Mean-field Spin Glass Research Articles

Mean-field type Ising spin glass in a diluted rare-earth intermetallic compound

We have investigated a phase transition of a long-range RKKY Ising spin glass DyxY1−xRu2Si2 by means of the dc and the ac magnetization measurements. The static nonlinear susceptibility χ2 exhibits a divergent behavior toward Tg = 1.925 K with the exponent of γ = 1.08. The characteristic relaxation time τ obtained from the ac magnetization measurements also diverges to the same Tg with the exponent of zν = 1.9 in zero static field . These results indicate the spin glass phase transition belonging to the mean-field universality class, in which the exponents of γMF and (zν)MF are 1 and 2 respectively. Moreover, we observed the divergence of τ in a finite field of 500 Oe toward Tg = 1.412 K with the exponent of zν = 2.1, being quite similar to that in zero field. It strongly suggests an existence of the spin glass phase transition in a finite field and the replica symmetry breaking (RSB) in the long-range Ising spin glass DyxY1−xRu2Si2.

New insights from one-dimensional spin glasses

The concept of replica symmetry breaking found in the solution of the mean-field Sherrington-Kirkpatrick spinglass model has been applied to a variety of problems in science ranging from biological to computational and even financial analysis. Thus it is of paramount importance to understand which predictions of the mean-field solution apply to non-mean-field systems, such as realistic short-range spin-glass models. The one-dimensional spin glass with random power-law interactions promises to be an ideal test-bed to answer this question: Not only can large system sizes—which are usually a shortcoming in simulations of high-dimensional short-range system—be studied, by tuning the power-law exponent of the interactions the universality class of the model can be continuously tuned from the mean-field to the short-range universality class. We present details of the model, as well as recent applications to some questions of the physics of spin glasses. First, we study the existence of a spin-glass state in an external field. In addition, we discuss the existence of ultrametricity in short-range spin glasses. Finally, because the range of interactions can be changed, the model is a formidable test-bed for optimization algorithms.

Large deviations of the free energy in diluted mean-field spin-glass

Sample-to-sample free-energy fluctuations in spin-glasses display a markedly different behaviour in finite-dimensional and fully connected models, namely Gaussian versus non-Gaussian. Spin-glass models defined on various types of random graphs are in an intermediate situation between these two classes of models and we investigate whether the nature of their free-energy fluctuations is Gaussian or not. It has been argued that Gaussian behaviour is present whenever the interactions are locally non-homogeneous, i.e. in most cases with the notable exception of models with fixed connectivity and random couplings . We confirm these expectations by means of various analytical results concerning the large deviations of the free energy. In particular we unveil the connection between the spatial fluctuations of the populations of fields defined at different sites of the lattice and the Gaussian nature of the free-energy fluctuations. In contrast, on locally homogeneous lattices the populations do not fluctuate over the sites and as a consequence the small deviations of the free energy are non-Gaussian and scale as in the Sherrington–Kirkpatrick model.

Predictive power of MCT: numerical testing and finite size scaling for a mean field spinglass

The aim of this paper is to test numerically the predictions from the mode-coupling theory(MCT) of the glass transition and study its finite size scaling properties in a model withan exact MCT transition, which we choose to be the fully connected randomorthogonal model. Surprisingly, some predictions are verified while others seem clearlyviolated, with inconsistent values of some MCT exponents. We show that this is dueto strong pre-asymptotic effects that disappear only in a surprisingly narrowregion around the critical point. Our study of finite size scaling (FSS) shows thatstandard theory valid for pure systems fails because of strong sample to samplefluctuations. We propose a modified form of FSS that accounts well for our results. Enpassant, we also give new theoretical insights into FSS in disordered systemsabove their upper critical dimension. Our conclusion is that the quantitativepredictions of MCT are exceedingly difficult to test even for models for whichMCT is exact. Our results highlight that some predictions are more robust thanothers. This could provide useful guidance when dealing with experimental data.

On the structure of quasi-stationary competing particle systems

We study point processes on the real line whose configurations $X$ are locally finite, have a maximum and evolve through increments which are functions of correlated Gaussian variables. The correlations are intrinsic to the points and quantified by a matrix $Q=\{q_{ij}\}_{i,j\in\mathbb{N}}$. A probability measure on the pair $(X,Q)$ is said to be quasi-stationary if the joint law of the gaps of $X$ and of $Q$ is invariant under the evolution. A known class of universally quasi-stationary processes is given by the Ruelle Probability Cascades (RPC), which are based on hierarchically nested Poisson--Dirichlet processes. It was conjectured that up to some natural superpositions these processes exhausted the class of laws which are robustly quasi-stationary. The main result of this work is a proof of this conjecture for the case where $q_{ij}$ assume only a finite number of values. The result is of relevance for mean-field spin glass models, where the evolution corresponds to the cavity dynamics, and where the hierarchical organization of the Gibbs measure was first proposed as an ansatz.

