Peierls Argument and Duality Transformations

Abstract

This chapter contains various exact results which apply to many lattice models. First we present the neat Peierls argument for the Ising model and prove the existence of two ordered phases at low temperatures. The argument can be extended to other spin models with short range interactions and discrete target spaces. We continue with the duality transformations which relate two lattice models. It maps one system at weak coupling or low temperature into another system at strong coupling or high temperature and thus leads to new insights into the strong-coupling regime of lattice systems. Duality transformations exist for many Abelian theories. In great detail we discuss the transformations for the Ising model, Z n spin models, Z n gauge models and U(1) gauge theory in various dimensions. The results are applied to compute critical temperatures of self-dual models or to relate the critical temperatures of dual models. The end-of-chapter problems deal with the duality transformations of Potts models.