Monte Carlo Studies of Connectivity Disorder

Abstract

The paradigm for studies of the effect of quenched, random disorder on uni-versal properties of critical phenomena are uncorrelated, randomly distributedcouplings [1–4]. This includes ferromagnetic random-bond models as well asthe physically very different case of spin glasses, where competing interactionscomplement disorder with frustration [2,5–9]. For a continuous phase transi-tion in the idealized pure system, the effect of random bonds has been convinc-ingly shown by renormalization group analyses as well as numerical investiga-tions to be able to induce a crossover to a new, disorder fixed point [3,10–14].Using phenomenological scaling theory, Harris [6] argued that such a crossovershould not occur for systems with a specific-heat exponent α 0 [10,11,15]. In the marginal case α = 0, realized, e.g.,by the Ising model in two dimensions, the regular critical behavior is merelymodified by logarithmic corrections [3]. Similarly, for systems exhibiting afirst-order phase transition in the regular case, the introduction of quencheddisorder coupling to the local energy density can weaken the transition tosecond (or even higher) order [9]. While this scenario has been rigorously es-tablished for the case of two dimensions and an arbitrarily small amount ofdisorder [7,8,16], the situation for higher-dimensional systems is less clear. Fora variety of systems in three dimensions, however, sufficiently strong disorderhas been shown numerically [17–19] to be able to soften the transition to acontinuous one.Spatial correlations of the disorder degrees of freedom lead to a modifica-tion of the fluctuations present in “typical” patches of the random system withrespect to the behavior expected from the central limit theorem for indepen-dent random variables, which is implicitly presupposed by Harris’ arguments.Such correlations for a random-bond model have been considered occasion-ally [20–23] and altered relevancecriteria have been proposed [20,24]. Luck [24]has considered a class of irregular systems not covered by the random-bond