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 ln L corrections (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 γ/ν 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 ξ 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 favour the Dotsenko-Shalaev picture of logarithmic corrections (enhancements) to pure system singularities, as opposed to the weak universality scenario.

The possible existence of self-organized criticality in Barkhausen noise is investigated theoretically through a single interface model, and experimentally from measurements in amorphous magnetostrictive ribbon Metglas 2605TCA under stress. Contrary to previous interpretations in the literature, both simulation and experiment indicate that the presence of a cutoff in the avalanche size distribution may be attributed to finite size effects. 05.40.+j, 75.60.Ej, 68.35.Rh The Barkhausen effect consists of magnetic noise caused by erratic jumps in the magnetization of a ferromagnetic material, under an increasing applied magnetic field [1]. A simple explanation for this effect is the combination of random pinning of domain walls by defects and the driving external field, which is essentially the same mechanism present in stick-slip processes [2]. Recently the statistical behavior of Barkhausen noise has attracted much interest, due to the possibility of providing an experimental realization of self-organized critical (SOC) behavior [3][4][5]. The subject is controversial, however. We concentrate here on the results obtained by Urbach et al, (UMM) [4] and Perković et al (PDS) [5].UMM measured the avalanche size probability distribution function in an Fe-Ni-Co alloy, and found power-law decay over approximately two decades, followed by an exponential cutoff. The same result was also observed in numerical simulations of the interface motion. The power-law behavior, obtained without any intentional fine-tuning of parameters, suggests that this system self-organizes into a critical state. On the other hand, PDS argued that such behavior can be explained without resource to SOC concepts: in their view, the power-law decay followed by a cutoff is evidence that the system is near but not quite at a conventional critical point. They performed simulations for the random-field Ising model (RFIM) under an external field, taking the local (pinning) fields to be gaussian-disordered with standard deviation R. The avalanche-size distribution is also characterized by a power law followed by a cutoff, and the power-law regime increases over several decades as R approaches a critical disorder R C .Although UMM and PDS approach the problem with apparently similar models, their conclusions regarding the critical nature of the Barkhausen noise are in contradiction. Here we show that in reality, the ingredients used in either model differ in crucial aspects where the onset of SOC is concerned, so it is not surprising that they end up with different findings.We investigate this question by using the simple model proposed by UMM [4] for the motion of a single domain wall in the Barkhausen noise regime. We find that the existence of a cutoff in the UMM model can be traced back to finite-size effects; experimental results, also to be described, bear out the idea that the cutoff to be found there originates from corresponding aspects in real systems.In UMM's model, the interface at time t is described, in space dimensionality d, by its heig...

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 strips. Estimates thus obtained are consistent with c = 1/2, the same as for the pure Ising model.

A transfer-matrix scaling technique is developed for randomly diluted systems and applied to the sitediluted 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 g at those limits, agree with exactly known results to within 1% error with largest strip widths of only L =5. The exponent g remains near the pure value ( 4 ) for all intermediate concentrations until it turns over to the percolation value at the threshold.

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We use transfer-matrix methods to calculate free energies and specific heats on long, finite-width strips of Ising spins with randomly distributed ferromagnetic couplings. By implementation of our code on a highly parallel computer, we have managed to generate high-quality data for strip widths up to L = 18 sites. An unequivocal trend towards a divergency of the specific heat in the thermodynamic limit can be discerned. Finite-size data appear to behave halfway between a single- ( ln L) and double- ( ln ln L) logarithmic dependence on strip width. This is in line with a crossover within the Dotsenko-Shalaev picture of logarithmic corrections to pure-system singularities.

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We consider long strips of finite width $L \leq 13$ sites of ferromagnetic Ising spins with random couplings distributed according to the binary distribution: $P(J_{ij})= {1 \over 2} ( \delta (J_{ij} -J_0) + \delta (J_{ij} -rJ_0) ) ,\ 0 < r < 1 $. Spin-spin correlation functions $ <\sigma_{0} \sigma_{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 $\ln <\sigma_{0} \sigma_{R}>$, 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 $\ln <\sigma_{0} \sigma_{R}>$ to be remarkably skewed at short distances, approaching a Gaussian only as $R/L \gg 1$ .Comment: uuencoded; when uudecoded will give RevTeX code for 13 pages, plus 10 Postscript figures; to appear in Physical Review