A phenomenological description of first-order phase transitions is proposed in terms of the solution of a certain linear, first-order, partial differential equation suggested by Fisher. In the asymptotic region of the critical point, the coefficient functions of the equation are uniquely determined by the scaling laws. The imposition of the further requirement that the phase boundary coincide with a characteristic curve of this equation is shown to lead to solutions that can reproduce the singular behavior---for example, the density and entropy discontinuities for the case of the liquid-gas transition---along the entire vapor-pressure curve. The important role played by the characteristic curves is stressed and confirmed for the special case of the ideal, $d$-dimensional, Bose gas. New relations involving discontinuities in the second derivatives of the thermodynamic functions are derived and shown to be useful in obtaining the coefficient functions near the phase boundary. Details are presented for the case of ferromagnetic and ferroelectric systems and include a comparison with experiment.
A phenomenological description of first-order phase transitions is proposed in terms of the solution of a certain linear, first-order, partial differential equation suggested by Fisher. In the asymptotic region of the critical point, the coefficient functions of the equation are uniquely determined by the scaling laws. The imposition of the further requirement that the phase boundary coincide with a characteristic curve of this equation is shown to lead to solutions that can reproduce the singular behavior---for example, the density and entropy discontinuities for the case of the liquid-gas transition---along the entire vapor-pressure curve. The important role played by the characteristic curves is stressed and confirmed for the special case of the ideal, d-dimensional, Bose gas. New relations involving discontinuities in the second derivatives of the thermodynamic functions are derived and shown to be useful in obtaining the coefficient functions near the phase boundary. Details are presented for the case of ferromagnetic and ferroelectric systems and include a comparison with experiment.
Scaling behavior for first-order phase transitions can be derived alternatively but consistently from renormalization-group, phenomenological, or finite-size considerations. A general analysis of densities at a renormalization-group fixed point demonstrates that if the coexistence of p distinct phases is possible, then p distinct eigenvalue exponents must equal the spatial dimensionality. This basic eigenvalue (or scaling) exponent condition can also be derived phenomenologically by various arguments not depending on detailed renormalization-group considerations. A scaling description of first-order phase transitions is presented and extended to finite systems with linear dimensions L, leading to a rounding proportional to ${L}^{\ensuremath{-}d}$, response-function maxima varying as ${L}^{d}\ensuremath{\propto}N$, and boundary-condition-dependent shifts which may be as large as $\ensuremath{\sim}{L}^{\ensuremath{-}1}$.
Relations between critical exponents, or scaling laws, at both continuous and discontinuous quantum phase transitions are derived and discussed. In general there are multiple dynamical exponents at these transitions, which complicates the scaling description. Some rigorous inequalities are derived, and the conditions needed for these inequalities to be equalities are discussed. Scaling laws involving the specific-heat exponents that are specific to quantum phase transitions are derived and contrasted with their counterparts at classical phase transitions. We also generalize the ideas of Fisher and Berker and others for applying (finite-size) scaling theory near a classical first-order transition to the quantum case. We then apply and illustrate all of these ideas by using the quantum ferromagnetic phase transition in metals as an explicit example. This transition is known to have multiple dynamical scaling exponents, and in general it is discontinuous in clean systems but continuous in disordered ones. Furthermore, it displays many experimentally relevant crossover phenomena that can be described in terms of fixed points, originally discussed by Hertz, that ultimately become unstable asymptotically close to the transition and give way to the asymptotic fixed points. These fixed points provide a rich environment for illustrating the general scaling concepts and exponent relations. We also discuss the quantum-wing critical point at the tips of the tricritical wings associated with the discontinuous quantum ferromagnetic transition from a scaling point of view.
