Universality and logarithmic corrections in two-dimensional random Ising ferromagnets

TL;DR

This paper investigates the effects of disorder on the two-dimensional random Ising model, proposing a finite-size scaling theory with logarithmic corrections, and finds results supporting the logarithmic correction scenario over weak universality.

Contribution

It introduces a finite-size scaling theory with explicit ln L corrections for the 2D random Ising model and tests it through transfer-matrix calculations, supporting the logarithmic correction scenario.

Findings

The ratio γ/ν remains the same as in the pure case.

Correlation lengths agree with the proposed theory.

Specific heat diverges in the thermodynamic limit.

Abstract

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 $\gamma/\nu$ 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 $\xi^{av}$, whose size dependence agrees very well with the proposed theory. We also examine the size dependence of the specific heat, which clearly suggestsa 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.