VI Scaling functions

[72.1.4.1] This section discusses how the general theory above may be used to obtain finite size scaling functions at the critical point.

[72.2.1.1] The finite size scaling function p~X⁢x,y for the probability density p⁢x,ξ,L of the observable X is defined through an equation analogous to (1.3) by

where dX is the anomalous dimension of X. [72.2.1.2] The ensemble limit yields explicit analytical expressions for the scaling functions p~X⁢x,y at the critical point. [72.2.1.3] This is seen from (4.11) as well as from (4.4) which become identical in the ensemble limit if ϖX<2. [72.2.1.4] If X is identified as the macroscopic (thermodynamic) equivalent of the microscopic observable Y then it follows from (4.4) and (4.11) that the finite ensemble scaling functions are given as

if 1<ϖX<2. [72.2.1.5] The superscript is a reminder for the ensemble limit. [72.2.1.6] The point ϖX=1 corresponding to first order transitions is singular and will not be discussed here. [72.2.1.7] For ϖX=2 on the other hand the thermodynamic form (4.4) yields a simple Gaussian while the fieldtheoretic form (4.11) gives

[72.2.1.8] This is the scaling function conjectured in [7] for the order parameter density on the basis of a Gaussian approximation. [72.2.1.9] Note that this scaling function, contrary to those for ϖX<2, does depend on the variable y separately. [72.2.1.10] Note also that the order parameter generally has anomalous dimension dΨ<d/2 and thus this scaling form for the order parameter distribution is expected to arise in the vicinity but not directly at the critical point.

[72.2.2.1] Another source for the dependence of the scaling function p~Ψ⁢x,y for the order parameter distribution on y is the appearance of the nonuniversal cutoff function R in the finite size scaling limit of equation (3.18). [72.2.2.2] With equation (3.18) and introducing the abbreviations R⁢x,L/ξ=R⁢x,∞,L/ξd,∞, h⁢x=h⁢x;ϖX,ζX,0,D and H⁢x=H⁢x;ϖX,ζX,0,D the analogue of equation (6.2) reads

for the finite size scaling limit. [72.2.2.3] Thus it is seen that the finite ensemble scaling function h corresponds to the universal part of the finite size scaling function which is independent of y while the cutoff function R is responsible [page 73, §0]    for the dependence on y and adds a nonuniversal part.

[73.1.1.1] The analytical expressions (3.5) and (3.12) for the universal part of critical finite size scaling functions can be employed to evaluate the scaling functions numerically. [73.1.1.2] In this effort the symmetry relation [28]

reduces the computational effort. [73.1.1.3] Moreover equation (6.5) suggests a relation with the phenomenon of spontaneous symmetry breaking within the present approach. [73.1.1.4] In this view the two scaling functions h⁢x;ϖX,±ζX,0,1 represent the two pure phases, and thus on general thermodynamic grounds the full scaling function is expected to become a convex combination

of two extremal phases. [73.1.1.5] The relation may be generalized to several phases or asymmetric situations.

[73.1.2.1] Consider now an ordinary critical point with a global symmetry such as in the Ising models. [73.1.2.2] Let X=Ψ be the order parameter which is assumed to be normalized such that D=1. [73.1.2.3] Then ϖX becomes ϖΨ=1+1/δ where δ is the equation of state exponent. [73.1.2.4] Abbreviating ζΨ as ζ the scaling function in equation (6.6) becomes

[73.1.2.5] For the symmetric case s=1/2 the function p~Ψ⁢x;δ,ζ,s is displayed in Figures 2, 3 and 4 for δ=3,5,15 and several choices of ζ. [73.1.2.6] The symmetrization s=1/2 in (6.7) corresponds to an “equal weight rule” which is known to apply for first order transitions [42]. [73.1.2.7] Figure 2 shows the case δ=3 which is the value for the universality class of mean field models. [73.1.2.8] The six values for ζ in Figure 2 through 4 are ζ=0.0,0.6,0.7,0.8,0.9,1.0. [73.1.2.9] The case ζ=1.0 corresponds to the double peak structure with the widest peak separation while the value ζ=0.0 corresponds to the singly peaked function whose maximum has the smallest height. [73.1.2.10] Figure 3 shows the case δ=5 which is close to the value of δ≈4.8 [16] for the threedimensional Ising model. [73.1.2.11] The value δ=15 in Figure 4 is the value for the two dimensional Ising universality class.

[73.1.2.12] The scaling functions displayed in Figures 2 through 4 are consistent with published data on critical scaling functions [7, 43, 44]. [73.1.2.13] Moreover it is seen that the universal shape parameter ζ is related to the type of boundary conditions. [73.1.2.14] Free boundary conditions apparently correspond to smaller values of the universal shape parameter ζ than periodic boundary conditions. [73.1.2.15] This correspondence between the value of ζ and the applied boundary conditions is not expected to be one to one. [73.1.2.16] The value of ζ may be influenced by other universal factors such as the type or symmetry of the pure phases. [page 74, §0]    [74.1.0.1] On the other hand the boundary conditions may also influence other parameters such as the value of the symmetrization s. [74.1.0.2] This is expected for boundary conditions which do not preserve the symmetry.

[74.1.1.1] Figure 5 shows that the scaling functions are not merely consistent but also in good quantitative agreement with Monte-Carlo simulations of the twodimensional Ising model [43, 44, 45] where the exact value of δ and the location of the critical point for the infinite system are known. [74.1.1.2] The open circles in Figure 5 represent the smooth interpolation through the data published in [43, 44, 45]. [74.1.1.3] The solid line is the analytical prediction shown in Figure 4 for ζ=1. [74.1.1.4] For the comparison the nonuniversal scaling factors which were chosen to yield unit norm and variance in [43, 44, 45] were matched to those of the theoretical curve. [74.1.1.5] The excellent agreement between theory and simulation suggests to identify the value ζ=1 with periodic boundary conditions. [74.1.1.6] It is however not clear whether this identification will hold more generally.