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Physical review. B, Condensed matter

Calculation of free energies for a three-dimensional Ising model by a modified Kadanoff's variational method

Yu-Ming Shih, Dong-Chung Jou, C. K. Pan, Wen‐Den Chen, H. M. Huang, Chin‐Kun Hu, Kuo-Gen Chen, W. S. Lee, D. S. Wan, Hsein Che Tseng

Published Jan 1, 1980• 4 citations
Publisher

Abstract

The modified Kadanoff's variational method recently considered by us is applied to calculate the free energies and their derivatives for a three-dimensional Ising model with bcc structure and without external magnetic field. We find a critical point at ${k}_{c}=0.162134$ and the associated critical indices $\ensuremath{\alpha}=0.12$, ${\ensuremath{\alpha}}^{\ensuremath{'}}=0.095$.

DOI
10.1103/physrevb.21.299

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The modified Kadanoff's variational method recently considered by us is applied to calculate the free energies and their derivatives for a three-dimensional Ising model with bcc structure and without external magnetic field. We find a critical point at kc=0.162134{k}_{c}=0.162134kc​=0.162134 and the associated critical indices \ensuremathα=0.12\ensuremath{\alpha}=0.12\ensuremathα=0.12, \ensuremathα\ensuremath′=0.095{\ensuremath{\alpha}}^{\ensuremath{'}}=0.095\ensuremathα\ensuremath′=0.095.
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)
An analytic method for deriving the free energy of a three-dimensional Ising-like system near the critical point in a homogeneous external field is developed in the the ρ6 model approximation. The mathematical description proposed for temperatures T>Tc (Tc is the phase-transition temperature in the absence of an external field) is valid for fields near , where the scaling variable is of the order of unity and power series in this variable are not effective. At the limiting field , the temperature and field effects on the system in the vicinity of the critical point are equivalent. The total free energy is obtained as a function of temperature, field and microscopic parameters of the system without using series expansions in the scaling variable. In addition to leading terms, the expression for the free energy includes the terms determining the temperature and field confluent corrections.