O(N) vector model in the ordered phase: Goldstone modes

Abstract

Abstract Chapter 10 examines the case of the spontaneous breaking of a continuous symmetry. When there is spontaneous symmetry breaking, the correlation length diverges at all temperatures in the ordered phase. Physical systems then display universal properties for all temperatures below Tc. This phenomenon is a consequence of the existence of Goldstone modes (massless bosons, in the quantum field theory terminology). Chapter 10 considers a lattice model with O(N) symmetry. In the continuum limit, it leads to the non–linear sigma model. A renormalization group can then be constructed and universal properties can be derived at and near dimension 2 in the framework of dimensional continuation and a d − 2 expansion. In dimension 2, the O(N) symmetric model illustrates asymptotic freedom with symmetry restoration and the Mermin–Wagner–Coleman theorem. In d > 2, a crossover scale appears, between infrared behaviour and critical behaviour.