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# 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 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 for the ensemble averages of in a finite and noncritical system is found from equations (4.4) and (3.16) as

 (7.1) (7.2)

where the amplitude of the finite, discrete and noncritical system is given as

 (7.3)

and the function is defined from equation (4.7) by replacing with and extracting a factor . [74.2.0.1] In the ensemble limit one obtains from this and (3.17) the result

 (7.4)

for the critical ensemble scaling amplitude of order 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 as

 (7.5)

which is valid for and . [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

 (7.6)

with in the ensemble limit. [74.2.0.6] Figure 6 shows a twodimensional plot of the ratio .

[74.2.1.1] If equation (7.5) is used to analytically continue beyond the regime the traditional fourth order cumulant for the order parameter is found to exhibit special problems if . [74.2.1.2] This is mainly due to the presence of the factor in (7.5). [74.2.1.3] The divergence must somehow become absorbed by the cutoff factor 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

 (7.7)

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

 (7.8)

[75.1.0.2] The interest in this formal expression is that it is still singular. [75.1.0.3] Within the domain it has simple poles along the lines

 (7.9)

and zeros along the lines

 (7.10)

[75.1.0.4] For the traditionally studied order parameter cumulant, i.e. setting , the pole at implies a divergence whenever , i.e. in mean field theory. [75.1.0.5] This result is consistent with the divergence found in conformal field theory [17]. [75.1.0.6] Note that the points along the singular mean field line 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 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 with periodic boundary conditions holds generally. [75.1.1.3] If the scaling functions with in Figures 2 through 4 are simply truncated sharply at , and subseqently rescaled to unit norm and variance, the order parameter cumulant may be calculated as usual, and it will depend upon the nonuniversal cutoff at . [75.2.0.1] The results of such a cutoff procedure are displayed in Figure 7 for the cases . [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 and some structure appears between and corresponding to the strong curvature in this region seen in Figures 2 and 3. [75.2.0.5] For the -Ising case the curve is flat up to about twice the maximal value 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 observed in simulations of the fivedimensional Ising model [11, 46] and the mean field calculation 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 . [75.2.0.8] The small difference in the cutoff 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 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 . [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 and be two independent measurements and their difference. [75.2.1.4] The limiting distribution function for and at criticality is given in equation (4.2). [75.2.1.5] Then the difference has the distribution function

 (7.11)

[page 76, §0]    in which the width is doubled, but has disappeared. [76.1.0.1] The fractional difference moment ratio is formed analogously to the moment ratio as

 (7.12)

and it has a universal value depending only on the scaling dimension of as long as . [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 as a function of length scale and temperature should then allow to extract the critical exponent.