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

Abstract

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

22 pages, 10 figures; minor changes with respect to v2

References in corpus (12)

• Gibbs States and the Set of Solutions of Random Constraint Satisfaction Problems
• Phase transition in the three dimensional Heisenberg spin glass: Finite-size scaling analysis
• The spin glass transition of the three dimensional Heisenberg spin glass
• Threshold values, stability analysis and high-q asymptotics for the coloring problem on random graphs
• Potts Glass on Random Graphs
• High q-State Clock Spin Glasses in Three Dimensions and the Lyapunov Exponents of Chaotic Phases and Chaotic Phase Boundaries
• Odd q-State Clock Spin-Glass Models in Three Dimensions, Asymmetric Phase Diagrams, and Multiple Algebraically Ordered Phases
• Statistical physics of nonlinear wave interaction
• One-dimensional infinite component vector spin glass with long-range interactions
• Generalised Bose-Einstein phase transition in large-$m$ component spin glasses
• Nonlinear $XY$ and $p$-clock models on sparse random graphs: mode-locking transition of localized waves
• Inherent Structures in m-component Spin Glasses

Cited by in corpus (3)

• Phase Transitions between Different Spin-Glass Phases and between Different Chaoses in Quenched Random Chiral Systems
• An improved Belief Propagation algorithm finds many Bethe states in the random field Ising model on random graphs
• Critical properties of disordered XY model on sparse random graphs