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Register a new userThe connection between statics and dynamics of spin glasses
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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 $T_c$ and the growth of correlations is observed. The spatial decay of the correlations at distances less than the dynamic correlation length $xi(t)$ 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 $xi(t) propto t^{1/z(T)}$ and find that it is compatible with the mean-field value of the critical dynamical exponent for short range spin glasses.
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We compare the critical behavior of the short-range Ising spin glass with a spin glass with long-range interactions which fall off as a power sigma of the distance. We show that there is a value of sigma of the long-range model for which the critical behavior is very similar to that of the short-range model in four dimensions. We also study a value of sigma for which we find the critical behavior to be compatible with that of the three dimensional model, though we have much less precision than in the four-dimensional case.
The statics-dynamics correspondence in spin glasses relate non-equilibrium results on large samples (the experimental realm) with equilibrium quantities computed on small systems (the typical arena for theoretical computations). Here we employ statics-dynamics equivalence to study the Ising spin-glass critical behavior in three dimensions. By means of Monte Carlo simulation, we follow the growth of the coherence length (the size of the glassy domains), on lattices too large to be thermalized. Thanks to the large coherence lengths we reach, we are able to obtain accurate results in excellent agreement with the best available equilibrium computations. To do so, we need to clarify the several physical meanings of the dynamic exponent close to the critical temperature.
We study zero-temperature, stochastic Ising models sigma(t) on a d-dimensional cubic lattice with (disordered) nearest-neighbor couplings independently chosen from a distribution mu on R and an initial spin configuration chosen uniformly at random. Given d, call mu type I (resp., type F) if, for every x in the lattice, sigma(x,t) flips infinitely (resp., only finitely) many times as t goes to infinity (with probability one) --- or else mixed type M. Models of type I and M exhibit a zero-temperature version of ``local non-equilibration. For d=1, all types occur and the type of any mu is easy to determine. The main result of this paper is a proof that for d=2, plus/minus J models (where each coupling is independently chosen to be +J with probability alpha and -J with probability 1-alpha) are type M, unlike homogeneous models (type I) or continuous (finite mean) mus (type F). We also prove that all other noncontinuous disordered systems are type M for any d greater than or equal to 2. The plus/minus J proof is noteworthy in that it is much less ``local than the other (simpler) proof. Homogeneous and plus/minus J models for d greater than or equal to 3 remain an open problem.
We present a mean field model for spin glasses with a natural notion of distance built in, namely, the Edwards-Anderson model on the diluted D-dimensional unit hypercube in the limit of large D. We show that finite D effects are strongly dependent on the connectivity, being much smaller for a fixed coordination number. We solve the non trivial problem of generating these lattices. Afterwards, we numerically study the nonequilibrium dynamics of the mean field spin glass. Our three main findings are: (i) the dynamics is ruled by an infinite number of time-sectors, (ii) the aging dynamics consists on the growth of coherent domains with a non vanishing surface-volume ratio, and (iii) the propagator in Fourier space follows the p^4 law. We study as well finite D effects in the nonequilibrium dynamics, finding that a naive finite size scaling ansatz works surprisingly well.
An Ashkin-Teller neural network, allowing for two types of neurons is considered in the case of low loading as a function of the strength of the respective couplings between these neurons. The storage and retrieval of embedded patterns built from the two types of neurons, with different degrees of (in)dependence is studied. In particular, thermodynamic properties including the existence and stability of Mattis states are discussed. Furthermore, the dynamic behaviour is examined by deriving flow equations for the macroscopic overlap. It is found that for linked patterns the model shows better retrieval properties than a corresponding Hopfield model.
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