11 1.2 1.4 1.6 1.8 2 0 1 2 3 4 5 T ˜ χ 1.4 1.5 1.6 1.7 1.8 0 0.2 0.4 0.6 T U q 0 2 4 6 8 10 0.4 0.5 0.6 0.7 0.8 P SG c β c a FIG. 12: (Color online) Results for the d$_{0}$ = 2 Edwards- Anderson model with long-range connections with a = J = 1 at c = 1. The top plot shows values of the Binder cumulant as a function of temperature and the bottom plot shows the temperature dependence of the magnetic susceptibility. The simulations were done for L = 8 ( ○ ), L = 16 ( × ), L = 32 (.) and L = 64 (+). In the top plot the vertical line is the theoretical prediction of the spin-glass critical temperature and the lines are polynomial fits of the Binder cumulants for each system size. In the lower plot the line is the theoretical prediction for the magnetic susceptibility. FIG. 11: (Color online) Phase diagram for the d$_{0}$ = 2 Edwards-Anderson model with long-range connections and coupling constant J = 1 . The curve is the value of β c a for the SG phase transition as obtained from equation (26) and our numerical results for the pure Ising model susceptibility for a system of side L = 128. T ( SG ) c = 1 . 6057. In Fig. 12 (top panel) we obtained crossings of the overlap parameter Binder cumulant for system sizes L = 8 , 16 and 32. The statistical error of the Binder cumulant, as expected, increases with system size and we excluded from the Binder cumulant intersection calculations the data for L = 64. Our numerical esti- mate of the critical temperature is T ( SG ) c = 1 . 6(1) still consistent with the expected critical temperature. Since, for an Ising model with zero coupling and zero external magnetic field ˜ χ$_{0}$ = 1, the prediction for the magnetization and the magnetic susceptibility is < | m | > = 0 and ˜ χ = [1 − c tanh( βJ )] − $^{1}$. In Fig. 12 (lower panel) we plot the magnetic susceptibility simu- lation results together with this theoretical prediction. We see that in the P phase the numerical estimates of the susceptibility approach the theoretical curve as the system size increases. theory is not exact in this case, there is a reasonable agreement between the predicted field dependence of the magnetization and susceptibility specially above the crit- ical temperature. For the case of J/J$_{0}$ = − 20 and c = 1 . 4 at d$_{0}$ = 1 we verified the existence of two second order phase transitions. A good agreement between the the- oretical temperature dependence of the susceptibility in the P phases and simulation results was observed. The location of the high temperature critical point was ex- plicitly verified using the cumulant crossing technique. For a one-dimensional model with J$_{0}$ = − 0 . 9, J = 0 . 5 and c = 10, we observed a first order phase transition as predicted by the theory but at a lower temperature. This may be a consequence of the fact that in this case, unlike the second-order phase transitions cases, the zero magnetization solution does not become unstable at the transition temperature and - consistently with the fact that the theory does not give exact results out of the pure P regions - the critical point obtained as the point where the two free-energy values equal is not exactly pre- dicted. Quite interestingly, we find that also the P-SG phase transition turns out to be first-order and, further- more, its critical point seems to coincide with the theo- retical one. In this work we have compared the predictions of an effective field theory for several Ising models on small world networks with Monte Carlo simulation results. All the predictions of the theory, where it is known to be exact, namely in the P region at zero external field, were confirmed by the simulation results. In particular we have checked the critical surfaces ( T, c ), the temperature dependence of the susceptibility, and the spin-spin corre- lation function. Furthermore, for the simplest version of the model, i.e. , the Viana-Bray model (where d$_{0}$ = 0), we stud- ied the effect of a non-zero external field. Although the V. CONCLUSIONS