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Physical review. B, Solid state

Renormalization Group and Critical Phenomena. II. Phase-Space Cell Analysis of Critical Behavior

Kenneth Geddes Wilson

Published Nov 1, 1971• 1,412 citations
Publisher

Abstract

A generalization of the Ising model is solved, qualitatively, for its critical behavior. In the generalization the spin ${S}_{\stackrel{\ensuremath{\rightarrow}}{\mathrm{n}}}$ at a lattice site $\stackrel{\ensuremath{\rightarrow}}{\mathrm{n}}$ can take on any value from $\ensuremath{-}\ensuremath{\infty} \mathrm{to} \ensuremath{\infty}$. The interaction contains a quartic term in order not to be pure Gaussian. The interaction is investigated by making a change of variable ${S}_{\stackrel{\ensuremath{\rightarrow}}{\mathrm{n}}}=\ensuremath{\Sigma}{m}^{}{\ensuremath{\psi}}_{m}(\mathrm{n}){S}_{m}^{\ensuremath{'}}$, where the functions ${\ensuremath{\psi}}_{m}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{n}})$ are localized wavepacket functions. There are a set of orthogonal wave-packet functions for each order-of-magnitude range of the momentum $\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}$. An effective interaction is defined by integrating out the wave-packet variables with momentum of order 1, leaving unintegrated the variables with momentum 0.5. Then the variables with momentum between 0.25 and 0.5 are integrated, etc. The integrals are computed qualitatively. The result is to give a recursion formula for a sequence of effective Landau-Ginsberg-type interactions. Solution of the recursion formula gives the following exponents: $\ensuremath{\eta}=0$, $\ensuremath{\gamma}=1.22$, $\ensuremath{\nu}=0.61$ for three dimensions. In five dimensions or higher one gets $\ensuremath{\eta}=0$, $\ensuremath{\gamma}=1$, and $\ensuremath{\nu}=\frac{1}{2}$, as in the Gaussian model (at least for a small quartic term). Small corrections neglected in the analysis may make changes (probably small) in the exponents for three dimensions.

DOI
10.1103/physrevb.4.3184

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A generalization of the Ising model is solved, qualitatively, for its critical behavior. In the generalization the spin Sn\ensuremath→{S}_{\stackrel{\ensuremath{\rightarrow}}{\mathrm{n}}}Sn\ensuremath→​ at a lattice site n\ensuremath→\stackrel{\ensuremath{\rightarrow}}{\mathrm{n}}n\ensuremath→ can take on any value from \ensuremath−\ensuremath∞to\ensuremath∞\ensuremath{-}\ensuremath{\infty} \mathrm{to} \ensuremath{\infty}\ensuremath−\ensuremath∞to\ensuremath∞. The interaction contains a quartic term in order not to be pure Gaussian. The interaction is investigated by making a change of variable Sn\ensuremath→=\ensuremathΣm\ensuremathψm(n)Sm\ensuremath′{S}_{\stackrel{\ensuremath{\rightarrow}}{\mathrm{n}}}=\ensuremath{\Sigma}{m}^{}{\ensuremath{\psi}}_{m}(\mathrm{n}){S}_{m}^{\ensuremath{'}}Sn\ensuremath→​=\ensuremathΣm\ensuremathψm​(n)Sm\ensuremath′​, where the functions \ensuremathψm(n\ensuremath→){\ensuremath{\psi}}_{m}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{n}})\ensuremathψm​(n\ensuremath→) are localized wavepacket functions. There are a set of orthogonal wave-packet functions for each order-of-magnitude range of the momentum k\ensuremath→\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}k\ensuremath→​. An effective interaction is defined by integrating out the wave-packet variables with momentum of order 1, leaving unintegrated the variables with momentum 0.5. Then the variables with momentum between 0.25 and 0.5 are integrated, etc. The integrals are computed qualitatively. The result is to give a recursion formula for a sequence of effective Landau-Ginsberg-type interactions. Solution of the recursion formula gives the following exponents: \ensuremathη=0\ensuremath{\eta}=0\ensuremathη=0, \ensuremathγ=1.22\ensuremath{\gamma}=1.22\ensuremathγ=1.22, \ensuremathν=0.61\ensuremath{\nu}=0.61\ensuremathν=0.61 for three dimensions. In five dimensions or higher one gets \ensuremathη=0\ensuremath{\eta}=0\ensuremathη=0, \ensuremathγ=1\ensuremath{\gamma}=1\ensuremathγ=1, and \ensuremathν=12\ensuremath{\nu}=\frac{1}{2}\ensuremathν=21​, as in the Gaussian model (at least for a small quartic term). Small corrections neglected in the analysis may make changes (probably small) in the exponents for three dimensions.
We apply the dynamical mean field theory (DMFT) approximation to the real, scalar ${\ensuremath{\varphi}}^{4}$ quantum field theory. By comparing to lattice Monte Carlo calculations, perturbation theory and standard mean field theory, we test the quality of the approximation in two, three, four and five dimensions. The quantities considered in these tests are the critical coupling for the transition to the ordered phase and the associated critical exponents $\ensuremath{\nu}$ and $\ensuremath{\beta}$. We also map out the phase diagram in the most relevant case of four dimensions. In two and three dimensions, DMFT incorrectly predicts a first-order phase transition for all bare quartic couplings, which is problematic, because the second-order nature of the phase transition of lattice ${\ensuremath{\varphi}}^{4}$ theory is crucial for taking the continuum limit. Nevertheless, by extrapolating the behavior away from the phase transition, one can obtain critical couplings and critical exponents. They differ from those of mean field theory and are much closer to the correct values. In four dimensions the transition is second order for small quartic couplings and turns weakly first order as the coupling increases beyond a tricritical value. In dimensions five and higher, DMFT gives qualitatively correct results, predicts reasonable values for the critical exponents and considerably more accurate critical couplings than standard mean field theory. The approximation works best for small values of the quartic coupling. We investigate the change from first- to second-order transition in the local limit of DMFT which is computationally much cheaper. We also discuss technical issues related to the convergence of the nonlinear self-consistency equation solver and the solution of the effective single-site model using Fourier-space Monte Carlo updates in the presence of a ${\ensuremath{\varphi}}^{4}$ interaction.
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 The approaches described so far in this book have given a broad phenomenological understanding of critical phenomena. However, although a substantial framework of results and connections has been built up, we have, as yet, no explanations for the following: What is needed is a theory, based on the physics of what is happening at the critical point. we argued in Chapter 1 that the special feature of criticality is that the correlation length is infinite and that the critical system is invariant on all length scales. The aim is to write down a (hopefully short, elegant, and comprehensible) mathematical theory which embodies this physics and explains all the observations listed above. A useful theory will also allow the calculation of critical exponents and transition temperatures, if not exactly, then within an accurate and well-controlled approximation scheme.
The critical behaviour of systems belonging to the three-dimensional Ising universality class is studied theoretically using the collective variables (CV) method.The partition function of a one-component spin system is calculated by the integration over the layers of the CV phase space in the approximation of the non-Gaussian sextic distribution of order-parameter fluctuations (the ρ 6 model).A specific feature of the proposed calculation consists in making allowance for the dependence of the Fourier transform of the interaction potential on the wave vector.The inclusion of the correction for the potential averaging leads to a nonzero critical exponent of the correlation function η and the renormalization of the values of other critical exponents.The contributions from this correction to the recurrence relations for the ρ 6 model, fixed-point coordinates and elements of the renormalization-group linear transformation matrix are singled out.The expression for a small critical exponent η is obtained in a higher non-Gaussian approximation.