The Ising Model at Criticality: Conformal Field Theory, the Random Current Representation, and Renormalization

Abstract

There is currently a large gap in the literature on the Ising model, one of the most significant models in statistical mechanics. Generally, articles focus on one specific technique used to analyze the model in a given dimensionality. The goal of this paper is to present a self-contained (to a reasonable extent) review of the various techniques used to analyze this model in all dimensions and the connections between the many different pictures. Discussions of the Ising model often present lattice solutions, renormalization theory, and conformal field theory techniques, but this paper will seek to incorporate the innovative random current formulation first discussed by Griffiths, Hurst, and Sherman, but more fully appreciated by M. Aizenman in his 1982 paper ”Geometric Analysis of φ\(^{4}\) Fields and Ising Models. Parts I and II." As an introduction, the one-dimensional model, as well as the simplicity and limitations of mean-field theory, will be presented. The main portion of this work will present a discussion of two lattice techniques for the two-dimensional model, namely the Kac-Ward solution and the random current approach; the connection to the free- fermion formulation and two-dimensional conformal field theory approach; and the power of the Callan-Syamanzik equation from renormalization theory in analyzing the critical Ising model in three dimensions and higher.