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 $\widetilde{p}_{X}(x,y)$ 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 $X$ . [72.2.1.2] The ensemble limit yields explicit analytical expressions for the scaling functions $\widetilde{p}_{X}(x,y)$ 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 $\varpi _{X}<2$ . [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

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

(6.2)

if $1<\varpi _{X}<2$ . [72.2.1.5] The superscript is a reminder for the ensemble limit. [72.2.1.6] The point $\varpi _{X}=1$ corresponding to first order transitions is singular and will not be discussed here. [72.2.1.7] 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)

[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 $\varpi _{X}<2$ , does depend on the variable $y$ separately. [72.2.1.10] 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.

[72.2.2.1] 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). [72.2.2.2] 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. [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]

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

(6.5)

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 $h(x;\varpi _{X},\pm\zeta _{X},0,1)$ 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. [73.1.1.5] 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.

[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 $X=\Psi$ be the order parameter which is assumed to be normalized such that $D=1$ . [73.1.2.3] Then $\varpi _{X}$ becomes $\varpi _{\Psi}=1+(1/\delta)$ where $\delta$ is the equation of state exponent. [73.1.2.4] 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)

[73.1.2.5] 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 $\zeta$ . [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 $\delta=3$ which is the value for the universality class of mean field models. [73.1.2.8] The six values for $\zeta$ in Figure 2 through 4 are $\zeta=0.0,0.6,0.7,0.8,0.9,1.0$ . [73.1.2.9] 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. [73.1.2.10] Figure 3 shows the case $\delta=5$ which is close to the value of $\delta\approx 4.8$ [16] for the threedimensional Ising model. [73.1.2.11] 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.

[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 $\zeta$ 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 $\zeta$ than periodic boundary conditions. [73.1.2.15] This correspondence between the value of $\zeta$ and the applied boundary conditions is not expected to be one to one. [73.1.2.16] The value of $\zeta$ 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.

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

[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 $\delta$ 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 $\zeta=1$ . [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 $\zeta=1$ with periodic boundary conditions. [74.1.1.6] 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.

Author:	R. Hilfer
originally published in:	Zeitschrift für Physik B 96, 63-77 (1994)
originally submitted:	February 1st, 1994