Viana-Bray Model Research Articles

Free-energy barriers in the Sherrington-Kirkpatrick model.

The free-energy landscape of the Sherrington-Kirkpatrick (SK) Ising spin glass is simple in the framework of the Thouless-Anderson-Palmer (TAP) equationsas each solution (which are minima of the free energy) has associated with it a nearby index-one saddle point. The free-energy barrier to escape the minimum is just the difference between the saddle point free energy and that at its associated minimum. This difference is calculated for the states with free energies f>f_{c}. It is very small for these states, decreasing as 1/N^{2}, where N is the number of spins in the system. These states are not marginally stable. We argue that such small barriers are why numerical studies never find these states when N is large. Instead, the states that are found are those that have marginal stability. For them the barriers are at least of O(1). f_{c} is the free energy per spin below which the states develop broken replica-symmetry-like overlaps with each other. In the regime f<f_{c} we can only offer some possibilities based around scaling arguments. One of these suggest that the barriers might become as large as N^{1/3}. That might be consistent with recent numerical studies on the Viana-Bray model, which were at variance with the expectations of Cugliandolo and Kurchan for the SK model.

Strong ergodicity breaking in aging of mean-field spin glasses

Out-of-equilibrium relaxation processes show aging if they become slower as time passes. Aging processes are ubiquitous and play a fundamental role in the physics of glasses and spin glasses and in other applications (e.g., in algorithms minimizing complex cost/loss functions). The theory of aging in the out-of-equilibrium dynamics of mean-field spin glass models has achieved a fundamental role, thanks to the asymptotic analytic solution found by Cugliandolo and Kurchan. However, this solution is based on assumptions (e.g., the weak ergodicity breaking hypothesis) which have never been put under a strong test until now. In the present work, we present the results of an extraordinary large set of numerical simulations of the prototypical mean-field spin glass models, namely the Sherrington-Kirkpatrick and the Viana-Bray models. Thanks to a very intensive use of graphics processing units (GPUs), we have been able to run the latter model for more than [Formula: see text] spin updates and thus safely extrapolate the numerical data both in the thermodynamical limit and in the large times limit. The measurements of the two-times correlation functions in isothermal aging after a quench from a random initial configuration to a temperature [Formula: see text] provides clear evidence that, at large times, such correlations do not decay to zero as expected by assuming weak ergodicity breaking. We conclude that strong ergodicity breaking takes place in mean-field spin glasses aging dynamics which, asymptotically, takes place in a confined configurational space. Theoretical models for the aging dynamics need to be revised accordingly.

Metastable minima of the Heisenberg spin glass in a random magnetic field

We have studied zero-temperature metastable minima in classical m-vector component spin glasses in the presence of m-component random fields for two models, the Sherrington-Kirkpatrick (SK) model and the Viana-Bray (VB) model. For the SK model we have calculated analytically its complexity (the log of the number of minima) for both the annealed case where one averages the number of minima before taking the log and the quenched case where one averages the complexity itself, both for fields above and below the de Almeida-Thouless (AT) field, which is finite for m>2. We have done numerical quenches starting from a random initial state (infinite temperature state) by putting spins parallel to their local fields until there is no further decrease of the energy and found that in zero field it always produces minima that have zero overlap with each other. For the m=2 and m=3 cases in the SK model the final energy reached in the quench is very close to the energy E_{c} at which the overlap of the states would acquire replica symmetry-breaking features. These minima have marginal stability and will have long-range correlations between them. In the SK limit we have analytically studied the density of states ρ(λ) of the Hessian matrix in the annealed approximation. Despite the fact that in the presence of a random field there are no continuous symmetries, the spectrum extends down to zero with the usual sqrt[λ] form for the density of states for fields below the AT field. However, when the random field is larger than the AT field, there is a gap in the spectrum, which closes up as the AT field is approached. The VB model behaves differently and seems rather similar to studies of the three-dimensional Heisenberg spin glass in a random vector field.

