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Journal of Physics C Solid State Physics

Critical behaviour of compressible dilute Ising systems-a Monte Carlo study

Bikas K. Chakrabarti

Published Nov 30, 1982• 5 citations
Publisher

Abstract

The critical behaviour of a compressible dilute Ising system has been studied here using Monte Carlo simulation techniques for a simple cubic lattice of size 20 3 . The discontinuity in the transition behaviour of the compressible Ising system is seen to be removed by the presence of lattice (spin) dilution. The observation agrees with the renormalisation group epsilon -expansion prediction.

DOI
10.1088/0022-3719/15/33/004

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The critical behaviour of a compressible dilute Ising system has been studied here using Monte Carlo simulation techniques for a simple cubic lattice of size 20 3 . The discontinuity in the transition behaviour of the compressible Ising system is seen to be removed by the presence of lattice (spin) dilution. The observation agrees with the renormalisation group epsilon -expansion prediction.
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The Monte Carlo renormalization-group approach is applied to investigate the critical behavior of the simple-cubic Ising model diluted randomly with nonmagnetic atoms of concentration x. For x=0.2 a value of \ensuremath{\nu}=0.688(13) is found which is well above the result \ensuremath{\nu}=0.629(4) for the pure model and above the exact lower bound of \ensuremath{\nu}=2/3, and the relative width of the asymptotic regime is definitely larger than 3\ifmmode\times\else\texttimes\fi{}${10}^{\mathrm{\ensuremath{-}}4}$. For x=0.1 the obtained value is below the exact lower bound, possibly indicating that the asymptotic critical regime has not been penetrated, that this regime has a width smaller than 1.3\ifmmode\times\else\texttimes\fi{}${10}^{\mathrm{\ensuremath{-}}4}$, and that it can hardly be explored by experiments or computer simulations.
Simulation of critical relaxation of the magnetization in a three-dimensional Ising model diluted by quenched nonmagnetic impurity atoms is carried out. The cubic systems with linear size L = 144 and spin concentrations p = 0.95, 0.8, 0.6, and 0.5 are studied. The dynamical critical exponent z is determined by the Monte Carlo method combined with the dynamical renormalization group method. The obtained results are: z (p): z (0.95) = 2.19 ± 0.07, z (0.8) = 2.29 ± 0.06, z (0.6) = 2.72 ± 0.08, and z (0.5) = 2.75 ± 0.08.
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.
Monte Carlo simulations of the critical behaviour of disordered Ising systems have led to concentration-dependent critical exponents which seem to violate universality. We apply the renormalization group to investigate the crossover effects which cause the observed behaviour. We improve the three-loop expansion of the Callan-Symanzik and Wilson functions by a Pade-Borel approximation and solve the flow equations for various initial points. The exponents found in the simulations can be related to regions in the space of coupling coefficients away from the fixed point.