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Physical Review A

Renormalization Group Equation for Critical Phenomena

Franz Wegner, Anthony Houghton

Published Jul 1, 1973• 729 citations
Publisher

Abstract

An exact renormalization equation is derived by making an infinitesimal change in the cutoff in momentum space. From this equation the expansion for critical exponents around dimensionality 4 and the limit $n=\ensuremath{\infty}$ of the $n$-vector model are calculated. We obtain agreement with the results of Wilson and Fisher, and with the spherical model.

DOI
10.1103/physreva.8.401

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An exact renormalization equation is derived by making an infinitesimal change in the cutoff in momentum space. From this equation the expansion for critical exponents around dimensionality 4 and the limit n=\ensuremath∞n=\ensuremath{\infty}n=\ensuremath∞ of the nnn-vector model are calculated. We obtain agreement with the results of Wilson and Fisher, and with the spherical model.
For the n-vector model of the general type a renormalization-group (RG) scheme in momentum space is developed. Using this scheme we obtain a generalized exact equation of the renormalization group. The main idea of this approach is not to allow the zero-order critical Hamiltonian to move away from the critical surface under the influence of the RG transformation. The equation found differs from the known ones in that all redundant operators are excluded. On the basis of the developed approach we argue that critical exponents are functionals of some function introducing the cutoff into the phase-transition theory.
A class of renormalization groups characterized as the general Gaussian renormalization group is defined. It includes the sharp cutoff renormalization group of Wilson, the linear renormalization group of Bell and Wilson, and the incomplete‐integration renormalization group of Wilson. The renormalization group approach to critical phenomena and the expansion around four dimensions is reviewed. The fixed point of the general Gaussian renormalization group is determined to order ε2, where ε=4−d (d is the dimensionality of the system), and the critical exponents are determined to the same order. The main conclusion is that the exponents do not depend on the cutoff functions of the renormalization group.
This paper presents an extension of Wilson's renormalization-group calculation of Ising-model critical exponents to include calculation of the critical exponent $\ensuremath{\eta}$. New recursion formulas are derived using the simplest set of consistent approximations which allow a nonzero $\ensuremath{\eta}$. They are intended to demonstrate, qualitatively, how nonzero values for $\ensuremath{\eta}$ are consistent with the renormalization-group approach; they do not represent systematic, quantitative improvements to Wilson's earlier calculation of the exponents $\ensuremath{\nu}$ and $\ensuremath{\gamma}$. The equations are solved both by $\ensuremath{\epsilon}$ expansion about four dimensions and by numerical integration in three dimensions. To order ${\ensuremath{\epsilon}}^{2}$ we obtain $\ensuremath{\eta}=0.05{\ensuremath{\epsilon}}^{2}$. Numerical results in three dimension are $\ensuremath{\eta}=0.058$, $\ensuremath{\nu}=0.588$, and $\ensuremath{\gamma}=1.14$. The relation $\ensuremath{\gamma}=(2\ensuremath{-}\ensuremath{\eta})\ensuremath{\nu}$ is confirmed.
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.
A scaling-field representation of Wilson's exact renormalization-group equation is derived and used for the approximate calculation of critical exponents for the continuous-spin Ising model in three dimensions. The truncation of the hierarchy of scaling-field equations that retains only the four most relevant scaling fields yields the critical exponents $\ensuremath{\nu}\ensuremath{\approx}0.60$, $\ensuremath{\eta}\ensuremath{\approx}0.06$, and ${\ensuremath{\Delta}}_{1}\ensuremath{\approx}\ensuremath{-}0.47$ (where ${\ensuremath{\Delta}}_{1}$ denotes Wortis's correction-to-scaling exponent).