Higher-order corrections for the quadratic Ising lattice susceptibility

Abstract

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 homogeIIeous differential approximants and Pade techniques. These three terms are found to vary as~t~' , t, and~t~'~with t =1 -T, /T, and their respective coefficients are obtained to within 0.01% accuracy. It is shown that, if terms of the form~t~' ln~t~and~t~'~ln~t~were present, their amplitudes would be less than 10 ' of the amplitudes of the~t~'~and ( t~'~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 10' with the known first eleven coefficients of this series is obtained.

Exact finite-size-scaling corrections to the critical two-dimensional Ising model on a torus

“…The problem, therefore, is to determine which one(s) of these assumptions are invalid, and why. Assumption (c) is extremely plausible from renormalization-group considerations, at least for periodic boundary conditions; and assumption (d) has been confirmed numerically through order (T − T c ) 3 at least as regards the bulk behavior of the susceptibility [13]. However, both numerical [15,16] and theoretical [17] evidence has recently emerged suggesting that irrelevant operators do contribute to the susceptibility at order (T − T c ) 4 .…”

“…Moreover, in the nearest-neighbor spin-1/2 2D Ising model, it has further been assumed that (d) There are no irrelevant operators [12,13].…”

The critical equation of state of the two-dimensional Ising model

“…This problem was recently addressed numerically in Refs. [13][14][15] where it was shown that, for rotationally invariant quantities like the free energy and its derivatives, the first correction due to the irrelevant operators in the squarelattice Ising model appears at order t 4 (hence no correction of order t 2 is present) and that (10) 102.0610(2) 0.36 −4.0056653 (10) 103.4119(2) 0.365 −4.028975 (1) 104.7673(2) 0.37 −4.052238 (1) 106.127(1) Table 1: Thermodynamic-limit results for χ n (t) t 15n/8−2 , for n = 4, 6. The quoted error bars are estimates of the systematic error of the extrapolation.…”

Study of the 2d Ising model with mixed perturbation

“…Firstly, we fit the the term proportional to t with the following function As we explain in the appendix A we set Y (2,1) (0, 0) = 0 according to the results of [25] and [26], moreover we can found Y (1,1) (0, 0) compatible with zero within the error in all our fits. The fit shows a non-zero correction due to the stress-energy tensor, in fact we are able to estimate the value of K 0 − 0.05828 < K 0 < −0.05820.…”

“…To achieve these results we compare the scaling function of the susceptibility along the thermal axis with the high precision estimates of [26] and [25], in this way we obtained These results are obtained for the thermal perturbation in the regime t = 0, h ℓ = 0, we observe that these results are valid also in our regime of interest, because the analytic continuation of Section 3.2.1 do not affect similar kinds of terms, in total agreement with our fits. Furthermore we obtain also the relations: In order to fulfill the above requirements, we set Y (2,1) (0, 0) = 0 because the other choice u = 0, a u = 0 was not consistent with our fits.…”