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Physical review. B, Condensed matter

Universal critical amplitude ratios for percolation

Amnon Aharony A

Published Jul 1, 1980• 105 citations
Publisher

Abstract

The hypothesis of universality implies that for every scaling relation among critical exponents there exists a universal ratio among the corresponding critical amplitudes. If one writes $B{|t|}^{\ensuremath{\beta}}$, ${{A}_{F}}^{\ifmmode\pm\else\textpm\fi{}}{|t|}^{2\ensuremath{-}\ensuremath{\alpha}}$, ${C}^{\ifmmode\pm\else\textpm\fi{}}{|t|}^{\ensuremath{-}\ensuremath{\gamma}}$, and ${\ensuremath{\xi}}_{0}{|t|}^{\ensuremath{-}\ensuremath{\nu}}$ [where $t=\frac{({p}_{c}\ensuremath{-}p)}{{p}_{c}}$, $p$ being the concentration of nonzero bonds, and +(-) stands for $p<{p}_{c}$ ($p>{p}_{c}$)] for the leading singular terms in the probability to belong to the infinite cluster, the mean number of clusters, the clusters' mean-square size, and the pair connectedness correlation length, then it is shown that the ratios $\frac{{{A}_{F}}^{+}}{{{A}_{F}}^{\ensuremath{-}}}$, $\frac{{C}^{+}}{{C}^{\ensuremath{-}}}$, ${{A}_{F}}^{+}{B}^{\ensuremath{-}2}{C}^{+}$, $\frac{{\ensuremath{\xi}}_{0}^{+}}{{\ensuremath{\xi}}_{0}^{\ensuremath{-}}}$, and ${{A}_{F}}^{+}{({\ensuremath{\xi}}_{0}^{+})}^{d}$ ($d$ is the dimensionality) are universal. Similar quantities are found for the behavior at $p={p}_{c}$ (as a function of a "ghost" field). All of these universal ratios are derived from a universal scaled equation of state, which is calculated to second order in $\ensuremath{\epsilon}=6\ensuremath{-}d$. The (extrapolated) results are compared with available information in dimensionalities $d=2, 3, 4, 5$, with reasonable agreements. The amplitude relations become exact at $d=6$, when logarithmic corrections appear. Additional universal ratios are obtained for the confluent correction to scaling terms.

