Approximate Methods and Landau Theory

In this chapter we introduce some basic approximation schemes to study phase transitions qualitatively. We start from the mean field approximation which we use to solve the Ising model and the q-Potts model which have, respectively, a second order and a first order phase transition provided the spatial dimension d is not to small. As a refinement of the mean field approach for the Ising model we also discuss in detail the Bethe-Peierls method. The mean field solution of the Ising and Potts models is used to motivate the phenomenological Landau theory of phase transitions: we analyze the four prototypical situations that may arise in this context. Then we introduce the Landau-Ginzburg refinement that takes care (to some extend) of the fluctuations in the order parameter. We digress on functional (path) integrals and the techniques to compute them, with special reference to the Gaussian ones and the corresponding correlation functions. As an illustration we discuss in detail the Ising model in d dimensions from the Landau-Ginzburg perspective. We close by introducing the Ginzburg criterion for the validity of the mean field approximation.