Phase diagram and large deviations in the free energy of mean-field spin glasses

We consider the probability distribution of large deviations in the spin-glass free energy for the Sherrington-Kirkpatrick mean field model, i.e. the exponentially small probability of finding a system with intensive free energy smaller than the most likely one. This result is obtained by computing $\Phi(n,T)=T \bar{Z^n}/ n$, i.e. the average value of the partition function to the power $n$ as a function of $n$. We study in full details the phase diagram of $\Phi(n,T)$ in the $(n,T)$ plane computing in particular the stability of the replica-symmetric solution. At low temperatures we compute $\Phi(n,T)$ in series of $n$ and $\tau=T_c-T$ at high orders using the standard hierarchical ansatz and confirm earlier findings on the $O(n^5)$ scaling. We prove that the $O(n^5)$ scaling is valid at all orders and obtain an exact expression for the coefficient in term of the function $q(x)$. Resumming the series we obtain the large deviations probability at all temperatures. At zero temperature the analytical prediction displays a remarkable quantitative agreement with the numerical data. A similar computation for the simpler spherical model is also performed and the connection between large and small deviations is discussed.

Real Spin Glasses Relax Slowly in the Shade of Hierarchical Trees

The Parisi solution of the mean-field spin glass has been widely accepted and celebrated. Its marginal stability in 3d and its complexity however raised the question of its relevance to real spin glasses. This paper gives a short overview of the important experimental results which could be understood within the mean-field solution. The existence of a true phase transition and the particular behaviour of the susceptibility below the freezing temperature, predicted by the theory, are clearly confirmed by the experimental results. The behaviour of the complex order parameter and of the Fluctuation Dissipation ratio are in good agreement with results of spontaneous noise measurements. The very particular ultrametric symmetry, the key feature of the theory, provided us with a simple description of the rejuvenation and memory effects observed in experiment. Finally, going a step beyond mean-field, the paper shortly discusses new analyses in terms of correlated domains characterized by their length scales, as well as new experiments on superspin glasses which compare well with recent theoretical simulations.

Universal structures in some mean field spin glasses and an application

We discuss a spin glass reminiscent of the random energy model (REM), which allows, in particular, to recast the Parisi minimization into a more classical Gibbs variational principle, thereby shedding some light into the physical meaning of the order parameter of the Parisi theory. As an application, we study the impact of an extensive cavity field on Derrida’s REM: Despite its simplicity, this model displays some interesting features such as ultrametricity and chaos in temperature.

Introduction to Special Issue: Statistical Mechanics on Random Structures

This Special Issue of Journal of Mathematical Physics on Statistical Mechanics on Random Structures aims to cover some recent results in the interplay of topics, such as disordered systems and, in particular, spin glasses, random graphs, and their common applications. The idea of this issue originated from an invitation of the Editor, Bruno Nachtergaele, which we enthusiastically accepted, and followed the organization by the present authors of the 2008 Young European Probabilists YEP meeting at EURANDOM Eindhoven, The Netherlands . The present issue aims to be an occasion for different scientific communities to have a common collection of new results: probabilists working on random graphs, physicists working on disordered systems, and mathematical physicists working on spin glasses. While the first community has developed along the years a detailed analysis of the topological properties of random graphs, mainly inspired by percolation, physicists first and mathematical physicists later at different levels of mathematical rigor developed the statistical mechanics analysis for systems of particles interacting over a random network. Three decades ago, statistical mechanics witnessed the introduction of disordered interactions in spin glass models. In addition to the description of condensed matter alloys, the proposed statistical mechanics formalism has proven to be an excellent method to study problems well beyond the original motivation and has indeed invaded fields as diverse as combinatorial optimization and neural networks. Interesting applications have been found in the biological, economical, and social sciences. Today, the rich mathematical structure of spin glasses is a leading research topic in probability theory. The introduction of random couplings can be seen as a first attempt to introduce random interactions in particle systems. Diluted spin glasses have moved the theory toward adding a new source of randomness in the connectivity property of the interaction network like those of Erdos-Renyi random graphs. Those are also relevant in computer science, since many random optimization problems are mapped in a natural way into the study of ground states of diluted mean-field spin glass models for example, the K-sat model has been solved within the framework of “one-step replica symmetry breaking” . Complex networks emerging in real systems found in biology, social sciences, and internet and power grids have revealed interesting properties of small world and scale free type, which led to an explosive development of the mathematical theory of random graphs. The modeling of complex networks has given rise to a large number of random graphs models, in which some correlations structure is introduced in the model, for example, the so-called “preferential attachment” type. To build a theory of interacting particles on random networks is a natural and necessary stop to be pursued in the near future. The present issue collects 20 selected contributions from about 40

Simple Glass Models and Their Quantum Annealing

We study first-order quantum phase transitions in mean-field spin glasses. We solve the quantum random energy model using elementary methods and show that at the transition the eigenstate suddenly projects onto the unperturbed ground state and that the gap between the lowest states is exponentially small in the system size. We argue that this is a generic feature of all "random first-order" models, which includes benchmarks such as random satisfiability. We introduce a two-time instanton to calculate this gap in general, and discuss the consequences for quantum annealing.

