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Physical review. B, Condensed matter

Monte Carlo renormalization-group study of the site-diluted simple-cubic Ising model

T. Holey, M. Fähnle

Published Jun 1, 1990• 28 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.41.11709

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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\ensuremath−4{10}^{\mathrm{\ensuremath{-}}4}10\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\ensuremath−4{10}^{\mathrm{\ensuremath{-}}4}10\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.
We study the diluted site-bond-correlated (SBC) Ising model within the scheme of the mean-field renormalization group. The SBC Ising model has been proposed to describe the phase diagram for KNip{\mathrm{KNi}}_{\mathit{p}}KNip​${\mathrm{Mg}}_{1\mathrm{\ensuremath{-}}\mathit{p}}$F3{\mathrm{F}}_{3}F3​. One treats the square (Z=4) and simple cubic (Z=6) lattices by using two different exchange interactions (denoted by models A and B). In the present paper, the phase diagram for some values of the correlation parameter \ensuremath{\alpha}, which range from zero to unity, is studied. We obtain two critical concentrations pc{\mathit{p}}_{\mathit{c}}pc​(\ensuremath{\alpha}=0) and pc{\mathit{p}}_{\mathit{c}}pc​(\ensuremath{\alpha}\ensuremath{\ne}0) in agreement with results from Monte Carlo methods. In addition, an interesting universal relation between the initial slope ${\mathrm{\ensuremath{\Delta}}}^{(\mathrm{\ensuremath{\mu}})}$(\ensuremath{\alpha}=1) and ${\mathrm{\ensuremath{\Delta}}}^{(\mathrm{\ensuremath{\mu}})}$(\ensuremath{\alpha}=0) is obtained (\ensuremath{\mu} denotes model A or B). From this relation, we can estimate the possible exact values to the initial slope on the completely correlated case. Our approach is valid for lattices with any coordination number Z.
Monte Carlo simulations of the short-time dynamic behavior are reported for three-dimensional weakly site-diluted Ising model with spin concentrations p=0.95 and 0.8 at criticality. In contrast to studies of the critical behavior of the pure systems by the short-time dynamics method, our investigations of site-diluted Ising model have revealed three stages of the dynamic evolution characterizing a crossover phenomenon from the critical behavior typical for the pure systems to behavior determined by the influence of disorder. The static and dynamic critical exponents are determined with the use of the corrections to scaling for systems starting separately from ordered and disordered initial states. The obtained values of the exponents demonstrate a universal behavior of weakly site-diluted Ising model in the critical region. The values of the exponents are compared to results of numerical simulations which have been obtained in various works and, also, with results of the renormalization-group description of this model.
The Monte Carlo renormalization group is applied to the three-dimensional Ising model on simple cubic lattices with 83{8}^{3}83, 163{16}^{3}163, 323{32}^{3}323, and 643{64}^{3}643 sites. The comparison of block-spin correlation functions from the largest lattices yields the nearest-neighbor critical coupling K1c=0.221654(6){K}_{1}^{c}=0.221654(6)K1c​=0.221654(6). After allowing for (i) interpolation to this best estimate for K1c{K}_{1}^{c}K1c​, (ii) an apparent finite-size effect in the renormalization-group transformation due to the measurement of correlation functions of too few (seven) operators, and (iii) the extrapolation for the effect of a slow transient towards the fixed point, the values $\ensuremath{\nu}=0.629(4)$ and $\ensuremath{\eta}=0.031(5)$ are obtained for the thermal and magnetic exponents. The correction-to-scaling exponent $\ensuremath{\omega}$ is estimated to be around 1; to obtain an accuracy competitive with other methods requires measurements with more than seven operators. We briefly review the problem of redundant operators and indicate the future prospects for this kind of calculation.
The thermal and magnetic properties of the simple cubic Ising model with random, quenched, site impurities have been determined by Monte Carlo calculations. The transition appears to remain sharp, and the critical temperature decreases monotonically towards zero as the impurity concentration is increased towards the percolation limit. No clear evidence is found for any change in the critical exponents away from the pure lattice values. These results are discussed in terms of Harris's predictions for the critical behavior of impure systems.