The Two-Dimensional Ising Model

Abstract

The two-dimensional Ising model is probably one of the most famous statistical models, and it has been extensively studied in the literature. Our aim in this chapter is to present a detailed study of its continuum limit, in the framework of conformally invariant (free fermionic or bosonic) field theories. After reviewing basic facts on the statistical-mechanical model, we concentrate on its continuum fermionic representation. This framework is particularly suitable for the computation of correlation functions of the energy operator on the plane. For correlations involving the spin operator, it is more convenient to consider a bosonic field theory, made of two independent Ising models. In this bosonic formulation, the spin operator has a simple realization in terms of the free field. To complete the study of correlators, we also present the solution of the continuum Ising model on the torus, and use it as an illustrative example of the general theory of conformal blocks covered in Chaps. 9 and 10.