Phase transitions are defined, and the concepts of order parameter and spontaneously broken symmetry are discussed. Simple models for magnetic phase transitions are introduced, together with some experimental examples. Critical exponents and the notion of universality are defined, and the consequences of the scaling assumptions are derived. Phase transitions and order parameters It is a fact of everyday experience that matter in thermodynamic equilibrium exists in different macroscopic phases. Indeed, it is difficult to imagine life on Earth without all three phases of water. A typical sample of matter, for example, has the temperature–pressure phase diagram presented in Fig. 1.1: by changing either of the two parameters the system may be brought into a solid, liquid, or gas phase. The change of phase may be gradual or abrupt. In the latter case, the phase transition takes place at well defined values of the parameters that determine the phase boundary. Phase transitions are defined as points in the parameter space where the thermodynamic potential becomes non-analytic. Such a non-analyticity can arise only in the thermodynamic limit, when the size of the system is assumed to be infinite. In a finite system the partition function of any system is a finite sum of analytic functions of its parameters, and is therefore always analytic. A sharp phase transition is thus a mathematical idealization, albeit one that describes the reality extremely well. Macroscopic systems typically contain ∼ 10 23 degrees of freedom, and as such are very close to being in the thermodynamic limit. The phase boundaries in Fig. 1.1, for example, for this reason represent reproducible physical quantities.
The usual static scaling laws are generalized to nonequilibrium phenomena by making assumptions on the behavior of time-dependent correlation functions near the critical point of second-order phase transitions. At any temperature different from Tc, the correlation functions are assumed to reflect the hydrodynamic behavior of the system, for sufficiently long wavelengths and low frequencies. As the critical temperature is approached, however, the range of spatial correlations in the system diverges, and the domain of applicability of hydrodynamics is reduced to a vanishingly small region of wavelengths and frequencies. The dynamic-scaling assumptions lead to predictions for the behavior of the hydrodynamic parameters near Tc, as well as for the form of the correlation functions for macroscopic distances and times, outside the hydrodynamic range. In particular, singularities are predicted to occur in the temperature dependence of transport coefficients, and anomalies are expected in the frequency spectrum of certain operators, which are observable by inelastic scattering of neutrons or light. A distinction is made between the restricted dynamic-scaling hypothesis, which refers to the order parameter only, and extended dynamic scaling, which applies to other operators and involves stronger assumptions. Applications are discussed to antiferromagnets, ferromagnets, the gas-liquid critical point, and the $\ensuremath{\lambda}$ transition in superfluid helium. Specific experiments are suggested to test the scaling assumptions, and existing experimental evidence is briefly reviewed. Finally, a comparison is made with other theories of dynamical behavior near critical points.
In Chapter 3, we examined two examples of phase transitions, the paramagnetic–ferromagnetic transition (Section 3.1.4) and the liquid–vapour transition (Section 3.5.4). In the latter case, thermodynamic functions such as entropy or specific volume are discontinuous at the transition. When, for example, one varies the temperature T at constant pressure, the transition takes place at a temperature T c where the Gibbs potentials of the two phases are equal, but the two phases coexist at T c and are either stable or metastable in the vicinity of T c . Each of the phases carries with it its own entropy, specific volume, etc., which are in general different at T = T c , hence the discontinuities. Such a phase transition is called a first order phase transition . The picture is quite different in the paramagnetic–ferromagnetic phase transition: the thermodynamic quantities are continuous at the transition, the transition is not linked to the crossing of two thermodynamic potentials and one never observes metastability. Such a transition is called a second order, or continuous, phase transition . The transition temperature is called the critical temperature and is denoted by T c . The new and remarkable feature which we shall encounter in continuous phase transitions is the existence of cooperative phenomena. To be specific, think of a spin system, for example an Ising or a Heisenberg model, where the interactions between spins are limited to nearest neighbours. More generally we shall consider short range interactions , which decrease rapidly as a function of the distance between spins, and we exclude from our study all long range interactions.
The thermodynamic properties of the noninteracting Bose gas in the neighborhood of its transition are examined in detail. The order parameter is a complex extensive variable, but the thermodynamic properties depend only on its amplitude under simple boundary conditions. As the dimensionality or the single-particle energy spectrum is varied, the critical singularity displays a variety of forms. The equation of state has a simple structure, different from the homogeneous form often discussed for critical systems but asymptotically reducing to the latter except when logarithmic singularities are involved. The correlation function in the critical region is a homogeneous function of the distance and a correlation length. Only for a quadratic energy spectrum is the Ornstein-Zernike theory result valid at the critical temperature. A precise correspondence is noted between the asymptotic properties of the ideal Bose gas transition and those of the spherical model of ferromagnetism.