Some comments on the Sherrington-Kirkpatrick model of spin glasses

In this paper the high-temperature phase of general mean-field spin glass models, including the Sherrington-Kirkpatrick (SK) model, is analyzed. The free energy in zero magnetic field is calculated explicitly for the SK model, and uniform bounds on quenched susceptibilities are established. It is also shown that, at high temperatures, mean-field spin glasses are limits of short-range spin glasses, as the range of the interactions tends to infinity.     

Interface free energies in p-spin glass models

The replica method is used to calculate the interface free energy associated with the change from periodic to antiperiodic boundary conditions in finite-dimensional p-spin glass models in the phase which at mean-field level has one-step replica symmetry breaking (1RSB). In any finite dimension the interface free energy is exponentially small for a large system. This result implies that, in finite dimensions, the 1RSB state does not exist, as it is destroyed by thermal excitation of arbitrarily large droplets. The implications of this for the theory of structural glasses are discussed.     

Spin glass models with ferromagnetically biased couplings on the Bethe lattice: analytic solutions and numerical simulations

We derive the zero-temperature phase diagram of spin glass models with a generic fraction of ferromagnetic interactions on the Bethe lattice. We use the cavity method at the level of one-step replica symmetry breaking (1RSB) and we find three phases: a replica-symmetric (RS) ferromagnetic one, a magnetized spin glass one (the so-called mixed phase), and an unmagnetized spin glass one. We are able to give analytic expressions for the critical point where the RS phase becomes unstable with respect to 1RSB solutions: we also clarify the mechanism inducing such a phase transition. Finally we compare our analytical results with the outcomes of a numerical algorithm especially designed for finding ground states in an efficient way, stressing weak points in the use of such numerical tools for discovering RSB effects. Some of the analytical results are given for generic connectivity.     

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."     

Critical fluctuations for quantum mean-field models

We propose a Ginzburg-Landau-type approximation for the local Gibbs states for quantum mean-field models that leads to the exact thermodynamics. Using this approach, we compute the spin fluctuations for some spin-1/2 models. At the critical temperature we find explicitly the distribution function showing abnormal fluctuations.On leave from the University of Torun, Poland.     

Competition between glassiness and order in a multispin glass

A mean-field multispin interaction spin glass model is analyzed in the presence of a ferromagnetic coupling. The static and dynamical phase diagrams contain four phases (paramagnet, spin glass, ordinary ferromagnet, and glassy ferromagnet) and exhibit reentrant behavior. The glassy ferromagnet phase has anomalous dynamical properties. The results are consistent with a nonequilibrium thermodynamics that has been proposed for glasses.     

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).     

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.      

Nonrealistic behavior of mean-field spin glasses

We study chaotic size dependence of the low-temperature correlations in the Sherrington-Kirkpatrick (SK) spin glass. We prove that as temperature scales to zero with volume, for any typical coupling realization, the correlations cycle through every spin configuration in every fixed observation window. This cannot happen in short-ranged models as there it would mean that every spin configuration is an infinite-volume ground state. Its occurrence in the SK model means that the commonly used "modified clustering" notion of states sheds little light on the replica symmetry breaking (RSB) solution of SK, and, conversely, the RSB solution sheds little light on the thermodynamic structure of Edwards-Anderson models.     

Rigorous Results for Hierarchical Models of Structural Glasses

We consider two non-mean-field models of structural glasses built on a hierarchical lattice. First, we consider a hierarchical version of the random energy model, and we prove the existence of the thermodynamic limit and self-averaging of the free energy. Furthermore, we prove that the infinite-volume entropy is positive in a high-temperature region bounded from below, thus providing an upper bound on the Kauzmann critical temperature. In addition, we show how to improve this bound by leveraging the hierarchical structure of the model. Finally, we introduce a hierarchical version of the \(p\) -spin model of a structural glass, and we prove the existence of the thermodynamic limit and self-averaging of the free energy.     

