Mean Field Approximation

Abstract

The simple mean field approximation is universally applicable and yields qualitatively correct results for many lattice systems. In some cases the approximation even produces exact results, for example for universal quantities. In the mean field approximation one replaces the interaction of a microscopic spin with its neighboring spins by an approximate interaction with an averaged spin. Hence the calculation of the free energy density or the order parameter reduces to a single spin problem. In this chapter we discuss the approximation for general spin models and lattice field theories in arbitrary dimensions. The derivation is based upon the variational principle for the effective action and also applies to models with non-trivial target spaces. On the way we introduce the critical exponents and calculate their mean field values. The mean field approximations for the Ising model, standard Potts models, clock models, lattice scalar models, non-linear O(N) models and antiferromagnetic systems are derived. The phenomenological Landau theory for phase transitions is outlined as well.