Self-Organized Criticality in Glassy Spin Systems Requires a Diverging Number of Neighbors

We investigate the conditions required for general spin systems with frustration and disorder to display self-organized criticality, a property which so far has been established only for the fully connected infinite-range Sherrington-Kirkpatrick Ising spin-glass model [Phys. Rev. Lett. 83, 1034 (1999)]. Here, we study both avalanche and magnetization jump distributions triggered by an external magnetic field, as well as internal field distributions in the short-range Edwards-Anderson Ising spin glass for various space dimensions between 2 and 8, as well as the fixed-connectivity mean-field Viana-Bray model. Our numerical results, obtained on systems of unprecedented size, demonstrate that self-organized criticality is recovered only in the strict limit of a diverging number of neighbors and is not a generic property of spin-glass models in finite space dimensions.

Spin glasses in the nonextensive regime

Spin systems with long-range interactions are "nonextensive" if the strength of the interactions falls off sufficiently slowly with distance. It has been conjectured for ferromagnets and, more recently, for spin glasses that, everywhere in the nonextensive regime, the free energy is exactly equal to that for the infinite range model in which the characteristic strength of the interaction is independent of distance. In this paper we present the results of Monte Carlo simulations of the one-dimensional long-range spin glasses in the nonextensive regime. Using finite-size scaling, our results for the transition temperatures are consistent with this prediction. We also propose and provide numerical evidence for an analogous result for dilutedlong-range spin glasses in which the coordination number is finite, namely, that the transition temperature throughout the nonextensive regime is equal to that of the infinite-range model known as the Viana-Bray model.

Notes on the Polynomial Identities in Random Overlap Structures

In these notes we review first in some detail the concept of random overlap structure (ROSt) applied to fully connected and diluted spin glasses. We then sketch how to write down the general term of the expansion of the energy part from the Boltzmann ROSt (for the Sherrington-Kirkpatrick model) and the corresponding term from the RaMOSt, which is the diluted extension suitable for the Viana-Bray model. From the ROSt energy term, a set of polynomial identities (often known as Aizenman-Contucci or AC relations) is shown to hold rigorously at every order because of a recursive structure of these polynomials that we prove. We show also, however, that this set is smaller than the full set of AC identities that is already known. Furthermore, when investigating the RaMOSt energy for the diluted counterpart, at higher orders, combinations of such AC identities appear, ultimately suggesting a crucial role for the entropy in generating these constraints in spin glasses.

Thermodynamics of the Lévy spin glass

We investigate the Lévy glass, a mean-field spin-glass model with power-law distributed couplings characterized by a divergent second moment. By combining extensively many small couplings with a spare random backbone of strong bonds the model is intermediate between the Sherrington-Kirkpatrick and the Viana-Bray models. A truncated version where couplings smaller than some threshold ε are neglected can be studied within the cavity method developed for spin glasses on locally treelike random graphs. By performing the limit ε→0 in a well-defined way we calculate the thermodynamic functions within replica symmetry and determine the de Almeida-Thouless line in the presence of an external magnetic field. Contrary to previous findings we show that there is no replica-symmetric spin-glass phase. Moreover we determine the leading corrections to the ground-state energy within one-step replica symmetry breaking. The effects due to the breaking of replica symmetry appear to be small in accordance with the intuitive picture that a few strong bonds per spin reduce the degree of frustration in the system.

Effective field theory for models defined over small-world networks: First- and second-order phase transitions

We present an effective field theory to analyze, in a very general way, models defined over small-world networks. Even if the exactness of the method is limited to the paramagnetic regions and to some special limits, it provides, yielding a clear and immediate (also in terms of calculation) physical insight, the exact critical behavior and the exact critical surfaces and percolation thresholds. The underlying structure of the nonrandom part of the model-i.e., the set of spins filling up a given lattice L0 of dimension d_{0} and interacting through a fixed coupling J0 -is exactly taken into account. When J_{0}> or = 0 , the small-world effect gives rise, as is known, to a second-order phase transition that takes place independently of the dimension d_{0} and of the added random connectivity c . When J0<0 , a different and novel scenario emerges in which, besides a spin-glass transition, multiple first- and second-order phase transitions may take place. As immediate analytical applications we analyze the Viana-Bray model (d_{0}=0) , the one-dimensional chain (d_{0}=1) , and the spherical model for arbitrary d_{0} .

