Scaling, Criticality and the Renormalization Group in Statistical Physics

Statistical mechanics is the branch of physics in which statistical methods are employed to understand how a large number of simple microscopic constituents of a system give rise to macroscopic properties. In this course we will study the universal features of phase transitions. We note that the course is complementary to the "Advanced Statistical Physics" lecture concentrating on more technical details and advanced topics in condensed matter systems.

The course focuses on the the renormalization group (RG) framework to describe a whole range of different paradigmatic systems. For example, we will learn how to apply the RG scheme to understand criticality and universal scaling in Percolation, the Ising model, the Phi-4 theory, the Kosterlitz-Thouless transition and in disordered systems.

The following topics are covered in this module:

• The general theory of the renormalization group
• Percolation theory as an introduction to the scaling hypothesis and RG
• Block spin RG and Widom scaling in the Ising model
• Effective field theory and perturbative RG (Wilson-Fisher fixed point)
• Non-linear sigma models
• XY model and the Kosterlitz-Thouless transition (Coulomb liquid RG)
• Disorder effects on phase transitions

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

• Understand the relationship between a diverging correlation length and scale invariance at a critical point
• Use the scaling hypothesis as the foundation of the renormalization group
• Derive the Widom scaling Ansatz from the real space RG of the Ising model
• Understand and apply momentum space renormalization group methods to effective field theories
• Analyse the stability of fixed points and how to calculate critical exponents via RG

No preconditions in addition to the requirements for the Master’s program in Physics.

The lecture will present the content via blackboard presentation with the occasional help of slides and animations. The lecture will focus on explaining basic methods via illustrative examples. Homework problems will cover important aspects of the subject and will help develop analytic skills of the students. The homework problems are discussed and partly presented in the exercise classes by the students themselves in order to develop the crucial skills to understand a physics problem and perform calculations.

Blackboard lectures, written notes for download, exercise sheets, all course material available via Moodle

• K. Christensen and N.R. Moloney, Complexity and Criticality. Imperial College Press, 2005.
• J.M. Yeomans, Statistical Mechanics of Phase Transition, OUP 1992.
• J. Cardy, Scaling and Renormalization in Statistical Physics, CUP 1996.
• M. Kardar, Statistical Physics of Particles, CUP 2007.
P.M. Chaikin and T.C. Lubensky, Principles of Condensed Matter Physics, CUP 1995.

There will be an oral exam of 30 minutes duration. Therein the achievement of the competencies given in section learning outcome is tested exemplarily at least to the given cognition level using calculation problems and comprehension questions.

For example an assignment in the exam might be:

• Derive the phase diagram and critical exponents via RG for 2D site percolation
• Describe the Kadanoff block spin RG procedure for the Ising model
• Discuss the idea of the epsilon expansion and the Wilson-Fisher fixed point

Participation in the exercise classes is strongly recommended since the exercises prepare for the problems of the exam and rehearse the specific competencies.