Correlation functions in the two-dimensional random-bond Ising model

TL;DR

This study analyzes spin-spin correlation functions in a 2D random-bond Ising model using transfer-matrix methods, revealing how their distributions scale with system size and distance, and comparing results to pure models and simulations.

Contribution

It introduces a transfer-matrix approach to compute correlation distributions in the 2D random-bond Ising model at criticality, and develops a scaling theory for their R-L dependence.

Findings

Average correlation functions are close to pure Ising model values.

Probability distribution of correlations is skewed at short distances.

Scaling variable R/L effectively describes distribution behavior.

Abstract

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 $R/L \gg 1$ .