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Physical Review A

Critical phenomena of fluids: Asymmetric Landau-Ginzburg-Wilson model

Jeffrey F. Nicoll

Published Oct 1, 1981• 149 citations
Publisher

Abstract

The critical behavior of simple single-component fluids near the liquid-vapor critical point (and of other systems in the same universality class) is examined via an asymmetric spin Hamiltonian in Landau-Ginzburg-Wilson form. The leading important odd terms to be added to an Ising-type system are a five-spin interaction $\ensuremath{\int}\ensuremath{\phi}^{5}(x)$ and a nonlocal cubic interaction $\ensuremath{-}\ensuremath{\int}\ensuremath{\phi}^{2}\ensuremath{\nabla}^{2}\ensuremath{\phi}$. These two operators can be combined into two different eigenoperators with distinct physical interpretations. One combination can be shown to be exactly equivalent to the mixing of field variables (e.g., chemical potential and temperature) in the temperaturelike variable of an Ising or Ising-type system. This mixing has been used phenomenologically, found in certain models, and justified on general geometric grounds. This is the first proof that all asymmetric models, assuming universality and barring accidents, will have such mixing; this implies the universal occurrence of a ${t}^{1\ensuremath{-}a}$ fluid diameter. The second combination is treated by renormalization-group methods and its contributions to the Helmholtz free energy, magnetic equation of state, and correlation length are calculated to $O({\ensuremath{\epsilon}}^{2}), \ensuremath{\epsilon}\ensuremath{\equiv}4\ensuremath{-}d$. There are several novel features of this interaction which persist to all orders in perturbation theory. In addition to the expected renormalized ${M}^{5}$ contribution to the free energy, there are also terms linear in $h$ (the magnetic field) and $M$ which could be described as nonanalytic fluctuation-induced shifts in the order parameter and field. These might appear experimentally as apparent shifts in the background terms for $h$ and $M$.

