The Structure of the Vacuum. I: Two-Dimensional Non-Linear O(3) Model

Abstract

Instanton contributions to the correlation functions in the two-dimensional O(3) non-liner σ model are carefully analyzed. Imposing the cluster property upon the vacuum, we conclude that the vacua are two-fold degenerate and that a new kind of topological symmetry breaking occurs. Applying this result to the classical O(3) Heisenberg spin model it is argued that the crossover from strong to weak coupling is due to a new kind of phase transition. An order parameter is the sign of the topological number density. As a numerical result we also obtain the correlation length in the weak coupling region, which coincides remarkably with the result by Shenker and Tobochnik.