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Journal of Physics A Mathematical and General

Non-universality in the dynamics at the percolation threshold

S. Jain

Published Aug 1, 1986• 19 citations
Publisher

Abstract

The dynamical behavior of the two-dimensional Ising model at the site percolation threshold on L*L lattices with L<or=64 is studied by Monte Carlo simulations. The results are consistent with the recently proposed singular dynamic scaling. The new dynamical exponents are estimated to be A s =0.62+or-0.12 and B s =3.92+or-0.89. Comparison with bond dilution strongly suggests that universality is violated for this model.

DOI
10.1088/0305-4470/19/11/004

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The dynamical behavior of the two-dimensional Ising model at the site percolation threshold on L*L lattices with L<or=64 is studied by Monte Carlo simulations. The results are consistent with the recently proposed singular dynamic scaling. The new dynamical exponents are estimated to be A s =0.62+or-0.12 and B s =3.92+or-0.89. Comparison with bond dilution strongly suggests that universality is violated for this model.
The critical relaxation of the magnetization in a 2D Ising model with quenched random non-magnetic impurities is studied by numerical simulation. A square lattice of size 400 2 and spin concentrations p = 1.0, 0.95, 0.9, 0.85, 0.8, 0.75 and 0.7 are considered. The dynamical critical exponent z is determined by the dynamical renormalization group Monte Carlo method. The values for z(p) are: z (1) = 2.24 ± 0.07, z (0.95) = 2.24 ± 0.06, z (0.9) = 2.24 ± 0.06, z (0.85) = 2.38 ± 0.05, z (0.8) = 2.51 ± 0.06, z (0.75) = 2.66 ± 0.07 and z (0.7) = 2.88 ± 0.06. A singular dynamic-scaling behaviour with z(p) = A '|ln( p − p c )| + B ' is found for systems with spin concentrations near the percolation threshold p c , where A ' = 0.56 ± 0.07 and B ' = 1.62 ± 0.07 are temperature-independent constants.
The critical relaxation of the magnetization in a two-dimensional Ising model with quenched random non-magnetic impurities has been studied by numerical simulation. A squared lattice of size 400 2 with spin concentrations p=1.0; 0.95, 0,9, 0.85, 0.8, 0.75 and 0.7 was considered. The dynamical critical exponent z was determined by the Monte Carlo method combined with the dynamical renormalization group method. The following values were received for z(p): z(1)=2.24+or-0.07, z(0.95)=2.24+or-0.06, z(0.9)=2.24+or-0.06, z(0.85)=2.38+or-0.05, z(0.8)=2.51+or-0.06, z(0.75)=2.66+or-0.07 and z(0.7)=2.88+or-0.06. A singular dynamic scaling behaviour with z=A' mod ln(p-p c ) mod +B' was found for systems with spin concentrations p<or=0.85 near the percolation threshold p c , where A'=0.56+or-0.07 and B'=1.62+or-0.07 are temperature-independent constants.
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 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 .