The authors reanalyse the Pade approximants for the susceptibility of different Ising and Heisenberg systems using the non-linear scaling variable t'=(T-T c )/T instead of t=(T-T c )/T c . The susceptibility data are well represented by the modified power law chi approximately (1/T)(t') - gamma up to high temperatures, whereas the conventional relation chi approximately t - gamma is valid only in a small temperature range. The trend of the deviations from the modified power law in terms of temperature, spin and lattice dimensionality looks quite sensible and physical. Altogether the domain where scaling ideas are useful appears to be much more extended than generally suspected.
We have calculated the low-temperature series expansions of the spontaneous magnetization and the zero-field susceptibility of the square-lattice ferromagnetic Ising model with first-neighbour interaction J1 and second-neighbour interaction J2 to the 30th and 26th order respectively by computer. Our results extend the previous calculations by Lee and Lin to six more orders. We use the Padé approximants to estimate the critical exponents and the critical temperature for different ratios of R = J2/J1. The estimated critical temperature as a function of R agrees with the estimation by Oitmaa from high-temperature series expansions.
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.
The method of developing exact power-series expansions for the partition function ZN and related thermodynamic functions for the Ising model valid below the critical point is generalized to include exchange interactions between first-, second-, and third-neighbor pairs. Expansions of the spontaneous magnetization M0(T) and zero field susceptibility χ0(T) are derived through to sixth order of perturbation for the s.q. lattice, and through to fifth order of perturbation for the Δ′r, b.c.c., s.c., and f.c.c. lattices, when interactions J1SizSjz and J2SkzSlz are present between first- and second-neighbor spins, respectively (second-neighbor model). These expansions have also been obtained for the case where interactions of equal magnitude (J1 = J2 = J3) are present between first-, second-, and third-neighbor pairs (third-equivalent-neighbor model); here expansions through to fifth order of perturbation are obtained for the s.q., Δ′r, b.c.c., and s.c. lattices and through to fourth order for the f.c.c. lattice. The Padé approximant·procedure is employed to discuss the effects of an extended but finite range of interaction on the behavior of M0(T) and χ0(T) for T→Tc− as characterized by the critical exponents β and γ′, respectively. For the second-equivalent-neighbor model lattices, it is found that 0.122 ≤ β ≤ 0.134 in two dimensions, and that 0.308 ≤ β ≤ 0.328 in three dimensions; from which it is concluded that β remains unchanged from its value in the nearest-neighbor model. The corresponding limits for γ′ in three dimensions are 1.18 ≤ γ′ ≤ 1.28; from this and the results for the b.c.c. lattice in particular, it is concluded that γ′ is probably 54 and hence the transition in χ0 is symmetrical about Tc (γ′ = γ). A repetition of this analysis for the third-equivalent-neighbor model three-dimensional lattices shows a marked shift in the estimated range of β and γ′; the results are 0.345 ≤ β ≤ 0.365, and 1.01 ≤ γ′ ≤ 1.14. In each of the above cases, the corresponding high-temperature (T > Tc) expansions of χ0(T) obtained previously have been analyzed to yield estimates of the critical exponent γ. The over-all results and in particular the estimates of γ for the s.q. and b.c.c. lattices suggest that this index is unaffected by extending the range of interaction, and that if γ is a rational fraction then it is the same fraction for the n.n. model and second- and third-equivalent neighbor models. Finally the high-temperature expansions of ZN in zero field, and of χ0(T) for the second-neighbor model are used to examine the dependence of the critical temperature Tc, the critical energy (E∞ − Ec)/kTc, and the critical entropy (S∞ − Sc)/k on the relative strengths of J1 and J2 for values of J2/J1 in the range 0 to 1. It is found that the variation of the critical point is well represented by Tc(α)=Tc(0)[1+m1α],where α = J2/J1 and lies in the range 0 ≤ α ≤ 1; and Tc(0) is the critical temperature of the nearest-neighbor model. The values of m1 are 0.61, 2.47, 0.84, 1.45, and 1.35 for the f.c.c., s.c., b.c.c., s.q., and Δ′r lattices, respectively. All these calculations are compared with the corresponding results for the Heisenberg model.
A simple systematic rule, inspired by high-temperature series expansion (HTSE) results, is proposed for optimizing the expression for thermodynamic observables of ferromagnets exhibiting critical behavior at
Tc. This ``extended scaling'' scheme leads to a protocol for the choice of scaling variables, $\ensuremath{\tau}=(T\ensuremath{-}{T}_{c})∕T$ or $({T}^{2}\ensuremath{-}{T}_{c}^{2})∕{T}^{2}$ depending on the observable instead of $(T\ensuremath{-}{T}_{c})∕{T}_{c}$, and more importantly to temperature dependent noncritical prefactors for each observable. The rule corresponds to scaling of the leading term of the reduced susceptibility above
Tc as ${\ensuremath{\chi}}_{c}^{*}(T)\ensuremath{\sim}{\ensuremath{\tau}}^{\ensuremath{-}\ensuremath{\gamma}}$ in agreement with standard practice with scaling variable $\ensuremath{\tau}$ and for the leading term of the second-moment correlation length as ${\ensuremath{\xi}}_{c}^{*}(T)\ensuremath{\sim}{T}^{\ensuremath{-}1∕2}{\ensuremath{\tau}}^{\ensuremath{-}\ensuremath{\nu}}$. For the specific heat in bipartite lattices, the rule gives ${C}_{c}^{*}(T)\ensuremath{\sim}{T}^{\ensuremath{-}2}{[({T}^{2}\ensuremath{-}{T}_{c}^{2})∕{T}^{2}]}^{\ensuremath{-}\ensuremath{\alpha}}$. The latter two expressions are not standard. The scheme can allow for confluent and noncritical correction terms. A stringent test of the extended scaling is made through analyses of high-precision numerical and HTSE data, or real data, on the three-dimensional canonical Ising,
XY, and Heisenberg ferromagnets. For the susceptibility $\ensuremath{\chi}(T)$ and the correlation length $\ensuremath{\xi}(T)$ of the three ferromagnets, their optimized expression, which consists of the leading terms [respectively, ${\ensuremath{\chi}}_{c}^{*}(T)$ and ${\ensuremath{\xi}}_{c}^{*}(T)$] and a quite limited number of confluent and noncritical correction terms, represents real data to surprisingly good approximations over the entire temperature range from
Tc to infinity. The temperature dependent prefactors introduced are of crucial importance not only in fixing the optimized expression at relatively high temperatures but also in determining appropriately the small amplitude correction terms. For the specific heat of the Ising ferromagnet,
Cc∗(T) combined with two noncritical correction terms which are calculated with no free parameters once the correlation length critical parameters are known reproduces real data nicely also over the whole temperature range.
By combining all available information it is possible to extend the numerical knowledge of the high- and low-temperature zero-field isothermal susceptibility of the Ising model. Estimates of the third most singular term of the ferromagnetic singularity are obtained, using both direct estimates and an extension of the Generalized Law of Corresponding States. A recent conjecture of Barouch, McCoy and Wu (1973) concerning the equality of the constant term in the asymptotic expansion of the high- and low-temperature ferromagnetic susceptibility is confirmed numerically. Finally, a non-physical critical exponent of the low-temperature susceptibility of the triangular lattice is identified.