The Gross-Neveu model and QCDs chiral phase transition

Quantum chromodynamics has a rather complicated phase structure. The finite temperature, chiral phase structure depends on the number of flavours and to a large extent on the particular values of the fermion masses. For two massless flavours there is a true second order transition. It has been argued that this transition belongs to the universality class of the three-dimensional O(4) spin model. The arguments have been questioned recently, and the transition was claimed to be mean field behaved. In this lecture we discuss this issue at the example of the three-dimensional, parity symmetric Gross-Neveu model at finite temperature, with a large number N of fermions. At zero temperature there is a phase where parity is spontaneously broken. At finite temperature, this model has a parity restoring second order transition. It reveals considerable similarity to the QCD chiral phase transition. There are related questions here concerning the universality class. We solve this problem essentially by means of the following methods: Large N expansion, dimensional reduction in the framework of quantum field theory, and high order convergent series expansions about disordered lattice systems.