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Journal of Physics A Mathematical and General

Critical dynamics at the percolation threshold

Anthony C. Maggs, Robin B. Stinchcombe

Published Feb 1, 1986• 3 citations
Publisher

Abstract

Dilution-induced criticality is discussed using exact and scaling descriptions. A simple exact argument is used to obtain the scaling form for the density of states in dilute chains. For higher-dimensional systems it is shown that the condition for the existence of a 'fraction' edge is equivalent to the violation of a scaling relation between the fracton dimension of the infinite cluster, d and various static exponents. In addition the result dd/d f is obtained for the fracton dimensionality of the finite clusters near the percolation threshold; d f is the fractal dimension of the infinite cluster.

DOI
10.1088/0305-4470/19/2/008

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Dilution-induced criticality is discussed using exact and scaling descriptions. A simple exact argument is used to obtain the scaling form for the density of states in dilute chains. For higher-dimensional systems it is shown that the condition for the existence of a 'fraction' edge is equivalent to the violation of a scaling relation between the fracton dimension of the infinite cluster, d and various static exponents. In addition the result dd/d f is obtained for the fracton dimensionality of the finite clusters near the percolation threshold; d f is the fractal dimension of the infinite cluster.
The critical dynamics of Heisenberg spin systems at the percolation threshold is obtained from the transformation of the equations of motion under a length scaling acheived by climination of sites. The method is applied exactly to ddd-dimensional nonrandom fractals representing the infinite-cluster backbone and approximatcly to a truly random twodimensional system to yield dynamic scaling forms for response function and characteristic frequency, and associated static and dynamic critical exponents.
The critical spin-wave dynamics of dilute ferromagnetic and antiferromagnetic Heisenberg systems near the percolation threshold is studied on a two-dimensional regular (fractal) model for bond percolation, which captures the relevant geometric features of the cluster structure. The dynamic critical exponent and spectral dimension for both the ferromagnetic and antiferromagnetic systems are calculated. The treatment of the dynamics in these systems requires the use of generalised scaling techniques for sublattice systems, which involve an extension of the parameter space.
Relying on the fractal character of the largest clusters at criticality, we employ a finite-size scaling analysis to obtain an accurate phase-diagram of the percolation transition in chains with bond concentration decaying as a power-law on the form 1/ r(1+sigma) . For the particular case of sigma=1, no percolation transition is observed to occur at a finite dilution, in contrast with the finite temperature Kosterlitz-Thouless transition exhibited in Ising and Potts chains with inverse square-law couplings. The fractal dimension of the critical percolation cluster is found to follow distinct dependencies on the decay exponent being numerically fitted by d(f) =0.35+4sigma/5 for 0<sigma<1/2 and d(f) = (1+sigma) /2 for 1/2<sigma<1 . We also compute average mass ratios of the two largest clusters at criticality.
The critical exponents which describe quantities which are measured per unit 'mass' of the infinite cluster near percolation are shown to be shifted by beta p (the exponent describing the probability of belonging to this cluster). The fractal dimensionality of the infinite cluster then replaces the Euclidean one in hyperscaling relations. The crossover exponent for the effects of random fields on dilute Ising models are zero temperature is then shown to be phi h= gamma p+ beta p. Similarly, that for random local concentrations is phi p= alpha p+ beta p.
Scaling curves for the density of states and localization length of fractons near a percolation threshold are obtained for dilute and superconducting lattices in both two and three dimensions. These are used to verify the predictions of single-parameter scaling and to show that the energy dependence of the inverse localization length is consistent with known values of the spectral dimensionalities of these structures. The results demonstrate that eigenstates of an infinite, percolating, dilute structure exhibit superlocalization of the form \ensuremath{\psi}(r)\ensuremath{\sim}${\mathit{e}}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\beta}}\mathit{r}\mathrm{\ensuremath{\delta}}}$ with values of \ensuremath{\delta}=1.13\ifmmode\pm\else\textpm\fi{}0.06 and 1.39\ifmmode\pm\else\textpm\fi{}0.07 in two and three dimensions, respectively.
Abstract In this chapter we apply the scaling theories developed in the one-dimensional system and the infinite-dimensional system to systems of finite dimensions. The lowest dimension with an interesting behavior is two dimensions. Here, we introduce effective ways to measure the cluster number density n(s,p)n(s,p)n(s,p) n ( s , p ) in two dimension. We develop the scaling theory for n(s,p)n(s,p)n(s,p) n ( s , p ) and demonstrate how to use data-collapse plots as an efficient method to measure the critical exponents. We also demonstrate how we can use the scaling theory for n(s,p)n(s,p)n(s,p) n ( s , p ) to derive expressions for the density of the spanning cluster, P, and the average cluster size, S. Finally, we demonstrate how the scaling theory provides scaling relations, that is, relations between exponents, and bounds for the values of the critical exponents.