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Physical Review Letters

Critical dynamics of diluted Ising systems

C K Harris, Robin B. Stinchcombe

Published Feb 24, 1986• 41 citations
Publisher

Abstract

Three different calculations are carried out to treat the low-temperature dynamics in diluted d-dimensional Ising systems near the percolation threshold. All three use recursive methods based on a domain-wall--diffusion picture. The standard form of bicritical dynamic scaling is found to be violated for d>1. A quadratic dependence of the logarithm of the characteristic decay time on the logarithm of the correlation length is found and related to recent experimental work.

DOI
10.1103/physrevlett.56.869

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Three different calculations are carried out to treat the low-temperature dynamics in diluted d-dimensional Ising systems near the percolation threshold. All three use recursive methods based on a domain-wall--diffusion picture. The standard form of bicritical dynamic scaling is found to be violated for d>1. A quadratic dependence of the logarithm of the characteristic decay time on the logarithm of the correlation length is found and related to recent experimental work.
The author presents an investigation of the dynamic critical behaviour near the percolation bicritical point of the 1D bond-diluted Glauber Ising model. The expected bicritical dynamic scaling form is rigorously established for the decay functions describing the decay of the equilibrium wavevector-dependent correlation function and the decay of the magnetisation from a (non-equilibrium) wavevector-dependent initial state. The corresponding characteristic decay times are evaluated exactly and exhibit a cross over from 'pure' to percolation dominated behaviour on increasing the dilution. The results in the percolation regime are in agreement with recent calculations in which domain boundary diffusion methods are applied to a wider class of kinetic Ising models.
Investigates the bicritical dynamics of one-dimensional bond-diluted kinetic Ising models in the percolation limit at criticality. A wide class of single-spin-flip dynamics is applied and domain boundary diffusion arguments are used to identify the dominant processes producing flipping of the ordered finite chain segments constituting the diluted chain. Exact expressions for the rate of flip as a function of length of segment are obtained and are used to derive corresponding expressions for the decay of equilibrium correlations, and of the magnetisation (from a non-equilibrium uniformly magnetised initial state) in the chain. The results are in agreement with those of earlier rigorous calculations restricted to the special case of Glauber dynamics.
The relaxation time \ensuremath{\tau} and the thermal correlation length ${\ensuremath{\xi}}_{\mathit{T}}$ in randomly diluted Ising magnets near the percolation threshold are related by the singular dynamic scaling ln\ensuremath{\tau}=f(ln${\ensuremath{\xi}}_{\mathit{T}}$), where f(x)=Ax2{\mathit{Ax}}^{2}Ax2+Bx+C, with constants A,B,C. We investigate the dimensionality dependence of A by Monte Carlo simulation and compare our observations with the theoretical predictions.
The authors use real-space renormalisation group techniques to study the kinetic Ising model (with Glauber dynamics) on bond-diluted d-dimensional hypercubic lattices, at the percolation threshold. The results confirm and generalise the recently proposed breakdown of dynamic scaling and they predict a dependence on dimensionality of the factors A and B on the temperature-dependent dynamic exponent z=A log xi +B (where xi is the thermal correlation length). These factors also depend on the lattice scaling factor (b); the authors suggest that the limit b=1 may be taken as the best choice for the important coefficient A, leading to A=d(d-1)/2 nu p .
A low-temperature approach for the critical Ising spin dynamics on percolation clusters is proposed. The average energy barrier to overturning a cluster of sss spins is shown to scale as lns\mathrm{ln}slns. This basic result is supported by exact numerical calculations of the energy-barrier distribution for two-and three-dimensional percolation clusters, and is argued to be a consequence of scale invariance. The resulting temperature as well as size dependence of the relaxation-time scales agrees very well with the results of our Monte Carlo simulations.