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arXiv:cond-mat/9407075 · Physical review. B, Condensed matter

Transfer-matrix scaling from disorder-averaged correlation lengths for diluted Ising systems

S. L. A. de Queiroz, Robin B. Stinchcombe

Published Oct 1, 1994• 25 citations
arXivHTMLPDF

Abstract

A transfer-matrix-scaling technique is developed for randomly diluted systems, and applied to the site-diluted Ising model on a square lattice in two dimensions. For each allowed disorder configuration between two adjacent columns, the contribution of the respective transfer matrix to the decay of correlations is considered only as far as the ratio of its two largest eigenvalues, allowing an economical calculation of a configuration-averaged correlation length. Standard phenomenological-renormalization procedures are then used to analyze aspects of the phase boundary which are difficult to assess accurately by alternative methods. For magnetic site concentration p close to ${\mathit{p}}_{\mathit{c}}$, the extent of exponential behavior of the ${\mathit{T}}_{\mathit{c}}$\ifmmode\times\else\texttimes\fi{}p curve is clearly seen for over two decades of variation of p-${\mathit{p}}_{\mathit{c}}$. Close to the pure-system limit, the exactly known reduced slope is reproduced to a very good approximation, though with nonmonotonic convergence. The averaged correlation lengths are inserted into the exponent-amplitude relationship predicted by conformal invariance to hold at criticality. The resulting exponent \ensuremath{\eta} remains near the pure value (1/4) for all intermediate concentrations until it crosses over to the percolation value at the threshold.

DOI
10.1103/physrevb.50.9976

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A transfer-matrix-scaling technique is developed for randomly diluted systems, and applied to the site-diluted Ising model on a square lattice in two dimensions. For each allowed disorder configuration between two adjacent columns, the contribution of the respective transfer matrix to the decay of correlations is considered only as far as the ratio of its two largest eigenvalues, allowing an economical calculation of a configuration-averaged correlation length. Standard phenomenological-renormalization procedures are then used to analyze aspects of the phase boundary which are difficult to assess accurately by alternative methods. For magnetic site concentration p close to pc{\mathit{p}}_{\mathit{c}}pc​, the extent of exponential behavior of the Tc{\mathit{T}}_{\mathit{c}}Tc​\ifmmode\times\else\texttimes\fi{}p curve is clearly seen for over two decades of variation of p-pc{\mathit{p}}_{\mathit{c}}pc​. Close to the pure-system limit, the exactly known reduced slope is reproduced to a very good approximation, though with nonmonotonic convergence. The averaged correlation lengths are inserted into the exponent-amplitude relationship predicted by conformal invariance to hold at criticality. The resulting exponent \ensuremath{\eta} remains near the pure value (1/4) for all intermediate concentrations until it crosses over to the percolation value at the threshold.
A transfer-matrix scaling technique is developed for randomly diluted systems and applied to the site-diluted Ising model on a square lattice. For each connected configuration between adjacent columns, the contribution of the respective transfer matrix to the decay of correlations is considered only as far as the ratio of the two largest eigenvalues, allowing an economical incorporation of configurational averages. Predictions for the phase boundary at and near the percolation and pure Ising limits, and for the correlation exponent \ensuremath{\eta} at those limits, agree with exactly known results to within 1% error with largest strip widths of only L=5. The exponent \ensuremath{\eta} remains near the pure value (1/4) for all intermediate concentrations until it turns over to the percolation value at the threshold.
We introduce a new approach to disordered two-dimensional Ising models based on the extension of the combinatorial solution to randomized supercells. Applying it to the site-diluted Ising model on the square lattice, we resolve the full phase boundary Tc(p)T_c(p)Tc​(p) from the pure-Ising point to the percolation limit Tc(pc)=0T_c(p_c)=0Tc​(pc​)=0 with, in principle, arbitrary precision. The critical eigenvalue governing the transition is found to follow a remarkably accurate linear interpolation between the Ising and percolation endpoints, whose small but systematic deviations reveal the nontrivial fine structure of the phase boundary. Near the percolation threshold, we confirm the crossover exponent ϕRSIM=1ϕ_{\rm RSIM}=1ϕRSIM​=1 and extract the nonuniversal amplitude αRSIM≃1.616{α_{\rm RSIM}\simeq 1.616}αRSIM​≃1.616.
The corrections to finite-size scaling in the critical two-point correlation function G(r) of 2D Ising model on a square lattice have been studied numerically by means of exact transfer-matrix algorithms. The systems of square geometry with periodic boundaries oriented either along <10> or along <11> direction have been considered, including up to 800 spins. The calculation of G(r) at a distance r equal to the half of the system size L shows the existence of an amplitude correction proportional to 1/L^2. A nontrivial correction proportional to 1/L^0.25 of a very small magnitude also has been detected in agreement with predictions of our recently developed GFD (grouping of Feynman diagrams) theory. A refined analysis of the recent MC data for 3D Ising, phi^4, and XY lattice models has been performed. It includes an analysis of the partition function zeros of 3D Ising model, an estimation of the correction-to-scaling exponent omega from the Binder cumulant data near criticality, as well as a study of the effective critical exponent eta and the effective amplitudes in the asymptotic expansion of susceptibility at the critical point. In all cases a refined analysis is consistent with our (GFD) asymptotic values of the critical exponents (nu=2/3, omega=1/2, eta=1/8 for 3D Ising model and omega=5/9 for 3D XY model), while the actually accepted "conventional" exponents are, in fact, effective exponents which are valid for approximation of the finite--size scaling behavior of not too large systems.
We consider the two-dimensional (2D) random Ising model on a diagonal strip of the square lattice, where the bonds take two values, J1>J2, with equal probability. Using an iterative method, based on a successive application of the star-triangle transformation, we have determined at the bulk critical temperature the correlation length along the strip xi(L) for different widths of the strip L</=21. The ratio of the two lengths xi(L)/L=A is found to approach the universal value A=2/pi for large L, independent of the dilution parameter J(1)/J(2). With our method we have demonstrated with high numerical precision, that the surface correlation function of the 2D dilute Ising model is self-averaging, in the critical point conformally covariant and the corresponding decay exponent is eta( ||)=1.
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.