p-Spin model in finite dimensions and its relation to structural glasses

The p-spin spin-glass model has been studied extensively at mean-field level because of the insights which it provides into the mode-coupling approach to structural glasses and the nature of the glass transition. We demonstrate explicitly that the finite-dimensional version of the three-spin model is in the same universality class as an Ising spin glass in a magnetic field. Assuming that the droplet picture of Ising spin glasses is valid we discuss how this universality may provide insights into why structural glasses are either "fragile" or "strong."     

Thermodynamics of the Farey Fraction Spin Chain

We consider the Farey fraction spin chain, a one-dimensional model defined on (the matrices generating) the Farey fractions. We extend previous work on the thermodynamics of this model by introducing an external field h. From rigorous and more heuristic arguments, we determine the phase diagram and phase transition behavior of the extended model. Our results are fully consistent with scaling theory (for the case when a “marginal” field is present) despite the unusual nature of the transition for h = 0, and the presence of long-range forces.     

Numerical results for spin glass ground states on Bethe lattices: Gaussian bonds

The average ground state energies for spin glasses on Bethe lattices of connectivities r = 3,...,15 are studied numerically for a Gaussian bond distribution. The Extremal Optimization heuristic is employed which provides high-quality approximations to ground states. The energies obtained from extrapolation to the thermodynamic limit smoothly approach the ground-state energy of the Sherrington-Kirkpatrick model for r ↦ ∞. Consistently for all values of r in this study, finite-size corrections are found to decay approximately with ~N^-4/5. The possibility of ~N^-2/3 corrections, found previously for Bethe lattices with a bimodal ± J bond distribution and also for the Sherrington-Kirkpatrick model, are constrained to the additional assumption of very specific higher-order terms. Instance-to-instance fluctuations in the ground state energy appear to be asymmetric up to the limit of the accuracy of our heuristic. The data analysis provides insights into the origin of trivial fluctuations when using continuous bonds and/or sparse networks.     

Ward Identities and Vanishing of the Beta Function for d = 1 Interacting Fermi Systems

We give a self consistent and simplified proof of the (asymptotic) vanishing of the Beta function in d=1 interacting Fermi systems as a consequence of a few properties deduced from the exact solution of the Luttinger model. Moreover, since the vanishing of the Beta function is usually “proved” in the physical literature through heuristic arguments based on Ward identities, we briefly discuss here also the possibility of exploiting this idea in a rigorous approach, by using a suitable Dyson equation. We show that there are serious difficulties, related to the presence of corrections (for which we get careful bounds), which are usually neglected.     

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 marginal 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 hopefully consistent with marginal stability.     

Applications of correlation inequalities to low density graphical codes

This contribution is based on the contents of a talk delivered at the Next-SigmaPhi conference held in Crete in August 2005. It is adressed to an audience of physicists with diverse horizons and does not assume any background in communications theory. Capacity approaching error correcting codes for channel communication known as Low Density Parity Check (LDPC) codes have attracted considerable attention from coding theorists in the last decade. Surprisingly strong connections with the theory of diluted spin glasses have been discovered. In this work we elucidate one new connection, namely that a class of correlation inequalities valid for Gaussian spin glasses can be applied to the theoretical analysis of LDPC codes. This allows for a rigorous comparison between the so called (optimal) maximum a posteriori and the computationaly efficient belief propagation decoders. The main ideas of the proofs are explained and we refer to recent works for the more lengthy technical details.     

Optimization with extremal dynamics

We explore a new general-purpose heuristic for finding high-quality solutions to hard discrete optimization problems. The method, called extremal optimization, is inspired by self-organized criticality, a concept introduced to describe emergent complexity in physical systems. Extremal optimization successively updates extremely undesirable variables of a single suboptimal solution, assigning them new, random values. Large fluctuations ensue, efficiently exploring many local optima. We use extremal optimization to elucidate the phase transition in the 3-coloring problem, and we provide independent confirmation of previously reported extrapolations for the ground-state energy of +/-J spin glasses in d = 3 and 4.     

Spin glass theory: Numerical and experimental results in three-dimensional systems

Here I will review the theoretical results that have been obtained for spin glasses. I will concentrate my attention on the predictions of the mean field approach in three dimensional systems and on its numerical and experimental verifications.     

