Here follow the hand-written lecture notes of a mini-course on infrared bounds, reflection positivity and long range order in lattice models of statistical physics with continuous internal symmetries, I gave at the Summer School on Mathematical Methods of Statistical Physics, held in Prague, in October 1996. (Thanks due to Roman Kotecky for organising this wonderful series of events.)
At the time this was one of the first (or very few) comprehensive courses on these topics. Likewise, these handwritten notes were somewhat unique at their time.
For more recent and more updated lecture notes on these issues see e.g., Marek Biskup's survey " Reflection Positivity and Phase Transitions in Lattice Spin Models" (in: Lecture Notes in Math., 1970, Springer 2009. pp 1-86).
An extended and updated (as of 2021) version of these notes is available at the web-page of my graduate course on STATISTICAL PHYSICS WITH CONTINUOUS SYMMETRIES: SYMMETRY BREAKING, LONG RANGE ORDER, PHASE TRANSITIONS.
The model and its symmetries
Mermin-Wagner theorem (classical case)
Fröhlich-Simon-Spencer theorem
The model and its symmetries
Quantum correlation inequalities: Bogoliubov's inequality; Falk-Bruch inequality
Mermin -Wagner theorem - the quantum setting
Dyson-Lieb-Simon theorem