Spontaneous breakdown of continuous symmetries near two dimensions

Abstract

The long-distance properties of classical Heisenberg ferromagnets below the transition point are related to a continuous-field theory, the nonlinear $\sigma$ model. The renormalizability of this model in two dimensions and its ultraviolet asymptotic freedom are used to derive renormalization-group equations valid above $d=2\mathrm{}$. It is argued that this model is renormalizable up to four dimensions. The scaling properties which incorporate critical and Goldstone singularities follow. Explicit calculations of exponents and of correlation functions in powers of $d-2\mathrm{}$ are given. A technique is proposed to make calculations in the symmetric phase applicable even in two dimensions.