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arXiv:cond-mat/9410054 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Correlation decay and conformal anomaly in the two-dimensional random-bond Ising ferromagnet

S. L. A. de Queiroz

Published Feb 1, 1995• 20 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreve.51.1030

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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 consider long, finite-width strips of Ising spins with randomly distributed couplings. Frustration is introduced by allowing both ferromagnetic and antiferromagnetic interactions. Free energy and spin-spin correlation functions are calculated by transfer-matrix methods. Numerical derivatives and finite-size scaling concepts allow estimates of the usual critical exponents $\ensuremath{\gamma}/\ensuremath{\nu},$ $\ensuremath{\alpha}/\ensuremath{\nu},$ and $\ensuremath{\nu}$ to be obtained, whenever a second-order transition is present. Low-temperature ordering persists for suitably small concentrations of frustrated bonds, with a transition governed by pure-Ising exponents. Contrary to the unfrustrated case, subdominant terms do not fit a simple, logarithmic-enhancement form. Our analysis also suggests a vertical critical line at and below the Nishimori point. Approaching this point along either the temperature axis or the Nishimori line, one finds nondiverging specific heats. A percolationlike ratio $\ensuremath{\gamma}/\ensuremath{\nu}$ is found upon analysis of the uniform susceptibility at the Nishimori point. Our data are also consistent with frustration inducing a breakdown of the relationship between correlation-length amplitude and critical exponents, predicted by conformal invariance for pure systems.
We address the issue of universality in two-dimensional disordered Ising systems, by considering long, finite-width strips of ferromagnetic Ising spins with randomly distributed couplings. We calculate the free energy and spin-spin correlation functions (from which averaged correlation lengths, ${\ensuremath{\xi}}^{\mathrm{av}}$, are computed) by transfer-matrix methods. An ansatz for the size dependence of logarithmic corrections to ${\ensuremath{\xi}}^{\mathrm{av}}$ is proposed. Data for both random-bond and site-diluted systems show that pure system behavior (with $\ensuremath{\nu}=1$) is recovered if these corrections are incorporated, discarding the weak-universality scenario.
The statistics of critical spin-spin correlation functions in Ising systems with nonfrustrated disorder are investigated on a strip geometry, via numerical transfer-matrix techniques. Conformal invariance concepts are used, in order to test for logarithmic corrections to pure power-law decay against distance. Fits of our data to conformal-invariance expressions, specific to logarithmic corrections to correlations on strips, give results with the correct sign, for the moments of order n = 0-4 of the correlation-function distribution. We find an interval of disorder strength along which corrections to pure-system behavior can be decomposed into the product of a known n -dependent factor and an approximately n -independent one, in accordance with predictions. A phenomenological fitting procedure is proposed, which takes partial account of subdominant terms of correlation-function decay on strips. In the low-disorder limit, it gives results in fairly good agreement with theoretical predictions, provided that an additional assumption is made.
We consider long strips of finite width L\ensuremath{\le}13 sites of ferromagnetic Ising spins with random couplings distributed according to the binary distribution P(Jij{\mathit{J}}_{\mathit{ij}}Jij​)=1/2[\ensuremath{\delta}(Jij{\mathit{J}}_{\mathit{ij}}Jij​-J0{\mathit{J}}_{0}J0​)+\ensuremath{\delta}(Jij{\mathit{J}}_{\mathit{ij}}Jij​- rJ0{\mathit{rJ}}_{0}rJ0​)], 0r1. Spin-spin correlation functions 〈${\mathrm{\ensuremath{\sigma}}}_{0}$${\mathrm{\ensuremath{\sigma}}}_{\mathit{R}}$〉 along the ``infinite'' direction are computed by transfer-matrix methods, at the critical temperature of the corresponding two-dimensional system, and their probability distribution is investigated. We show that, although in-sample fluctuations do not die out as strip length is increased, averaged values converge satisfactorily. These latter are very close to the critical correlation functions of the pure Ising model, in agreement with recent Monte Carlo simulations. A scaling approach is formulated, which provides the essential aspects of the R and L dependence of the probability distribution of ln〈${\mathrm{\ensuremath{\sigma}}}_{0}$${\mathrm{\ensuremath{\sigma}}}_{\mathit{R}}$〉, including the result that the appropriate scaling variable is R/L. Predictions from scaling theory are borne out by numerical data, which show the probability distribution of ln〈${\mathrm{\ensuremath{\sigma}}}_{0}$${\mathrm{\ensuremath{\sigma}}}_{\mathit{R}}$〉 to be remarkably skewed at short distances, approaching a Gaussian only as R/L\ensuremath{\gg}1. \textcopyright{} 1996 The American Physical Society.
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.
We address the question of weak versus strong universality scenarios for the random-bond Ising model in two dimensions. A finite-size scaling theory is proposed, which explicitly incorporates lnL\mathrm{ln} LlnL corrections (L(L(L is the linear finite size of the system) to the temperature derivative of the correlation length. The predictions are tested by considering long, finite-width strips of Ising spins with randomly distributed ferromagnetic couplings, along which free energy, spin-spin correlation functions, and specific heats are calculated by transfer-matrix methods. The ratio $\ensuremath{\gamma}/\ensuremath{\nu}$ is calculated and has the same value as in the pure case; consequently conformal invariance predictions remain valid for this type of disorder. Semilogarithmic plots of correlation functions against distance yield average correlation lengths ${\ensuremath{\xi}}^{\mathrm{av}}$, whose size dependence agrees very well with the proposed theory. We also examine the size dependence of the specific heat, which clearly suggests a divergency in the thermodynamic limit. Thus our data consistently favor the Dotsenko-Shalaev picture of logarithmic corrections (enhancements) to pure system singularities, as opposed to the weak universality scenario.