Phase Transitions and Critical Phenomena: Classical Theories

Abstract

(d) What is the asymptotic expression of the curve of coexistence of phases in the immediate vicinity of the critical point? (e) Use your results to obtain the critical exponents ß, ì, 8, and a. 5. Consider the Curie-Weiss equation for ferromagnetism, m = tanh(ßH + ßÀm) , Obtain an asymptotic expression for the isothermal susceptibility, X (T, H), at T = Tc for H --O. Obtain asymptotic expressions for the spontaneous magnetization for T .:.: Tc (that is, for T --0) and T ;: Tc (that is, for t --0-). 13 The Ising Model Most of the experiments in the neighborhood of critical points indicate that critical exponents assume the same universal values, far from the predic-tions of the "classical theories" (as represented by Landau's phenomenol-ogy, 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 phys-ical interest, the microscopic interactions are of short range. We wil see that statistical systems with long-range microscopic interactions lead to the set of classical critical exponents. Owing to the universal behavior of critical exponents, it is enough to ana-lyze very simple (but nontrivial) models in order to construct a microscopic theory of the critical behavior. The Ising model, including short-range in-teractions between spin variables on the sites of a d-dimensional lattice, plays the role of a prototypical system. The Ising spin Hamiltonian is given 258 13, The Ising Model 13. The Ising Model 259 by In one dimension, it is relatively easy to obtain an expression for this free energy. We wil use the technique of the transfer matrices, which can also be written in higher dimensions, to obtain a solution for the Ising chain. However, as shown by Ising in 1925, tils one-dimensional solution is quite deceptive, since the free energy is an analytic function of T and H (ex-cept at the trivial point T = H = 0), which precludes the existence of a spontaneous magnetization (and of any phase transition). Several approximate techniques have been developed to solve the Ising model in two and three dimensions. Some of them are quite simple and useful, and may lead to reasonable qualitative results for the phase dia-grams (besides providing useful tools to investigate more complex model systems). However, as pointed out before, phase transitions are associated with a nonanalytic behavior of the free energy in the thermodynamic limit. As a consequence, we should be warned against any truncations or pertur-bative expansions around the critical point. Indeed, most of the approxi-mate schemes can be written as a Landau expansion, leading to classical critical exponents. In a mathematical "tour de force," Lars Onsager, in 1944, obtained an analytical solution for the Ising model on a square lattice, with nearest-neighbor interactions, in the absence of an external field. For T -7 Te, the specific heat diverges according to a logarithmic asymptotic form, N 1í = -f2':aiaj -HLai, (ij) i=l where ai is a random variable assuming the values :f1 on the sites i = 1,2, ..., N of ad-dimensional hypercubic lattice. The first term, where the sum is over pairs of nearest-neighbor sites, represents the interaction ener-gies introduced to bring about ~n ordered ferromagnetic state (if J ~ 0). The second term, involving the interaction between the applied field Hand the spin system, is of a purely paramagnetic character (as we have already seen in previous chapters of this book). Since it was proposed by Lenz and solved in one dimension by Ernst Ising in 1925, the Ising model has gone through a long history ¡see, for example, the paper by S. G. Brush, in Rev. Mod. Phys. 39, 883 (1967)J. The Ising model can represent the main features of distinct physical systems. In the usual magnetic interpretation, the Ising spin variables are taken as spin components (that may be pointing either up or down, along the direction of the applied field) of crystalline magnetic ions. We may also consider a binary alloy of type AB. In this case, the spin variables indicate