The local potential approximation of the renormalization group and its applications

In the local potential approximation, the renormalization group is reduced to a differential equation. We study the general properties of the equation and in particular we show that the RG flow is the gradient of a scalar function. Then, the differential equation is solved numerically for two classes of models. The first one is that of the usual n-component Heisenberg models and serves as a quantitative test of the approximation. More challenging are the Stiefel non-linear σ-models Vn,2 which are used to describe a phase transition with a symmetry O(n) broken down to O(n-2). For these models, the usual RG perturbative expansions fail. Reliable three-dimensional critical behaviors are obtained using the local potential approximation. In particular, the model V3,2 in three dimensions is of physical interest: it possesses an almost second order transition with ν= 0.63. ((orig.))