Large Deviations in the Free Energy of Mean-Field Spin Glasses

We compute analytically the probability distribution of large deviations in the spin-glass free energy for the Sherrington-Kirkpatrick mean-field model; i.e., we compute the exponentially small probability of finding a system with intensive free energy smaller than the most likely one. This result is obtained by computing the average value of the partition function to the power n as a function of n. At zero temperature this absolute prediction displays a remarkable quantitative agreement with the numerical data.

Universality of the REM for Dynamics of Mean-Field Spin Glasses

We consider a version of Glauber dynamics for a p-spin Sherrington– Kirkpatrick model of a spin glass that can be seen as a time change of simple random walk on the N-dimensional hypercube. We show that, for all p ≥ 3 and all inverse temperatures β > 0, there exists a constant γ β ,p > 0, such that for all exponential time scales, exp(γ N), with γ < γ β ,p , the properly rescaled clock process (time-change process) converges to an α-stable subordinator where α = γ/β 2 < 1. Moreover, the dynamics exhibits aging at these time scales with a time-time correlation function converging to the arcsine law of this α-stable subordinator. In other words, up to rescaling, on these time scales (that are shorter than the equilibration time of the system) the dynamics of p-spin models ages in the same way as the REM, and by extension Bouchaud’s REM-like trap model, confirming the latter as a universal aging mechanism for a wide range of systems. The SK model (the case p = 2) seems to belong to a different universality class.

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

Critical behavior of mean-field spin glasses on a dilute random graph

We provide a rigorous strategy to find the critical exponents of the overlaps for dilute spin glasses, in the absence of an external field. Such a strategy is based on the expansion of a suitably perturbed average of the overlaps, which is used in the formulation of the free energy as the difference between a cavity part and the derivative of the free energy itself, considered as a function of the connectivity of the model. We assume the validity of certain reasonable approximations, equivalent to assuming a second-order transition, e.g. that higher powers of overlap monomials are of smaller magnitude near the critical point, of which we do not provide a rigorous proof.

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.

On one property of Derrida–Ruelle cascades

We present new results about Poisson–Dirichlet point processes and Derrida–Ruelle cascades that are motivated by applications to mean-field spin glass models and, in particular, by the attempt to express Guerra's interpolation in the Sherrington–Kirkpatrick model entirely in the language of the cascades. To cite this article: D. Panchenko, M. Talagrand, C. R. Acad. Sci. Paris, Ser. I 345 (2007).

Comment on “Spherical2+pspin-glass model: An analytically solvable model with a glass-to-glass transition”

Guided by old results on simple mode-coupling models displaying glass-glass transitions, we demonstrate, through a crude analysis of the solution with one step of replica symmetry breaking (1RSB) derived by Crisanti and Leuzzi for the spherical $s+p$ mean-field spin glass [Phys. Rev. B 73, 014412 (2006)], that the phase behavior of these systems is not yet fully understood when $s$ and $p$ are well separated. First, there seems to be a possibility of glass-glass transition scenarios in these systems. Second, we find clear indications that the 1RSB solution cannot be correct in the full glassy phase. Therefore, while the proposed analysis is clearly naive and probably inexact, it definitely calls for a reassessment of the physics of these systems, with the promise of potentially interesting developments in the theory of disordered and complex systems.

Extreme value problems in random matrix theory and other disordered systems

We review some applications of central limit theorems and extreme values statistics in thecontext of disordered systems. We discuss several problems, in particular concerningrandom matrix theory and the generalization of the Tracy–Widom distribution when thedisorder has ‘fat tails’. We underline the relevance of power-law tails for directed polymersand mean-field spin glasses and we point out various open problems and conjectures onthese matters. We find that, in many instances, the assumption of Gaussian disordercannot be taken for granted.

Mean-Field Spin Glass in the Observable Representation

The state space for the N-spin mean-field (Sherrington-Kirkpatrick) spin glass - nominally an N-cube - is embedded in a low dimensional continuous space in such a way that metastable and stable phases can easily be discerned, a concept of nearness of configurations defined, and peaks in the Parisi q-parameter overlap distribution identified. The dynamical and partly hierarchical interrelation of these phases can be directly imaged.

Competitive agents in a market: Statistical physics of the minority game

A brief review is presented of the minority game, a simple frustrated many-body system stimulated by considerations of a market of competitive speculative agents. Its cooperative behaviour exhibits phase transitions and both ergodic and non-ergodic regimes. It provides novel challenges to statistical physics, reminiscent of those of mean-field spin glasses.