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arXiv:cond-mat/0009475 · Europhysics Letters (EPL)

Finite-size scaling at the dynamical transition of the mean-field 10-state Potts glass

C Brangian, Walter Kob, Kurt Binder

Published Mar 1, 2001• 28 citations
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Abstract

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.

DOI
10.1209/epl/i2001-00216-4

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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.
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.
Finite-size effects in the mean-field Ising spin glass and the mean-field three-state Potts glass are investigated by Monte Carlo simulations. In the thermodynamic limit, each model is known to exhibit a continuous phase transition into the ordered state with a full and a one-step replica-symmetry breaking (RSB), respectively. In the Ising case, the Binder parameter g calculated for various finite sizes remains positive at any temperature and crosses at the transition point, while in the Potts case g develops a negative dip without showing a crossing in the g>0 region. By contrast, non-self-averaging parameters always remain positive and show a clear crossing at the transition temperature in both cases. Our finding suggests that care should be taken in interpreting the numerical data of the Binder parameter, particularly when the system exhibits a one-step-like RSB.
We study the critical behavior of the short-range ppp-state Potts spin glass in three and four dimensions using Monte Carlo simulations. In three dimensions, for p=3p=3p=3, a finite-size scaling analysis of the correlation length shows clear evidence of a transition to a spin-glass phase at ${T}_{\mathrm{c}}\ensuremath{\simeq}0.273$ for a Gaussian distribution of interactions and ${T}_{\mathrm{c}}\ensuremath{\simeq}0.377$ for a $\ifmmode\pm\else\textpm\fi{}J$ (bimodal) distribution. These results indicate that the lower critical dimension of the three-state Potts glass is below 3. By contrast, the correlation length of the ten-state (p=10)(p=10)(p=10) Potts glass in three dimensions remains small even at very low temperatures and thus shows no sign of a transition. In four dimensions we find that the p=3p=3p=3 Potts glass with Gaussian interactions has a spin-glass transition at ${T}_{\mathrm{c}}\ensuremath{\simeq}0.536$.
Abstract We present results of numerical simulations on a one-dimensional Ising spin glass with long-range interactions. Parameters of the model are chosen such that it is a proxy for a short-range spin glass above the upper critical dimension (i.e. in the mean-field regime). The system is quenched to a temperature well below the transition temperature and the growth of correlations is observed. The spatial decay of the correlations at distances less than the dynamic correlation length agrees quantitatively with the predictions of a static theory, the ‘metastate’, evaluated according to the replica symmetry breaking (RSB) theory. We also compute the dynamic exponent z ( T ) defined by and find that it is compatible with the mean-field value of the critical dynamical exponent for short range spin glasses.
For small lattices with linear dimension L ranging from L=3 to L=8 we obtain the distribution function P(q) of the overlap q between two real replicas of the three-state Potts-glass model with symmetric nearest-neighbor interaction with a Gaussian distribution. A finite-size-scaling analysis suggests a zero-temperature transition to occur with an exponentially diverging correlation length ${\ensuremath{\xi}}_{\mathrm{SG}}$\ensuremath{\sim}exp(C/${\mathit{T}}^{\mathrm{\ensuremath{\sigma}}}$). This implies that d=3 is the lower critical dimension.