Universal Finite-size Scaling Functions Research Articles

Recent developments in the Monte Carlo approach to percolation problems

Using a Monte Carlo method, Hu, Lin and Chen found that bond and site percolation models on planar lattices have universal finite-size scaling functions for the probability W m for the appearance of m percolating clusters, which implies that the average number of percolating clusters, C, is a universal quantity. Hu and Lin found that C increases linearly with the aspect ratio, R, of the lattice for large R, i.e. in this case C= aR with a constant a. Hu and Lin also found that a is apparently independent of the boundary conditions. For the periodic boundary conditions in both horizontal and vertical directions, Ziff et al. found that the number of clusters per lattice site, n, for percolation on planar lattices of linear dimensions L can be written as n= n c + b/ N, where n c is n in the limit L→∞, b is a constant, and N is the number of lattice sites. Ziff et al. found that b is universal and argued that b is the number of percolating clusters so that its universality may be related to the universality of C. Hu found that for large R, b= b c R, but b c ≠ a. In this paper, we use a cluster Monte Carlo simulation method to calculate the number of clusters per site, n, of a q-state bond-correlated percolation model (QBCPM) which is equivalent to the q-state Potts on L× L square lattices. We find that for q≥2 the slopes of n as a function of 1/ L 2 are negative. For q=2, i.e. the Ising model, we find that our data can be well represented by n=n c−c/L+b/L 2+⋯ , where b can be calculated exactly from conformal field theory, c>0 and can be calculated exactly from the critical internal energy of the Ising model.

Test of renormalization predictions for universal finite-size scaling functions.

We calculate universal finite-size scaling functions for systems with an n-component order parameter and algebraically decaying interactions. Just as previously found for short-range interactions, this leads to a singular epsilon expansion, where epsilon is the distance to the upper critical dimension. Subsequently, we check the results by numerical simulations of spin models in the same universality class. Our systems offer the essential advantage that epsilon can be varied continuously, allowing an accurate examination of the region where epsilon is small. The numerical calculations turn out to be in striking disagreement with the predicted singularity.

Cluster analysis and finite-size scaling for Ising spin systems.

Based on the connection between the Ising model and a correlated percolation model, we calculate the distribution function for the fraction (c) of lattice sites in percolating clusters in subgraphs with n percolating clusters, f(n)(c), and the distribution function for magnetization (m) in subgraphs with n percolating clusters, p(n)(m). We find that f(n)(c) and p(n)(m) have very good finite-size scaling behavior and that they have universal finite-size scaling functions for the model on square, plane triangular, and honeycomb lattices when aspect ratios of these lattices have the proportions 1:square root[3]/2:square root[3]. The complex structure of the magnetization distribution function p(m) for the system with large aspect ratio could be understood from the independent orientations of two or more percolation clusters in such a system.

Universal Finite-Size Scaling Functions in the 3D Ising Spin Glass

We study the three-dimensional Edwards-Anderson model with binary interactions by Monte Carlo simulations. Direct evidence of finite-size scaling is provided, and the universal finite-size scaling functions are determined. Monte Carlo data are extrapolated to infinite volume with an iterative procedure up to correlation lengths xi \approx 140. The infinite volume data are consistent with a conventional power law singularity at finite temperature Tc. Taking into account corrections to scaling, we find Tc = 1.156 +/- 0.015, nu = 1.8 +/- 0.2 and eta = -0.26 +/- 0.04. The data are also consistent with an exponential singularity at finite Tc, but not with an exponential singularity at zero temperature.

Universal scaling functions and quantities in percolation models

We briefly review recent work on universal finite-size scaling functions (UFSSFs) and quantities in percolation models. The topics under discussion include: (a) UFSSFs for the existence probability (also called crossing probability) E p , the percolation probability P, and the probability W n of the appearance of n percolating clusters, (b) universal slope for average number of percolating clusters, (c) UFSSFs for a q-state bond-correlated percolation model corresponding to the q-state Potts model. We also briefly mention some very recent related developments and discuss implications of our results.

Universal finite-size scaling functions for critical systems with tilted boundary conditions

We calculate finite-size scaling functions (FSSF's) of Binder parameter g and magnetization distribution function $p(m)$ for the Ising model on ${L}_{1}\ifmmode\times\else\texttimes\fi{}{L}_{2}$ square lattices with periodic boundary conditions in the horizontal ${L}_{1}$ direction and tilted boundary conditions in the vertical ${L}_{2}$ direction such that the $i\mathrm{th}$ site in the first row is connected with the $\mathrm{mod}{(i+cL}_{1}{,L}_{1})\mathrm{th}$ site in the ${L}_{2}$ row of the lattice, where $1l~il~{L}_{1}.$ For fixed sets of $(a,c)$ with ${a=L}_{1}{/L}_{2},$ the FSSF's of g and $p(m)$ are universal and in such cases ${a/(c}^{2}{a}^{2}+1)$ is an invariant. For percolation on lattices with fixed a, the FSSF of the existence probability (also called spanning probability) is not affected by c.