On the mean-field spin glass transition

In this paper we analyze two main prototypes of disordered mean-field systems, namely the Sherrington-Kirkpatrick (SK) and the Viana-Bray (VB) models, to show that, in the framework of the cavity method, the transition from the annealed regime to a broken replica symmetry phase can be thought of as the failure of the saturability property (detailed explained along the paper) of the overlap fluctuations which act as the order parameters of the theory. We show furthermore how this coincides with the lacking of the commutativity of the infinite volume limit with respect to a, suitably chosen, vanishing perturbing field inducing the transition as prescribed by standard statistical mechanics. This is another step towards a complete theory of disordered systems. As a well known consequence it turns out that the annealed and the replica symmetric regions must coincide, implying that the averaged overlap is zero in this phase. Within our framework the finding of the values of the critical point for the SK and line for the VB becomes available straightforwardly and the method is of a large generality and applicable to several other mean field models

Some Observations for Mean-Field Spin Glass Models

We obtain bounds to show that the pressure of a two-body, mean-field spin glass is a Lipschitz function of the underlying distribution of the random coupling constants, with respect to a particular semi-norm. This allows us to re-derive a result of Carmona and Hu, on the universality of the SK model, by a different proof, and to generalize this result to the Viana–Bray model. We also prove another bound, suitable when the coupling constants are not independent, which is what is necessary if one wants to consider “canonical” instead of “grand canonical” versions of the SK and Viana–Bray models. Finally, we review Viana–Bray type models, using the language of Lévy processes, which is natural in this context.

Random quantum Ising chains with competing interactions

In this paper we discuss the criticality of a quantum Ising spin chain with competing random ferromagnetic and antiferromagnetic couplings. Quantum fluctuations are introduced via random local transverse fields. First we consider the chain with couplings between first and second neighbors only and then generalize the study to a quantum analog of the Viana-Bray model, defined on a small world random lattice. We use the Dasgupta-Ma decimation technique, both analytically and numerically, and focus on the scaling of the lattice topology, whose determination is necessary to define any infinite disorder transition beyond the chain. In the first case, at the transition the model renormalizes towards the chain, with the infinite disorder fixed point described by Fisher. This corresponds to the irrelevance of the competition induced by the second neighbors couplings. As opposed to this case, this infinite disorder transition is found to be unstable towards the introduction of an arbitrary small density of long range couplings in the small world models.

Random multi-overlap structures for optimization problems

We extend to the K-SAT and p-XOR-SAT optimization problems the results recently achieved, by introducing the concept of random multi-overlap structure, for the Viana-Bray model of diluted mean field spin glass. More precisely we can prove a generalized bound and an extended variational principle for the free energy per site in the thermodynamic limit. Moreover a trial function implementing ultrametric breaking of replica symmetry is exhibited. The ultrametric structure exhibits the same factorization property as the optimal structures for the Viana-Bray model and the Sherrington-Kirkpatrick nondiluted model.

Long-range frustration in finite connectivity spin glasses: a mean-field theory and its application to the random K-satisfiability problem

A mean-field theory of long-range frustration is constructed for spin glass systems with quenched randomness of vertex–vertex connections and of spin–spin coupling strengths. This theory is applied to a spin glass model of the random K-satisfiability (K-SAT) problem (K=2 or K=3). The satisfiability transition in a random 2-SAT formula occurs when the clauses-to-variables ratio α approaches αc(2)=1. However, long-range frustration among unfrozen variable nodes builds up only when α>αR(2)=4.4588. For the random 3-SAT problem, we find a long-range frustrated mean-field solution when α>αR(3)=4.1897. The long-range frustration order parameter R of this solution jumps from zero to a finite positive value at αR(3), while the energy density increases only gradually from zero as a function of α. The SAT–UNSAT transition point of this solution is lower than the value of αc(3)=4.267 obtained by the survey propagation algorithm. Two possible reasons for this discrepancy are suggested. The zero-temperature phase diagram of the ±J Viana–Bray model is also determined, which is identical to that of the random 2-SAT problem. The predicted phase transition between a non-frustrated and a long-range frustrated spin glass phase might also be observable in real materials at a finite temperature.

Random Multi-Overlap Structures and Cavity Fields in Diluted Spin Glasses

We introduce the concept of Random Multi-Overlap Structure (RaMOSt) as a generalization of the one introduced by Aizenman, Sims and Starr for non-diluted spin glasses. We use such concept to find generalized bounds for the free energy of the Viana-Bray model of diluted spin glasses and to formulate and prove the Extended Variational Principle that implicitly provides the free energy of the model. Then we exhibit a theorem for the limiting RaMOSt, analogous to the one found by F. Guerra for the Sherrington–Kirkpatrick model, that describes some stability properties of the model. Last, we show how our technique can be used to prove the existence of thermodynamic limit of the free energy. The present work paves the way to a revisited Parisi theory for diluted spin systems.

