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Journal of Physics C Solid State Physics

The breakdown of dynamic scaling of the bond-diluted Ising model at the percolation threshold for hypercubic lattices

J M Nunes da Silva, E J S Laga

Published May 20, 1987• 7 citations
Publisher

Abstract

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 .

DOI
10.1088/0022-3719/20/14/003

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Journal of Physics A Mathematical and General•May 11, 1985•6 citations
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 .
The critical dynamics of the diluted two-dimensional Ising model below the percolation threshold is studied. The dynamics exhibits a crossover from a concentration-dependent behaviour for xi T >> xi p to a percolating lattice type behaviour for xi T << xi p , where xi T and xi p are the thermal and percolation correlation lengths, respectively. As a consequence, the dynamical exponents A and B are shown to be the limits of dynamical functions A(x) and B(x), respectively; here, x=ln xi p /ln xi T . Extensive Monte Carlo simulations support our predictions.
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.
A two-step renormalisation group approach-a Migdal-Kadanoff (or decimation) transformation (MK) followed by a mean-field renormalisation group (MFRG)-has been applied to random bond Ising models on d-dimensional hypercubic lattices. Calculations of the critical couplings and percolation concentrations for two and three dimensions show much improvement, even for small size clusters, compared with the results of separate MK or MFRG calculations. Critical exponents nu T and nu p are also improved in some cases. Estimates of critical curves in the concentration-temperature plane are also obtained for the two- and three-dimensional models.