CRITICAL PROPERTIES OBTAINED BY A CONFORMAL TRANSFORMATION METHOD

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CRITICAL PROPERTIES OBTAINED BY A CONFORMAL TRANSFORMATION METHOD

M. Lee, H. Stanley

To cite this version:

M. Lee, H. Stanley. CRITICAL PROPERTIES OBTAINED BY A CONFORMAL TRANS- FORMATION METHOD. Journal de Physique Colloques, 1971, 32 (C1), pp.C1-352-C1-353.

�10.1051/jphyscol:19711120�. �jpa-00213937�

JOURNAL DE PHYSIQUE CoIIoque C I, suppldment au no 2-3, Tome 32, Fkvriei--Mars 1971, page C 1 - ³⁵²

CRITICAL PROPERTIES OBTAINED

BY A CONFORMAL TRANSFORMATION METHOD

M. H. LEE and H. E. STANLEY

Physics Department, Massachusetts Institute of Technology Cambridge, Massachusetts, U. S. A.

RBsumk. - Les propri6t6s critiques de tous les modkles 5 trois dimensions du magnktisme sont obtenues par des mkthodes d'extrapolation de skrie. Une incertitude. considerable existe dans les valeurs critiques estim6es dans les cas oh les coefficients d'une s6rie donnte ne se comportent pas rkgulikrement. Nous prouvons que le compor- tement irrkgulier de beaucoup de s6ries peut 6tre compris en termes de singularites non-physiques qui se trouvent proches d'une singularit6 physique et de ce fait ⁽⁽ interfkrent ⁾⁾ avec elle. De plus, nous trouvons que les valeurs critiques de ces series peuvent &tre estimees valablement en faisant une transformation conforme des singularitks non physiques, loin de la singularitk physique, avec pour ~6sultat que t( les shies transformkes ^)> se comportent de faqon regulikre, Nous pouvons ensuite analyser ces s6r1es transformees en utilisant les m6thodes standard.

Nous avons applique cette technique a l'obtent~on des propriktks critiques pour des hamiltoniens &Heisenberg, d'lsing et XY et pour a la fois des reseaux cubiques et spinelles. Un r6sultat particulikrement int6ressant est que I'exposant de la susceptibilit6 pour le modde &Heisenberg S = 112, sur des r6seaux cubiques, a la yaleur

y = 1,36 f 0,04. Cette valeur est un peu plus basse que la valeur acceptke antkrieurement y = 1,43 & 0,01, mais elle est coherente avec la prediction de la loi d'tchelle que y (5') est indkpendant du nombre quantique de spin S.

Abstract. - Critical properties for all realistic three-dimensional models of magnetism are obtained by series extra- polation methods. Considerable uncertainty exists in the estimated critical values for those cases in which the coefficients of a given series do not behave regularly. However, the critical values for these series can be estimated reliably by confor- mally mapping the nonphysical singularities away from the physical singularity.

I. Introduction. - Estimates for the critical point and exponent are made from series expansions usually by ratio and Padt approximant techniques. When the expansion coefficients of a given series do not behave regularly,considerableuncertainty existsin theestimated critical values and in some cases no useful estimates can be made at all.

We describe here a transformation method which is capable of analyzing high temperature series expan- sions with irregular coefficients.

11. Transformation method. - Consider the high temperature expansion for the susceptibility series x(K), where K r J / k , T. For K < Kc EZ J/k, Tc, the reduced susceptibility 2 has a power series about the origin in the form

CO

Z K ) = C ^{an Kn ,}

n = O (1)

where a finite number of the expansion coefficients a,, are exactly determined for various lattices of several useful models. Near the critical point the susceptibility is assumed to diverge as

j j ( K ) - ( K c - K ) - Y K + K i (3

where y is the critical exponent.

As an illustration we shall consider the susceptibility for the S = c~ Heisenberg ferromagnet on the nearest- neighbor B-site spinel lattice. Presently known are 8 coefficients for this function [I, 21. The ratios of these coefficients are, however, not very regular. This irregular behavior of the coefficients may be shown to be due to interfering effects of nonphysical singularities of the susceptibility, located close f o the circle of convergence. The radius of convergence is determined by a positive real pole which is the ferromagnetic critical point. PadB analysis shows that the principal singularities of the susceptibility for this Heisenberg

ferromagnet on the spinel lattice are located in the complex K-plane at

Kl ^G Kc c 0.75, K2 ^N 0.30 + ^0.80 i and

K, -- ^{0.30 - 0.80 i} .

