Advanced Statistical Physics

summer term 2023
Lecturer: Prof. Dr. Tilman Enss
Link to LSF
22 participants

This advanced theory lecture builds on the statistical physics course (MKTP1) and introduces paradigmatic models of statistical physics and their critical properties near phase transitions.  In particular, we shall discuss the Heisenberg and O(N) vector models, the nonlinear sigma model, the XY model, the Sine-Gordon model, and the spherical model.  By computing their critical behavior, one can understand the phase transitions in many different systems in statistical physics, condensed matter physics and beyond, which belong to the same universality classes.  We will use field theoretic methods and introduce renormalization, epsilon expansion, and duality transformation.


  1. Landau theory and O(N) vector model
  2. Renormalization group and universality
  3. Nonlinear sigma model and epsilon expansion
  4. Topological excitations in the XY and Sine-Gordon models and the Kosterlitz-Thouless transition
  5. Spherical model and quantum phase transitions

Date and Location

Lecture Thursday 11.15-13.00h, Philosophenweg 19 seminar room


2023-04-20: Lecture 1, Landau theory and mean field ansatz (lecture notes pages AST-1-1 to 1-4)

2023-04-27: Lecture 2, Fluctuations (1-5 to 1-8)

2023-05-04: Lecture 3, O(N) model and renormalization (1-9 to 2-2) (watch here)

2023-05-11: Lecture 4, RG equations (2-2 to 2-6)

2023-05-25: Lecture 5, Universality and multiple fixed points (2-6 to 2-10)

2023-06-01: Lecture 6, Crossover phenomena, scaling relations and Goldstone modes (2-11 to 3-1)

2023-06-08: holiday

2023-06-15: Lecture 7, Nonlinear sigma model and its renormalization (3-2 to 3-6)

2023-06-22: Lecture 8, XY model: spin-wave excitations (3-7 to 4-4)

2023-06-29: Lecture 9, XY model: vortices and Coulomb gas (4-5 to 4-8)

2023-07-06: Lecture 10, Sine-Gordon model and BKT transition (4-9 to 4-12)

2023-07-13: Lecture 11, Quantum phase transitions and spherical model (4-13 to 5-3)

2023-07-20: Lecture 12, Quantum critical scaling (5-4 to 5-6)


In this lecture we use the field theoretical language; for a recap see for instance Mudry chapter 1.

For starters:

  • Cardy, Scaling and Renormalization in Statistical Physics, Cambridge University Press (1996)
  • Mudry, Lecture Notes on Field Theory in Condensed Matter Physics, World Scientific (2014)

Further reading:

  • Altland and Simons, Condensed Matter Field Theory, Cambridge University Press (2010)
  • Kadanoff, Statistical Physics: statics, dynamics and renormalization, World Scientific (2000)
  • Negele and Orland, Quantum Many-Particle Systems, Addison-Wesley (1988)
  • Zinn-Justin, Phase Transitions and Renormalization Group, Oxford University Press (2007)

Practice groups

Advanced Statistical Physics
summer term 2023
T. Enss
Link zum LSF
22 participants