The phase diagrams of a variety of physical systems, among them certain antiferromagnets and ternary fluid mixtures, may include a critical line intersecting a line of first-order transitions at a critical end point rather than at the familiar tricritical point. At the critical end point the order parameter has critical fluctuations in the paramagnetic phase and a finite jump. Within Landau theory this phenomenon appears with the introduction of a negative ${\ensuremath{\varphi}}^{6}$ coupling; here we extend that theory to include fluctuations by a momentum-shell renormalization-group analysis close to four dimensions. In previous real-space analyses the critical end point was associated with a new fixed point. In contrast, the Wilson-Fisher fixed point which characterizes the entire critical line describes the critical end point here. We derive the $O(\ensuremath{\epsilon})$ corrections to the mean-field exponents which, in Landau theory, describe the approach to the critical end point from high temperatures. At the same time, the discontinuity in $〈\ensuremath{\varphi}〉$ at the critical end point predicted by Landau theory is essentially unaltered by fluctuations. It emerges from a renormalization-group calculation of the Gibbs free energy for the Ginzburg-Landau-Wilson model with a negative ${\ensuremath{\varphi}}^{6}$ term and a positive ${\ensuremath{\varphi}}^{8}$ term.
The phase diagrams of a variety of physical systems, among them certain antiferromagnets and ternary fluid mixtures, may include a critical line intersecting a line of first-order transitions at a critical end point rather than at the familiar tricritical point. At the critical end point the order parameter has critical fluctuations in the paramagnetic phase and a finite jump. Within Landau theory this phenomenon appears with the introduction of a negative \ensuremathφ6 coupling; here we extend that theory to include fluctuations by a momentum-shell renormalization-group analysis close to four dimensions. In previous real-space analyses the critical end point was associated with a new fixed point. In contrast, the Wilson-Fisher fixed point which characterizes the entire critical line describes the critical end point here. We derive the O(\ensuremathϵ) corrections to the mean-field exponents which, in Landau theory, describe the approach to the critical end point from high temperatures. At the same time, the discontinuity in 〈\ensuremathφ〉 at the critical end point predicted by Landau theory is essentially unaltered by fluctuations. It emerges from a renormalization-group calculation of the Gibbs free energy for the Ginzburg-Landau-Wilson model with a negative \ensuremathφ6 term and a positive \ensuremathφ8 term.
An isolated critical point on a line of first-order transitions (Landau point) is studied by means of the low-temperature renormalization group and N1 expansion. Critical exponents are computed in dimension d=3, using polynomials fitting the first two terms of the expansion about d=2 and the previously calculated first three terms of the $\ensuremath{\epsilon}$ expansion about d=4. The results are ${\ensuremath{\lambda}}_{3}=1.081$ and ${\ensuremath{\lambda}}_{6}=\ensuremath{-}1.617$. The conclusions for the shape of the phase diagram are drawn from these values.
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.
Using the local potential approximation of the exact renormalization group (RG) equation, we show the various domains of values of the parameters of the O(1)-symmetric scalar Hamiltonian. In three dimensions, in addition to the usual critical surface Sc (attraction domain of the Wilson-Fisher fixed point), we explicitly show the existence of a first-order phase transition domain Sf separated from Sc by the tricritical surface St (attraction domain of the Gaussian fixed point). Sf and Sc are two distinct domains of repulsion for the Gaussian fixed point, but Sf is not the basin of attraction of a fixed point. Sf is characterized by an endless renormalized trajectory lying entirely in the domain of negative values of the ϕ4-coupling. This renormalized trajectory exists also in four dimensions making the Gaussian fixed point ultra-violet stable (and the ϕ44 renormalized field theory asymptotically free but with a wrong sign of the perfect action). We also show that very retarded classical-to-Ising crossover may exist in three dimensions (in fact below four dimensions). This could be an explanation of the unexpected classical critical behavior observed in some ionic systems.
Using the local potential approximation of the exact renormalization group (RG) equation, we show various domains of values of the parameters of the O(1) -symmetric scalar Hamiltonian.In three dimensions, in addition to the usual critical surface S c (attraction domain of the Wilson-Fisher fixed point), we explicitly show the existence of a first-order phase transition domain S f separated from S c by the tricritical surface S t (attraction domain of the Gaussian fixed point).S f and S c are two distinct domains of repulsion for the Gaussian fixed point, but S f is not the basin of attraction of a fixed point.S f is characterized by an endless renormalized trajectory lying entirely in the domain of negative values of the ϕ 4 -coupling.This renormalized trajectory also exists in four dimensions making the Gaussian fixed point ultra-violet stable (and the ϕ 4 4 renormalized field theory asymptotically free but with a wrong sign of the perfect action).We also show that a very retarded classical-to-Ising crossover may exist in three dimensions (in fact below four dimensions).This could be an explanation of the unexpected classical critical behaviour observed in some ionic systems.
Abstract In Chapter 14, the singular behavior of ferromagnetic systems with O(N) symmetry and short-range interactions, near a second order phase transition has been determined in the mean-field approximation, which is also a quasi-Gaussian approximation. The mean-field approximation predicts a set of universal properties, properties independent of the detailed structure of the microscopic Hamiltonian, the dimension of space, and, to a large extent, of the symmetry of systems. However, the leading corrections to the mean-field approximation, in dimensions smaller than or equal to four, diverge at the critical temperature, and the universal predictions of the mean-field approximation cannot be correct. Such a problem originates from the non-decoupling of scales and leads to the question of possible universality. In Chapter 9, the question has been answered in four dimensions using renormalization theory, and related renormalization group (RG) equations. Moreover, below four dimensions, in an expansion around the mean-field, the most singular terms near criticality can be also formally recovered from a continuum, low-mass φ4 field theory. More generally, following Wilson, to understand universality beyond the mean-field approximation, it is necessary to build a general renormalization group in the form of flow equations for effective Hamiltonians and to find fixed points of the flow equations. Near four dimensions, the flow equations can be approximated by the renormalization group of quantum field theory (QFT), and the fixed points and critical behaviours derived within the framework of the Wilson-Fisher ϵ expansion.
Wilson's momentum-shell transformation with the concomitant expansion around the upper critical dimension is defined. The basic notions of relevant and irrelevant couplings, renormalization flow, and fixed points are introduced. The origins of scaling and of universality are explained, and corrections to mean-field values of critical exponents are computed. The field theoretic renormalization group is briefly discussed and used to calculate the anomalous dimension. Idea We found that the direct perturbation theory in the Ginzburg–Landau–Wilson theory breaks down below the upper critical dimension because the perturbation parameter grows with the correlation length, and so becomes arbitrarily large as the critical point is approached. If the system were finite, on the other hand, the correlation length would be bound by its size, and perturbation theory could succeed. Singular thermodynamic behavior near the critical point comes from the thermodynamic limit, or, more precisely, from those modes that have arbitrary low energies in an infinitely large system. This is called the infrared singularity . This observation suggests the following strategy to avoid the problem of direct perturbation theory. First, note that mass m only provides the energy scale in Eq. (2.36). It is practical to rescale it out by absorbing it into the chemical potential, rede- fined as 2 m μ/ħ 2 → μ, and into the interaction coupling, as 2 m λ/ħ 2 → λ. Similarly, near the critical point, the temperature T ≈ T c may be replaced by the critical temperature, and then eliminated by rescaling the action as 2 m κ B T c S /ħ 2 → S.