DOI
10.1103/physreva.24.2203

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Physical review. B, Condensed matter•Jan 1, 1982•33 citations
The critical behavior of simple single-component fluids near the liquid-vapor critical point (and of other systems in the same universality class) is examined via an asymmetric spin Hamiltonian in Landau-Ginzburg-Wilson form. The leading important odd terms to be added to an Ising-type system are a five-spin interaction \ensuremath∫\ensuremathϕ5(x)\ensuremath{\int}\ensuremath{\phi}^{5}(x)\ensuremath∫\ensuremathϕ5(x) and a nonlocal cubic interaction \ensuremath−\ensuremath∫\ensuremathϕ2\ensuremath∇2\ensuremathϕ\ensuremath{-}\ensuremath{\int}\ensuremath{\phi}^{2}\ensuremath{\nabla}^{2}\ensuremath{\phi}\ensuremath−\ensuremath∫\ensuremathϕ2\ensuremath∇2\ensuremathϕ. These two operators can be combined into two different eigenoperators with distinct physical interpretations. One combination can be shown to be exactly equivalent to the mixing of field variables (e.g., chemical potential and temperature) in the temperaturelike variable of an Ising or Ising-type system. This mixing has been used phenomenologically, found in certain models, and justified on general geometric grounds. This is the first proof that all asymmetric models, assuming universality and barring accidents, will have such mixing; this implies the universal occurrence of a t1\ensuremath−a{t}^{1\ensuremath{-}a}t1\ensuremath−a fluid diameter. The second combination is treated by renormalization-group methods and its contributions to the Helmholtz free energy, magnetic equation of state, and correlation length are calculated to O(\ensuremathϵ2),\ensuremathϵ\ensuremath≡4\ensuremath−dO({\ensuremath{\epsilon}}^{2}), \ensuremath{\epsilon}\ensuremath{\equiv}4\ensuremath{-}dO(\ensuremathϵ2),\ensuremathϵ\ensuremath≡4\ensuremath−d. There are several novel features of this interaction which persist to all orders in perturbation theory. In addition to the expected renormalized M5{M}^{5}M5 contribution to the free energy, there are also terms linear in hhh (the magnetic field) and MMM which could be described as nonanalytic fluctuation-induced shifts in the order parameter and field. These might appear experimentally as apparent shifts in the background terms for hhh and MMM.
A Landau-Ginzburg-Wilson model is derived for a single-component fluid whose particles interact via a two-body potential. The effective Hamiltonian contains both even and odd powers in the order parameter (local fluid density). We study the effect of quintic interactions absent in the Ising model, and characterized by a new exponent ${\ensuremath{\varphi}}_{5}$, on various singular thermodynamic functions near the critical point. Mean-field, scaling, and renormalization-group (in $d=4\ensuremath{-}\ensuremath{\epsilon}$ dimensions) theories are used to evaluate the non-Ising effects due to ${\ensuremath{\varphi}}_{5}$. One of the main results of the asymmetry in this model is that it produces the leading singularity of the liquid-vapor coexistence diameter near the critical point: ${|t|}^{\ensuremath{\beta}\ensuremath{-}{\ensuremath{\varphi}}_{5}}$, where $\ensuremath{\beta}$ is the usual order-parameter exponent and ttt is the reduced tem$\stackrel{\mathrm{\ifmmode \dot{}\else \.{}\fi{}}}{\mathrm{p}}$erature. The singularity ${|t|}^{1\ensuremath{-}\ensuremath{\alpha}}$ ($\ensuremath{\alpha}$ being the specific heat exponent) predicted by earlier phenomenological theories is not present in this theory. The problems associated with observing the non-Ising effects are briefly discussed.
We review results concerning the critical behavior of spin systems at equilibrium. We consider the Ising and the general O(N)-symmetric universality class. For each of them, we review the estimates of the critical exponents, of the equation of state, of several amplitude ratios, and of the two-point function of the order parameter. We report results in three and two dimensions. We discuss the crossover phenomena that are observed in this class of systems. In particular, we review the field-theoretical and numerical studies of systems with medium-range interactions. Moreover, we consider several examples of magnetic and structural phase transitions, which are described by more complex Landau-Ginzburg-Wilson Hamiltonians, such as N-component systems with cubic anisotropy, O(N)-symmetric systems in the presence of quenched disorder, frustrated spin systems with noncollinear or canted order, and finally, a class of systems described by the tetragonal Landau-Ginzburg-Wilson Hamiltonian with three quartic couplings. The results for the tetragonal Hamiltonian are original, in particular we present the six-loop perturbative series for the β-functions and the critical exponents.
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 question of a possible difference between the universality classes of fluids and Ising-like magnets is addressed by perturbation theory and the renormalization group. The most dangerous possibility is that of an ${\ensuremath{\varphi}}^{5}$ addition to the usual ${\ensuremath{\varphi}}^{4}$ theory. We show that no ${\ensuremath{\varphi}}^{5}$ fixed point exists in the framework of an expansion around d=103d=\frac{10}{3}d=310​. Further we show that to $O({\ensuremath{\epsilon}}_{4}^{2})$, ${\ensuremath{\epsilon}}_{4}\ensuremath{\equiv}4\ensuremath{-}d$, the ordinary ${\ensuremath{\varphi}}^{4}$ fixed point is stable against the perturbations that mix with ${\ensuremath{\varphi}}^{5}$. Two new correction-to-scaling exponents are found. One of the exponents, ${\ensuremath{\Delta}}_{5}$, is poorly determined with a range of values from 0.5 to 1.0 compatible with the $O({\ensuremath{\epsilon}}_{4}^{2})$ result. However, its positivity rules out a separate fluid fixed point, indicating fluid-magnet asymptotic universality. The second exponent, ${\ensuremath{\Delta}}_{3}$, can be determined exactly: ${\ensuremath{\Delta}}_{3}=1\ensuremath{-}\ensuremath{\alpha}\ensuremath{-}\ensuremath{\beta}$. This implies the universal existence of a contribution to the fluid diameter scaling like the internal energy.
Nonuniversal scale transformations of the physical fields are extended to pure quantum fluids and used to calculate the susceptibility, the specific heat, and the order parameter density along the critical isochore of near its liquid-vapor critical point. Within the so-called preasymptotic domain, where the Wegner expansion restricted to the first term of confluent corrections to scaling is expected to be valid, the results are in agreement with the experimental measurements and recent predictions, either based on the minimal-substraction renormalization and the massive renormalization schemes within phi (4)(d = 3) (n = 1)the model, or based on the crossover parametric equation of state for Ising-like systems.
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.