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Physical review. B, Condensed matter

New method for calculating series expansions of correlation functions in the d=2 Ising model

Ranjan K. Ghosh, Robert E. Shrock

Published Jul 1, 1984• 6 citations
Publisher

Abstract

We exhibit a new method for calculating high-and low-temperature series expansions for diagonal spin-spin correlation functions in the $d=2$ Ising model. The method makes use of a differential equation obeyed by these functions.

DOI
10.1103/physrevb.30.19

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Progress of Theoretical Physics•Jun 1, 1984•35 citations
We exhibit a new method for calculating high-and low-temperature series expansions for diagonal spin-spin correlation functions in the d=2d=2d=2 Ising model. The method makes use of a differential equation obeyed by these functions.
Abstract We consider spin–spin correlation functions for spins along a row, R n = ⟨ σ 0,0 σ n ,0 ⟩, in the two-dimensional Ising model. We discuss a method for calculating general- n expressions for coefficients in high-temperature and low-temperature series expansions of R n and apply it to obtain such expressions for several higher-order coefficients. In addition to their intrinsic interest, these results could be useful in the continuing quest for a nonlinear ordinary differential equation whose solution would determine R n , analogous to the known nonlinear ordinary differential equation whose solution determines the diagonal correlation function ⟨ σ 0,0 σ n , n ⟩ in this model.
A method is described for obtaining perturbation expansions of the pair correlation function about the ferromagnetically ordered state of the S=1/2 Ising model. The expansions take the form of a double power series in temperature and field variables and may be used in any region of the T-H plane of the ferromagnetic and in the disordered region of the antiferromagnet. Coefficients are calculated for the face-centred cubic lattice which allow the low-temperature expansions in zero field to be obtained through u31 and the high-field expansions along the critical isotherm to be obtained through mu 6. Agreement is obtained with existing high-temperature expansions.
High-temperature series expansions have been derived for pair correlations in the square Ising model with nearest-neighbor and diagonal interactions. Enting has pointed out that constraints on the series expansions can be obtained from exact results on the disorder line of the model. The constraints are imposed on the pair correlation series and turn out to be satisfied by the published series coefficients. The coefficients thus pass a rather severe test. In addition, the constraints allow the prediction of a few extra coefficients.