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Physical review. B, Condensed matter

Mean-field renormalization group: Approach to the diluted site-bond-correlated Ising model

J. Ricardo de Sousa, Douglas F. de Albuquerque

Published Dec 1, 1993• 8 citations
Publisher

Abstract

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 ${\mathrm{KNi}}_{\mathit{p}}$${\mathrm{Mg}}_{1\mathrm{\ensuremath{-}}\mathit{p}}$${\mathrm{F}}_{3}$. 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 ${\mathit{p}}_{\mathit{c}}$(\ensuremath{\alpha}=0) and ${\mathit{p}}_{\mathit{c}}$(\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.

DOI
10.1103/physrevb.48.16484

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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 {\mathrm{KNi}}_{\mathit{p}}$${\mathrm{Mg}}_{1\mathrm{\ensuremath{-}}\mathit{p}}$${\mathrm{F}}_{3}. 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 \ensuremathΔ(\ensuremathμ){\mathrm{\ensuremath{\Delta}}}^{(\mathrm{\ensuremath{\mu}})}\ensuremathΔ(\ensuremathμ)(\ensuremath{\alpha}=1) and \ensuremathΔ(\ensuremathμ){\mathrm{\ensuremath{\Delta}}}^{(\mathrm{\ensuremath{\mu}})}\ensuremathΔ(\ensuremathμ)(\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.
A unified effective-field renormalization-group framework (EFRG) for both quenched bond- and site-diluted Ising models is herein developed by extending recent works. The method, as in the previous works, follows up the same strategy of the mean-field renormalization-group scheme (MFRG), and is achieved by introducing an alternative way for constructing classical effective-field equations of state, based on rigorous Ising spin identities. The concentration dependence of the critical temperature, Tc(p), and the critical concentrations of magnetic atoms, pc, at which the transition temperature goes to zero, are evaluated for several two- and three-dimensional lattice structures. The obtained values of Tc and pc and the resulting phase diagrams for both bond and site cases are much more accurate than those estimated by the standard MFRG approach. Although preserving the same level of simplicity as the MFRG, it is shown that the present EFRG method, even by considering its simplest size-cluster version, provides results that correctly distinguishes those lattices that have the same coordination number, but differ in dimensionality or geometry.
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.
We consider a diluted Ising model in which the absence of a spin affects the exchange coupling of a nearest-neighbor pair along the line joining the three spins; that is, it acquires the value $\ensuremath{\alpha}J$, where $\ensuremath{\alpha}$ is a phenomenological parameter $\ensuremath{\epsilon}[0,1]$. This model has been proposed to explain the experimental phase diagram for $\mathrm{KN}{\mathrm{i}}_{x}{\mathrm{Mg}}_{1\ensuremath{-}x}{\mathrm{F}}_{3}$. A position-space renormalization-group analysis clearly distinguishes two percolation thresholds depending on whether $\ensuremath{\alpha}=0$ or $\ensuremath{\alpha}>0$, though both cases seem to be in the same universality class. Further, thermal fluctuations dominate over the geometrical ones as in the uncorrelated case, and the critical curve (critical temperature versus concentration of magnetic sites) displays an upward curvature for intermediate degrees of correlation $0<\ensuremath{\alpha}<1$, as experimentally observed.
We study the transverse-field Ising model on a square lattice with bond- and site-dilution at zero temperature by stochastic series expansion quantum Monte Carlo simulations. Tuning the transverse field hhh and the dilution ppp, the quantum phase diagram of both models is explored. Both quantum phase diagrams show long-range order for small hhh and small ppp. The ordered phase of each is separated from the disordered (quantum) Griffiths phase by second-order phase transitions on two critical lines touching at a multi-critical point. Using Binder ratios we locate quantum critical points with high accuracy. The order-parameter critical exponent βββ and the average correlation-length exponent νavν_{\mathrm{av}}νav​ are determined along the critical lines and at the multi-critical points via finite-size scaling. We find three internally consistent sets of critical exponents and compare them with potentially connected universality classes. The quantum Griffiths phase in the vicinity of the phase transition lines is analyzed through the local susceptibility. Our results indicate that activated scaling occurs not only at the percolation transition, but also at the phase transition line for ppp smaller than the percolation threshold pcp_cpc​.