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Physica A Statistical Mechanics and its Applications

Dynamical susceptibility from simulations of a mean field Potts glass

Claudio Brangian

Published Feb 21, 2004• 3 citations
Publisher
DOI
10.1016/j.physa.2003.12.057

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We investigate by means of Monte Carlo simulations the fully connected p -state Potts model for different system sizes in order to see how the static and dynamic properties of a finite model compare with the, exactly known, behaviour of the system in the thermodynamic limit. Using p = 10 we are able to study the equilibrium dynamics for system sizes as large as N = 2560. We find that the static quantities, such as the energy, the entropy, the spin glass susceptibility as well as the distribution of the order parameter P ( q ) show very strong finite-size effects. From P ( q ) we calculate the fourth-order cumulant g 4 ( N , T ) and the Guerra parameter G ( N , T ) and show that these quantities cannot be used to locate the static transition temperature for the system sizes investigated. Also the spin-autocorrelation function C ( t ) shows strong finite-size effects in that it does not show a plateau even for temperatures around the dynamical critical temperature T D . We show that the dependence on N and T of the α-relaxation time can be understood by means of a dynamical finite-size scaling ansatz. C ( t ) does not obey the time-temperature superposition principle for temperatures around T D , but does so for significantly lower T . Finally we study the relaxation dynamics of the individual spins and show that their dependence on time depends strongly on the chosen spin, i.e. that the system is dynamically very heterogeneous, which explains the non-exponentiality of C ( t ).
We present the results of Monte Carlo simulations of two different three-dimensional-Potts glass models with short-range random interactions. In the first model, a ± J -distribution of the bonds is chosen, in the second model a Gaussian distribution. In both cases, the first two moments of the distribution are chosen to be J 0 = −1, Δ J = +1, so that no ferromagnetic ordering of the Potts spins can occur. We find that for all temperatures investigated the spin glass susceptibility remains finite, that the spin glass order parameter remains zero, and that the specific heat has only a smooth Schottky-like peak. These results can be understood quantitatively by considering small but independent clusters of spins. Hence, we have evidence that there is no static phase transition at any nonzero temperature. Consistent with these findings, only very minor size effects are observed, which implies that all correlation lengths of the models remain very short. We also compute for both models the time auto-correlation function C ( t ) of the Potts spins. While in the Gaussian model C ( t ) shows a smooth uniform decay, the correlator for the ± J model has several distinct steps. These steps correspond to the breaking of bonds in small clusters of ferromagnetically coupled spins (dimers, trimers, etc). The relaxation times follow simple Arrhenius laws, with activation energies that are readily interpreted within the cluster picture, giving evidence that the system does not have a dynamic transition at a finite temperature. Hence we find that for the present models, all the transitions known for the mean-field version of the model are completely wiped out. Finally, we also determine the time auto-correlation functions of individual spins, and show that the system is dynamically very heterogeneous.
We use Monte Carlo simulations to study the static and dynamical properties of a Potts glass with infinite-range Gaussian-distributed exchange interactions for a broad range of temperature and system size up to N = 2560 spins. The results are compatible with a critical divergence of the relaxation time τ at the theoretically predicted dynamical transition temperature T D , τ ∝ ( T − T D ) −Δ with Δ ≈ 2. For finite N a further power law at T = T D is found, τ( T = T D ) ∝ N z ⋆ with z ⋆ ≈ 1.5 and for T > T D dynamical finite-size scaling seems to hold. The order parameter distribution P ( q ) is qualitatively compatible with the scenario of a first-order glass transition as predicted from one-step replica symmetry breaking schemes.