Approximating the XY model on a random graph with a q-state clock model

摘要

Numerical simulations of spin glass models with continuous variables set the problem of a reliable but efficientdiscretization of such variables. In particular, the main question is how fast physical observables computed in thediscretized model converge toward the ones of the continuous model when the number of states of the discretizedmodel increases. We answer this question for the XY model and its discretization, the q-state clock model, inthe mean-field setting provided by random graphs. It is found that the convergence of physical observables isexponentially fast in the number q of states of the clock model, so allowing a very reliable approximation of theXY model by using a rather small number of states. Furthermore, such an exponential convergence is found tobe independent from the disorder distribution used. Only at T = 0, the convergence is slightly slower (stretchedexponential). Thanks to the analytical solution to the q-state clock model, we compute accurate phase diagrams inthe temperature versus disorder strength plane. We find that, at zero temperature, spontaneous replica symmetrybreaking takes place for any amount of disorder, even an infinitesimal one. We also study the one step of replicasymmetry breaking (1RSB) solution in the low-temperature spin glass phase.