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Physical review. B, Solid state

Scaling behavior of second-order phase transitions

R. P. Van Royen

Published May 1, 1976• 6 citations
Publisher

Abstract

Using the Landau Hamiltonian for the description of second-order phase transitions, we give a proof of scaling for any continuous number of dimensions below four. The proof is based on a summation of diagrams having a power-law divergence and standard renormalization-group methods. The proof is constructive in that it leads to an unambiguous calculation for the critical exponents $\ensuremath{\eta}$ and $\ensuremath{\gamma}$. We present in this paper a detailed discussion of the proof; we also compare our method with the $\ensuremath{\epsilon}$ expansion leading to an interesting aspect of that theory: We find that the contribution to the critical exponents of order $\ensuremath{\epsilon}$ can be gotten without any calculation of diagrams. In this paper we have only made a lowest-order calculation in three dimensions. To this order we are of course unable to locate the relevant fixed point, but it leads to a relation between $\ensuremath{\eta}$ and $\ensuremath{\gamma}$---also to lowest order---which is such that if $\ensuremath{\gamma}$ is fixed to be 1.25, then $\ensuremath{\eta}$ turns out to be 0.12.

DOI
10.1103/physrevb.13.4079

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Using the Landau Hamiltonian for the description of second-order phase transitions, we give a proof of scaling for any continuous number of dimensions below four. The proof is based on a summation of diagrams having a power-law divergence and standard renormalization-group methods. The proof is constructive in that it leads to an unambiguous calculation for the critical exponents \ensuremathη\ensuremath{\eta}\ensuremathη and \ensuremathγ\ensuremath{\gamma}\ensuremathγ. We present in this paper a detailed discussion of the proof; we also compare our method with the \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ expansion leading to an interesting aspect of that theory: We find that the contribution to the critical exponents of order \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ can be gotten without any calculation of diagrams. In this paper we have only made a lowest-order calculation in three dimensions. To this order we are of course unable to locate the relevant fixed point, but it leads to a relation between \ensuremathη\ensuremath{\eta}\ensuremathη and \ensuremathγ\ensuremath{\gamma}\ensuremathγ---also to lowest order---which is such that if \ensuremathγ\ensuremath{\gamma}\ensuremathγ is fixed to be 1.25, then \ensuremathη\ensuremath{\eta}\ensuremathη turns out to be 0.12.
Different perturbation theory treatments of the Ginzburg-Landau phase transition model are discussed. This includes a criticism of the perturbative renormalization group (RG) approach and a proposal of a novel method providing critical exponents consistent with the known exact solutions in two dimensions. The usual perturbation theory is reorganized by appropriate grouping of Feynman diagrams of φ4 model with O(n) symmetry. As a result, equations for calculation of the two-point correlation function are obtained which allow to predict possible exact values of critical exponents in two and three dimensions by proving relevant scaling properties of the asymptotic solution at (and near) the criticality. The new values of critical exponents are discussed and compared to the results of numerical simulations and experiments.
Abstract In this work, we define and calculate critical exponents associated with higher order thermodynamic phase transitions. Such phase transitions can be classified into two classes: with or without a local order parameter. For phase transitions involving a local order parameter, we write down the Landau theory and calculate critical exponents using the saddle point approximation. Further, we investigate fluctuations about the saddle point and demarcate when such fluctuations dominate over saddle point calculations by introducing the generalized Ginzburg criteria. We use Wilsonian renormalization group (RG) to derive scaling forms for observables near criticality and obtain scaling relations between the critical exponents. Afterwards, we find out fixed points of the RG flow using the one-loop beta function and calculate critical exponents about the fixed points for third and fourth order phase transitions.
An isolated critical point on a line of first-order transitions (Landau point) is studied by means of the low-temperature renormalization group and 1N\frac{1}{N}N1​ expansion. Critical exponents are computed in dimension d=3d=3d=3, using polynomials fitting the first two terms of the expansion about d=2d=2d=2 and the previously calculated first three terms of the $\ensuremath{\epsilon}$ expansion about d=4d=4d=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.
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].
A fundamental issue in the renormalization-group (RG) theory of critical phenomena concerns the allowed values of critical exponents that are consistent with the continuous nature of a phase transition. Here we conjecture a lower bound for the length-scale exponent ννν, which should hold for the large class of continuous transitions associated with ddd-dimensional Landau-Ginzburg-Wilson (LGW) Φ4Φ^4Φ4 theories with a multicomponent scalar field φφφ and a unique φ⋅φφ\cdot φφ⋅φ quadratic term (including some extensions with fermionic and gauge fields), describing many universality classes of critical phenomena. If Δφ=(d−2+η)/2Δ_φ=(d-2+η)/2Δφ​=(d−2+η)/2 is the dimension of the order-parameter field φφφ, and Δε=d−1/νΔ_\varepsilon=d-1/νΔε​=d−1/ν is the RG dimension of the energy operator ε\varepsilonε, which can be identified with [φ⋅φ][φ\cdot φ][φ⋅φ] (the squared field with a proper subtraction of the mixing with the identity), we conjecture the inequality Δε≥2ΔφΔ_\varepsilon \ge 2 Δ_φΔε​≥2Δφ​, which implies ν≥(2−η)−1ν\ge (2-η)^{-1}ν≥(2−η)−1 and γ=(2−η)ν≥1γ= (2-η)ν\ge 1γ=(2−η)ν≥1. These inequalities are supported by general arguments for ferromagnetic lattice models, by εεε-expansion results for generic LGW Φ4Φ^4Φ4 theories close to four dimensions, exact relations for two-dimensional minimal conformal field theories, and are consistent with all further known (numerical, perturbative, and exact) results for LGW Φ4Φ^4Φ4 theories. In particular, since unitarity requires η≥0η\ge 0η≥0, the above inequality implies ν≥1/2ν\ge 1/2ν≥1/2 for unitary theories. This lower bound is more restrictive than ν>1/dν> 1/dν>1/d, derived by noting that ν=1/dν=1/dν=1/d characterizes the singular finite-size behavior at first-order transitions.
Different perturbation theory treatments of the Ginzburg-Landau phase transition model are discussed. This includes a criticism of the perturbative renormalization group (RG) approach and a proposal of a novel method providing critical exponents consistent with the known exact solutions in two dimensions. The usual perturbation theory is reorganized by appropriate grouping of Feynman diagrams of φ4 model with O(n) symmetry. As a result, equations for calculation of the two-point correlation function are obtained which allow to predict possible exact values of critical exponents in two and three dimensions by proving relevant scaling properties of the asymptotic solution at (and near) the criticality. The new values of critical exponents are discussed and compared to the results of numerical simulations and experiments.