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VI Scaling functions

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

[] The finite size scaling function \widetilde{p}_{X}(x,y) for the probability density p(x,\xi,L) of the observable X is defined through an equation analogous to (1.3) by

p(x,L,\xi)=L^{{d(d_{X}-d^{*})/(d-d^{*})}}\:\widetilde{p}_{X}(xL^{{d(d_{X}-d^{*})/(d-d^{*})}},L/\xi _{{d^{*}}}) (6.1)

where d_{X} is the anomalous dimension of X. [] The ensemble limit yields explicit analytical expressions for the scaling functions \widetilde{p}_{X}(x,y) at the critical point. [] This is seen from (4.11) as well as from (4.4) which become identical in the ensemble limit if \varpi _{X}<2. [] 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

\widetilde{p}_{X}^{{ES}}(x,y)=\widetilde{p}_{Y}^{{ES}}(x,y)=h(x;\varpi _{X},\zeta _{X},0,D) (6.2)

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

\widetilde{p}_{X}^{{ES}}(x,y)=\frac{1}{\sqrt{4\pi Dy^{{2d_{X}-d}}}}\;\exp\left(-\frac{x^{2}}{4Dy^{{2d_{X}-d}}}\right). (6.3)

[] This is the scaling function conjectured in [7] for the order parameter density on the basis of a Gaussian approximation. [] Note that this scaling function, contrary to those for \varpi _{X}<2, does depend on the variable y separately. [] Note also that the order parameter generally has anomalous dimension d_{\Psi}<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.

[] Another source for the dependence of the scaling function \widetilde{p}_{\Psi}(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). [] With equation (3.18) and introducing the abbreviations R(x,L/\xi)=R(x,\infty,(L/\xi)^{d},\infty), h(x)=h(x;\varpi _{X},\zeta _{X},0,D) and H(x)=H(x;\varpi _{X},\zeta _{X},0,D) the analogue of equation (6.2) reads

\widetilde{p}_{X}^{{FSS}}(x,y)=\left\{\begin{array}[]{rl}R(x,y)\, h(x)+H(x)\dfrac{\displaystyle\partial R(x,y)}{\displaystyle\partial x}&\text{\ \ \ \ :\ \ \ \  for\ \ }x\leq 0\\
R(x,y)\, h(x)-\left(1-H(x)\right)\dfrac{\displaystyle\partial R(x,y)}{\displaystyle\partial x}&\text{\ \ \ \ :\ \ \ \ \text{for}\ \ }x>0\end{array}\right. (6.4)

for the finite size scaling limit. [] 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.

[] 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. [] In this effort the symmetry relation [28]

h(-x,\varpi _{X},\zeta _{X},0,1)=h(x;\varpi _{X},-\zeta _{X},0,1) (6.5)

reduces the computational effort. [] Moreover equation (6.5) suggests a relation with the phenomenon of spontaneous symmetry breaking within the present approach. [] In this view the two scaling functions h(x;\varpi _{X},\pm\zeta _{X},0,1) represent the two pure phases, and thus on general thermodynamic grounds the full scaling function is expected to become a convex combination

\widetilde{p}_{X}(x)=\widetilde{p}_{X}^{{ES}}(x,y)=s\, h(x;\varpi _{X},\zeta _{X},0,D)+(1-s)\, h(x;\varpi _{X},-\zeta _{X},0,D) (6.6)

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

Figure 2: Universal part of the finite size scaling functions \widetilde{p}_{\Psi}(x;3,\zeta,1/2) for the order parameter probability density function for the mean field universality class corresponding to \delta=3 for the equation of state exponent (or \varpi _{\Psi}=1+(1/\delta)=4/3). All curves have width D=1, and symmetrization s=1/2. Different curves correspond to different choices of the universal symmetry or shape parameter \zeta=0.0,0.6,0.7,0.8,0.9,1.0. The curves for \zeta=0.0 and \zeta=1.0 are labelled explicitly, the curves for other values of \zeta interpolate between them.

[] Consider now an ordinary critical point with a global symmetry such as in the Ising models. [] Let X=\Psi be the order parameter which is assumed to be normalized such that D=1. [] Then \varpi _{X} becomes \varpi _{\Psi}=1+(1/\delta) where \delta is the equation of state exponent. [] Abbreviating \zeta _{\Psi} as \zeta the scaling function in equation (6.6) becomes

\widetilde{p}_{\Psi}(x;\delta,\zeta,s)=s\, h\left(x;1+1/\delta,\zeta,0,1\right)+(1-s)\, h\left(x;1+1/\delta,-\zeta,0,1\right). (6.7)

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

Figure 3: Same as Figure 2 with \delta=5 close to the d=3 Ising (\delta\approx 4.8) universality class.

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

Figure 4: Same as Figure 2 with \delta=15 corresponding to the d=2 Ising universality class.

[] 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 \delta and the location of the critical point for the infinite system are known. [] The open circles in Figure 5 represent the smooth interpolation through the data published in [43, 44, 45]. [] The solid line is the analytical prediction shown in Figure 4 for \zeta=1. [] 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. [] The excellent agreement between theory and simulation suggests to identify the value \zeta=1 with periodic boundary conditions. [] It is however not clear whether this identification will hold more generally.

Figure 5: Comparison between the scaling function \widetilde{p}_{\Psi}(x;15,1,1/2) (solid line) for the order parameter density of the two dimensional Ising universality class (\delta=15) with a smoothed interpolation through the simulation results of Refs.[43, 44, 45], (open circles) under the assumption that \zeta=1 corresponds to periodic boundary conditions.