The Ising Model

Abstract

Most of the experiments in the neighborhood of critical points indicate that critical exponents assume the same universal values, far from the predictions of the “classical theories” (as represented by Landau’s phenomenology, for example). We now recognize that the universal values of the critical exponents depend on a just few ingredients:

• (i) The dimension of physical systems. Usual three-dimensional systems are associated with a certain class of critical exponents. There are experimental realizations of two-dimensional systems, whose critical behavior is characterized by another class of distinct and well-defined critical exponents.

• (ii) The dimension of the order parameter. For simple fluids and uniaxial ferromagnets, the order parameter is a scalar number. For an isotropic ferromagnet, the critical parameter is a three-dimensional vector.

• (iii) The range of the microscopic interactions. For most systems of physical interest, the microscopic interactions are of short range. We will see that statistical systems with long-range microscopic interactions lead to the set of classical critical exponents.