Universality and logarithmic corrections in two-dimensional random Ising ferromagnets

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 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.

Critical behavior of the two-dimensional spin-diluted Ising model via the equilibrium ensemble approach

“…12. The variation of ν as a function of ρ at constant η and constant γ/ν would qualify the observed non-universalityif it must be upheld -to be of the weak form.…”

“…These models have been widely studied, both because the corresponding pure system is well understood and because they constitute a marginal case in the Harris criterion 1 which assesses whether disorder constitutes a relevant or irrelevant perturbation for the critical behaviour of the pure system. For models of this type, the discussion currently appears to narrow down on two conflicting scenarios, namely the logarithmic corrections [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17] versus the weak universality [18][19][20][21] scenario, though a broader spectrum of alternatives had been discussed earlier [22][23][24] (the interested reader will find a comprehensive report on the literature up to approximately 1982 in an early review by Stinchcombe 25 ). We will describe these scenarios and discuss these (and related) results in greater detail later on.…”

“…with the same critical exponents) as that of the pure 2d Ising model. These findings are almost all concerned with the ferromagnetic bond disordered situation, either theoretically, [3][4][5]17 or via numerical 6,7,[9][10][11][12][13][14]16 and series expansion 15 approaches, but the comparison with the spin diluted model is possible supposing -as it is generally believed to be the case -that the two systems are in the same universality class. 25,17 In discussing our results, we have to address two main issues.…”

Scaling analysis of the site-diluted Ising model in two dimensions

“…(2.12), are found in Refs. [15,16,17,18] and Refs. [18,19,20] for the bond-disordered and random-site Ising models, respectively (see also Refs.…”

Critical behavior of the spin-3/2 Blume-Capel model on a random two-dimensional lattice

“…Recently, the predicted softening effect at first-order phase transitions has been confirmed for 3D q-state Potts models with q ≥ 3 using Monte Carlo [5][6][7] and high temperature series expansion [8] techniques. The overall picture is even better in two dimensions (2D) where several models with α pure > 0 [9][10][11][12] and the marginal (α pure = 0) [13][14][15][16][17] have been investigated.…”