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Physical review. B, Solid state

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

Sheldon L. Katz, J. D. Gunton

Published Sep 1, 1977• 5 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.16.2163

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Journal of Physics A Mathematical and General•Jul 1, 1977•13 citations
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.
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.
Higher{order vertices at zero external momenta for the scalar eld theory, describing the critical behaviour of the Ising model, are studied within the eld{theoretical renormalization group (RG) approach in three dimensions.Dimensionless six{point g6 and eight{point g8 eective coupling constants are calculated in the three{loop approximation.Their numerical values, universal at criticality, are estimated by means of the Pad e and Pad e{Borel summation of the RG expansions found and by putting the renormalized quartic coupling constant equal to its universal xed{point value known from six{loop RG calculations.The values of g 6 obtained are compared with their analogs resulting from the {expansion, Monte Carlo simulations, the Wegner{Houghton equations and the linked cluster expansion series.The eld{theoretical estimates for g 6 are shown to be in a good agreement with each other, diering considerably from the values given by other methods.
A variational approximation to the Ising model in a magnetic field (T< TcT<~{T}_{c}T< Tc​), based on the zero-field solution, yields an upper bound on the free energy, an approximate equation of state, and a lower bound on the magnetization, all having the correct critical indices. In two dimensions, these are numerically evaluated, and compared with the results of series expansions.
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.