Corrections to scaling in disordered systems with competing interactions

Using a mean-field effective-field model for disordered systems with competing interactions we have (a) obtained explicit expressions for the corrections to scaling for both ferromagnets and spin glasses up to order h³ in the magnetization, and (b) performed numerical calculations of the non-linear susceptibility of both ferromagnets and spin glasses as a function of reduced field and temperature. Scaling plots constructed from the numerical data show that the departures from universality are much more significant for spin glasses than they are for ferromagnets and that, while the discrepancy can be reduced through a redefinition of the scaling variables, a knowledge of the proper non-universal scaling correction is essential for spin glasses if a reasonable degree of universality is to be achieved.     

Aging, rejuvenation and memory phenomena in spin glasses

In this paper, we review several important features of the out-of-equilibrium dynamics of spin glasses. Starting with the simplest experiments, we discuss the scaling laws used to describe the isothermal aging observed in spin glasses after a quench down to the low-temperature phase. We report in particular new results on the sub-aging behaviour of spin glasses. We then discuss the rejuvenation and memory effects observed when a spin glass is submitted to temperature variations during aging, from the point of view of both energy landscape pictures and real-space pictures. We highlight the fact that both approaches point out the necessity of hierarchical processes involved in aging. Finally, we report an investigation of the effect of small temperature variations on aging in spin glass samples with various anisotropies which indicates that this hierarchy depends on the spin anisotropy.     

Direct-interaction approximation by the modal-interaction perturbation technique

An alternate method is presented of obtaining the direct-interaction equations by combining the heuristic and rigorous derivations of Kraichnan. Within the framework of the model dynamic representation of Kraichnan's rigorous theory, we have developed the irreducible diagram expansion systematically * by formalizing the perturbation argument of his heuristic derivation. It is hoped that the present work will provide a further insight into the analytical structure of the irreducible diagram expansion and bridge the gap apparent in the two original derivations of the direct-interaction equations given by Kraichnan.     

On the p-spin interaction transverse Ising spin-glass model without replicas

The p-spin interaction Ising spin glass model in the presence of a transverse field is studied in the large p-limit by means of a convenient operator extension of the cavity fields method avoiding replicas and the Trotter-Suzuki transformation. The results appear quite consistent with those recently obtained for the same model using conventional treatments within the replica trick. This gives additional support to the cavity fields approach as a promising tool towards a general theory of classical and quantum spin glasses.     

Electricity and Magnetism,Volume 1: Third Edition,Electricity and Magnetism,Volume 2: Third Edition,by B.I. Bleaney and B. Bleaney

During the last decade the term ‘spin glass’ has become prominent in the literature on magnetism. It refers to magnetic alloys where the spins on the impurities become locked or frozen into random orientations below a characteristic temperature T ₀. In this article the properties of spin glasses are described with particular reference to the two archetypal examples AuFe and CuMn. Interest in spin glasses was mainly stimulated by some a.c. susceptibility measurements which showed sharp, cusp-like peaks, accurately defining T ₀ and suggesting that some type of phase transition was occurring. The Mossbauer effect and the anomalous Hall effect also showed clear features at T ₀ supporting this viewpoint. But measurements of the electrical resistivity and ‘specific heat’, here usually meaning the molar heat capacity, also the remanence, magnetic hysteresis and time-dependent effects observed in spin glasses were difficult to reconcile with a phase transition approach. This article discusses the results obtained from the very wide variety of experimental techniques which have been used to investigate spin glasses, and also deals with some of the important theoretical concepts which have arisen out of these studies. Then follows a short account of the many systems which have been found to exhibit spin glass behaviour and which suggest that it is a widespread magnetic state of matter. Lastly, an example is given which shows that some of the ideas of spin glasses are applicable to problems outside the sphere of magnetic alloys.     

Replica Symmetry Breaking in Short-Range Spin Glasses: Theoretical Foundations and Numerical Evidences

We discuss replica symmetry breaking (RSB) in spin glasses. We update work in this area, from both the analytical and numerical points of view. We give particular attention to the difficulties stressed by Newman and Stein concerning the problem of constructing pure states in spin glass systems. We mainly discuss what happens in finite-dimensional, realistic spin glasses. Together with a detailed review of some of the most important features, facts, data, and phenomena, we present some new theoretical ideas and numerical results. We discuss among others the basic idea of the RSB theory, correlation functions, interfaces, overlaps, pure states, random field, and the dynamical approach. We present new numerical results for the behaviors of coupled replicas and about the numerical verification of sum rules, and we review some of the available numerical results that we consider of larger importance (for example, the determination of the phase transition point, the correlation functions, the window overlaps, and the dynamical behavior of the system).     

