VII Amplitude ratios

[74.1.2.1] This section discusses universal amplitudes such as those defined in (1.6) and their ratios. [74.1.2.2] In numerical simulations of critical phenomena amplitude ratios such as (1.7) are used routinely to extract critical parameters Π→c and exponents from simulations of finite systems. [74.1.2.3] It is then of interest to analyze finite size amplitude ratios within the present framework.

[74.1.3.1] The absolute moment of order σ for the ensemble averages of X in a finite and noncritical system is found from equations (4.4) and (3.16) as

where the amplitude X~⁢σ;a,ξ,L of the finite, discrete and noncritical system is given as

and the function r⁢x;ξ/a,L/ξ,a⁢L/ξ2 is defined from equation (4.7) by replacing Ψ with X and extracting a factor h⁢x;ϖX,ζX,0,D. [74.2.0.1] In the ensemble limit one obtains from this and (3.17) the result

for the critical ensemble scaling amplitude of order σ in an infinite system. [74.2.0.2] The subscript is again a reminder for the ensemble scaling limit. [74.2.0.3] The integral in (7.4) can be evaluated for D=1 as

which is valid for -1<Re⁢σ<ϖX,1<ϖX<2 and -1<ζX<1. [74.2.0.4] A derivaton of this result is given in Appendix B. [74.2.0.5] This allows to calculate the general moment ratios

with -1<σ1,σ2<ϖX in the ensemble limit. [74.2.0.6] Figure 6 shows a twodimensional plot of the ratio g⁢3/4,1/4;ϖX,ζX.

[74.2.1.1] If equation (7.5) is used to analytically continue g⁢σ1,σ2;ϖX,ζX beyond the regime -1<σ1,σ2<ϖX the traditional fourth order cumulant g⁢4,2;ϖΨ,ζΨ for the order parameter is found to exhibit special problems if ϖΨ<2. [74.2.1.2] This is mainly due to the presence of the factor sin⁡π⁢σ/2 in (7.5). [74.2.1.3] The divergence must somehow become absorbed by the cutoff factor r⁢0;∞,c,0 in the finite size scaling limit. [74.2.1.4] Assuming that this is indeed the case it is then of interest to consider the quantity

[page 75, §0]    in the finite size scaling limit assuming that it exists. [75.1.0.1] Then the traditional finite size cumulant becomes

[75.1.0.2] The interest in this formal expression is that it is still singular. [75.1.0.3] Within the domain 1<ϖX<2,-1<ζX<1 it has simple poles along the lines

and zeros along the lines

[75.1.0.4] For the traditionally studied order parameter cumulant, i.e. setting X=Ψ, the pole at 4/3 implies a divergence whenever δ=3, i.e. in mean field theory. [75.1.0.5] This result is consistent with the divergence g∞⁢0∝η-1 found in conformal field theory [17]. [75.1.0.6] Note that the points ζ=±1/2 along the singular mean field line ϖΨ=4/3 are intersection points with a line of zeros.

[75.1.1.1] Irrespective of these problems it is of interest to estimate values for the traditional order parameter cumulant ratio g∞⁢0 because much previous work has focussed on it. [75.1.1.2] Within the present approach this is possible from the knowledge of the scaling functions if it is assumed that the identification of ζ=1 with periodic boundary conditions holds generally. [75.1.1.3] If the scaling functions with ζ=1 in Figures 2 through 4 are simply truncated sharply at ±xm⁢a⁢x, and subseqently rescaled to unit norm and variance, the order parameter cumulant g∞⁢0 may be calculated as usual, and it will depend upon the nonuniversal cutoff at xm⁢a⁢x. [75.2.0.1] The results of such a cutoff procedure are displayed in Figure 7 for the cases δ=3,5,15. [75.2.0.2] It is seen that the cumulant is distinctly cutoff dependent. [75.2.0.3] Note that all curves appear to diverge as the cutoff increases.

[75.2.0.4] For the cases δ=3 and δ=5 some structure appears between xm⁢a⁢x=2 and 3 corresponding to the strong curvature in this region seen in Figures 2 and 3. [75.2.0.5] For the 2⁢d-Ising case the curve is flat up to about twice the maximal value 1.39 for the simulations of Bruce and coworkers [43, 45]. [75.2.0.6] Figure 5 provides a possible explanation for the poor agreement between the value g∞⁢0=2.042±0.05 observed in simulations of the fivedimensional Ising model [11, 46] and the mean field calculation g∞⁢0=2.188... from [15]. [75.2.0.7] The simulation result is indicated as the solid arrow, the analytical result as the dashed arrow pointing to the curve δ=3. [75.2.0.8] The small difference in the cutoff xm⁢a⁢x corresponding to these values suggests that the discrepancy may result from different nonuniversal (but most likely smooth) cutoffs in the two estimates.

[75.2.1.1] Finally, the fact that the value of the universal shape parameter ζX appears to be related to the choice of bondary conditions suggests a method of constructing critical amplitude ratios which do not depend on boundary conditions, or other factors influencing ζX. [75.2.1.2] The basic idea is to use the difference of two independent observations of ensemble averages or sums. [75.2.1.3] Let XM⁢N and XM⁢N′ be two independent measurements and YM⁢N=XM⁢N-XM⁢N′ their difference. [75.2.1.4] The limiting distribution function PXM⁢N⁢x for XM⁢N and XM⁢N′ at criticality is given in equation (4.2). [75.2.1.5] Then the difference YM⁢N has the distribution function

[page 76, §0]    in which the width is doubled, but ζX has disappeared. [76.1.0.1] The fractional difference moment ratio Δ⁢σ1,σ2,ϖX is formed analogously to the moment ratio g as

and it has a universal value depending only on the scaling dimension of X as long as σ1,σ2<ϖX. [76.1.0.2] If the scaling dimension is universal then the fractional difference moment ratio is independent of boundary conditions. [76.1.0.3] Plotting Δ⁢σ1,σ2,ϖX as a function of length scale and temperature should then allow to extract the critical exponent.