SERIES EXPANSIONS FOR THE SPHERICAL AND ISING MODELS WITH LARGE LATTICE DIMENSIONALITY

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 at 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 > 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