If scaling variables and scaling expressions are chosen judiciously, ``critical'' scaling analyses do not need to be restricted to a narrow range of temperatures near the critical point. The standard scaling variable $(T\ensuremath{-}{T}_{c})/{T}_{c}$ is inadapted to wide ranges of temperature because it diverges at high temperatures. With the variable $\ensuremath{\tau}=(T\ensuremath{-}{T}_{c})/T$, in the high-dimension (mean-field) limit, the reduced susceptibility and the second-moment correlation length of the Ising ferromagnet depend on temperature as $\ensuremath{\chi}(T)={\ensuremath{\tau}}^{\ensuremath{-}1}$ and $\ensuremath{\xi}(T)={T}^{\ensuremath{-}1/2}{\ensuremath{\tau}}^{\ensuremath{-}1/2}$ exactly over the entire temperature range above the critical temperature
Tc. For the canonical two-dimensional square lattice near-neighbor Ising ferromagnet, it is shown that compact ``extended scaling'' expressions analogous to the high-dimensional limit form but with the appropriate exponents give accurate approximations to the true temperature dependencies, again over the entire temperature range from
Tc to infinity. Within this approach, for near-neighbor interaction systems, there is no crossover temperature above which mean-field-like behavior sets in.
We have extended, from order 12 through order 25, the high-temperature series expansions (in zero magnetic field) for the spin-spin correlations of the spin-
S Ising models on the square, simple-cubic and body-centered-cubic lattices. On the basis of this large set of data, we confirm accurately the validity of the scaling and universality hypotheses by resuming several tests which involve the correlation function, its moments and the exponential or the second-moment correlation lengths.
It is often assumed that for treating numerical (or experimental) data on continuous transitions the formal analysis derived from the renormalization-group theory can only be applied over a narrow temperature range, the ``critical region''; outside this region correction terms proliferate rendering attempts to apply the formalism hopeless. This pessimistic conclusion follows largely from a choice of scaling variables and scaling expressions, which is traditional but very inefficient for data covering wide temperature ranges. An alternative ``extended scaling'' approach can be made where the choice of scaling variables and scaling expressions is rationalized in the light of well established high-temperature series expansion developments. We present the extended scaling approach in detail, and outline the numerical technique used to study the three-dimensional (
3D) Ising model. After a discussion of the exact expressions for the historic
1D Ising spin chain model as an illustration, an exhaustive analysis of high quality numerical data on the canonical simple cubic lattice
3D Ising model is given. It is shown that in both models, with appropriate scaling variables and scaling expressions (in which leading correction terms are taken into account where necessary), critical behavior extends from
Tc up to infinite temperature.
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.
From a consideration of high temperature series expansions in ferromagnets and in spin glasses, we propose an extended scaling scheme involving a set of scaling formulas which expresses to leading order the temperature (T) and the system size (L) dependences of thermodynamic observables over a much wider range of T than the corresponding one in the conventional scaling scheme. The extended scaling, illustrated by data on the canonical 2d ferromagnet and on the 3d bimodal Ising spin glass, leads to consistency in the estimates of critical parameters obtained from scaling analyses for different observables.
Extended high-temperature series expansions are presented for the spin-s Ising model on the face-centred cubic lattice. Series coefficients for the zero-field internal energy (U 0 ) and the second moment ( mu 2 ) of the correlation function are derived, for 1 / 2 <or=s<or= 9 / 2 , to order 14 in the high-temperature expansion variable K(=J/kT). For s= 1 / 2 , the fourth field derivative of the free energy is calculated to order K 14 and the magnetisation series is given to order 13 and 27 in the temperature and magnetic field variables respectively. For s=1 to 9 / 2 , the zero-field susceptibility series are derived to order K 14 .