Thermodynamic functions in the vicinity of an ordinary critical point are expected to obey asymptotic scaling laws as $t=\frac{(T\ensuremath{-}{T}_{c})}{{T}_{c}}$ and the ordering field,
h, approach zero. However, the optimal scaling variables are the nonlinear scaling fields, ${g}_{t}=t+{b}_{t}{h}^{2}+{c}_{t}{t}^{2}+\ensuremath{\cdots}$ and ${g}_{h}=h(1+{c}_{h}t+\ensuremath{\cdots})$. The nonlinearities yield correction factors to the leading power-law (and scaling) variation of thermodynamic quantities,
L, of the form ($1+{a}_{L}t+{b}_{L}{t}^{2}+\ensuremath{\cdots}$), etc., where the correction amplitudes
aL,bL,…, are uniquely determined by the nonlinear scaling-field coefficients. It follows that "analytic" corrections to, e.g., the susceptibility, are directly related to those for the free energy and magnetization (in zero field). The term
bth2 also generates nonanalytic contributions such as an additive, energylike term, varying as ${|t|}^{1\ensuremath{-}\ensuremath{\alpha}}$ in the zero-field susceptibility, and factors like ($1+{c}_{L}{|h|}^{2\ensuremath{-}\frac{1}{\ensuremath{\Delta}}}$) on the critical isotherm,
t=0. Irrelevant scaling fields yield further, in general distinct, nonanalytic corrections, and cause shifts in
Tc and various amplitudes although "universal" ratios remain constant.
Four terms in the expansion for the zero-field susceptibility of the quadratic Ising lattice at criticality are known exactly. We have computed three additional terms in this expansion by analyzing Nickel's high-temperature series for this lattice with second-order homogeneous differential approximants and Pad\'e techniques. These three terms are found to vary as \ensuremath{\Vert}t${\ensuremath{\Vert}}^{1/4}$,t, and \ensuremath{\Vert}t${\ensuremath{\Vert}}^{5/4}$ with t=1-
Tc/T, and their respective coefficients are obtained to within 0.01% accuracy. It is shown that, if terms of the form \ensuremath{\Vert}t${\ensuremath{\Vert}}^{1/4}$ln\ensuremath{\Vert}t\ensuremath{\Vert} and \ensuremath{\Vert}t${\ensuremath{\Vert}}^{5/4}$ln\ensuremath{\Vert}t\ensuremath{\Vert} were present, their amplitudes would be less than ${10}^{\mathrm{\ensuremath{-}}5}$ of the amplitudes of the \ensuremath{\Vert}t${\ensuremath{\Vert}}^{1/4}$ and \ensuremath{\Vert}t${\ensuremath{\Vert}}^{5/4}$ terms, respectively. The results are in accord with the predictions of the renormalization group if irrelevant variables are neglected and indicate that, if singularities due to irrelevant variables are present, their associated exponents must exceed 2. The results are also used to predict the coefficients of the low-temperature series for this system. Agreement to as much as one part in
105 with the known first eleven coefficients of this series is obtained.
We have made substantial advances in elucidating the properties of the susceptibility of the square lattice Ising model. We discuss its analyticity properties, certain closed form expressions for subsets of the coefficients, and give an algorithm of complexity O(N6) to determine its first N coefficients. As a result, we have generated and analyzed series with more than 300 terms in both the high- and low-temperature regime. We quantify the effect of irrelevant variables to the scaling-amplitude functions. In particular, we find and quantify the breakdown of simple scaling, in the absence of irrelevant 1 scaling fields, arising first at order |T −Tc | 9/4, though high-low temperature symmetry is still preserved. At terms of order |T −Tc | 17/4 and beyond, this symmetry is no longer present. The short-distance terms are shown to have the form (T − Tc) p (log |T − Tc|) q with p ≥ q2. Conjectured exact expressions for some correlation functions and series coefficients in terms of elliptic theta functions also foreshadow future developments.
Partial-differential approximants are used to study the critical behavior of the susceptibility, \ensuremath{\chi}(x,y), of the Klauder and double-Gaussian scalar spin, or O(1) models on a square lattice using two-variable series to order
x21 where x\ensuremath{\propto}J/
kBT while y serves to interpolate analytically from the Gaussian or free-field model at y=0 to the standard spin-(1/2) Ising model at y=1. The pure Ising critical point at y=1 appears to be the only non-Gaussian multisingularity in the range 0<y\ensuremath{\le}1. It is concluded that the exponent \ensuremath{\theta} characterizing the leading irrelevant corrections to scaling lies in the range \ensuremath{\theta}=1.35\ifmmode\pm\else\textpm\fi{}0.25. This supports the validity of Nienhuis's conjecture \ensuremath{\theta}=(4/3) but it is argued that, contrary to normal expectations, this (rational) value entails only logarithmic corrections to pure Ising critical behavior. The existence of strong crossover effects for 0.1\ensuremath{\lesssim}y\ensuremath{\lesssim}0.6 and the appearance of an effective exponent, ${\ensuremath{\gamma}}_{\mathrm{eff}\mathrm{\ensuremath{\simeq}}1.9}$ to 2.0, is discussed and related to work on the \ensuremath{\lambda}
cphi4 model.
Although the two-dimensional spin- 1 / 2 Ising model was solved in zero field in 1944, no exact results are yet available for the spin-1 model. This model should have the same dominant critical exponents as the spin- 1 / 2 model, but appears to exhibit non-analytic corrections to scaling as expected for a phi 4 model and unlike the spin- 1 / 2 model. In this paper new 45-term low temperature series for the magnetisation, specific heat and susceptibility of the spin-1 model on the square lattice are presented and analysed. A new 24-term staggered susceptibility series for the hard square model is presented and the extant order parameter series for this model (which is also in the phi 4 universality class) are also considered. For both models a non-analytic confluent correction with exponent 1.0< Delta 1 <1.3 is found. The validity of this result is enhanced by a comparison with the S= 1 / 2 case.