are closely related to solvable lattice models in statistical physics, knot invariants and more. The idea of this project is to study one or more of such modern quantum versions of classical objects from algebra and how they can be used with great effect to describe, study and solve important models in modern mathematical physics.

Valence-bond solid states in quantum statistical physics (Jakob Björnberg). Valence-bond solids (VBS) are states in quantum statistical physics which display many interesting properties. While ultimately just vectors in a large tensor product, they play a fundamental role for our understanding of several aspects of quantum statistical physics. They arise as ground-states (i.e.\ eigenvectors for the lowest eigenvalue) for certain antiferromagnetic Hamiltonians (Hermitian operators). The original motivation for studying VBS states was to prove a version of a famous conjecture of Haldane (the original conjecture is still open) and they have proved valuable far beyond this. The key idea for the construction of VBS states is a simple observation about the representation theory for SU(2), but as it turns out many interesting properties of VBS states can be studied using combinatorial / probabilistic methods. Thus, the topic has a nice combination of mathematical techniques. Part of the project will be to review the subject area, which may include the basics of quantum spin systems, their ground-states, and Haldane's conjecture. From there, the project can be taken in various directions based on generalizations of the original VBS state and their properties and applications.