THE KAC LIMIT FOR DILUTED SPIN GLASSES

We study diluted spin glass models in arbitrary dimension, where each spin interacts with a finite number of other spins chosen at random with a probability decaying to zero over some distance γ-1. For systems with pairwise interactions we show that the infinite-volume free energy converges to that of the mean-field Viana–Bray model,1 in the Kac limit γ→0. For p-spin like models we get only one bound: the free-energy is bounded from above by the one of the mean-field diluted p-spin.

On the entropy of the Viana-Bray model

The entropy of the Viana-Bray model at zero temperature and external field is calculated within the solution which takes into account only delta functions for the global order parameter P(h). It is shown that such solution is unsatisfactory both from the viewpoint of stability analysis and for not reproducing the well known Sherrington-Kirkpatrick result in the large connectivity limit thus pointing out the relevance of considering solutions with continuous part in P(h) for such model and possibly related models.

Ageing in spin-glasses in three, four and infinite dimensions

The spin update engine (SUE) machine is used to extend, by a factor of 1000, the timescale of previous studies of the ageing, out-of-equilibrium dynamics of the Edwards–Anderson model with binary couplings, on large lattices (L = 60). The correlation function, C(t + tw, tw), tw being the time elapsed under a quench from high-temperature, follows nicely a slightly-modified power law for t > tw. Very small (logarithmic), yet clearly detectable deviations from the full-ageing t/tw scaling can be observed. Furthermore, the t < tw data show clear indications of the presence of more than one time sector in the ageing dynamics. Similar results are found in four dimensions, but a rather different behaviour is obtained in the infinite-dimensional z = 6 Viana–Bray model. Most surprisingly, our results in infinite dimensions seem incompatible with dynamical ultrametricity. A detailed study of the link correlation function is presented, suggesting that its ageing properties are the same as for the spin correlation function.

The High Temperature Region of the Viana–Bray Diluted Spin Glass Model

In this paper, we study the high temperature or low connectivity phase of the Viana-Bray model. This is a diluted version of the well known Sherrington-Kirkpatrick mean field spin glass. In the whole replica symmetric region, we obtain a complete control of the system, proving annealing for the infinite volume free energy, and a central limit theorem for the suitably rescaled fluctuations of the multi-overlaps. Moreover, we show that free energy fluctuations, on the scale 1/N, converge in the infinite volume limit to a non-Gaussian random variable, whose variance diverges at the boundary of the replica-symmetric region. The connection with the fully connected Sherrington-Kirkpatrick model is discussed.

Replica bounds for optimization problems and diluted spin systems

In this paper we generalize to the case of diluted spin models and random combinatorial optimization problems a technique recently introduced by Guerra (cond-mat/0205123) to prove that the replica method generates variational bounds for disordered systems. We analyze a family of models that includes the Viana–Bray model, the diluted p-spin model or random XOR-SAT problem, and the random K-SAT problem, showing that the replica/cavity method, at the various levels of approximation, provides systematic schemes to obtain lower bounds of the free-energy at all temperatures and of the ground state energy. In the case of K-SAT and XOR-SAT it thus gives upper bounds of the satisfiability threshold. Our analysis underlines deep connections with the cavity method which are not evident in the long range case.

Cluster expansions in dilute systems: applications to satisfiability problems and spin glasses.

We develop a systematic cluster expansion for dilute systems in the highly dilute phase. We first apply it to the calculation of the entropy of the K-satisfiability problem in the satisfiable phase. We derive a series expansion in the control parameter, the average connectivity, that is identical to the one obtained by using the replica approach with a replica symmetric (RS) ansatz, when the order parameter is calculated via a perturbative expansion in the control parameter. As a second application we compute the free energy of the Viana-Bray model in the paramagnetic phase. The cluster expansion allows one to compute finite-size corrections in a simple manner, and these are particularly important in optimization problems. Importantly enough, these calculations prove the exactness of the RS ansatz below the percolation threshold, and might require its revision between this and the easy-to-hard transition.