DOI
10.1103/physrevb.22.400

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The hypothesis of universality implies that for every scaling relation among critical exponents there exists a universal ratio among the corresponding critical amplitudes. If one writes B∣t∣\ensuremathβB{|t|}^{\ensuremath{\beta}}B∣t∣\ensuremathβ, AF\ifmmode±\else\textpm\fi∣t∣2\ensuremath−\ensuremathα{{A}_{F}}^{\ifmmode\pm\else\textpm\fi{}}{|t|}^{2\ensuremath{-}\ensuremath{\alpha}}AF​\ifmmode±\else\textpm\fi∣t∣2\ensuremath−\ensuremathα, C\ifmmode±\else\textpm\fi∣t∣\ensuremath−\ensuremathγ{C}^{\ifmmode\pm\else\textpm\fi{}}{|t|}^{\ensuremath{-}\ensuremath{\gamma}}C\ifmmode±\else\textpm\fi∣t∣\ensuremath−\ensuremathγ, and \ensuremathξ0∣t∣\ensuremath−\ensuremathν{\ensuremath{\xi}}_{0}{|t|}^{\ensuremath{-}\ensuremath{\nu}}\ensuremathξ0​∣t∣\ensuremath−\ensuremathν [where t=(pc\ensuremath−p)pct=\frac{({p}_{c}\ensuremath{-}p)}{{p}_{c}}t=pc​(pc​\ensuremath−p)​, ppp being the concentration of nonzero bonds, and +(-) stands for p<pcp<{p}_{c}p<pc​ (p>pcp>{p}_{c}p>pc​)] for the leading singular terms in the probability to belong to the infinite cluster, the mean number of clusters, the clusters' mean-square size, and the pair connectedness correlation length, then it is shown that the ratios AF+AF\ensuremath−\frac{{{A}_{F}}^{+}}{{{A}_{F}}^{\ensuremath{-}}}AF​\ensuremath−AF​+​, C+C\ensuremath−\frac{{C}^{+}}{{C}^{\ensuremath{-}}}C\ensuremath−C+​, AF+B\ensuremath−2C+{{A}_{F}}^{+}{B}^{\ensuremath{-}2}{C}^{+}AF​+B\ensuremath−2C+, \ensuremathξ0+\ensuremathξ0\ensuremath−\frac{{\ensuremath{\xi}}_{0}^{+}}{{\ensuremath{\xi}}_{0}^{\ensuremath{-}}}\ensuremathξ0\ensuremath−​\ensuremathξ0+​​, and AF+(\ensuremathξ0+)d{{A}_{F}}^{+}{({\ensuremath{\xi}}_{0}^{+})}^{d}AF​+(\ensuremathξ0+​)d (ddd is the dimensionality) are universal. Similar quantities are found for the behavior at p=pcp={p}_{c}p=pc​ (as a function of a "ghost" field). All of these universal ratios are derived from a universal scaled equation of state, which is calculated to second order in \ensuremathϵ=6\ensuremath−d\ensuremath{\epsilon}=6\ensuremath{-}d\ensuremathϵ=6\ensuremath−d. The (extrapolated) results are compared with available information in dimensionalities d=2,3,4,5d=2, 3, 4, 5d=2,3,4,5, with reasonable agreements. The amplitude relations become exact at d=6d=6d=6, when logarithmic corrections appear. Additional universal ratios are obtained for the confluent correction to scaling terms.
We estimate, at d=3d=3d=3, from ${\ensuremath{\varphi}}^{4}$ field theory the universal ratios of confluent critical correction amplitudes relative to the correlation length, the susceptibility, and the specific heat above Tc{T}_{c}Tc​. Our central values are comparable to these available from analysis of high-temperature series and the $\ensuremath{\epsilon}$ expansion. Furthermore, experiments in progress on a binary mixture are in good agreement with one of our results, the ratio of the confluent corrections for the specific heat and susceptibility which have not been previously estimated. We also present, in detail, a derivation of the universality of these quantities in the framework of field theory.
Writing thermodynamic quantities near a critical point as ${f}_{i}\ensuremath{\sim}{A}_{i}{|t|}^{\ensuremath{-}{\ensuremath{\lambda}}_{i}}(1+{a}_{i}{|t|}^{\ensuremath{\Delta}})$, where $t=\frac{(T\ensuremath{-}{T}_{c})}{{T}_{c}}$, we show that to leading order in $\ensuremath{\epsilon}=4\ensuremath{-}d$ one has $\frac{{a}_{i}}{{a}_{j}}=\frac{({\ensuremath{\lambda}}_{i}\ensuremath{-}{{\ensuremath{\lambda}}_{i}}^{0})}{({\ensuremath{\lambda}}_{j}\ensuremath{-}{{\ensuremath{\lambda}}_{j}}^{0})}$, where ${{\ensuremath{\lambda}}_{i}}^{0}$ is the mean-field value of ${\ensuremath{\lambda}}_{i}$. This ratio is also equal to $\frac{({\ensuremath{\lambda}}_{i, \mathrm{eff}}\ensuremath{-}{\ensuremath{\lambda}}_{i})}{({\ensuremath{\lambda}}_{j, \mathrm{eff}}\ensuremath{-}{\ensuremath{\lambda}}_{j})}$, where ${\ensuremath{\lambda}}_{i, \mathrm{eff}}$ is derived from a fit of data with ${f}_{i}\ensuremath{\sim}{|t|}^{\ensuremath{-}{\ensuremath{\lambda}}_{i, \mathrm{eff}}}$. Various experiments are analyzed and compared to these predictions.
A general method is given whereby mmm-connectedness correlation functions can be studied in the percolation problem. The method for m=1m=1m=1 involves calculating the kth power of the correlation function $ 〈\ensuremath{\sigma}(x)\ensuremath{\sigma}({x}^{\ensuremath{'}})〉 $ for a randomly dilute Ising model at nonzero temperature and subsequently averaging over configurations. The final step is to take the limit $k\ensuremath{\rightarrow}0$. This method is tested by reproducing the standard results for the percolation problem from an extension of the calculation of Stephen and Grest. For m>1m>1m>1 an additional "color" index is introduced and a Hamiltonian is constructed in which different colors repel one another, thereby giving an exact prescription for mmm-connectedness. Order parameters for mmm-connectedness