O(N) models and the Absence of Continuous Symmetry Breaking

Abstract: Two weeks ago, we observed that the Ising Model in Z^d exhibits discrete symmetry breaking at low temperatures in the sense that its Gibbs measures are not individually invariant under a global spin flip. In this talk, we introduce a more general framework to study models whose spins can be acted on by SO(N) and ask when the Gibbs measures are invariant under these global rotations. We give some intuition as to why a Peierls-type argument cannot hold in the continuous setting, and then we prove a result of Mermin and Wagner which says that the Gibbs measures will be SO(N)-invariant provided that d=1 or 2 and the dimension N of the space of spins is at least 2. The proof will rely crucially on the existence of spin waves in higher N-dimensions, which allow us to alter spins inside our domain at arbitrarily low energy cost.