Math Courses

Infrared bounds, reflection positivity and continuous symmetry breaking in classical and quantum spin systems

by Prof. Bálint Tóth (Bristol & Budapest)


List of topics covered in the lectures: 

        0. Quick survey of the Ising model – for reference and comparison.

  1. The Classical Heisenberg (a.k.a. O(N)) model and some variations.
  2. The Classical Mermin-Wagner Theorem:
    No continuous symmetry  breaking at positive temperatures in d=2.  

  3. Reflection positivity, Gaussian domination, infrared bounds.
    Fröhlich-Simon-Spencer Theorem: Continuous symmetry breaking at positive temperatures, in d>=3.  

  4. The Quantum Heisenberg (XXZ) models and their internal symmetries.  
  5. Bogoliubov's inequality and the Quantum Mermin-Wagner Theorem.
    No LRO at positive temperatures in d=2.

  6. Other quantum correlation inequalities (Bogoliubov, Röpstorff, Falk-Bruch)
  7. Reflection positivity - the quantum case.  Dyson-Lieb-Simon Theorem:
    Néel order in antiferromagnetic systems at positive temperatures, in d>=3.

        8. Bose lattice gas and the problem of Bose Einstein Condensation.  (time permitting)

Lecture notes covering the topics above are available below. The time schedule of the course may be found at the following webpage.

The lectures will be held at the GSSI Auditorium, and also streamed online through zoom. To receive the coordinates for the streaming, you may contact Serena Cenatiempo at serena dot cenatiempo at