Mathematical Statistical Physics

The module covers the following topics: Gibbs measures: DLR conditions, existence, uniqueness (Dobrushin's theorem), phase transitions, absence of spontaneous symmetry breaking in two dimensions. Ising model: high temperature phase, Peierls argument, cluster expansion, Fortuin-Kasteleyn representation, FKG inequality, spontaneous symmetry breaking in continuous models. Non-equilibrium model systems: Exclusion processes, matrix product ansatz, interacting particle systems.

The main goal of this module is to acquire a deeper mathematical and physical understanding of phase transitions and collective phenomena that occur in macroscopic interacting particle systems.

After successful completion of the module the students are able to:

1. apply notions to handle infinite systems in the thermodynamic limit in a rigorous way

The primary goal is an in depth mathematical and physical understanding of macroscopic interacting particle systems their collective phenomena like phase transitions.

No prerequisites in addition to the requirements for the Master’s program in Quantum Science and Technology. Familiarity with quantum mechanics is assumed, at the level of an introductory course from a Bachelor degree in physics. Basic notions of probability theory and functional analysis can be helpful.

The module consists of a lecture series (4 SWS) and exercise classes (2 SWS), comprising two lecture sessions and one exercise session per week. The main teaching material is presented on the blackboard or by beamer. Lectures are supplemented by weekly problem sets, deepening the understanding of core concepts through concrete calculations. Solutions to the problem sets are discussed in the exercise sessions. Participation in the exercise classes is strongly recommended, since the exercises are aids for acquiring a deeper understanding of the core tools of condensed matter many-body physics and field theory and for practicing to solve typical exam problems.

Blackboard presentations, slides.

Standard textbooks on many-body theory, e.g.:

• Sacha Friedli and Yvan Velenik: “Statistical Mechanics of Lattice Systems: a Concrete Mathematical Introduction”
• Ola Bratteli, Derek Robinson: “Operator Algebras and Quantum Statistical Mechanics”

There will be a written exam of 180 minutes duration. Therein the achievement of the competencies given in section learning outcome is tested exemplarily at least to the given cognition level using conceptual questions and computational tasks.

For example an assignment in the exam might be:

• Demonstrate the existence of the pressure in a perturbed Ising model
• Show that the set of DLR measures is invariant under local perturbations
• Prove that a KMS state with a symmetry implies that the time evolution has the same symemtry