  Almost all known high temperature superconductors are cuprates, which can be
suitably modelled by the two dimensional Hubbard model. To understand the
interplay of various long range properties as antiferromagnetism and
superconductivity, one can calculate the phase diagram of the Hubbard model in
the charge density-temperature plane. This analysis is conveniently carried out
by means of exact renormalization group equations that we apply in the
formalism of the effective average action. For this purpose, we derive an
equivalent version of the Hubbard model that takes the form of a Yukawa theory.
  From this modified model long range order in various channels can be
extracted by simple calculation of bosonic expectation values. We are able to
reproduce the main features of the phase diagram of high temperature
superconductors. Furthermore, our analysis shows how the Mermin-Wagner theorem
can be reconciled with the existence of antiferromagnetic long range order at
non vanishing temperature and how the inclusion of different kinds of bosonic
fluctuations affect the phase diagram.
