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arXiv:cond-mat/9405047 · Physical Review Letters

Critical Behavior of the Randomly Spin Diluted 2D Ising Model: A Grand Ensemble Approach

Reimer Kühn

Published Oct 17, 1994• 62 citations
arXivHTMLPDF

Abstract

The critical behavior of the 2D spin diluted Ising model is investigated by a new method that combines a grand ensemble approach to disordered systems with phenomenological renormalization. We observe a continuous variation of critical exponents with the density $\ensuremath{\rho}$ of magnetic impurities, respecting, however, weak universality in the sense that $\ensuremath{\eta}$ and $\frac{\ensuremath{\gamma}}{\ensuremath{\nu}}$ do not depend on $\ensuremath{\rho}$ while $\ensuremath{\gamma}$ and $\ensuremath{\nu}$ separately do. Our results are in complete agreement with a recent Monte Carlo study.

DOI
10.1103/physrevlett.73.2268

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The critical behavior of the 2D spin diluted Ising model is investigated by a new method that combines a grand ensemble approach to disordered systems with phenomenological renormalization. We observe a continuous variation of critical exponents with the density \ensuremathρ\ensuremath{\rho}\ensuremathρ of magnetic impurities, respecting, however, weak universality in the sense that \ensuremathη\ensuremath{\eta}\ensuremathη and \ensuremathγ\ensuremathν\frac{\ensuremath{\gamma}}{\ensuremath{\nu}}\ensuremathν\ensuremathγ​ do not depend on \ensuremathρ\ensuremath{\rho}\ensuremathρ while \ensuremathγ\ensuremath{\gamma}\ensuremathγ and \ensuremathν\ensuremath{\nu}\ensuremathν separately do. Our results are in complete agreement with a recent Monte Carlo study.
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 equilibrium ensemble approach to disordered systems is used to investigate the critical behavior of the two-dimensional Ising model in the presence of quenched random site dilution. The numerical transfer matrix technique in semi-infinite strips of finite width, together with phenomenological renormalization and conformal invariance, is particularly suited to putting the equilibrium ensemble approach to work. A method by which to extract with great precision the critical temperature of the model is proposed and applied. A more systematic finite-size scaling analysis than in previous numerical studies has been performed. A parallel investigation, along the lines of the two main scenarios currently under discussion, namely, the logarithmic corrections scenario (with critical exponents fixed in the Ising universality class) versus the weak universality scenario (critical exponents varying with the degree of disorder), is carried out. In interpreting our data, maximum care is constantly taken to be open to both possibilities. A critical discussion shows that an unambiguous discrimination between the two scenarios is still not possible on the basis of the available finite-size data.
The critical behavior of the disordered ferromagnetic Ising model is studied numerically by the Monte Carlo method in a wide range of variation of concentration of nonmagnetic impurity atoms. The temperature dependences of correlation length and magnetic susceptibility are determined for samples with various spin concentrations and various linear sizes. The finite-size scaling technique is used for obtaining scaling functions for these quantities, which exhibit a universal behavior in the critical region; the critical temperatures and static critical exponents are also determined using scaling corrections. On the basis of variation of the scaling functions and values of critical exponents upon a change in the concentration, the conclusion is drawn concerning the existence of two universal classes of the critical behavior of the diluted Ising model with different characteristics for weakly and strongly disordered systems.
We consider two-dimensional Ising models with randomly distributed ferromagnetic bonds and study the local critical behavior at defect lines by extensive Monte Carlo simulations. Both for ladder and chain type defects, non-universal critical behavior is observed: the critical exponent of the defect magnetization is found to be a continuous function of the strength of the defect coupling. Analyzing corresponding stability conditions, we obtain new evidence that the critical exponent ννν of the bulk correlation length of the random Ising model does not depend on dilution, i.e. ν=1ν=1ν=1.
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.