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Physical Review Letters

Critical Dynamics at the Percolation Threshold by Fractal and Scaling Approaches

C K Harris, Robin B. Stinchcombe

Published May 2, 1983• 60 citations
Publisher

Abstract

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 $d$-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.

DOI
10.1103/physrevlett.50.1399

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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.
Critical spin wave dynamics in the dilute Heisenberg chain near the percolation threshold is treated by two complementary approaches. The first exploits the renormalisation group transformation of parameters under a length scaling achieved by decimation on the equations of motion of the random system, the second obtains from the contributions from chain segments of all possible sizes the average dynamic response using a continuum approach valid for small wavevector k and long percolation correlation length xi p. Both approaches yield identical dynamic scaling forms for the dynamic response, with dynamic exponent z=2, and details of the crossover of characteristic frequency between hydrodynamic and critical form as k xi p varies from 0 to infinity . A detailed expression for the scaling function for the dynamic response is also obtained.
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.
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.
Ising and Heisenberg ferromagnets and spin-glasses are studied at the percolation threshold for power-law bond distributions, P(J)\ensuremath{\sim}\ensuremath{\Vert}J${\ensuremath{\Vert}}^{\ensuremath{\alpha}\mathrm{\ensuremath{-}}1}$ for J\ensuremath{\rightarrow}0. Nonuniversal, \ensuremath{\alpha}-dependent, critical behavior is found. A hierarchical model for the ``backbone'' yields, via finite-size scaling, expressions for the exponent t in the relation \ensuremath{\Upsilon}\ensuremath{\propto}(p-pc{p}_{c}pc​)t{)}^{t})t for the generalized stiffness coefficient, and for the percolation-to-thermal crossover exponent \ensuremath{\varphi}.