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physica status solidi (b)

An Improved Mean Field Approximation

J. R. Faleiro Ferreira

Published Nov 1, 1988• 2 citations
Publisher

Abstract

Abstract An upper bound for the free energy may generate an approximation which preserves short order range effects even when a sum of single spins is used as the auxiliary Hamiltonian. Unlike the usual mean field approximation it may distinguish the cubic lattice from the planar triangular lattice. Its application to the Ising chain recovers some exact results.

DOI
10.1002/pssb.2221500131

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The critical values of coupling strength of the non-self-dual Ising lattices under decimation transformation

Tuncer Kaya

In this work, the values of critical coupling strengths of the Ising lattices which are changing their lattice structure (or non-self-dual) under decimation transformations, such as the honeycomb, the triangular and the body centered cubic Ising lattices, are obtained by a modified real space renormalization group approach (RSRG). This modification is necessary to obtain a proper relation between the coupling strengths of the original and the decimated lattices. Indeed, we have achieved to obtain a proper renormalized coupling strength relation for honeycomb and triangular lattices readily, since the decimation transformation of the honeycomb lattice produces the triangular lattice or vice versa. Here, the problem of having physically untractable interactions between degrees of freedom in the renormalized Hamiltonian, which leads eventually to inevitable approximations in the treatment, except for the 1D Ising chain, has been solved with a proper approximation. Especially for the 3D Ising lattices, the physically untractable interactions appearing in the renormalized Hamiltonian make the mathematical treatment too cumbersome. As a result, there is not enough research dealing with the 3D Ising lattices using RG theory. Our approximation is based on using the simple relation [Formula: see text], which is, of course, a very relevant first-order approximation, if [Formula: see text]. With the help of this approximation, decimation transformation process produces only pairwise interactions in the renormalized Hamiltonian instead of having four spins, six spins, or even eight spin interactions which appear naturally if all the terms are kept in the renormalized Hamiltonians of the Ising lattices in 2D and higher dimensions. Without this approximation, one cannot apply analytic RG treatment feasibly to even simple cubic lattice, let alone applying it to the body centered cubic lattice. Using this modified RG approach, the values of critical coupling strengths of the honeycomb, the triangular and the body centered cubic Ising lattices are obtained analytically as 0.6470, 0.2754 and 0.1589, respectively. Apparently, these estimations are really close to the results obtained from cumbersome exact treatments which are 0.6585, 0.2747 and 0.1575 for the honeycomb, the triangular and the body centered cubic lattices.

Modern Physics Letters B•Dec 30, 2021•2 citations
Abstract An upper bound for the free energy may generate an approximation which preserves short order range effects even when a sum of single spins is used as the auxiliary Hamiltonian. Unlike the usual mean field approximation it may distinguish the cubic lattice from the planar triangular lattice. Its application to the Ising chain recovers some exact results.
Abstract Very few statistical mechanical models have been solved exactly. In three dimensions not even the nearest neighbour spin-1/2 Ising model is tractable. Therefore it is necessary to resort to approximation methods. One of the most widely used of these is mean-field theory. A systematic way of deriving the mean-field theory for a given microscopic Hamiltonian is to start from the Bogoliubov inequality This gives the best possible approximation to the true free energy for a given choice of Ho because the inequality (4.1) insists that the meanfield free energy cannot fall below the true free energy. (This is analogous to the variational principle in quantum mechanics.) The usual choice for Ho is a free Hamiltonian -one with no interactions-as this enables explicit calculation of the right-hand side of (4.1).
We study a modified mean-field approximation for the Ising Model in arbitrary dimension. Instead of taking a "central" spin, or a small "drop" of fluctuating spins coupled to the effective field of their nearest neighbors as in the Mean-Field or the Bethe-Peierls-Weiss methods, we take an infinite chain of fluctuating spins coupled to the mean field of the rest of the lattice. This results in a significative improvement of the Mean-Field approximation with a small extra effort.
The article provides a tutorial review on how to treat Ising models within mean-field (MF), effective-field (EF) and exact methods. MF solutions of the spin-1 Blume-Capel (BC) model and the mixed-spin Ising model demonstrate a change of continuous phase transitions to discontinuous ones at a tricritical point. A quantum phase transition of the spin-S Ising model driven by a transverse field is explored within MF method. EF theory is elaborated within a single- and two-spin cluster approach to demonstrate an efficiency of this approximate method. The long-standing problem of this method concerned with a self-consistent determination of the free energy is addressed. EF theory is adapted for the spin-1/2 Ising model, the spin-S BC model and the transverse Ising model. The particular attention is paid to continuous and discontinuous transitions. Exact results for the spin-1/2 Ising chain, spin-1 BC chain and mixed-spin Ising chain are obtained using the transfer-matrix method, the crucial steps of which are reviewed for a spin-1/2 Ising square lattice. Critical points of the spin-1/2 Ising model on several lattices are rigorously obtained with the help of dual, star-triangle and decoration-iteration transformations. Mapping transformations are adapted to obtain exact results for the mixed-spin Ising model on planar lattices. An increase in the coordination number of the mixed-spin Ising model on decorated planar lattices gives rise to reentrant transitions, while the critical temperature of the mixed-spin Ising model on a regular honeycomb lattice is always greater than that of two semi-regular archimedean lattices. The effect of selective site dilution of the mixed-spin Ising model on a honeycomb lattice upon phase diagrams is examined. The review affords a brief account of the Ising models solved within MF, EF and exact methods along with a few comments on their future applicability.
In this work, the values of critical coupling strengths of the Ising lattices which are changing their lattice structure (or non-self-dual) under decimation transformations, such as the honeycomb, the triangular and the body centered cubic Ising lattices, are obtained by a modified real space renormalization group approach (RSRG). This modification is necessary to obtain a proper relation between the coupling strengths of the original and the decimated lattices. Indeed, we have achieved to obtain a proper renormalized coupling strength relation for honeycomb and triangular lattices readily, since the decimation transformation of the honeycomb lattice produces the triangular lattice or vice versa. Here, the problem of having physically untractable interactions between degrees of freedom in the renormalized Hamiltonian, which leads eventually to inevitable approximations in the treatment, except for the 1D Ising chain, has been solved with a proper approximation. Especially for the 3D Ising lattices, the physically untractable interactions appearing in the renormalized Hamiltonian make the mathematical treatment too cumbersome. As a result, there is not enough research dealing with the 3D Ising lattices using RG theory. Our approximation is based on using the simple relation [Formula: see text], which is, of course, a very relevant first-order approximation, if [Formula: see text]. With the help of this approximation, decimation transformation process produces only pairwise interactions in the renormalized Hamiltonian instead of having four spins, six spins, or even eight spin interactions which appear naturally if all the terms are kept in the renormalized Hamiltonians of the Ising lattices in 2D and higher dimensions. Without this approximation, one cannot apply analytic RG treatment feasibly to even simple cubic lattice, let alone applying it to the body centered cubic lattice. Using this modified RG approach, the values of critical coupling strengths of the honeycomb, the triangular and the body centered cubic Ising lattices are obtained analytically as 0.6470, 0.2754 and 0.1589, respectively. Apparently, these estimations are really close to the results obtained from cumbersome exact treatments which are 0.6585, 0.2747 and 0.1575 for the honeycomb, the triangular and the body centered cubic lattices.