Critical exponents of the n-component model via renormalisation group recursion formulae for dimension between 2 and 4

Abstract

The authors study the n-component phi 4 model in space dimensions 2<or=d<or=4, for various values of n using the Wilson recursion formula. In two dimensions, asymptotic freedom is seen when n>or=3, whereas a phase transition occurs for n=1, with the Ising-type value for the exponent nu . Approaching two-dimensions, the model is shown to tend to the corresponding nonlinear sigma -model Sn-1. The value of the exponent nu compares well with the first-order d-2 expansion. An extension due to Golner (1973), in the scalar case, of the recursion formula and giving an exponent eta not=0, is generalised to the vectorial case. It is solved numerically for 2<or=d<or=4. In three dimensions the value obtained for eta is the same within the error bars as the one computed by field theoretic techniques or high-temperature expansions. The value of eta in the Ising case, when d=2, compares well with the Onsager value.