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Physical review. B, Solid state

Some remarks on Kadanoff's variational approximations for renormalization-group transformations

Michel Droz, Sheldon L. Katz, J. D. Gunton

Published Jan 1, 1977• 7 citations
Publisher

Abstract

The recursion equations corresponding to the one-hypercube approximation of Kadanoff are studied in more detail for the Wilson-Fisher $n$-vector model in $4\ensuremath{-}\ensuremath{\epsilon}$ dimensions and the $d$-dimensional Ising model. The free energy for the one-dimensional Ising model is solved exactly in this approximation and is shown to give the correct singular behavior near zero temperature and the exact free energy in zero field. The corrections to scaling exponents are shown to be exact to first order in $\ensuremath{\epsilon}$ for the Wilson-Fisher model and are given for the $d=2,3, \mathrm{and} 4$ dimensional Ising models. The amplitude of the specific-heat divergence is calculated for the two-dimensional Ising model and is shown to be in reasonable agreement with the exact result.

DOI
10.1103/physrevb.15.237

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The recursion equations corresponding to the one-hypercube approximation of Kadanoff are studied in more detail for the Wilson-Fisher nnn-vector model in 4\ensuremath−\ensuremathϵ4\ensuremath{-}\ensuremath{\epsilon}4\ensuremath−\ensuremathϵ dimensions and the ddd-dimensional Ising model. The free energy for the one-dimensional Ising model is solved exactly in this approximation and is shown to give the correct singular behavior near zero temperature and the exact free energy in zero field. The corrections to scaling exponents are shown to be exact to first order in \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ for the Wilson-Fisher model and are given for the d=2,3,and4d=2,3, \mathrm{and} 4d=2,3,and4 dimensional Ising models. The amplitude of the specific-heat divergence is calculated for the two-dimensional Ising model and is shown to be in reasonable agreement with the exact result.
The Kadanoff lower-bound renormalization transformation is used to obtain approximate values of the critical exponents for the spin-1/2 Ising model on a simple hypercubic ddd-dimensional lattice for all integral values of z=2dz={2}^{d}z=2d between 3 and 16. The results for $\ensuremath{\gamma}, \ensuremath{\delta}$, and the thermal and magnetic correction to scaling exponents are compared with corresponding results obtained from the $\ensuremath{\epsilon}$ expansion and series calculations.
We present a systematic procedure for the direct calculation of the free energy and its first and second derivatives for the Ising model in one to three dimensions with a wide class of symmetry properties. This renormalization group method is based on the modified Kadanoff's variational method (MKUM). This method is not only general but also very accurate numerically both near and far from the critical point. Further it includes correction to scaling effects not present in the standard linearized renormalization group treatment. This work describes the techniques and presents some illustrative results for the square latttice and body-centered cubic lattice with nearest neighbor interactions. A full treatment of our results will be given elsewhere. (Author)
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 In Chapter 14, the singular behavior of ferromagnetic systems with O(N) symmetry and short-range interactions, near a second order phase transition has been determined in the mean-field approximation, which is also a quasi-Gaussian approximation. The mean-field approximation predicts a set of universal properties, properties independent of the detailed structure of the microscopic Hamiltonian, the dimension of space, and, to a large extent, of the symmetry of systems. However, the leading corrections to the mean-field approximation, in dimensions smaller than or equal to four, diverge at the critical temperature, and the universal predictions of the mean-field approximation cannot be correct. Such a problem originates from the non-decoupling of scales and leads to the question of possible universality. In Chapter 9, the question has been answered in four dimensions using renormalization theory, and related renormalization group (RG) equations. Moreover, below four dimensions, in an expansion around the mean-field, the most singular terms near criticality can be also formally recovered from a continuum, low-mass φ4 field theory. More generally, following Wilson, to understand universality beyond the mean-field approximation, it is necessary to build a general renormalization group in the form of flow equations for effective Hamiltonians and to find fixed points of the flow equations. Near four dimensions, the flow equations can be approximated by the renormalization group of quantum field theory (QFT), and the fixed points and critical behaviours derived within the framework of the Wilson-Fisher ϵ expansion.
The use of variational approximations in the study and application of renormalization groups is discussed. In particular, a simple approximation is derived which yields an upper bound to the free energy of Ising models on d-dimensional lattices. The optimal transformation, which yields the least upper bound, is determined analytically. The criterion proposed by Kadanoff (1975) to determine the 'best' approximation to the fixed point is found to fail in this case. The reasons for this failure and several of the basic problems posed by variational approximations are discussed.