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.     

Uniqueness of Ground States for Short-Range Spin Glasses in the Half-Plane

We consider the Edwards-Anderson Ising spin glass model on the half-plane \mathbbZ ×\mathbbZ⁺{\mathbb{Z} \times \mathbb{Z}^+} with zero external field and a wide range of choices, including mean zero Gaussian for the common distribution of the collection J of i.i.d. nearest neighbor couplings. The infinite-volume joint distribution K(J,a){\mathcal{K}(J,\alpha)} of couplings J and ground state pairs α with periodic (respectively, free) boundary conditions in the horizontal (respectively, vertical) coordinate is shown to exist without need for subsequence limits. Our main result is that for almost every J, the conditional distribution K(a | J){\mathcal{K}(\alpha\,|\,J)} is supported on a single ground state pair.     

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).     

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.     

Short-Range Ordering in Amorphous Structures

A numerical investigation is made of some details of the short-range order in model amorphous structures with free boundaries and a number of different interaction potentials. Five- and seven-member rings — typical features of amorphous structures — are found out in all two-dimensional cross-sections of the generated clusters. Histograms of distribution of the nearest-neighbor numbers are constructed and an analysis of the positions of atoms with different coordination numbers within a model cluster volume is performed. An evaluation is made of the effects of the boundary on the coordination number distribution. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 42–45, September, 2005.     

高温超导体中的短程反铁磁自旋关联

传统超导体具有大的相干长度 ,借助于由隧穿测量给出的电流 -电压关系 ,可以反演出电声子相互作用的谱密度α2 Ｆ(ω) .对于高温超导体来说 ,情况完全不同 ,它的相干长度很短 ,为要研究其中的相互作用谱密度 ,最好的办法是采用红外光反射率测量或进行中子非弹性散射实验 .最近 ,美国橡树岭 (ＯａｋＲｉｄｇｅ)国家实验室的Ｐ .Ｃ .Ｄａｉ等 ,对ＹＢａ2 Ｃｕ3Ｏ6 ｘ(ＹＢＣＯ)系列样品 ,给出了中子非弹性散射研究的新数据 .结果表明 ,短程反铁磁自旋关联 ,在超导形成中起着重要的作用 .中子散射与Ｘ射线散射的区别主要在于 :Ｘ射线散射的强…     

Interaction-Flip Identities in Spin Glasses

We study the properties of fluctuation for the free energies and internal energies of two spin glass systems that differ for having some set of interactions flipped. We show that their difference has a variance that grows like the volume of the flipped region. Using a new interpolation method, which extends to the entire circle the standard interpolation technique, we show by integration by parts that the bound imply new overlap identities for the equilibrium state. As a side result the case of the non-interacting random field is analyzed and the triviality of its overlap distribution proved.     

Duality in Finite-Dimensional Spin Glasses

We present an analysis leading to a conjecture on the exact location of the multicritical point in the phase diagram of spin glasses in finite dimensions. The conjecture, in satisfactory agreement with a number of numerical results, was previously derived using an ansatz emerging from duality and the replica method. In the present paper we carefully examine the ansatz and reduce it to a hypothesis on analyticity of a function appearing in the duality relation. Thus the problem is now clearer than before from a mathematical point of view: The ansatz, somewhat arbitrarily introduced previously, has now been shown to be closely related to the analyticity of a well-defined function.     

Local Energy Statistics in Spin Glasses

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. We review some rigorous results confirming the validity of this conjecture. In the context of the SK models, we analyse the limits of the validity of the conjecture for energy levels growing with the volume of the system. In the case of the Generalised Random energy model, we give a complete analysis for the behaviour of the local energy statistics at all energy scales. In particular, we show that, in this case, the REM conjecture holds exactly up to energies E _N < β _c N, where β _c is the critical temperature. We also explain the more complex behaviour that sets in at higher energies. 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.     

Convergence to Equilibrium for Spin Glasses

We derive estimates on the thermalisation times for dynamical spin glasses at high temperature. Received: 4 November 1999 / Accepted: 15 May 2000     

A Unified Stability Property in Spin Glasses

Gibbs’ measures in the Sherrington-Kirkpatrick type models satisfy two asymptotic stability properties, the Aizenman-Contucci stochastic stability and the Ghirlanda-Guerra identities, which play a fundamental role in our current understanding of these models. In this paper we show that one can combine these two properties very naturally into one unified stability property.     

Overlap Synchronisation in Multipartite Random Energy Models

In a multipartite random energy model, made of a number of coupled generalised random energy models (GREMs), we determine the joint law of the overlaps in terms of the ones of the single GREMs. This provides the simplest example of the so-called overlap synchronisation.     

Thermodynamics and Universality for Mean Field Quantum Spin Glasses

We study aspects of the thermodynamics of quantum versions of spin glasses. By means of the Lie-Trotter formula for exponential sums of operators, we adapt methods used to analyze classical spin glass models to answer analogous questions about quantum models.     

A Quantitative Clustering Approach to Ultrametricity in Spin Glasses

We discuss the problem of ultrametricity in mean field spin glasses by means of a hierarchical clustering algorithm. We complement the clustering approach with quantitative testing: we discuss both in some detail. We show that the elimination of the (in this context accidental) spin flip symmetry plays a crucial role in the analysis, since the symmetry hides the real nature of the data. We are able to use in the analysis disorder averaged quantities. We are able to exhibit a number of features of the low T phase of the mean field theory, and to claim that the full hierarchical structure can be observed without ambiguities only on very large lattice volumes, not currently accessible by numerical simulations.