Multiple bifurcation problems for chemical reactors

This paper presents a mathematical strategy for exploring the dynamical behavior of chemical reactors. Our intent is to stimulate further development of the mathematics with which we approach these problems as well as to illustrate how it can be usefully applied to models of chemical reactors. In the process of accomplishing this task, we extend and correct published work on the models we consider. At the same time, we point to the mathematical problems which arise. Many of the results presented rely upon computer studies that hopefully will be surplanted by rigorous arguments in the future.     

How does the structural inhomogeneity influence the shear band behaviours of metallic glasses

ABSTRACT

Here we propose a model to understand the influence of structural inhomogeneity on the shear band behaviours of metallic glasses. By considering the inhomogeneous structure and stress concentration, the model predicts that the strain for shear band nucleation in metallic glasses can be variable and far below the theoretical elastic limit. During sliding, the shear band will approach a dynamic equilibrium state of balanced free volume generation and annihilation. By considering the accumulation of irreversible structure change, the shear band will finally develop into fracture. Under fluctuating load, the shear band shows an ‘activate-arrest’ behaviour, which results from a delayed response to the external load change. These results reasonably explain and correlate the physics behind the elastic limit, stick-slip shear band behaviour, implicit shear events, and shear band fracture in metallic glasses. The study can provide another perspective and platform to understand the correlations between structural inhomogeneity and shear band behaviours in metallic glasses, and further explore other shear band related phenomena not only in metallic glasses but also in the class of shear-softened materials.     

What Does It Take to Solve the 3D Ising Model? Minimal Necessary Conditions for a Valid Solution

An exact solution of the Ising model on the simple cubic lattice is one of the long-standing open problems in rigorous statistical mechanics. Indeed, it is generally believed that settling it would constitute a methodological breakthrough, fomenting great prospects for further application, similarly to what happened when Lars Onsager solved the two-dimensional model eighty years ago. Hence, there have been many attempts to find analytic expressions for the exact partition function Z, but all such attempts have failed due to unavoidable conceptual or mathematical obstructions. Given the importance of this simple yet paradigmatic model, here we set out clear-cut criteria for any claimed exact expression for Z to be minimally plausible. Specifically, we present six necessary—but not sufficient—conditions that Z must satisfy. These criteria will allow very quick plausibility checks of future claims. As illustrative examples, we discuss previous mistaken “solutions”, unveiling their shortcomings.     

The temperature dependence of the magnetic susceptibility of spin glasses

The temperature dependence of the magnetic susceptibility of spin glasses is calculated exactly on the basis of a magnetic cluster model, following Néel. The susceptibility of a single cluster below its blocking temperature is taken to be negligibly small compared to its value above. A distribution of blocking temperatures may be obtained from observations on spin glasses. Conditions for the sharpness of the susceptibility peak are discussed and the low-temperature variation of the susceptibility is also obtained.     

The Percolation Signature of the Spin Glass Transition

Magnetic ordering at low temperature for Ising ferromagnets manifests itself within the associated Fortuin–Kasteleyn (FK) random cluster representation as the occurrence of a single positive density percolating network. In this paper we investigate the percolation signature for Ising spin glass ordering—both in short-range (EA) and infinite-range (SK) models—within a two-replica FK representation and also within the different Chayes–Machta–Redner two-replica graphical representation. Based on numerical studies of the ±J EA model in three dimensions and on rigorous results for the SK model, we conclude that the spin glass transition corresponds to the appearance of two percolating clusters of unequal densities.     

Local Energy Statistics in Disordered Systems: A Proof of the Local REM Conjecture

Recently, Bauke and Mertens conjectured that the local statistics of energies in random spin systems with discrete spin space should, in most circumstances, be the same as in the random energy model. Here we give necessary conditions for this hypothesis to be true, which we show to be satisfied in wide classes of examples: short range spin glasses and mean field spin glasses of the SK type. We also show that, under certain conditions, the conjecture holds even if energy levels that grow moderately with the volume of the system are considered. Research supported in part by the DFG in the Dutch-German Bilateral Research Group ``Mathematics of Random Spatial Models from Physics and Biology' and by the European Science Foundation in the Programme RDSES.