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SERIES EXPANSIONS FOR THE SPHERICAL AND ISING MODELS WITH LARGE LATTICE
DIMENSIONALITY
S. Miloševij, H. Stanley
To cite this version:
S. Miloševij, H. Stanley. SERIES EXPANSIONS FOR THE SPHERICAL AND ISING MODELS WITH LARGE LATTICE DIMENSIONALITY. Journal de Physique Colloques, 1971, 32 (C1), pp.C1- 346-C1-348. �10.1051/jphyscol:19711117�. �jpa-00213933�
JOURNAL DE PHYSIQUE Colloque C 1, supplkment au no 2-3, Tome 32, Fgvrier-Mars 1971, page C 1 - 346
SERIES EXPANSIONS FOR THE SPHERICAL AND ISING MODELS WITH LARGE LATTICE DIMENSIONALITY (")
by S. MILOSEVI~ (**) and H. E. STANLEY
Physics Department, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
RBsumk. - On peut considerer le modele sphbrique, ou bien comme une approximation calculable exactement du modele d'Ising ou bien comme un modele de spins interagissant de maniere isotope lorsque la dimension de spin v tend vers l'infini. Nous calculons ici 80 termes dans le developpement du modele sphkrique valable a haute temperature et comparons les rbultats B ceux qu'on obtiendrait avec. 10 termes seulement (comme dans le cas oh v est fini). On trouve que les exposants au point critique que l'on obtiendra~t~avec un dkveioppement court sont extrememerit douteux lorsque d > 3 (d Ctant la dimension du reseau). En particulier, 11 est tres possible que les exposants moyens soient valables pour toute valeur de v lorsque d > 3.
Abstract. - The spherical model can be regarded either as an exactly-soluble approximation to the Ising model or as a model of isotropically interacting spins in which the spin dimensionality v approaches infinity. In this note we cal- culate 80 terms in the high-temperature expansions for the spherical model and compare with predictions which would have been obtained if only - 10 terms were known (as in the case for finite v). We find that the critical point exponents deduced from short serles are extremely doubtful when d > 3, where d 1s the lattlce dimens~onality. In particular, we note that it is quite possible that the mean field exponents are valid for all v if d > 3.
There is neither general proof of the existence of the critical point exponents nor clear evidences of a set of physical parameters on which these exponents are mainly dependent. The facts are that the charac- teristics of phase transitions predominantly depend on the dimensionality d of the lattices and that so far the scaling law hypothesis [ l ] 121 is the only unifying theory of the critical point exponents for various physical systems. Hence, it is desirable to investigate these facts for mathematically tractable model systems.
The two-dimensional Ising model and the three- dimensional spherical model [3] are the only two soluble models of magnetic systems which exibit phase transition. The spherical model is a modification of the Ising model but a t high temperatures and at high- density limit the properties of these two are very similar [4] [ 5 ] . Recently it has been conjectured the breakdown of one of the predictions of the scaling law hypothesis for -the three-dimensional Ising model
[6] [7]. Therefore we are led to examine the similar
predictions for the spherical model and because of the mentioned similarity of the two models we will consi- der the critical exponents above the transition point T, of the spherical model with nearest-neihbor inter- actions for d 2 3 (for d < 3 there is no phase transi- tion [3], i. e. T, = 0).
Since the mentioned (( breakdown >> of the scaling hypothesis was established on the basis of high- temperature expansions we have firstly performed this type of calculations. For the Ising model Fisher and Gaunt [8] have calculated 11 coefficients in the corres- ponding expansions, but for the spherical model we have been able to calculate 80 coefficients. The results for the zero-field susceptibility are displayed in figure 1.
The similarity in the asymptotic behavior of the two
(*) Supported by the National Science Foundation under contract No GP-15 428.
(**) On leave of absence from the Institute of Physics, Belgrade, Yugoslavie.
