The theory of static critical phenomena in systems at equlibrium is discussed. The nature of the classical phenomenological theory of Landau is explained. Critical point exponents β, γ, γ', δ, . . . obtained from the Landau theory differ from the results of the theory for Ising model and experimental results. Statistical evaluation of the order parameter is obtained by taking into account the fluctuation. Then the correlation effect is considered. These approaches make implicit use of the assumption that the free energy if expandable in power series in the order parameter. From the calculation of spatial fluctuation and long range correlation effect the validity of the Landau theory is discussed.
The theory of static critical phenomena in systems at equlibrium is discussed. The nature of the classical phenomenological theory of Landau is explained. Critical point exponents β, γ, γ', δ, . . . obtained from the Landau theory differ from the results of the theory for Ising model and experimental results. Statistical evaluation of the order parameter is obtained by taking into account the fluctuation. Then the correlation effect is considered. These approaches make implicit use of the assumption that the free energy if expandable in power series in the order parameter. From the calculation of spatial fluctuation and long range correlation effect the validity of the Landau theory is discussed.
The theory of critical phenomena in systems at equilibrium is reviewed at an introductory level with special emphasis on the values of the critical point exponents α, β, γ,..., and their interrelations. The experimental observations are surveyed and the analogies between different physical systems - fluids, magnets, superfluids, binary alloys, etc. - are developed phenomenologically. An exact theoretical basis for the analogies follows from the equivalence between classical and quantal `lattice gases' and the Ising and Heisenberg-Ising magnetic models. General rigorous inequalities for critical exponents at and below T c are derived. The nature and validity of the `classical' (phenomenological and mean field) theories are discussed, their predictions being contrasted with the exact results for plane Ising models, which are summarized concisely. Padé approximant and ratio techniques applied to appropriate series expansions lead to precise critical-point estimates for the three-dimensional Heisenberg and Ising models (tables of data are presented). With this background a critique is presented of recent theoretical ideas: namely, the `droplet' picture of the critical point and the `homogeneity' and `scaling' hypotheses. These lead to a `law of corresponding states' near a critical point and to relations between the various exponents which suggest that perhaps only two or three exponents might be algebraically independent for any system.
Abstract Whether a compound experiences a phase transition is a matter of thermodynamics. According to the Landau theory, the Gibbs free energy can be expanded to a Taylor series with respect to an order parameter η. For a continuous phase transition, only even powers of η may appear in the Taylor series. For a temperature-driven phase transition, η is proportional to [(Tc − T)/Tc]β, T being the temperature, Tc the critical temperature, and β the critical exponent. Landau theory shows discrepancies with experimental findings, especially when there are short-range atomic interactions. An improvement has been achieved with the renormalization-group theory, where fluctuations are considered.
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.
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.
The Landau theory on the phase transition derived statistical-mechanically on the basis of the symmetry breaking potential method with the use of inverse temperature series expansion is examined. In that method an estimate of the mean field is improved systematically by taking account of short range correlations. The results at low temperatures in relation to the expansion parameter are investigated. The critical temperature obtained is equal to that in the [n, 0] Padé approximation applied to the inverse of susceptibility. By making use of many results obtained up to now in the calculation of susceptibility with the use of inverse temperature series expansion, the critical temperatures of the Ising, Heisenberg and XY models are determined. The possibility of long range ordering in low dimensional systems and the case of the symmetry breaking potential incommutative with the Hamiltonian are also investigated.
These lectures describe the physics of phase transitions and critical phenomena. After introducing the thermodynamics of phase transitions, these general considerations are made explicit by presenting statistical mechanical calculations on a lattice gas model in the mean-field approximation. Then the Landau theory of phase transitions is presented. The form of free energy in the Landau theory is motivated by an explicit mean field calculation for the Ising model. This is followed by a discussion of spatial correlations and the breakdown of the mean field theory due to increase of spatial fluctuations near the critical point. This calculation also brings out the role of dimension of the system for the critical behaviour. The basic ideas of Renormalization Group (RG) are then introduced and illustrated by a simple RG calculation for the one-dimensional Ising model. Real-space RG calculations are then applied to the two-dimensional Ising model on the triangular lattice, which, in fact, brings forth several key features of the RG technique. A discussion of more general aspects of the RG formalism, such as scaling for spatial correlation functions, relevant and irrelevant couplings, universality, upper critical dimension, etc. are introduced.