are identified. The m=2m=2m=2 order parameter couples through a trilinear term to the m=1m=1m=1 order parameter. The main result is that ${\ensuremath{\beta}}^{(m)}$ the exponent for mmm-connectedness is given by ${\ensuremath{\beta}}^{(m)}=m\ensuremath{\beta}+\ensuremath{\nu}{\ensuremath{\psi}}^{(m)}$, where $\ensuremath{\beta}$ and $\ensuremath{\nu}$ are the usual exponents for percolation and ${\ensuremath{\psi}}^{(m)}$ is a new crossover exponent which, to lowest order in $\ensuremath{\epsilon}=6\ensuremath{-}d$, is given by ${\ensuremath{\psi}}^{(m)}=\frac{m(m\ensuremath{-}1){\ensuremath{\epsilon}}^{2}}{49}$. This result implies that the fractal dimensionality of the biconnected part of the critically percolating cluster is given in terms of the percolation critical exponents as $\frac{\ensuremath{\gamma}}{\ensuremath{\nu}}\ensuremath{-}{\ensuremath{\psi}}^{(2)}$. If "nodes" are defined as triconnected points, then ${\ensuremath{\beta}}^{(3)}$ is the critical exponent associated with their density in the infinite cluster. We also discuss evidence that the "node-link" model of Skal-Shklovskii-de Gennes breaks down for ddd less than some critical value $\stackrel{^}{d}$. Numerically we adduce evidence that $\stackrel{^}{d}$ may be larger than 3.
We have shown that several critical amplitude ratios are universal in the critical phenomena near the surface. The general statement on the number of the scaling relations among surface critical exponents and the surface universal amplitude ratios is given. The universal scaling function for the equation of state in the presence of surface is discussed. We have calculated the amplitude ratios and the scaling function for the ordinary, special, and extraordinary transitions in the mean-field approximation.
The thermodynamic functions of an nnn-component ddd-dimensional ${\ensuremath{\sigma}}^{4}$ theory are written in the form ${f}_{i}={A}_{i}{|t|}^{\ensuremath{-}{\ensuremath{\lambda}}_{i}}(1+{a}_{i}{|t|}^{\ensuremath{\Delta}})$, where $t=\frac{(T\ensuremath{-}{T}_{c})}{{T}_{c}}$. The correction-to-scaling amplitudes ai{a}_{i}ai​ are calculated to leading order in 1n\frac{1}{n}n1​ for 2<d<42<d<42<d<4, and the universality of their ratios is explicitly shown. The effective exponents ${\ensuremath{\lambda}}_{i,\mathrm{eff}}={\ensuremath{\lambda}}_{i}\ensuremath{-}{a}_{i}\ensuremath{\Delta}{|t|}^{\ensuremath{\Delta}}$ violate the thermodynamic scaling laws already at this leading order. The amplitudes ai{a}_{i}ai​ have singularities (as functions of ddd) at the dimensionalities ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{d}}_{l}=2+\frac{4}{l},l=3,4,5,\dots{}$. Above Tc{T}_{c}Tc​, the specific heat and the susceptibility correction amplitudes ac+{a}_{c}^{+}ac+​ and ${a}_{\ensuremath{\chi}}^{+}$ are discontinuous at ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{d}}_{l}$ and at ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{d}}_{2l}(l\ensuremath{\ge}2)$, respectively. Below Tc{T}_{c}Tc​, the magnetization and the stiffness correction amplitudes are zero, ${a}_{M}={a}_{{\ensuremath{\rho}}_{s}}=0$, while ${a}_{c}^{\ensuremath{-}}$ diverges at ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{d}}_{2l+1}(l\ensuremath{\ge}1)$ but is continuous at d2l{d}_{2l}d2l​ (in particular at d=3d=3d=3). Results are also given for universal ratios among corrections-to-scaling amplitudes for the dependence of thermo-dynamic functions on the magnetization at T=TcT={T}_{c}T=Tc​ and at a nonzero field.
Abstract Universality, encompassing critical exponents, scaling functions, and dimensionless quantities, is fundamental to phase transition theory. In finite systems, universal behaviors are also expected to emerge at the pseudocritical point. Focusing on two-dimensional percolation, we show that the size distribution of the largest cluster asymptotically approaches to a Gumbel form in the subcritical phase, a Gaussian form in the supercritical phase, and transitions within the critical finite-size scaling window. Numerical results indicate that, at consistently defined pseudocritical points, this distribution exhibits a universal form across various lattices and percolation models (bond or site), within error bars, yet differs from the distribution at the critical point. The critical polynomial, universally zero for two-dimensional percolation at the critical point, becomes nonzero at pseudocritical points. Nevertheless, numerical evidence suggests that the critical polynomial, along with other dimensionless quantities such as wrapping probabilities and Binder cumulants, assumes fixed values at the pseudocritical point that are independent of the percolation type (bond or site) but vary with lattice structures. These findings imply that while strict universality breaks down at the pseudocritical point, certain extreme-value statistics and dimensionless quantities exhibit quasi-universality, revealing a subtle connection between scaling behaviors at critical and pseudocritical points.