Exact Results for TwoDimensional Ising Models

Authors: David A Lavis George M Bell

Publish Date: 1999

Volume: , Issue: , Pages: 205-240

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Abstract

We already have exact results for the onedimensional Ising model Chap 2 and Sect 41 and exact transcriptions between models for instance from the zerofield Ising ferromagnet to the zerofield antiferromagnet Sect 42 or from the Ising model to the simple lattice fluid or twocomponent mixture Chaps 5 and 6 In Sects 8286 dual and startriangle transformations are derived connecting the partition function for the zerofield Ising model on a given lattice to that of the zerofield Ising model on a related lattice at a different temperature1 Some associated relations connecting nearestneighbour correlations are obtained a particularly important one being derived in Sect 86 We then derive exact results for critical conditions and thermodynamic functions for the twodimensional zerofield Ising model There are a large number of approaches to this all mathematically intricate2 The original method of Onsager 1944 Kaufman 1949 and Kaufman and Onsager 1949 depended on transfer matrices of which the 2 × 2 matrix appearing in the theory of the onedimensional Ising model in Sect 24 is a simple example Other workers have employed the Pfaffian see Volume 2 Chap 8 Green and Hurst 1964 McCoy and Wu 1973 or combinatorial methods Kac and Ward 1952 Vdovichenko 1965a 1965b We shall use a method due to Baxter and Enting 1978 which is closely related to the preceding transformation theory and has the advantage of giving results simultaneously for the square triangular and honeycomb lattices The exact expressions for configurational energy and heat capacity depend on elliptic integrals and these protean quantities appear in the literature in a number of different forms We have tried to use expressions valid for the whole temperature range and closely related to the formulae given for the partition function or free energy

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