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arXiv:cond-mat/0211195 · Journal of Physics A Mathematical and General

Statics and dynamics of the ten-state nearest-neighbour Potts glass on the simple-cubic lattice

Claudio Brangian, Walter Kob, Kurt Binder

Published Oct 15, 2003• 22 citations
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Abstract

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.

DOI
10.1088/0305-4470/36/43/012

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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 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 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 ${T}_{\mathrm{c}}\ensuremath{\simeq}0.536$.
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.
By a recursive method numerically exact free energies and correlation functions are calculated for finite two- and three-dimensional Ising lattices with bonds chosen randomly either ±J or from a gaussian distribution. While (dynamic) Monte Carlo simulations of precisely the same systems show the onset of Edwards-Anderson ordering at rather sharply defined nonzero freezing temperature Tf, the exact static calculations imply that there is no static order parameter qEA = [<Si≳2T]av at nonzero temperature. The correlation length, which would become infinite when qEA would start to become nonzero, is found at Tf to be of the order of a few lattice spacings only. As a result, the freezing-in of these models is identified as a dynamic phenomenon, and not a phase transition associated with static Edwards-Anderson order. The specific heat of the gaussian spin glass is found to vary nearly linear with temperature. On the other hand, in two-dimensional Ising systems with random fields a nonzero (Mattis-like) spin glass order is found.
The static and dynamic magnetic response of the two-dimensional Edwards-Anderson model with a nearest-neighbor Gaussian exchange distribution is investigated by Monte Carlo simulation. A plateau in the equilibrium (slowly field cooled) susceptibility MeqH\frac{{M}_{\mathrm{eq}}}{H}HMeq​​ is found, which diverges for small fields with a power law $\frac{{M}_{\mathrm{eq}}}{H}\ensuremath{\propto}{H}^{\frac{1}{\ensuremath{\Delta}}}$, $\ensuremath{\Delta}=3.5\ifmmode\pm\else\textpm\fi{}0.5$. The boundary of the plateau Hceq(T){H}_{c}^{\mathrm{eq}}(T)Hceq​(T) tends to zero temperature as ${H}_{c}^{\mathrm{eq}}(T)\ensuremath{\propto}{T}^{\ensuremath{\Delta}}$, consistent with a scaling description appropriate for a static phase transition at T=0T=0T=0, and the associate scaling function is estimated. Surprisingly the data also are consistent with a scaling representation with nonzero freezing temperatures Tf{T}_{f}Tf​, in striking similarity to experimental data, but Tf=0{T}_{f}=0Tf​=0 is shown to be the correct choice. The zero-field-cooled susceptibility starts to differ from MeqH\frac{{M}_{\mathrm{eq}}}{H}HMeq​​ below a certain critical field Hc(t){H}_{c}(t)Hc​(t), ttt being the time scale over which the field is applied. In the $H\ensuremath{-}T$ plane Hc(t){H}_{c}(t)Hc​(t) extrapolates to a time-dependent freezing temperature Tc(t){T}_{c}(t)Tc​(t), and closely resembles the de Almeida---Thouless line. A tentative interpretation of these findings is attempted by combining scaling considerations with the reorientation of correlated clusters which have the size of the correlation length describing Edwards-Anderson order. The reduction of the free-energy barriers due to the magnetic field is treated in analogy with the interface free-energy reduction of ferromagnets in random fields.
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.