Explicit generation of the branching tree of states in spin glasses

We present a numerical method to generate explicit realizations of the tree of states in mean-field spin glasses. The resulting study illuminates the physical meaning of the full replica symmetry breaking solution and provides detailed information on the structure of the spin-glass phase. A cavity approach ensures that the method is self-consistent and permits the evaluation of sophisticated observables, such as correlation functions. We include an example application to the study of finite-size effects in single-sample overlap probability distributions, a topic that has attracted considerable interest recently.Comment: Version accepted for publication in JSTA

Parisi States in a Heisenberg Spin-Glass Model in Three Dimensions

We have studied low-lying metastable states of the $\pm J$ Heisenberg model in two ($d=2$) and three ($d=3$) dimensions having developed a hybrid genetic algorithm. We have found a strong evidence of the occurrence of the Parisi states in $d=3$ but not in $d=2$. That is, in $L^d$ lattices, there exist metastable states with a finite excitation energy of $\Delta E \sim O(J)$ for $L \to \infty$, and energy barriers $\Delta W$ between the ground state and those metastable states are $\Delta W \sim O(JL^{\theta})$ with $\theta > 0$ in $d=3$ but with $\theta < 0$ in $d=2$. We have also found droplet-like excitations, suggesting a mixed scenario of the replica-symmetry-breaking picture and the droplet picture recently speculated in the Ising SG model.Comment: 4 pages, 6 figure

On the origin of ultrametricity

In this paper we show that in systems where the probability distribution of the the overlap is non trivial in the infinity volume limit, the property of ultrametricity can be proved in general starting from two very simple and natural assumptions: each replica is equivalent to the others (replica equivalence or stochastic stability) and all the mutual information about a pair of equilibrium configurations is encoded in their mutual distance or overlap (separability or overlap equivalence).Comment: 13 pages, 1 figur

The mean field theory of spin glasses: the heuristic replica approach and recent rigorous results

The mathematically correct computation of the spin glasses free energy in the infinite range limit crowns 25 years of mathematic efforts in solving this model. The exact solution of the model was found many years ago by using a heuristic approach; the results coming from the heuristic approach were crucial in deriving the mathematical results. The mathematical tools used in the rigorous approach are quite different from those of the heuristic approach. In this note we will review the heuristic approach to spin glasses in the light of the rigorous results; we will also discuss some conjectures that may be useful to derive the solution of the model in an alternative way.Comment: 12 pages, 1 figure; lecture at the Flato Colloquia Day, Thursday 27 November, 200

On the Four-Dimensional Diluted Ising Model

In this letter we show strong numerical evidence that the four dimensional Diluted Ising Model for a large dilution is not described by the Mean Field exponents. These results suggest the existence of a new fixed point with non-gaussian exponents.Comment: 9 pages. compressed ps-file (uufiles

Violation of the Fluctuation Dissipation Theorem in Finite Dimensional Spin Glasses

We study the violation of the fluctuation-dissipation theorem in the three and four dimensional Gaussian Ising spin glasses using on and off equilibrium simulations. We have characterized numerically the function X(C) that determine the violation and we have studied its scaling properties. Moreover we have computed the function x(C) which characterize the breaking of the replica symmetry directly from equilibrium simulations. The two functions are numerically equal and in this way we have established that the conjectured connection between the violation of fluctuation dissipation theorem in the off-equilibrium dynamics and the replica symmetry breaking at equilibrium holds for finite dimensional spin glasses. These results point to a spin glass phase with spontaneously broken replica symmetry in finite dimensional spin glasses.Comment: 13 pages, 4 figures, also available at http://chimera.roma1.infn.it/index_papers_complex.htm

The marginally stable Bethe lattice spin glass revisited

Bethe lattice spins glasses are supposed to be marginally stable, i.e. their equilibrium probability distribution changes discontinuously when we add an external perturbation. So far the problem of a spin glass on a Bethe lattice has been studied only using an approximation where marginally stability is not present, which is wrong in the spin glass phase. Because of some technical difficulties, attempts at deriving a marginally stable solution have been confined to some perturbative regimes, high connectivity lattices or temperature close to the critical temperature. Using the cavity method, we propose a general non-perturbative approach to the Bethe lattice spin glass problem using approximations that should be hopeful consistent with marginal stability.Comment: 23 pages Revised version, hopefully clearer that the first one: six pages longe

4D Spin Glasses in Magnetic Field Have a Mean Field like Phase

By using numerical simulations we show that the 4D $J=\pm 1$ Edwards Anderson spin glass in magnetic field undergoes a mean field like phase transition. We use a dynamical approach: we simulate large lattices (of volume $V$) and work out the behavior of the system in limit where both $t$ and $V$ go to infinity, but where the limit $V \to \infty$ is taken first. By showing that the dynamic overlap $q$ converges to a value smaller than the static one we exhibit replica symmetry breaking. The critical exponents are compatible with the ones obtained by mean field computations.Comment: Physrev format, 5 ps figures include

Replica equivalence in the Edwards-Anderson model

After introducing and discussing the "link-overlap" between spin configurations we show that the Edwards-Anderson model has a "replica-equivalent" quenched equilibrium state, a property introduced by Parisi in the description of the mean-field spin-glass phase which generalizes ultrametricity. Our argument is based on the control of fluctuations through the property of stochastic stability and works for all the finite-dimensional spin-glass models.Comment: 12 pages, few remarks added. To appear in Journal of Physics A: Mathematical and Genera

Small Window Overlaps Are Effective Probes of Replica Symmetry Breaking in 3D Spin Glasses

We compute numerically small window overlaps in the three dimensional Edwards Anderson spin glass. We show that they behave in the way implied by the Replica Symmetry Breaking Ansatz, that they do not qualitatively differ from the full volume overlap and do not tend to a trivial function when increasing the lattice volume. On the contrary we show they are affected by small finite volume effects, and are interesting tools for the study of the features of the spin glass phase.Comment: 9 pages plus 5 figure