Phase Transitions and Lattice Systems

A central topic in Statistical Mechanics is the theory of phase transitions. In this chapter we introduce the problem and discuss some basic rigorous results. Our main theoretical laboratory is the Ising model in diverse dimensions. After discussing a few generalities on phase transitions, we present the Yang-Lee theory and the Peierls argument for the existence of spontaneous magnetization at low temperature. We introduce the formalism of the transfer matrix and use it to characterize the pure equilibrium states using Frobenius complete reducibility of non-negative matrices. The one-dimensional lattice models are solved explicitly. After introducing the convenient language of lattice gauge theory, we prove the order-disorder self-duality of the two-dimensional Ising model in the context of general electromagnetic dualities in Abelian gauge theories. We construct disorder and Fermi operators for the two-dimensional Ising model. Finally we discuss the Mermin-Wagner theorem on the absence of spontaneous breaking of continuous symmetries in two dimensions. In the last section we introduce the notions of universality, critical exponents, and critical dimensions.