FIG. 1. -Evidence from series expansions for the value of the susceptibility exponent y. Only the Ising model (v = 1) and the spherical model ( v = oo) are shown, where v is the spin dimensionality. Analogous plots for other values of v lie in between the v = 1 and v = oo curves, and extrapolations support the hypothesis [17] that the critical index y is a mono- tonic function of v. The function shown is given by
where tn,n-1 = npm - (n - 1) p n - I , p n = a n / a i ~ n - i and a n are the coefficients in the susceptibility series with expansion para-
meter x E JIkT.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19711117
SERIES EXPANSIONS FOR THE SPHERICAL AND ISING MODELS WITH LARGE C 1 - 347
models can be observed and besides a danger of obtaining erroneous results from the high-temperature calculations is noticeable. Namely, it is evident that if there were results only for 11 coefficients for the spherical model, then one might conjecture spurious values for the corresponding critical exponents y, however, from the presented numerical results we conclude that y is equal 1 for the spherical model with d > 3 and that the conjecture [9] that the four- dimensional Ising model has mean-field critical indices is a plausible one. We have supported our conclusion for the spherical model by direct calculations of y.
These calculations are based on an idea first utilized by Joyce [lo].
The zero-field susceptibility per particle for the spherical model with the simple cubic lattice and the nearest-neighbor interaction is determined [3] [lo] by
where pO and J are the elementary magnetic moment and exchange parameter respectively, while z is the saddle point of an integral related to the partition function. For the given temperature T the value of z is to be found from the equation
It is impossible to solve this integral and thus to find a n explicit expression for z as a function of tempera- ture unless d = 1 [3]. Hence we have to perform some asymptotic expansion of the integral in terms of (z - d) in order to find behavior of the susceptibility in the vicinity of the critical point. The convenient expansion is a generalization of the work by Maradu- din et al. [ll]. Having performed such an expansion we can find from eq. (2) a relation between ( z - a) and ( T - T,), where T, is the critical temperature, and then it is not difficult to get the critical exponent y from the eq. (1). We found that y = 2 for d = 3 and y = 1 for d > 3.
In order to consider predictions of the scaling law hypothesis we have calculated some additional expo- nents. The so-called gap exponent [12] A is to be determined by the asymptotic behavior of the fourth derivative of the free energy with respect to the external magnetic field H, at the value H = 0. We have firstly found this derivative for the free energy per particle Y
...I:, d o 1 d ... dw, z - C cos oj
j = 1 I ('I
where z is solution of eq. (2). Accoraing t o the defini-
tion of A, it should have the asymptotic behavior of the form
{z]H=o
- ( T - (4)We have secondly generalized the above mentioned expansion t o the case of the integral in eq. (3) and hence we have obtained its expansion in terms of ( z - d). Knowing asymptotic relation between ( z - d)
and ( T - T,) from eq. (2) we found that A = 512 for d = 3 and A = 3/2 for d > 3.
In a similar way fe found that the critical exponent a for the specific heat is equal zero for d 2 3 but that its substitute [I21 a, is equal to - (d - 4)/2 for d > 3 and a, = - 1 for d = 3. The results for y, A, a, and a, are displayed in Table 1. We should notice that these results support the recently noted similarity between the spherical model and the ideal Bose gas [I31 in the respective critical regions and in addition they agree with the very recent analysis done by Theumann from a different point of view [14].
TABLE I
Some critical point exponents for d-dimensional spherical model
d 3 4 n
-- - - -
Y 2 1 1
A 512
0 3/2
0 312
CI 0
as - 1 0 - ( a - 4 ) / 2
One of the predictions of the scaling law hypothesis is the following relation [I51 between the critical point exponents
A = 1 + $(y - a3 (9
By using the values from Table I we see that this relation is not satisfied for d > 4 with the exponent a,, whereas it is satisfied with a. Among the various formulations of the scaling hypothesis the Fisher dro- plet model approach [16] is only one which contains the explicit statement that the relations like (5) should be satisfied with a, primary. Hence there is the break- down of the relation (5) for the spherical model with d > 4. We should also notice that the acceptance of a, = - 1 for d = 3 is rather formal [I21 since there is no singularity in the specific heat for the three-dimen- sional spherical model. Thus even for d = 3 there is breakdown of the relation (5).
In summary, the conclusions of our work are (1) one must be very careful in deducing critical point exponents from the small number of coefficients available in most high-temperature expansions, (2) the difference between using a and a, in the scaling law relations is well-illustrated by the spherical model and (3) it is quite possible that the Jsing model critical exponents acquire respective mean-field values for lattices with d > 3.
We wish to thank W. K Theumann and G. Stell for helpful discussions.
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