Universal finite-size scaling functions for percolation on three-dimensional lattices

Using a histogram Monte Carlo simulation method (HMCSM), Hu, Lin, and Chen found that bond and site percolation models on planar lattices have universal finite-size scaling functions for the existence probability ${E}_{p},$ the percolation probability $P,$ and the probability ${W}_{n}$ for the appearance of $n$ percolating clusters in these models. In this paper we extend above study to percolation on three-dimensional lattices with various linear dimensions $L.$ Using the HMCSM, we calculate the existence probability ${E}_{p}$ and the percolation probability $P$ for site and bond percolation on a simple-cubic (sc) lattice, and site percolation on body-centered-cubic and face-centered-cubic lattices; each lattice has the same linear dimension in three dimensions. Using the data of ${E}_{p}$ and $P$ in a percolation renormalization group method, we find that the critical exponents obtained are quite consistent with the universality of critical exponents. Using a small number of nonuniversal metric factors, we find that ${E}_{p}$ and $P$ have universal finite-size scaling functions. This implies that the critical ${E}_{p}$ is a universal quantity, which is $0.265\ifmmode\pm\else\textpm\fi{}0.005$ for free boundary conditions and $0.924\ifmmode\pm\else\textpm\fi{}0.005$ for periodic boundary conditions. We also find that ${W}_{n}$ for site and bond percolation on sc lattices have universal finite-size scaling functions.

Failure of universal finite-size scaling above the upper critical dimension

Previous theories have predicted that O( n) symmetric systems in a finite cubic geometry with periodic boundary conditions have universal finite-size scaling functions near criticality in d>4 dimensions. On the basis of exact results for the O( n) symmetric ϕ 4 model in the large n limit we show that universal finite-size scaling does not hold in the predicted form because of significant cut-off and lattice effects for d>4. It is shown that finite-size scaling is valid with two reference lengths which turn out to be identical with the amplitudes of the bulk correlation length. For the ϕ 4 field theory the finite-size scaling functions are shown to be non-universal, i.e., to depend explicitly on the cut-off and on the bare four-point coupling constant, whereas for a ϕ 4 lattice model the finite-size scaling functions have a different form that is independent of the lattice spacing and the four-point coupling constant.

Universality in dynamic critical phenomena

We use heat bath dynamics to evaluate the dynamic critical exponent z and the dynamic finite-size scaling function of an Ising model on square, planar triangular, and honeycomb lattices. We find convincing evidence that z is universal and, by choosing an aspect ratio and a nonuniversal metric factor for the scaled time of each lattice, we can obtain a universal dynamic finite-size scaling function for the Ising model on the planar lattices. Our results suggest many interesting problems for further research. @S1063-651X~97!15408-3#

Specific heat of critical films, the Casimir force, and wetting films near critical end points.

Close to continuous phase transitions with a diverging correlation length \ensuremath{\xi}, the thermal singularities of films with thickness L are governed by a universal finite-size scaling function \ensuremath{\Theta}(L/\ensuremath{\xi}) of their free energy. We propose several experiments for magnetic and fluid films as well as numerical simulations which allow one to determine \ensuremath{\Theta}(L/\ensuremath{\xi}). Based on field-theoretical results for \ensuremath{\Theta}(L/\ensuremath{\xi}), we make quantitative predictions for the specific heat of magnetic films, of $^{4}\mathrm{He}$ films, and of films composed of $^{3}\mathrm{\ensuremath{-}}^{4}$He mixtures close to the onset of superfluidity. The critical fluctuations within fluid films lead to a long-range Casimir force between the confining walls which can be studied by an atomic-force microscope. We discuss the strength and the temperature dependence of this force for various fluid systems. The study of wetting films close to critical end points gives direct access to the full scaling function \ensuremath{\Theta}(L/\ensuremath{\xi}). In this respect we discuss in detail the wetting of a substrate by $^{4}\mathrm{He}$ and $^{3}\mathrm{\ensuremath{-}}^{4}$He mixtures close to their \ensuremath{\lambda} transitions as well as by binary liquid mixtures close to their critical end points.

Finite size effects in phase transitions

We develop a systematic approach to the calculations of finite size effects in phase transitions. The method consists of constructing an effective hamiltonian for the homogeneous modes, obtained by tracing out all other degrees of freedom. These modes are obtained by averaging the order parameter over the finite dimensions of the system. These techniques, together with the renormalization group, lead to explicit calculations of universal finite size scaling functions, under the form of (2+ϵ) or singular (4-ϵ) expansions. Some simple universal results above the upper critical dimension are presented. Simple and universal properties of the rounding of first order transitions are derived.