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A Study of the Free Energy of the Lenz-Ising Model Using the Cluster Variation Method of Morita

J. J. Condon, Joann M. Harnden, Henning Leidecker

Published Oct 6, 1987• 0 citations

Abstract

Abstract : The Cluster Variation Method proposed by Morita is shown to yield a development of the entropy into an alternating series when applied to the Lenz- Ising Model. Truncation of this series to its first N terms gives an N-th order variationally based approximation to the exact statistical behavior of this system. The first four orders of this sequence of approximations have been carried out in this work. curious behaviors such as complex Curie temperatures have been observed by others and this work shows that such behavior is due to the alternating sign of the terms in the entropy expansion. The effect is large enough to cause a non-physical global minimum to appear on the boundary of the variation space, which is certainly unphysical. A modification of the Cluster Variation Method is effective in suppressing these undesired effects. The (2n+1) -cluster results are then an improvement over those of the (2n+1)-cluster approximation. The (2n)-cluster results are less affected by spurious minima. Keywords: Statistical mechanics; Ising model; Critical point; Ferromagnetism; Order disorder; Thermodynamics; Probability theory; Minimization; Free energy; Cooperative behavior; Curie temperature; Chaos; Variation methods; Entropy; Phase transitions; Clusters.

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Methods of statistical physics

友安田中

1. Laws of thermodynamics 2. Thermodynamic relations 3. The ensemble theory 4. System Hamiltonians 5. The density matrix 6. The cluster variation method 7. Infinite series representations of correlation functions 8. The extended mean-field approximation 9. The exact Ising lattice identities 10. Propagation of short range order 11. Phase transition of the two-dimensional Ising model Appendix 1. The gamma function Appendix 2. The critical exponent in the tetrahedron approximation Appendix 3. Programming organization of the cluster variation method Appendix 4. A unitary transformation applied to the Hubbard Hamiltonian Appendix 5. Exact Ising identities on the diamond lattice References Subject index.

Cambridge University Press eBooks•Jan 1, 2002•0 citations

Abstract : The Cluster Variation Method proposed by Morita is shown to yield a development of the entropy into an alternating series when applied to the Lenz- Ising Model. Truncation of this series to its first N terms gives an N-th order variationally based approximation to the exact statistical behavior of this system. The first four orders of this sequence of approximations have been carried out in this work. curious behaviors such as complex Curie temperatures have been observed by others and this work shows that such behavior is due to the alternating sign of the terms in the entropy expansion. The effect is large enough to cause a non-physical global minimum to appear on the boundary of the variation space, which is certainly unphysical. A modification of the Cluster Variation Method is effective in suppressing these undesired effects. The (2n+1) -cluster results are then an improvement over those of the (2n+1)-cluster approximation. The (2n)-cluster results are less affected by spurious minima. Keywords: Statistical mechanics; Ising model; Critical point; Ferromagnetism; Order disorder; Thermodynamics; Probability theory; Minimization; Free energy; Cooperative behavior; Curie temperature; Chaos; Variation methods; Entropy; Phase transitions; Clusters.

The cluster-variation method was used to calculate the critical temperature for the fcc Ising ferromagnet in three different cluster approximations. A scheme for determing a set of independent cluster variables is presented, which considerably simplifies the minimization of the free energy. The system of equations arising from such minimization is solved, in the disordered state, by means of a simple iteration technique. The highest level of approximation treated in this work yields a critical temperature which is only 1.5% above the one estimated by the exact high-temperature expansion of the zero-field susceptibility. Furthermore, the high-temperature expansion for the specific heat gives four exact coefficients and the fifth is determined to within 0.4%.

Motivated by a recent proposal of a Bethe approximation for the triangular Ising antiferromagnet [Phys. Rev. B 56, 8241 (1997)], which seems to predict a disordered phase at any temperature in zero field, we analyze in some detail several mean-field-like approximations for this model, namely, the Bethe approximation itself, the cluster variation method, and the hard-spin mean-field theory. We show (i) that the disordered phase predicted by the Bethe approximation is unphysical at low enough temperature because of a negative entropy; (ii) how the results of the cluster variation method (namely, zero-temperature entropy and critical temperature of the spurious transition) converge to the exact ones for increasing cluster size; (iii) that it is possible to construct a cluster variation approximation which yields a disordered phase which is stable down to zero temperature; (iv) a few, so far unknown, zero-temperature results (entropy and internal energy) of the hard-spin mean-field theory.

A linked-cluster expansion technique for spin systems has been used to obtain the free energy, the magnetization, and the susceptibility series for three Ising systems: (1) The Ising model; (2) The Blume-Capel model; and (3) The Blume-Emery-Griffiths (BEG) model. Each Hamiltonian is first divided into a single-ion potential and a term describing the interaction of the spin fluctuations. Only the latter is treated as perturbation in the series expansion. In the parametric phase our series reduces to the exact high-temperature series. However, in the ordered phase the same series can be used to study the low temperature behavior, such as the variation of the magnetization with temperature. The analysis of the eighth-order series for systems (1) and (2) show results in excellent agreement with the previous ones obtained from the high-temperature and low-temperature series. The series analysis for the BEG model shows quite different behavior from that predicted by the mean-field approximation.

A general perturbation theory of the statistics of spin interactions is developed in the form of a linked cluster expansion with particular emphasis on the Ising model. The theory applies to the evaluation of the expectation value of arbitrary spin functions as well as of the free energy. The thermodynamical consistency of the perturbation expansion is shown to arise from (1) variational principles satisfied by the free energy after a "renormalization procedure" has been carried out and (2) a generalized "Ward Identity" between renormalized quantities. These results are used to discuss inconsistencies in recent high-density theories of ferromagnetism and an improved theory obtained by the summation of all the convolution diagrams (nodal expansion) is briefly presented. The applicability of the method to general quantum mechanical many-body problems, including the theory of the Heisenberg model of ferromagnetism, is shown.

Motivated by a recent proposal of a Bethe approximation for the triangular Ising antiferromagnet [Phys. Rev. B {\bf 56}, 8241 (1997)], which seems to predict a disordered phase at any temperature in zero field, we analyze in some detail several mean-field like approximations for this model, namely the Bethe approximation itself, the cluster variation method and the hard-spin mean-field theory. We show: (i) that the disordered phase predicted by the Bethe approximation is unphysical at low enough temperature because of a negative entropy; (ii) how the results of the cluster variation method (namely, zero temperature entropy and critical temperature of the spurious transition) converge to the exact ones for increasing cluster size; (iii) that it is possible to construct a cluster variation approximation which yields a disordered phase which is stable down to zero temperature; (iv) a few, so far unknown, zero temperature results (entropy and internal energy) of the hard-spin mean-field theory.

1. Laws of thermodynamics 2. Thermodynamic relations 3. The ensemble theory 4. System Hamiltonians 5. The density matrix 6. The cluster variation method 7. Infinite series representations of correlation functions 8. The extended mean-field approximation 9. The exact Ising lattice identities 10. Propagation of short range order 11. Phase transition of the two-dimensional Ising model Appendix 1. The gamma function Appendix 2. The critical exponent in the tetrahedron approximation Appendix 3. Programming organization of the cluster variation method Appendix 4. A unitary transformation applied to the Hubbard Hamiltonian Appendix 5. Exact Ising identities on the diamond lattice References Subject index.