I Introduction

[page 63, §1]
[63.2.1.1] Analysis of finite size effects [1] has become an indispensible tool in the numerical simulation of critical phenomena [2, 3, 4, 5]. [63.2.1.2] According to the nonrigorous renormalization group derivations of finite size scaling [6] the singular part of the free energy $f_{{sing}}(t,h,u,L)$ and the correlation length $\xi$ have the scaling form

$\displaystyle f_{{sing}}(t,h,u,L)$	$\displaystyle=$	$\displaystyle L^{{-d}}\;\widetilde{f}(tL^{{y_{t}}},hL^{{y_{h}}},uL^{{y_{u}}})$		(1.1)
$\displaystyle\xi(t,h,u,L)$	$\displaystyle=$	$\displaystyle L\;\widetilde{\xi}(tL^{{y_{t}}},hL^{{y_{h}}},uL^{{y_{u}}})$		(1.2)

where $t$ denotes the reduced temperature $t=(T-T_{c})/T_{c}$ relative to the critical temperature $T_{c}$ of the infinite system, $h$ is the field conjugate to the order parameter, $u$ is an irrelevant variable, $L$ the system size, $d$ the spatial dimension, and $y_{t},y_{h}>0$ and $y_{u}<0$ are the renormalization group eigenvalues for $t,h$ and $u$ .

[63.2.2.1] More heuristically there are several possibilities to introduce finite size scaling through a scaling hypothesis. [63.2.2.2] One such method [7, 8] assumes that the probability density $p(\psi,L)$ for the order parameter $\Psi$ of the transition can be written as

$p(\psi,L,\xi)=L^{{d(d_{\Psi}-d^{*})/(d-d^{*})}}\:\widetilde{p}_{\Psi}(\psi L^{{d(d_{\Psi}-d^{*})/(d-d^{*})}},L/\xi _{{d^{*}}})$

(1.3)

where $d_{\Psi}$ is the anomalous or scaling dimension of the order parameter, $d^{*}$ is Fishers anomalous dimension of the vacuum [9], and $\xi _{{d^{*}}}$ is Binders thermodynamic length [8]. [63.2.2.3] If hyperscaling holds then $d^{*}=0$ , the thermodynamic length becomes the correlation length, $\xi _{0}=\xi$ , and the exponent in (1.3) reduces to the familiar form $d_{\Psi}=\beta/\nu$ where $\beta$ is the order parameter exponent and $\nu$ the correlation length exponent. [63.2.2.4] The finite size scaling Ansatz (1.3) can be extended to arbitrary composite operators, an important case being the energy density ${\mathcal{E}}$ for which the exponent becomes $d_{{\mathcal{E}}}=(1-\alpha)/\nu$ if hyperscaling holds. [63.2.2.5] All finite size scaling relations (1.1)–(1.3) are assumed to hold in the finite size scaling limit [page 64, §0]

$L\rightarrow\infty,\xi\rightarrow\infty$

(1.4)

where $L/\xi=c$ is kept constant.

[64.1.1.1] Despite their very plausible and seemingly general character finite size scaling relations are not generally valid [10]. [64.1.1.2] Violations of finite size scaling are closely related to violations of hyperscaling relations [10, 11]. [64.1.1.3] These violations have been rationalized via the so called mechanism of “dangerous irrelevant” variables [12] or by saying that the correlation length $\xi$ is not the only relevant length [5]. [64.1.1.4] “Dangerous irrelevant” variables are relevant to critical behaviour because by definition they induce a singularity in one or both of the scaling functions $\widetilde{f}(x,y,z)$ and $\widetilde{\xi}(x,y,z)$ as $z\rightarrow 0$ . [64.1.1.5] The mechanism of dangerous irrelevant variables does not give general modelindependent criteria for the validity or violation of hyperscaling and finite size scaling. [64.1.1.6] The present paper attempts to establish positivity of the specific heat exponent as such a general criterion for the validity of hyperscaling relations.

[64.1.2.1] Given the scaling Ansatz (1.3) another well known problem with present finite size scaling theory concerns integrals of the scaling function appearing in (1.3). [64.1.2.2] To see this calculate the finite size scaling form for the absolute moments of order $\sigma$ from (1.3) for the case $d^{*}=0$ as

$\langle|\Psi|^{\sigma}\rangle(L,\xi)=L^{{-\sigma\beta/\nu}}\widetilde{\Psi}_{\sigma}(L/\xi)$

(1.5)

where the new scaling function $\widetilde{\Psi}_{\sigma}(z)$ is given in terms of $\widetilde{p}_{\Psi}(x,y)$ as

