Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Physics A Mathematical and General

Upper bounds to free energies by renormalization group methods

Michael N. Barber

Published Jul 1, 1977• 13 citations
Publisher

Abstract

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.

DOI
10.1088/0305-4470/10/7/016

Related papers

Variational Principles and Approximate Renormalization Group Calculations

Leo P. Kadanoff

Approximate recursion relations which give upper and lower bounds on the free energy are described. "Optimal" calculations of the free energy can then be obtained by treating parameters within the renormalization equations variationally. As an example, simple lower-bound relations are defined for the two- and three-dimensional Ising models. At the fixed point, a parameter is set variationally, and then critical indices are calculated.

Physical Review Letters•Apr 21, 1975•172 citations

Lower-bound approximation to the free energy of the three-dimensional Ising model

Sheldon L. Katz, J. D. Gunton

We use the one-hypercube variational approximation to the renormalization-group equations to calculate a lower bound for the zero-field free energy of an Ising model on a bcc lattice. Our results for the free energy, energy and specific heat are in good agreement with those obtained by series expansions. Estimates are given for the critical exponents for the situation in which the variational parameter is varied in linearizing around the fixed point.

Physical review. B, Solid state•Sep 1, 1977•5 citations

Analysis of the lower-bound renormalization-group approximation

Naeem Jan

The lower-bound variational renormalization-group approximation of Kadanoff is reformulated and applied to the triangular Ising lattice. The basic inconsistencies inherent in the method are removed and the results thus obtained are in fair agreement with the exact values.

Physical review. B, Condensed matter•Sep 1, 1979•0 citations

Direct calculation of the derivatives of the free energy for Ising models by a modified Kadanoff's variational method

Chin‐Kun Hu, Peter H. Kleban

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)

Unknow•Feb 1, 1982•0 citations

Lower bound approximation to the free energy of an Ising model on a simple cubic lattice

Chin‐Kun Hu, Wen‐Den Chen, Yu-Ming Shih +7 more

We apply a modified Kadanoff's variational method to calculate the lower bound zero-field free energies and their derivatives for an Ising model on the simple cubic lattice. We find a critical point at Kc = 0.2393769 with precision ±10−7.

Canadian Journal of Physics•Oct 1, 1981•1 citation

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

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

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.

Physical review. B, Solid state•Jan 1, 1977•7 citations
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.
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)
The recursion equations corresponding to the one-hypercube approximation of Kadanoff are studied in more detail for the Wilson-Fisher nnn-vector model in $4\ensuremath{-}\ensuremath{\epsilon}$ 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 $\ensuremath{\epsilon}$ 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.