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.
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.
The use of series expansions to determine the nature of critical point singularities, especially the values of the critical exponents, is reviewed.Ratio techniques, based on extrapolation of the ratios of successive expansion coefficients, and Padé approximant methods, utilizing logarithmic derivatives, are explained, illustrated, and compared. Introduction.In this lecture I will survey briefly the phenomena observed near the critical point of a system.The range of behavior is very varied and we shall only examine one or two typical examples.The main aim will be to establish the existence of the so-called "critical point exponents" which describe the nature of the mathematical singularities exhibited by the different thermodynamic functions as the critical point is approached.The theory of these singularities may be developed via suitable statistical-mechanical models which, while caricatures of a realistic physical system, retain such vital features as the dimensionality of space, the finite range of the interactions, and certain basic symmetry properties.These models in turn are usually intractable as regards exact mathematical analysis.However, general methods can be developed which yield power series expansions for many functions of interest; frequently between 8 to 30 leading coefficients of such power series can be obtained.The problem is then a numerical one of extrapolating the series in such a way as to estimate the nature of the particular critical point singularity and the behavior of the function in its vicinity.I will show how this can, in favorable cases, be handled by studying the ratios of successive expansion coefficients.The method was pioneered by Domb and his coworkers, especially Sykes and Fisher [1], [2].The reliability of the techniques can be tested against the relatively few closed form mathematical results, in particular, Onsager's renowned solution of the two-dimensional Ising models [3].In less favorable examples this approach fails, and then, as first demonstrated by Baker [3], Padé approximants have a vital role to play.In a number of important cases the intelligent application of Padé approximants has been strikingly successful [3], [4].Such techniques have thus established a central place for themselves in the
A two-series approach to partial differential approximant analysis of power series is presented. Instead of double series, $f(x,y)=\ensuremath{\Sigma}{c}_{\mathrm{ij}}{x}^{i}{y}^{j}$, our approach uses two one-variable series in x, f and $\frac{\ensuremath{\partial}f}{\ensuremath{\partial}y}$, and has the efficiency and stability of one-variable methods. 21-term high-temperature series are analyzed for the susceptibility and correlation length squared for double-Gaussian Ising models on the bcc lattice. Critical exponents are $\ensuremath{\gamma}=1.2378(6)$, $2\ensuremath{\nu}=1.2623(6)$, and $\ensuremath{\eta}=0.0375(5)$; correction-to-scaling exponents are ${\ensuremath{\theta}}_{\ensuremath{\chi}}=0.52(3)$ and ${\ensuremath{\theta}}_{{\ensuremath{\xi}}^{2}}=0.49(4)$; and the subdominant critical amplitude ratio is $\frac{{a}_{\ensuremath{\xi}}}{{a}_{\ensuremath{\chi}}}=0.83(5)$.
According to the available publications, the field theoretical renormalization group approach in the two-dimensional case gives the critical exponents that differ from the known exact values. This property is associated with the existence of nonanalytic contributions in the renormalization group functions. The situation is analyzed in this work using a new algorithm for summing divergent series that makes it possible to determine the dependence of the results for the critical exponents on the expansion coefficients for the renormalization group functions. It has been shown that the exact values of all the exponents can be obtained with a reasonable form of the coefficient functions. These functions have small nonmonotonic sections or inflections, which are poorly reproduced in natural interpolations. It is not necessary to assume the existence of singular contributions in the renormalization group functions.
Exact power series expansions provide a powerful method for studying the critical behaviour of many systems. Efficient analysis methods are essential to fully exploit the series approach. A particularly challenging problem in the analysis of these expansions is the elucidation of the critical behaviour in cases where critical temperature (or threshold) as well as dominant and correction exponent is unknown. We describe a scheme (which we call Visualization of Graphical methods for Series analysis, or VGS) developed explicitly for this situation and discuss its high precision implementation in a workstation environment. Our approach involves visualization of multiple approximants in a three-dimensional space. Several examples from Ising models, percolation and exactly solved systems are given.
First, a new method proposed by Baker and Rushbrooke is used to study the simple ferromagnetic Ising model at and below the Curie temperature. Of course, the properties of the Ising model are already well known, so that the main aim here is to assess the potential and reliability of the new method, since it has wide applicability to other models which have not been otherwise studied. Between 8 and 16 coefficients of exact high-temperature expansions for fixed values of the magnetization are derived for various two- and three-dimensional lattices. A Pad\'e-approximant analysis of these expansions at the critical isotherm and magnetic phase boundary enables us to estimate the critical exponents $\ensuremath{\beta}$, ${\ensuremath{\gamma}}^{\ensuremath{'}}$, and $\ensuremath{\delta}$, and plot the spontaneous magnetization. The results are in good agreement with previous calculations. Secondly, an analysis of the exact series expansions provides no support for the conjecture that the phase boundary is a line of essential singularities. However, the same expansions strongly suggest the existence of a "spinodal" curve, whose properties are in reasonable agreement with the predictions of various heuristic arguments (based essentially upon analyticity at the phase boundary and one-phase homogeneity in the critical region). Finally, structure and a mild extension of the proven analyticity of the free energy are used to show the $\ensuremath{\Delta}\ensuremath{\le}{\ensuremath{\Delta}}^{\ensuremath{'}}$, $\ensuremath{\gamma}\ensuremath{\le}{\ensuremath{\gamma}}^{\ensuremath{'}}$.
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.