Suppose the susceptibility x(K) is conformally transformed into, say, x(K*), such that the correspond- ing nonphysical singularities KT and K? are now farther removed from the new circle of convergence whose radius is determined by K:. Then the transformed series will be asymptotically dominated by the nearest and strongest pole K? and this will enable us to obtain the asymptotic properties more reliably by the ratio technique.

If we have a completely convergent series expansion, there may be in principle a variety of conformal trans- formations which can be used for our purpose. But in practice thermodynamic functions such as the suscepti- bility are never given by more than a finite number of expansion coefficients. As a result, there are certain restrictions to the kind of transformation one may use.

The essential one is that the n initial coefficients of the transformed series shall depend only on the n exactly known coefficients of the original series. One simple transformation which meets the requirement is the Euler transform

where A and B are real numbers. The values for A and B are dictated by the nature of singularities of a func- tion under consideration. Once the singularities are known, best values for A and B can be easily deter- mined by trial.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19711120

CRITICAL PROPERTIES OBTAINED BY A CONFORMAL TRANSFORMATION METHOD C 1 - ³⁵³

111. New results. - We shall apply the Euler transform (3) with A ⁼ 1 and B = 2 to the susceptibi- lity series for the S = co Heisenberg ferromagnet on the spinel lattice [I, 21. The resulting series for the susceptibility is then

Pad6 analysis of the transformed series (4) shows that the singularities K 1 , K2 and K3 are, respectively, mapped into K: ^cx 0.30, K: N 0.35 + 0.15 i and K: N 0.35 - 0.15 i. In the transformed plane the nonphysical singularities are a little farther removed from the circle of convergence than in the original plane. In the new plane the radius of convergence is still determined by the physical pole K:. Hence the transformed series (4) is now expected to be more satisfactory for ratio analysis, as indeed a ratio plot of (4) bears out. Our analysis yields :

+ o 10

K: = 0.300 0 + 0.001 and y = 1.24-0:05 .

By (3) we get Kc = 0.750 + 0.006. We note that the susceptibility exponent for the spinel lattice is considerably smaller than y - 1.38 for regular cubic lattices [3].

By applying similar bilinear transformations to the susceptibility series for the S = 3 Ising and the S = c~

XY ferromagnets on the nearest-neighbor B-site

spinel lattice, we have obtained critical values which are given in Table I. We have also given estimates for the correlation range exponent v, similarly obtained from the high temperature expansion of second moment of the spin-spin correlation function [2].

The transformation method is also applied to high temperature series expansions of the S = + Heisenberg ferromagnet on regular cubic lattices. These series are notoriously irregular and estimates for critical values

Critical values of the S = 3 Ising S ^{=' c~} XY, and S ⁼ co Heisenberg ferromagnets on the spinel lattice obtained from the exact high temperature series expansions by the transformation method.

Ising XY Heisenberg

- -

Kc 0.231 Jr 0.005 0.685 0 & 0.005 0.750 & 0.006

y 1.1 1.23 1.24

v l 0.53 0.67 & 0.06 0.70 f 0.06

(e. g., y = 1.43 + 0.01) are provided mainly by Pad&

approximant technique alone [4]. We believe that these values are somewhat too large. Our estimates for the critical exponent y = 1.36 -1- 0.04, while not in agreement with the value of Baker et al., are consistent with the arguments of Bowers and Woolf [5]. Our value is also close to the susceptibility exponent for the classical ( S = co) Heisenberg ferromagnet on regular cubic lattices [3, 51.

The general concept of transformation methods seems to have been proposed over a decade ago by Danielian and Stevens [6], and more recently by Guttmann [7]. Similar ideas are known to have been considered by M. Wortis and D. D. Betts [a].

References STANLEY (H. E.), KAPLAN (T. A.), J. Appl. Phys., 1967,

38, 977.

JASNOW (D.), MOORE (M. A.), Phys. Rev., 1968,176,751.

STANLEY ( H . E.), Phys. Rev., 1967, 158, 546 ; J. Appl.

Phys., 1969, 40, 1272 and references cited therein.

BAKER Jr (G. A.), GILBERT (H. E.), EVE (J.), RUSH-

BROOKE (G. S.), P h y ~ . Rev., 1967, 164, 800.

BOWERS (R. G.), WOOLF (M. E.), Phys. Rev., 1969, 117, 917.

DANIELIAN (A.), STEVENS (K. W. H.), Pmc. Phys.

Soc., (London) 1957, B 70., 326.

GUTTMANN (A. J.), Thesis, University of New South Wales, 1969 (unpublished). See also GUTTMANN (A. J.), THOMPSON (C. J.), Phys. Letters, 1969, 28A, 679.

BETTS (D. D.) et AL., to be published.