A probabilistic approach to the models of spin glasses (original) (raw)
Physical Review B
A Landau-Ginzburg description of a spin-glass which incorporates naturally the concept of "frustration, " or the incompatibility of different local stable spin configurations in neighboring regions, is presented. For a planar spin, the effective Hamiltonian has a form analogous to that of the Landau-Ginzburg functional for a superconductor in a magnetic field, except that the role of the vector potential is taken by a quenched random variable Q(x) which represents the wave vector of the spin-density wave of minimum local free energy. The model is thus a simple transcription to a Landau-Ginzburg picture of the basic notion of a spinglass as a material whose properties are determined by competition between ferromagnetic and antiferromagnetic interactions. The probability distribution of Q(x) is chosen not to depend on Q(x) directly (in order not to favor any particular value of Q), but to be Gaussian in the curl of Q(x). The variance f of this distribution, the mean-square vorticity in Q(z), is a measure of the degree of frustration. [Any longitudinal part of Q(x) is gauged away by rotating the local spin axes appropriately. ] For a classical vector (Heisenberg) spin system, the analogous description is a Hamiltonian of O(3) Yang-Mills form, again with the gauge random variable. Two calculations are presented. The first tests the stability of the f = 0 theory (thermodynamically identical to an ordinary ferromagnet) against the introduction of a small amount of frustration. The result is that the f = 0 fixed point is unstable, and no new fixed point (of order 4d) appears. Thus the spin-glass transition does not appear to be related to any normal sort of critical point with a particular local-spin-density configuration as a "hidden" order parameter. The second is a mean-field analysis of a transition to a state characterized by an Edwards-Anderson order parameter; its qualitative features are similar to those of mean-field theories for other models for spin-glasses. The conditions for the thermodynamic stability of such a state remain unknown,
On a classical spin glass model
Zeitschrift f�r Physik B Condensed Matter, 1983
A simple, exactly soluble, model of a spin-glass with weakly correlated disorder is presented. It includes both randomness and frustration, but its solution can be obtained without replicas. As the temperature T is lowered, the spin-glass phase is reached via an equilibrium phase transition at T--T I. The spin-glass magnetization exhibits a distinct S-shape character, which is indicative of a field-induced transition to a state of higher magnetization above a certain threshold field. For suitable probability distributions of the exchange interactions. (a) A mixed phase is found where spin-glass and ferromagnetism coexist. (b) The zero-field susceptibility has a flat plateau for 0_<T_< T~ and a Curie-Weiss behaviour for T > T I. (c) At low temperatures the magnetic specific heat is linearly dependent on the temperature. The physical origin of the dependence upon the probability distributions is explained, and a careful analysis of the ground state structure is given.
Random free energies in spin glasses
1985
Abstract The free energies of the pure states in the spin glass phase are studied in the mean field theory. They are shown to be independent random variables with an exponential distribution. Physical implications concerning the fluctuations from sample to sample are worked out. The physical nature of the mean field theory is fully characterized. Résumé Les énergies libres des états purs sont étudiées dans la théorie de champ moyen.
Some comments on the Sherrington-Kirkpatrick model of spin glasses
Communications in Mathematical Physics, 1987
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 shortrange spin glasses, as the range of the interactions tends to infinity.
Universality-class dependence of energy distributions in spin glasses
Physical Review B, 2005
We study the probability distribution function of the ground-state energies of the disordered onedimensional Ising spin chain with power-law interactions using a combination of parallel tempering Monte Carlo and branch, cut, and price algorithms. By tuning the exponent of the power-law interactions we are able to scan several universality classes. Our results suggest that mean-field models have a non-Gaussian limiting distribution of the ground-state energies, whereas non-meanfield models have a Gaussian limiting distribution. We compare the results of the disordered onedimensional Ising chain to results for a disordered two-leg ladder, for which large system sizes can be studied, and find a qualitative agreement between the disordered one-dimensional Ising chain in the short-range universality class and the disordered two-leg ladder. We show that the mean and the standard deviation of the ground-state energy distributions scale with a power of the system size. In the mean-field universality class the skewness does not follow a power-law behavior and converges to a nonzero constant value. The data for the Sherrington-Kirkpatrick model seem to be acceptably well fitted by a modified Gumbel distribution. Finally, we discuss the distribution of the internal energy of the Sherrington-Kirkpatrick model at finite temperatures and show that it behaves similar to the ground-state energy of the system if the temperature is smaller than the critical temperature.
Some considerations of finite-dimensional spin glasses
Journal of Physics A: Mathematical and Theoretical, 2008
In talk I will review the theoretical results that have been obtained for spin glasses, paying a particular attention to finite dimensional spin glasses. I will concentrate my attention on the formulation of the mean field approach and on its numerical and experimental verifications. I will mainly considered equilibrium properties at zero magnetic field, where the situation is clear and it should be not controversial. I will present the various hypothesis at the basis of the theory and I will discuss their physical status.
Short-Range Spin Glasses: The Metastate Approach
Encyclopedia of Mathematical Physics, 2006
We discuss the metastate, a probability measure on thermodynamic states, and its usefulness in addressing difficult questions pertaining to the statistical mechanics of systems with quenched disorder, in particular short-range spin glasses. The possible low-temperature structures of realistic (i.e., short-range) spin glass models are described, and a number of fundamental open questions are presented.
