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Scaling, Criticality and Renormalization: Percolation, Ising model and beyond

Module PH2292

This module handbook serves to describe contents, learning outcome, methods and examination type as well as linking to current dates for courses and module examination in the respective sections.

Basic Information

PH2292 is a semester module in English language at Master’s level which is offered in winter semester.

This Module is included in the following catalogues within the study programs in physics.

  • Specific catalogue of special courses for condensed matter physics
  • Complementary catalogue of special courses for nuclear, particle, and astrophysics
  • Complementary catalogue of special courses for Biophysics
  • Complementary catalogue of special courses for Applied and Engineering Physics
  • Specialization Modules in Elite-Master Program Theoretical and Mathematical Physics (TMP)

If not stated otherwise for export to a non-physics program the student workload is given in the following table.

Total workloadContact hoursCredits (ECTS)
300 h 90 h 10 CP

Responsible coordinator of the module PH2292 is Johannes Knolle.

Content, Learning Outcome and Preconditions


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. We will investigate the phenomenon of a phase transition, that is, a sudden change in a global feature that characterises a system.

Atomic physics and quantum mechanics determine the physics of matter on the microscopic scale, but how do the short-ranged interactions that apply at the nanoscale give rise to macroscopic phenomena, for example in condensed matter physics, such as the freezing of liquids, the magnetisation of a ferromagnet, and the onset of superfluidity and superconductivity? 

It turns out that systems near phase transitions have 'universal' properties that do not depend on microscopic details. In this course we will focus on canonical basic models of condensed matter physics displaying phase transitions: the percolation model and the Ising model. We will see how fluctuations destroy order at a phase transition, turning the system into a fractal structure. The course will start with simple effective theories to describe percolation and magnetic ordering. It will introduce the basic Landau theory of phase transitions. It will then proceed to see how such theories break down near critical points and how renormalization group ideas are needed to describe the system at criticality. The mathematical framework developed will be a powerful framework applicable also outside the realm of equilibrium physics.

The following topics are covered in this module:

  • Phase transitions and spontaneous symmetry breaking
  • Percolation theory as an introduction to the scaling hypothesis
  • Scale invariance and fractal dimension
  • Scale invariance of critical points
  • Ising model as a simple statistical mechanics model of magnetism
  • Divergent fluctuations and susceptibility at critical points
  • Critical exponents and universality classes
  • Mean-field theory
  • Renormalization group methods
  • Effective field theories and renormalization

Learning Outcome

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

  • Describe the notion of an order parameter in a phase transition
  • Understand the relationship between a diverging correlation length and scale invariance at a critical point
  • Understand scale invariance in a statistical fractal structure
  • Use the scaling hypothesis to deduce scaling relations among critical exponents
  • Use the scaling hypothesis as the foundation of the renormalization group
  • Derive exact solutions of Percolation in one dimension and on the Bethe lattice for the mean cluster size, cluster size distribution, and strength of the percolating cluste
  • Define magnetisation, magnetic susceptibility, the spin correlation function and the
    spin correlation length for the Ising model
  • Use a mean-field theory of the Ising model to calculate basic observables
  • Understand and apply real space renormalization group methods of the Ising model
  • Understand and apply momentum space renormalization group methods to effective field theories


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

Courses, Learning and Teaching Methods and Literature

Learning and Teaching Methods

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 presented in the exercise classes by the students themselves under supervision in order to develop the the crucial skills to understand a physics problem and perform calculations.


Blackboard lectures, written notes for download, exercise sheets, course homepage


  • 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

Module Exam

Description of exams and course work

There will be a written exam of 90 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 exact solutions in one dimension and on the Bethe lattice for the mean cluster size, cluster size distribution, and strength of the percolating cluster
  • Describe the phase diagram of the Ising model in terms of temperature and applied field
  • Implement an approximate single-parameter decimation scheme for the real space renormalisation group for the 2D Ising model in zero field; explain the need for approximation; give physical interpretations of the fixed points; give an estimate of the correlation length exponent.

In the exam the following learning aids are permitted: hand-written sheet with formulas, double-sided

Exam Repetition

The exam may be repeated at the end of the semester.

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