Order-parameter fluctuations (OPF) in spin glasses: Monte Carlo simulations and exact results for small sizes

The use of parameters measuring order-parameter fluctuations (OPF) has been encouraged by the recent results reported in referenece [2,3] which show that two of these parameters, G and G _c, take universal values in the . In this paper we present a detailed study of parameters measuring OPF for two mean-field models with and without time-reversal symmetry which exhibit different patterns of replica symmetry breaking below the transition: the Sherrington-Kirkpatrick model with and without a field and the Ising p-spin glass (p = 3). We give numerical results and analyze the consequences which replica equivalence imposes on these models in the infinite volume limit. We give evidence for the transition in each system and discuss the character of finite-size effects. Furthermore, a comparative study between this new family of parameters and the usual Binder cumulant analysis shows what kind of new information can be extracted from the finite T behavior of these quantities. The two main outcomes of this work are: 1) Parameters measuring OPF give better estimates than the Binder cumulant for T _c and even for very small systems they give evidence for the transition. 2) For systems with no time-reversal symmetry, parameters defined in terms of connected quantities are the proper ones to look at. Received 20 September 2000 and Received in final form 10 January 2001     

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.     

On the connection between spin glasses and gauge field theories

A simple connection between Ising spin glasses and the Z₂ lattice gauge theory, at negative plaquette temperatures, is presented. It is first shown that annealed models give useful lower bounds on the free energy and ground-state energy of spin glasses. However, they have unphysical low temperature properties (e.g. a negative entropy), which are related to a temperature dependence of the frustration. A restricted annealing scheme is presented which remedies this deficiency through the introduction of a pure gauge coupling counterterm. The possible phase diagrams of the lattice gauge system and their relevance to spin glass transitions are discussed.     

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.     

Measuring overlaps in mesoscopic spin glasses via conductance fluctuations

We consider the electronic transport in a mesoscopic metallic spin glass. We show that the distribution of overlaps between spin configurations can be inferred from the reduction of the conductance fluctuations by the magnetic impurities. Using this property, we propose new experimental protocols to probe spin glasses directly through their overlaps.     

Electron spectral function and algebraic spin liquid for the normal state of underdoped high T(c) superconductors

We propose to describe the spin fluctuations in the normal state (spin-pseudogap phase) of underdoped high T(c) cuprates as a manifestation of an algebraic spin liquid. Within the slave boson implementation of spin-charge separation, the normal state is described by massless Dirac fermions, charged bosons, and a gauge field. The gauge interaction, as an exact marginal perturbation, drives the mean-field free-spinon fixed point to a new spin-quantum fixed point-the algebraic spin liquid. Luttinger-liquid-like line shapes for the electron spectral function are obtained in the normal state, and we show how a coherent quasiparticle peak appears as spin and charge recombine.     

The Genealogical Tree of a Chromosome

We consider a simple neutral model to describe the genealogy of chromosomes by taking into account the effects of both recombination and coalescence. Seen as a statistical physics problem, the model looks like an inverse problem: A number of properties such as pair or three-point correlations can be computed easily, but the prediction of global properties, in particular the average number of ancestors, remains difficult. In the absence of exact solutions, these global properties can nevertheless be estimated by the usual approximations: series expansions, Monte Carlo simulations, mean-field theory. Simulations exhibit also non-self-averaging properties similar to those of mean-field spin glasses.     

On the nature of the low-temperature phase in discontinuous mean-field spin glasses

The low-temperature phase of discontinuous mean-field spin glasses is generally described by a one-step replica symmetry breaking (1RSB) ansatz. The Gardner transition, i.e. a very-low-temperature phase transition to a full replica symmetry breaking (FRSB) phase, is often regarded as an inessential, and somehow exotic phenomenon. In this paper we show that the metastable states which are relevant for the out-of-equilibrium dynamics of such systems are always in a FRSB phase. The only exceptions are (to the best of our knowledge) the p-spin spherical model and the random energy model (REM). We also discuss the consequences of our results for aging dynamics and for local search algorithms in hard combinatorial problems. Received 10 February 2003 Published online 20 June 2003 RID="a" ID="a"e-mail: Federico.Ricci@roma1.infn.it RID="b" ID="b"UMR 8549, Unité Mixte de Recherche du Centre National de la Recherche Scientifique et de l' école Normale Supérieure     

Construction of analytically tractable mean-field theories for quantum models

A general theoretical framework for the construction of maximally complex, yet analytically tractable mean-field theories for quantum-mechanical models is presented. These mean-field theories fulfil several strict conditions which are derived from analogous theories in classical statistical mechanics. In particular, they are thermodynamically consistent, conserving approximations and provide exact bounds on the free energy of the original model. The formalism is used to construct a mean-field theory for the Hubbard model in thestrong-coupling limit.