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 ν approaches infinity. In this note we calculate 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 ν). We find that the critical point exponents deduced from short series are extremely doubtful when d > 3, where d is the lattice dimensionality. In particular, we note that it is quite possible that the mean field exponents are valid for all ν if d > 3.
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 ν approaches infinity. In this note we calculate 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 ν). We find that the critical point exponents deduced from short series are extremely doubtful when d > 3, where d is the lattice dimensionality. In particular, we note that it is quite possible that the mean field exponents are valid for all ν if d > 3.
A model Hamiltonian H=−JΣ(ij)Si·Sj is considered, where Si are isotropically interacting D-dimensional classical spins. Thus the S=½ Ising, classical planar, and classical Heisenberg models are obtained if D=1, 2, and 3, respectively; moreover, as D→∞, the free energy corresponding to H approaches the free energy of the exactly-soluble spherical model of Berlin and Kac. H can also be solved exactly providing the lattice dimensionality d=1. For lattices of higher dimensionality, we must resort to approximation techniques; hence we calculate high-temperature expansions for arbitrary D and d of the susceptibility, internal energy (specific heat), and the second moment. (With a few exceptions, such expansions have been available in the past only for D=1, 3 and d=1–3.) Somewhat smoother series to facilitate more reliable extrapolation are then obtained by re-expanding all of the series in terms of the nearest-neighbor spin correlation function for a one-dimensional lattice. We then study the dependence on spin and lattice dimensionality of the critical temperatures Tc(D, d) and the critical exponents γ(D, d) [(χ∼(T-Tc)−γ], α(D, d) [C∼(T-Tc)−α], ν(D, d) [inverse correlation range κ∼(T-Tc)−v]. These ``critical properties'' are all found to vary monotonically (and in most cases, smoothly) with D and d. The assumptions behind making critical phenomena predictions from a dozen or so terms (of an infinite series!) are given increased plausibility by the fact that we have extended the various expansions for the spherical model to 80–100 terms and find that none of the predictions based upon ∼10 terms is changed [e.g., Tc(∞, 2)=0, γ(∞, 3)=2, etc.].
The finite-lattice method of series expansion is generalized to the q-state Potts model on the simple cubic lattice. It is found that the computational effort grows exponentially with the square of the number of series terms obtained, unlike two-dimensional lattices where the computational requirements grow exponentially with the number of terms. For the Ising (q=2) case the authors have extended the low-temperature series for the partition functions, magnetization and zero-field susceptibility to u 26 from u 20 . The high-temperature series for the zero-field partition function is extended from nu 18 to nu 22 . Subsequent analysis gives critical exponents in agreement with those from field theory.
Recent lattice Hamiltonian strong-coupling calculations for the (1+1)-dimensional Ising, O(2), O(3), and O(4) models are extended to d+1 dimensions and to general O(n) symmetry. This allows a nontrivial numerical check on the rate of convergence of the strong-coupling expansions as the multidimensional models all have standard second-order phase transitions which have been well studied by other methods. Dlog-Pad\'e analysis of the fourth-order mass gap series for the (2+1)-dimensional Ising model show the critical exponent $\ensuremath{\nu}$ to be calculated to about the 10% level of accuracy. The behavior of $\ensuremath{\nu}$ as a function of d and n is exhibited and compared against known results. Mean-field behavior is recovered exactly in the $d\ensuremath{\rightarrow}\ensuremath{\infty}$ limit, while the Berlin-Kac spherical model is obtained in the $n\ensuremath{\rightarrow}\ensuremath{\infty}$ limit.
We have calculated the high-temperature series expansion of the zero-field susceptibility of the square-lattice Ising model with first and second neighbour interactions to the 20th order by computer. Our results extend the previous calculation by Hsiao and Lin to two more orders. We use the Padé approximants to estimate the critical exponent γ and the critical temperature. Our result 1.747 < γ < 1.753 supports the universality conjecture that all two-dimensional Ising models have the same critical exponent γ = 1.75.
Using the vertex renormalized form of the technique developed by Englert,1 high-temperature series expansions for the spin-spin correlation function of the classical anisotropic Heisenberg model are calculated for various lattices and anisotropies through order T−8 (close-packed lattices) and T−9 (loose-packed lattices). These series are combined and then extrapolated to give the high-temperature critical indices γ (zero-field susceptibility), ν (correlation range), and α (specific heat) as functions of anisotropy. The results are consistent with the hypothesis that the critical indices only change when there is a change in the symmetry of the system, e.g., in interpolating between the Ising and isotropic Heisenberg models, indices remain Ising-like until the system becomes isotropic, at which point they appear to change discontinuously. Previous results for the limiting cases are confirmed and extended. Series for the limiting cases (spin-infinity Ising, XY and isotropic Heisenberg models) as well as for the spin-½ Ising model2 have been derived for the B-site spinel lattice. Analysis of these series shows that the anomalies in the critical indices, reported on the basis of six-term expansions,3 disappear with the use of longer series.
We have extended through ${\ensuremath{\beta}}^{23}$ the high-temperature expansion of the second field derivative of the susceptibility for Ising models of general spin, with nearest-neighbor interactions, on the simple cubic and the body-centered cubic lattices. Moreover, the expansions for the nearest-neighbor correlation function, the susceptibility, and the second correlation moment have been extended up to ${\ensuremath{\beta}}^{25}.$ Taking advantage of these new data, we can improve the accuracy of direct estimates of critical exponents and of hyperuniversal combinations of critical amplitudes such as the renormalized four-point coupling gr or the quantity usually denoted by ${R}_{\ensuremath{\xi}}^{+}.$ In particular, we obtain $\ensuremath{\gamma}=1.2371(1),$ $\ensuremath{\nu}=0.6299(2),$ ${\ensuremath{\gamma}}_{4}=4.3647(20),$ gr=1.404(3), and ${R}_{\ensuremath{\xi}}^{+}=0.2668(5).$ We have used a variety of series extrapolation procedures and, in some of the analyses, we have assumed that the leading correction-to-scaling exponent $\ensuremath{\theta}$ is universal and roughly known. We have also verified, to high precision, the validity of the hyperscaling relation and of the universality property both with regard to the lattice structure and to the value of the spin.