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Progress of Theoretical Physics

On the Expansion Theory of Critical Exponents

Masuo Suzuki

Published Aug 1, 1973• 4 citations
Publisher

Abstract

General aspects of expansions of critical exponents are investigated with emphasis on matching conditions to determine them. There are given several suggestings and some applications to typical models of critical phenomena.
DOI
10.1143/ptp.50.393

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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
The modern techniques of field theory applied to critical phenomena, are briefly discussed, with particular emphasis on the use of the e-expansion to extract the asymptotic behaviour in the critical region. The dimensional regularisation and minimal subtraction method of renormalisation, Introduced by 't Hooft and Veltman, is outlined. These methods, along with a Monte Carlo simulation on a highly parallel computer, are utilised in extracting information pertaining to the behaviour near criticality of the random, site-diluted Ising model. The universal ratio of the susceptibility amplitude above and below the critical temperature is determined to one higher order than previously in an ɛ-expansion, within the framework of the replica formalism. The calculation is performed using a Taylor expansion, in the number of replicas, of the transverse propagator appearing In the Hamiltonian, about the longitudinal. This higher order correction goes in the right direction for agreement with experiment. An extension of the formalism to embrace the presence of two sets of replica spins is used to derive the o(ɛ¹/²) term in an expansion of the additional static - correlation function, C⁽ˢ⁾(q), arising from the non-interchangeability of the thermal and configurational averages In the random model. The appearance of C(sl(q) in the structure factor Implies that results of neutron scattering experiments to measure the susceptibility amplitude ratio must be re-Interpreted. Finally, analysis of the properties of block spin variables In a Monte Carlo simulation Is used to determine the phase diagram and critical exponents In two-dimensions. Possible Interpretations of the observed evolution of the exponent ratio, ß/υ, under Increasing dilution are discussed.