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

Abstract

The equilibrium ensemble approach to disordered systems is used to investigate the critical behaviour of the two dimensional Ising model in presence of quenched random site dilution. The numerical transfer matrix technique in semi-infinite strips of finite width, together with phenomenological renormalization and conformal invariance, is particularly suited to put the equilibrium ensemble approach to work. A new method to extract with great precision the critical temperature of the model is proposed and applied. A more systematic finite-size scaling analysis than in previous numerical studies has been performed. A parallel investigation, along the lines of the two main scenarios currently under discussion, namely the logarithmic corrections scenario (with critical exponents fixed in the Ising universality class) versus the weak universality scenario (critical exponents varying with the degree of disorder), is carried out. In interpreting our data, maximum care is constantly taken to be open in both directions. A critical discussion shows that, still, an unambiguous discrimination between the two scenarios is not possible on the basis of the available finite size data.

Universal dependence on disorder of two-dimensional randomly diluted and random-bond±JIsing models

“…Finally, note that there are no analytic redefinitions of g that map Eq. (28) in an identical equation with…”

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

“…The difficulties in unambiguously discriminating between the weak and strong scenarios on the basis of finite-size data were highlighted in Refs. [28,37,39]. Indeed, in Ref.…”

Kühn and Mazzeo Reply:

“…These measures agree on an increasing set of moments with the measure characterizing the quenched disorder. Moreover, at each level of the approximation studied in [2,4] the variational problem is convex (also in the thermodynamic limit), hence its solution unique. In this sense, our approximation scheme is perfectly well defined, and it is legitimate to investigate its efficiency in analyzing unsolved problems in the physics of systems with quenched randomness.…”

“…Given the representation of the approximate measures studied in [2,4], it is perfectly conceivable that they could approach the joint measure of the quenched system, if one were able to carry the approximative scheme to its end. While a formal proof of this statement may be difficult, it nonetheless indicates that differences between the limiting measure of our scheme and the weakly Gibbsian measure it attempts to approximate in the low temperature phases might be rather subtle.…”

“…The joint measure related to the 2D spin-diluted Ising model studied in [2,4] is Gibbsian in the hightemperature phase, while only weakly Gibbsian in the ferromagnetic low-temperature phases (for definitions see the literature cited in [1]). Given the representation of the approximate measures studied in [2,4], it is perfectly conceivable that they could approach the joint measure of the quenched system, if one were able to carry the approximative scheme to its end.…”