$\widetilde{\Psi}_{\sigma}(y)=\int\;|x|^{\sigma}\widetilde{p}_{\Psi}(x,y)\; dx.$

(1.6)

[64.1.2.3] From these moments one finds for the ratio related to the renormalized coupling constant the result

$g(L,\xi)=\frac{\langle\Psi^{4}\rangle}{\langle\Psi^{2}\rangle^{2}}=\frac{\widetilde{\Psi}_{4}(L/\xi)}{\widetilde{\Psi}_{2}^{2}(L/\xi)}.$

(1.7)

which implies that

$g_{\infty}(c)=\lim _{{\genfrac{}{}{0.0pt}{}{L,\xi\rightarrow\infty}{L/\xi=c}}}g(L,\xi)=\frac{\widetilde{\Psi}_{4}(c)}{\widetilde{\Psi}_{2}^{2}(c)}$

(1.8)

in the finite size scaling limit for which $L/\xi=c$ is a constant. [64.1.2.4] While the value $g_{\infty}(\infty)=3$ for the trivial high temperature fixed point is universal, the value $g_{\infty}(0)$ for the nontrivial fixed point is found to depend on seemingly nonuniversal factors. [64.1.2.5] Moreover, numerical difficulties arise in different methods of estimating $g_{\infty}(0)$ [7, 13, 14, 15, 16]. [64.1.2.6] The problem is particularly apparent for the mean field universality class. [64.1.2.7] Twodimensional conformal field theory predicts that $g_{\infty}(0)\propto\eta^{{-1}}$ in the limit $\eta\rightarrow 0$ [17]. [64.1.2.8] Here $\eta$ is the correlation function exponent, and $\eta=0$ in mean field theory. [64.1.2.9] Similarly for the $n$ -vector models above four dimensions $g_{\infty}(0)$ becomes $n$ -dependent [15] in stark contrast to the “superuniversality” of mean field exponents and amplitude ratios. [64.1.2.10] The numerical agreement with Monte-Carlo simulations is poor and the authors of Ref. [13] have called for further studies to clarify the discrepancy. [64.1.2.11] The present paper attempts to contribute to this point.

[64.2.1.1] Let me summarize the objectives of this work resulting from the above exposition of two problems with current finite size scaling theory. [64.2.1.2] The first objective is to provide general criteria for the validity or violation of finite size scaling. [64.2.1.3] The second objective is to investigate the finite size scaling functions and finite size amplitude ratios in the ensemble limit.

[64.2.2.1] Methodically, the results of this paper follow directly from a recently introduced classification theory of phase transition [18, 19, 20, 21, 22, 23]. [64.2.2.2] Let me briefly outline the basic idea. Within the classification theory it was shown that each phase transition in thermodynamics as well as in statistical mechanics is characterized by a set of generalized Ehrenfest orders plus a set of slowly varying functions. [64.2.2.3] This classification is macroscopic in the sense that it involves only thermodynamic averages while conformal field theory focusses on microscopic higher order correlation functions. [64.2.2.4] The classification in thermodynamics [18] is based upon the application of fractional calculus, the one in statistical mechanics [22] rests upon the theory of limit distributions for sums of independent random variables. [64.2.2.5] The latter theory, which cannot be employed in the traditional way of performing the scaling limit, became applicable by introducing a fundamentally new scaling limit, which was called ensemble limit. [64.2.2.6] In the ensemble limit critical systems decompose into an infinite ensemble of infinitely large, yet uncorrelated blocks. [64.2.2.7] The classification schemes in thermodynamics and statistical mechanics are mathematically very different but can be related to each other by studying the fluctuations in the ensemble of blocks. [64.2.2.8] The difference between the classification schemes is found to be related to violations of hyperscaling. [64.2.2.9] Moreover, a thermodynamic form of scaling, called finite ensemble scaling, emerges from the classification. [64.2.2.10] The basic idea of this paper is to regard finite ensemble scaling as a macroscopic or thermodynamic form of finite size scaling. [64.2.2.11] Thus the limit distributions in the classification theory ought to be related to the probability distributions, such as $p(\psi,L,\xi)$ , appearing in finite size scaling theory. [64.2.2.12] To show that this expectation is indeed borne out it is first necessary to discuss in some detail the different scaling limits and finite ensemble scaling. [64.2.2.13] Subsequently the classification approach can be related to the theory of finite size scaling, hyperscaling and general scaling at critical points. [64.2.2.14] In the last two sections critical finite size scaling functions and amplitude ratios are discussed and compared with Monte Carlo simulations. [64.2.2.15] The comparison of the predicted universal part of the finite size scaling functions for the order parameter distribution at criticality with Monte Carlo simulations for Ising models shows good quantitative agreement.

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