Spin glasses and fragile glasses: Statics, dynamics, and complexity

Abstract

In this paper I will briefly review some theoretical results that have been obtained in recent years for spin glasses and fragile glasses. I will concentrate my attention on the predictions coming from the so called broken replica symmetry approach and on their experimental verifications. I will also mention the relevance or these results for other fields, and in general for complex systems.broken replica symmetry ͉ complex systems ͉ nonequilibrium ͉ taxonomy S pin glasses have been intensively studied in the last 30 years. They are very interesting for many reasons:Y Spin glasses are the simplest example of glassy systems. There is a highly nontrivial mean field approximation that can be used to derive some of the main properties of glassy systems, e.g., history-dependent response (1-3); this property, in the context of mean field approximation, is related to the existence of many equilibrium states. (Much ink has been used to discuss the precise mathematical meaning of this last sentence; for a careful discussion see refs. 4 and 5.) Y The study of spin glasses opens an important window for studying off-equilibrium behavior. Aging (6) and the related violations of the equilibrium fluctuation dissipation relations emerge in a natural way and they can be studied in a simple setting (7-10). Many of the ideas developed in this context can be used in other physical fields such as fragile glasses, colloids, granular materials and combinatorial optimization problems (and also for other complex systems). Y The theoretical concepts and the tools developed in the study of spin glasses are based on two logically equivalent, but very different methods: the algebraic broken replica symmetry method and the probabilistic cavity approach. They have a wide domain of applications. Some of the properties that appear in the mean field approximation, such as ultrametricity, are unexpected and counterintuitive. Y Spin glasses also provide a testing ground for a more mathematically inclined probabilistic approach: the rigorous proof of the correctness of the solution of the mean field model came out after 20 years of efforts where new ideas [e.g., stochastic stability (11-13)], and new variational principles (14, 15) were at the basis of a recent rigorous proof (16).In this paper I will present a short review of some of the results that have been obtained by using this approach. General ResultsThe simplest spin glass Hamiltonian is of the form:where the Js are quenched (i.e., time independent) random variables located on the links connecting two points of the lattice and the s are Ising variables (i.e., Ϯ1). The total number of points is denoted with N and it goes to infinity in the thermodynamic limit. We can consider four models of increasing complexity:Y The Sherrington-Kirkpatrick (SK) model (17) As far as the free energy is concerned, one can prove the following rigorous results:The SK model is thus a good starting point for studying also the finite-dimensional case with short-range interaction, which is the most realistic and d...

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Density of States for a Specified Correlation Function and the Energy Landscape

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Fixed-point distributions of short-range Ising spin glasses on hierarchical lattices

“…A m ong many controversial points, an identification o f those characteristics from this m ean-field solution that should persist in the limit o f short-range (e.g., nearest-neighbor) interactions represents one o f the m ajor questions in the Ising SG problem nowadays. M otivated by this, a lot o f w ork has been pursued on nearest-neighbor-interaction three-dim ensional Ising SG m odels, for w hich it is generally accepted now adays that a SG phase occurs at finite tem peratures [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]; it should be em phasized that some o f this w ork has required extensive com putational efforts.…”

“…From the analytical point o f view, most approaches were carried on its mean-field form ulation, defined in term s o f infinite-range interactions. The appropriate meanfield solution is expressed in term s o f an infinite num ber o f order param eters, characterizing an order-param eter function, m anifesting the property o f replica-sym m etry breaking [1][2][3][4][5][6]. A m ong many controversial points, an identification o f those characteristics from this m ean-field solution that should persist in the limit o f short-range (e.g., nearest-neighbor) interactions represents one o f the m ajor questions in the Ising SG problem nowadays.…”