First- and second-order phase transitions in Ising models on small world networks, simulations and comparison with an effective field theory arXiv:1001.1342v2 [cond-mat.dis-nn] 28 Jul 2010 A. L. Ferreira, 1 J. F. F. Mendes, 1 and M. Ostilli 1, 2 $^{1}$Departamento de F´ısica and I3N, Universidade de Aveiro, 3810-193 Aveiro, Portugal 2 Statistical Mechanics and Complexity Center (SMC), INFM-CNR SMC, Italy We perform simulations of random Ising models defined over small-world networks and we check the validity and the level of approximation of a recently proposed effective field theory. Simulations confirm a rich scenario with the presence of multicritical points with first- or second-order phase transitions. In particular, for second-order phase transitions, independently of the dimension d$_{0}$ of the underlying lattice, the exact predictions of the theory in the paramagnetic regions, such as the location of critical surfaces and correlation functions, are verified. Quite interestingly, we verify that the Edward-Anderson model with d$_{0}$ = 2 is not thermodynamically stable under graph-noise. PACS numbers: 05.50.+q, 64.60.aq, 64.70.-p, 64.70.P- Disordered systems represent one of the most impor- tant fields of statistical mechanics. Disorder is at the base of many interesting phenomena whose understanding is often far from being trivial or immediate. Its use varies from applications in condensed matter physics to com- puter science and, more recently, to the broad range of natural, artificial, and social networks. One of the most important analytical tool to study disordered systems is represented by suitable mean-field theories. Originally developed to understand spin glass models, the replica method (RM) and the cavity method (CM) are nowadays largely used in many other fields of statistical physics and computer science and represent the most powerful ana- lytical methods to investigate the intricate nature of the spin glass and other disordered phases [1–3]. Among disordered models whose dimensionality - in a broad sense - can be considered infinite [4], one can sin- gle out a “hierarchical” family of models of increasing difficulty such as: fully connected models (correspond- ing to infinite connectivity in the thermodynamic limit) like the Sherrington-Kirkpatrick model [5], finite connec- tivity models like the Viana-Bray model [6], and mod- els defined over small-world networks [7, 8]. This lat- ter class of models has been introduced in recent years and represents an important development in modeling more realistic situations in which the spins, besides inter- acting through a random finite connectivity, distributed according to some given distribution with average de- gree c , interact also through short-range connections. In other words, small-world models constitute an interplay between purely random and regular finite-dimensional models. It is known that, despite the underlying finite dimensionality d$_{0}$ present in these kind of graphs, in the thermodynamic limit, models defined on them manifest a mean field behavior. This fact, far from being trivial, to be rigorously proved, may lead to hope that the RM or the CM could be used to solve small-world models. Indeed such methods have been already successfully ex- ploited in [9–11] for d$_{0}$ = 1 small-world models defined upon adding a Poisson distributed random connectivity c to the underlying regular one-dimensional chain. How- ever, if we take a look at the mathematical structure of these methods we recognize the following. For what con- cerns the RM, we need to know analytically the two lead- ing eigenvectors of the transfer matrix of the Ising model without the shortcuts but immersed in a random external field; whereas for what concerns the CM, it is essential that the underlying graph L$_{0}$ had a tree-like structure, i.e. , no loops, at least in the thermodynamic limit. As a consequence, both methods seem hardly applicable to small-world models if d$_{0}$ > 1. The effective field theory we have recently developed in [12], based on mapping a generic random model onto a non random one (the “pure model”) [4, 13, 14] is instead applicable to these models. In fact, though this theory is able to give exact answers only in the paramagnetic (P) regions, there is no limita- tion in the underlying dimension d$_{0}$ . Whereas the theory can be fully treated analytically for d$_{0}$ ≤ 1, for d$_{0}$ > 1, we can still apply it semi-analytically [34]. All we need to apply the theory is to solve - analytically or numerically - the pure model in d$_{0}$ dimension in the absence of the ran- dom shortcuts and in the presence of a uniform external field. The values of a certain observable O$_{0}$ so obtained una tantum will be then used to get the corresponding value O for the model in the presence of the shortcuts and for any choice of the disorder parameters (couplings and connectivity). This feature, together with the fact that the effective field equations of the theory have a very simple structure and a more immediate physical interpre- tation compared to the equations of the RM method (in which the introduction of several coupled auxiliary fields is necessary), makes this effective field theory particu- larly interesting to all those applications in which d$_{0}$ > 1 or else the number of parameters of the model is high. Of course one has to pay such an advantage with the impos- sibility to get exact results out of the P region. However, as we shall show in this paper, also in the other regions of the phase diagram the theory succeeds in giving effective approximations allowing us to obtain important insights on the frozen states, even if we do not have a direct ac- I. INTRODUCTION