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Physical review. B, Condensed matter

Field-theoretical approach to critical phenomena

C. Bagnuls, Claude Bervillier

Published Jan 1, 1990• 153 citations
Publisher

Abstract

The study of the Wegner-Houghton equation in the local-potential approximation yields infrared-stable renormalization-group trajectories that are rejected by field theory. The most stable one corresponds to an approach to the infrared-stable fixed point from the unusual side, yielding negative correction-to-scaling amplitudes as already observed in a binary mixture. We also show how this affects the analysis made by Baker and Kincaid, who concluded that the field-theoretic framework fails in describing the critical behavior of the spin-(1/2 Ising model at d=3.

DOI
10.1103/physrevb.41.402

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The study of the Wegner-Houghton equation in the local-potential approximation yields infrared-stable renormalization-group trajectories that are rejected by field theory. The most stable one corresponds to an approach to the infrared-stable fixed point from the unusual side, yielding negative correction-to-scaling amplitudes as already observed in a binary mixture. We also show how this affects the analysis made by Baker and Kincaid, who concluded that the field-theoretic framework fails in describing the critical behavior of the spin-(1/2 Ising model at d=3.
The large distance behaviour, near the critical temperature, of continuous phase transitions with short range interactions can be reproduced by an effective local field theory. Within this framework, the universal properties of the large distance can be determined by finding fixed points of general, functional renormalization group (RG) equations (Wilson, Wegner). For dimension D > 4, the large scale behavior is governed by the Gaussian fixed point (related to mean field theory), while for D < 4 the Gaussian fixed point is unstable. Near D = 4, the RG equations can be solved in the form of an ǫ = 4 − D expansion, a new stable fixed point is found and this leads to the famous Wilson-Fisher expansion for critical exponents and other universal quantities. Moreover, one realizes that, in this framework, the effective field theory can be reduced to a renormalizable field theory and, at leading order in the critical domain, the much simpler quantum field theory renormalization group can be used. A number of physicists, including the Saclay group (Brezin, Le Guillou, Zinn-Justin) have developed sophisticated field theory techniques, to prove general scaling laws and to calculate exponents to higher orders [1].
We give an approximate renormalization-group formulation which parallels that of Wilson. The group generator represents the momentum-independent limit of the differential generator of Wegner and Houghton. The eigenfunctions near the Gaussian point are computed for all spin dimensions nnn and lattice dimensions ddd, including d=2d=2d=2. The nontrivial fixed-point Hamiltonian in dimensions near $d=\frac{2\mathcal{O}}{(\mathcal{O}\ensuremath{-}1)}$, together with the eigenvalues near that nontrivial fixed point, are found explicitly to first order in ${\ensuremath{\epsilon}}_{\mathcal{O}}\ensuremath{\equiv}\mathcal{O}(2\ensuremath{-}d)+d$ for all values of nnn and the order O\mathcal{O}O. Odd-dominated Ising systems and corresponding expansions in ${\ensuremath{\epsilon}}_{\mathcal{O}\ensuremath{-}\frac{1}{2}}$ are also treated.
A numerical algorithm is used to solve the Wegner-Houghton equation for the scalar phi^4 model without any limiting ansatz for the local potential. Non-perturbative critical exponents are obtained in the infrared scaling regime of the symmetric theory. A precursor of the symmetry breaking, a dynamical Maxwell-cut is found which reveals itself in the approximate degeneracy of the blocked action for modes at the sufficiently low cutoff. A mechanism of Quantum Censorship is conjectured which is supposed to be operational for large enough quantum fluctuations and prevents the exact degeneracy of the Wilsonian action together with the tree-level renormalization.
The conventional renormalization-group analysis of the tricritical behavior of a metamagnet is completed by a description of the first-order transition below the tricritical temperature, and the conditions for the discontinuity in the magnetization are explicitly verified in an Ising spin model. The singularity structure of the crossover scaling function is derived from relations between the scaling fields for the critical and tricritical fixed points. Numerical calculations illustrating various aspects of the theory are given for a square-lattice Ising model in a four-cell cluster approximation.
Abstract Chapter 6 describes how the perturbative renormalization group (RG) of quantum field theory has made it possible to derive universal properties of continuous macroscopic phase transitions. The RG initially emerged as a consequence of the necessity to cancel infinities that appear in the perturbative expansion (the renormalization procedure) and the possibility of defining the parameters of the renormalized theory at different momentum scales. Although the field theory RG is now understood to be an asymptotic form, it has made it possible to confirm the Wilson–Fisher fixed point and led to an understanding of universality for a large class of critical phenomena. In the framework of dimensional continuation, zeros of RG beta functions, which correspond to Wilson–Fisher’s fixed points, have made it possible to recover Wilson–Fisher’s epsilon expansion, proving scaling relations and calculating critical exponents. Series summation methods have then been used to generate precise values of exponents.
Wegner's paradigm for constructing renormalization groups in statistical mechanics is applied to fluids. The coordinates of the fraction ($1\ensuremath{-}y$) of the molecules are integrated over and a system of interactions describing the remaining fraction yyy is constructed with the aid of the McMillan-Mayer solution theory. Renormalized Ursell functions are defined in terms of which the group operations are shown to be linear. The group operations commute with a scaling operation and it is found that a series of renormalization groups depending on a parameter xxx can be constructed by combining the operation of integrating over coordinates with the scaling operation. By comparing the renormalized cluster functions with the form of the critical cluster functions given by scaling it is shown that a fixed point of one of the series of groups corresponds to the critical point. The parameter xxx turns out to be the scaling exponent for the chemical potential. The question of the universality of critical phenomena is expressed as the question of the discreteness or continuity of the spectrum of values of the parameter xxx for which the linear fixed-point equations have a nontrivial solution. It is suggested that this question can be answered definitively only when a precise formulation is made of the function space in which the sequence of renormalized Ursell functions must lie.