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Ferroelectrics

Critical phenomena at phase transitions in fluids and model ferromagnets

Robert B. Griffiths

Published Jan 1, 1974• 1 citation
Publisher

Abstract

Various concepts used in discussing critical phenomena in fluids and ferromagnets are reviewed: order parameter, correlation length, critical exponents, scaling, universality, crossover effects, tricritical points, and the effects of long-range forces.
DOI
10.1080/00150197408237952

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A theory for tricritical phenomena proposed by Wegner and the author is described on an elementary level. The emphasis of the discussion is on (i) the differences between critical and tricritical phase transitions, (ii) the crossover phenomena that occur near tricritical points, and (iii) the determination of tricritical exponents, scaling fields and scaling functions. The results have been obtained by renormalization‐group techniques and a scaling field approach to crossover phenomena. The elements of these techniques are briefly explained.
To set the stage for our subsequent thorough discussion of dynamic critical phenomena, we first review the theoretical description of second-order equilibrium phase transitions. (Readers already well acquainted with this material may readily move on to Chapter 2.) To this end, we compare the critical exponents following from the van-der-Waals equation of state for weakly interacting gases with the results from the Curie–Weiss mean-field approximation for the ferromagnetic Ising model. We then provide a unifying description in terms of Landau–Ginzburg theory, i.e., a long-wavelength expansion of the effective free energy with respect to the order parameter. The Gaussian model is analyzed, and a quantitative criterion is established that defines the circumstances when non-linear fluctuations need to be taken into account properly. Thereby we identify d c = 4 as the upper critical dimension for generic continuous phase transitions in thermal equilibrium. The most characteristic feature of a critical point turns out to be the divergence of the correlation length that renders microscopic details oblivious. As a consequence, not only the correlation functions, but remarkably the thermodynamics as well of a critical system are governed by an emergent unusual symmetry: scale invariance. A simple scaling ansatz is capable of linking different critical exponents; as an application, we introduce the basic elements of finite-size scaling. Finally, a brief sketch of Wilson's momentum shell renormalization group method is presented, intended as a pedagogical preview of the fundamental RG ideas. Exploiting the scale invariance properties at the critical point, the scaling forms of the free energy and the order parameter correlation function are derived.
This article outlines a tutorial review of some recent progress in the theory of phase transitions and criticality. Topics mentioned are: |φ|6 field theories in (d=3)-dimensional space which correspond to the limits of tricriticality in spherical models; the desirability of finding real systems displaying critical endpoints, etc., in a magnetic field; scaling theories for first-order magnetic transitions in block and cylinder geometries, including spin-wave contributions, ϵ=4 - d expansions for finite-size effects at and near criticality; the universal correlation length amplitude for critical-point correlations in a finite cylinder and its evaluation in d = 2 dimensions on the basis of conformal covariance; the predictions of conformal covariance for other finite-size and surface critical-point properties; the implications, stemming from field theory, of conformal covariance and unitarity for general criticality in two dimensions, including infinitely many sets of exact critical exponents for Ising-like, three-state Potts-like, … systems; the discovery of new, exactly soluble two-dimensional, “restricted solid-on-solid” models which display all the predicted conformally covariant multicritical points and a second infinite set as well!
Abstract This first chapter is an elementary introduction to the models and methods that will be used later. The ferromagnetic (Curie) transition serves as our prototype for all second-order phase transitions, and serves also to introduce the notions of broken symmetry and of order parameter. The basic model of ferromagnetism, namely the Ising model, is described in Section 1.2, and solved for an elementary case. The reason why this model is so important is that it has an exact solution in two dimensions, which exhibits spontaneous magnetization. Section 1.3 is devoted to a first approach to mean-field methods, which remain amongst the most frequently exploited approximations even today. They will lead us to define a first set of critical exponents; the listing of such exponents will then be completed in Section 1.4, once correlation functions have been defined. Finally, in Section 1.5 a qualitative description of critical phenomena will enable us, in a first approximation, to define the concept of scale invariance and its relation to the behaviour of the theory under dilatations.
Overview As discussed in Chapter 16 , when we approach the critical temperature of a phase coexistence line, several fascinating things start to happen which together have come to be known as “critical phenomena.” It is observed that the spatial correlation between relevant order parameters like density, magnetization, etc. becomes infinitely large. This is reflected in the divergence of inverse susceptibility of the respective order parameters. Landau’s theory explains this in terms of the flattening of the free energy surface but fails to unravel many of the other mysteries of critical phenomena that have kept scientists engaged now for about a century. Critical phenomena are characterized by certain singular properties, like the universality of the critical exponents. Several aspects of the critical phenomena are discussed in this chapter with an aim to maintain continuity with earlier chapters. A simple theory (like that of Landau) cannot explain many of the observed features and this problem remained unsolved for a long time until K. G. Wilson could develop a sophisticated theory that explained almost everything. At the core of Wilson’s method is the renormalization group technique. Physically, this technique employs the expected invariance of the Hamiltonian due to a transformation that removes local interactions and replaces them by effective longer range interactions. Because of the existence of long range correlations, the transformation is expected to converge to a fixed point. The procedure is quite mathematical and is the only robust method to obtain the critical exponents. Here we discuss the method briefly, as an introduction to the subject.
The Ising fluid we have investigated is a system of hard spheres which carry an Ising spin. A short range ferromagnetic potential acts between particles. A renormalization group study of the model, based on the hierarchical reference theory of fluids, gives quantitative predictions on the phase diagram and the phase boundaries. The model can show two distinct phase transitions: a magnetic order-disorder transition and a liquid-vapor phase separation which meet at a tricritical point. The critical properties of the magnetic transition have been studied by Monte Carlo simulations at two densities in systems with up to 4000 particles. The locations of the critical points are in good agreement with the theoretical estimates. However, a finite-size scaling analysis shows effective critical exponents which deviate from the expected Ising values. These discrepancies resemble the analogous result of a previous numerical study of Heisenberg fluids. Here they are interpreted as due to crossover phenomena, partly related to Fisher exponent renormalization induced by the presence of noncritical density fluctuations.