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Journal of physics. A, Proceedings of the Physical Society. General

On a new method of series analysis in lattice statistics

Anthony John Guttmann, G S Joyce

Published Sep 1, 1972• 99 citations
Publisher

Abstract

A new recurrence relation method for analysing the singular behaviour of series expansions is described. It is shown using plausible nonrigorous arguments that the method can resolve logarithmic singularities and finite cusp singularities.
DOI
10.1088/0305-4470/5/9/001

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Several new general methods of series analysis are introduced and tested on an assortment of known functions. These methods are then applied to several of the series expansions known for the thermodynamic properties of the spin-\textonehalf{} Ising model. We are able to obtain an unbiased analysis of the high-temperature specific-heat singularity, and find $\ensuremath{\alpha}\ensuremath{\approx}0.13\ifmmode\pm\else\textpm\fi{}0.01$. The application of our methods in other cases generally agrees with that of previous workers. Our methods in practice amount to procedures for looking more closely at the dominant singularity, at the expense of accuracy for competing singularities. For certain types of function, thought to occur in the study of critical phenomena, our methods are noticeably more effective than those previously employed.
The authors propose a new method of series analysis in which the available series coefficients are fitted to an algebraic equation. Such a proposal restricts solutions to have algebraic singularities with rational critical exponents. The advantage of the method over the method of differential approximants is that in those cases where the two methods are exact, the algebraic approximant method usually (though not invariably) requires far fewer series coefficients for its exact specification, and hence discovery. The method is therefore potentially valuable in the study of two-dimensional critical systems in particular.
A simple method is proposed for determining characteristic parameters of critical-point singularities from power-series expansions. It applies to a broad class of functions of physical interest and provides acceptable, simultaneous approximations to the critical point, exponent, and amplitude. Several examples of low- and high-temperature series expansions for thermodynamic properties of spin-(1/2) Ising models are studied, and quite accurate results are obtained for both close- and loose-packed arrays.
Abstract Exact power series expansions for thermodynamic functions have in the past proved an invaluable aid to understanding the critical behaviour of insoluble models. Indeed, the first suggestions of power law singularities at criticality were based on such analyses. Immediately before the renormalization group was proposed, work using series expansions had led to a large body of evidence that exponents had universal properties, that they were the same above and below the critical temperature, and that mean-field values set in abruptly above four dimensions.