Critical Behavior of the Randomly Spin Diluted 2D Ising Model: A Grand Ensemble Approach

Abstract

The critical behaviour of the randomly spin-diluted Ising model in two space dimensions is investigated by a new method which combines a grand ensemble approach to disordered systems proposed by Morita with the phenomenological renormalization group scheme of Nightingale. Accurate approximations for the phase diagram and for the connectivity length exponent of the percolation transition are obtained. Our results suggest that the thermal phase transition of the disordered system might be different from that of the pure system: we observe a continuous variation of critical exponents with the density ρ of magnetic impurities, respecting, however, weak universality in the sense that η and γ/ν do not depend on ρ while γ and ν separately do. Our results are in qualitative and quantitative agreement with a recent Monte-Carlo study.

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.…”

“…[18,19,20] for the RBIM and the RSIM, respectively. However, in [28] it was claimed that such apparent double-logarithmic FSS behaviour does not necessarily imply [29] [30] and theoretical support for finite C ∞ (t) [31,32] [33] or numerical support for finite C ∞ (t) [28,34] [ 35,36,37,38] divergence of the specific heat, and numerically based counter-claims that the specific heat remains finite (so that α < 0 or α = 0 andα < 0) in the random-bond [28,34] and random-site models [35,36,37,38] also exist (see also Refs. [29,30,31,32,33]).…”

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

“…Let us first consider the quartic cumulantŪ 4 defined in Eq. (14). At fixed R ξ ,Ū 4 (L) is expected to behave asŪ …”

Continuous spin models on annealed generalized random graphs

“…It is worthwhile to contrast this constructive result which allows us to describe the system obtained by annealing over graphs at fixed weights as a mean-field Gibbs model with joint potential U with a "negative" result in a related but slightly different situation: In the so-called Morita-approach to disordered systems of theoretical physics [18,28,34] one tries to interpret a quenched model as a formal Gibbsian model with a new Hamiltonian depending both on disorder variables and spin variables. The original motivation to do so stems from non-rigorous renormalization group theory with an aim to determine critical exponents of the random system.…”