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arXiv:cond-mat/9312017

The Energy-Energy Correlation Function of the Random Bond Ising Model in Two Dimensions

K. Ziegler

Published Dec 3, 1993• 0 citations
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Abstract

The energy-energy correlation function of the two-dimensional Ising model with weakly fluctuating random bonds is evaluated in the large scale limit. Two correlation lengths exist in contrast to one correlation length in the pure 2D Ising model: one is finite whereas the other is divergent at the critical points. The corresponding exponent of the divergent correlation length is $ν_e=1/2$ in contrast to the pure system where $ν_e=1$. The calculation is based on a previously developed effective field theory for the energy density fluctuations.

DOI
10.48550/arxiv.cond-mat/9312017

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The energy-energy correlation function of the two-dimensional Ising model with weakly fluctuating random bonds is evaluated in the large scale limit. Two correlation lengths exist in contrast to one correlation length in the pure 2D Ising model: one is finite whereas the other is divergent at the critical points. The corresponding exponent of the divergent correlation length is νe=1/2ν_e=1/2νe​=1/2 in contrast to the pure system where νe=1ν_e=1νe​=1. The calculation is based on a previously developed effective field theory for the energy density fluctuations.
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 effects of random impurities on the various thermodynamic functions are studied with the aid of the two-dimensional Ising model. Of special interest are the limiting forms of these functions near the critical point. This problem is approached by a Green's function formulation as developed by Kadanoff. Since experiments only measure the average effects of impurities, we consider the average of the Green's function over all distributions of impurities and calculate it by perturbation theory. From this function we find the location of the new critical temperature and the temperature dependence of the coherence length. Thermodynamic and correlation functions are calculated by means of perturbation expansions and their functional dependence on the coherence length investigated. It is found that the magnetic properties such as the magnetization, susceptibility, and spin-spin correlations retain the same functional dependence on the coherence length. The specific heat is found to have a finite value at the critical point, and the T = Tc form of the energy-energy correlation c function is changed. The thermodynamic functions exhibit new critical indices very close to Tc. These renormalized exponents have been predicted c in literature and are found to satisfy the scaling laws predictions.
A 2D Ising model in which the bonds K fluctuate randomly about Kc{\mathrm{K}}_{\mathrm{c}}Kc​, the critical value of the pure system, is considered. The ensemble average of the square of the two-point function, s0{\mathrm{s}}_{0}s0​sR{\mathrm{s}}_{\mathrm{R}}sR​〉2{\mathrm{〉}}^{2}〉2〉Av{\mathrm{〉}}_{\mathrm{Av}}〉Av​, is shown to decay as (lnR)1/4{)}^{1/4})1/4${\mathrm{R}}^{\mathrm{\ensuremath{-}}1/2}$ aat the critical point. This implies that s0{\mathrm{s}}_{0}s0​sR{\mathrm{s}}_{\mathrm{R}}sR​Av{}_{\mathrm{Av}}Av​ is bounded above by (lnR)1/8{)}^{1/8})1/8${\mathrm{R}}^{\mathrm{\ensuremath{-}}1/4}$ in disagreement with the exp [-(ln lnR)2{)}^{2})2] decay law found by Dotsenko and Dotsenko by a different method. On the other hand, the present calculation reproduces their specific-heat singularity C\ensuremath{\sim}ln\ensuremath{\Vert}ln\ensuremath{\tau}\ensuremath{\Vert} (\ensuremath{\tau}=K-Kc{\mathrm{K}}_{\mathrm{c}}Kc​).
The two-dimensional random-bond Ising model is numerically studied on long strips by transfer-matrix methods. It is shown that the rate of decay of correlations at criticality, as derived from averages of the two largest Lyapunov exponents plus conformal invariance arguments, differs from that obtained through direct evaluation of correlation functions. The latter is found to be, within error bars, the same as in pure systems. Our results confirm field-theoretical predictions. The conformal anomaly c is calculated from the leading finite-width correction to the averaged free energy on the strips. Estimates thus obtained are consistent with c=1/2, the same as for the pure Ising model.
We investigate and contrast, via the Wang-Landau (WL) algorithm, the effects of quenched bond randomness on the self-averaging properties of two Ising spin models in 2d. The random bond version of the superantiferromagnetic (SAF) square model with nearest- and next-nearest-neighbor competing interactions and the corresponding version of the simple ferromagnetic Ising model are studied. We find that, the random bond SAF model shows a strong violation of self-averaging, much stronger than that observed in the case of the random bond Ising model. Our analysis of the asymptotic scaling behavior of the variance of the distribution of the sample-dependent pseudocritical temperatures is found to be consistent with the renormalization group prediction of Aharony and Harris. Using this alternative approach, we find estimates of the correlation length exponent ννν in agreement with results obtained from the usual finite-size scaling (FSS) methodology.