Non-equilibrium Relations for Spin Glasses with Gauge Symmetry
2010
We study the applications of non-equilibrium relations such as the Jarzynski equality and fluctuation theorem to spin glasses with gauge symmetry. It is shown that the exponentiated free-energy difference appearing in the Jarzynski equality reduces to a simple analytic function written explicitly in terms of the initial and final temperatures if the temperature satisfies a certain condition related to gauge symmetry. This result is used to derive a lower bound on the work done during the non-equilibrium process of temperature change. We also prove identities relating equilibrium and non-equilibrium quantities. These identities suggest a method to evaluate equilibrium quantities from non-equilibrium computations, which may be useful to avoid the problem of slow relaxation in spin glasses.
Some non-perturbative calculations on spin glasses
Journal of Physics A: Mathematical and General, 1995
Models of spin glasses are studied with a phase transition discontinuous in the Parisi order parameter. It is assumed that the leading order corrections to the thermodynamic limit of the high temperature free energy are due to the existence of a metastable saddle point in the replica formalism. An ansatz is made on the form of the metastable point and its contribution to the free energy is calculated. The Random Energy Model is considered along with the p-spin and the p-state Potts Models in their p < ∞ expansion. Please contact me if you want the original pictures as poscript files.
Gauge Theory for Quantum Spin Glasses
Journal of the Physical Society of Japan, 2006
The gauge theory for random spin systems is extended to quantum spin glasses to derive a number of exact and/or rigorous results. The transverse Ising model and the quantum gauge glass are shown to be gauge invariant. For these models, an identity is proved that the expectation value of the gauge invariant operator in the ferromagnetic limit is equal to the one in the classical equilibrium state on the Nishimori line. As a result, a set of inequalities for the correlation function are proved, which restrict the location of the ordered phase. It is also proved that there is no long-range order in the two-dimensional quantum gauge glass in the ground state. The phase diagram for the quantum XY Mattis model is determined.
We review recent theoretical progress on glassy dynamics, with special emphasis on the importance and universality of the aging regime, which is relevant to many experimental situations. The three main subjects which we address are: (i) Phenomenological models of aging (coarsening, trap models), (ii) Analytical results for the low-temperature dynamics of mean-field models (corresponding to the mode-coupling equations); and (iii) Simple non-disordered models with glassy dynamics. We discuss the interrelation between these approaches, and also with previous work in the field. Several open problems are underlined -in particular the precise relation between mean-field like (or mode-coupling) descriptions and finite dimensional problems.
Physical Review B, 2013
At the mean-field level, on fully connected lattices, several disordered spin models have been shown to belong to the universality class of "structural glasses" with a "random first-order transition" (RFOT) characterized by a discontinuous jump of the order parameter and no latent heat. However, their behavior in finite dimensions is often drastically different, displaying either no glassiness at all or a conventional spin-glass transition. We clarify the physical reasons for this phenomenon and stress the unusual fragility of the RFOT to short-range fluctuations, associated, e.g., with the mere existence of a finite number of neighbors. Accordingly, the solution of fully connected models is only predictive in very high dimension, whereas despite being also mean-field in character, the Bethe approximation provides valuable information on the behavior of finite-dimensional systems. We suggest that before embarking on a full blown account of fluctuations on all scales through computer simulation or renormalization-group approach, models for structural glasses should first be tested for the effect of short-range fluctuations and we discuss ways to do it. Our results indicate that disordered spin models do not appear to pass the test and are therefore questionable models for investigating the glass transition in three dimensions. This also highlights how nontrivial is the first step of deriving an effective theory for the RFOT phenomenology from a rigorous integration over the short-range fluctuations.
TOWARD A MEAN FIELD THEORY FOR SPIN GLASSES
We find an approximate solution of the Sherrington-Kirkpatrick model for spin glasses; the internal energy and the specific heat are in very good agreement with the computer simulations, the zero temperature entropy is unfortunately negative, although it is very small.
On the Number of Metastable States in Spin Glasses
Europhysics Letters (EPL), 1995
In this letter, we show that the formulae of Bray and Moore for the average logarithm of the number of metastable states in spin glasses can be obtained by calculating the partition function with m coupled replicas with the symmetry among these explicitly broken according to a generalization of the 'two-group' ansatz. This equivalence allows us to find solutions of the bm equations where the lower 'band-edge' free energy equals the standard static free energy. We present these results for the Sherrington-Kirkpatrick model, but we expect them to apply to all mean-field spin glasses.
2012
In these notes, we continue our investigation of classical toy models of disordered statistical mechanics, through techniques recently developed and tested mainly on the paradigmatic Sherrington-Kirkpatrick spin glass. Here, we consider the p-spin-glass model with Ising spins and interactions drawn from a normal distribution N [0, 1]. After a general presentation of its properties (e.g. self-averaging of the free energy, existence of a suitable thermodynamic limit), we study its equilibrium behavior within the Hamilton-Jacobi framework and the smooth cavity approach. Through the former we find both the RS and the 1-RSB expressions for the free-energy, coupled with their self-consistent relations for the overlaps. Through the latter, we recover these results as irreducible expression, and we study the generalization of the overlap polynomial identities suitable for this model; a discussion on their deep connection with the structure of the internal energy and the entropy closes the investigation.
Study of a simple hypothesis for the mean-field theory of spin-glasses
1981
Abstract A detailed study is presented of the consequences of a simple hypothesis for the Sherrington-Kirkpatrick model, namely that in the spin-glass phase the entropy is independent of the applied magnetic field. This hypothesis leads to predictions in excellent agreement with the results of available Monte-Carlo simulations, for the ground-state energy, the entropy in zero field, the energy and spontaneous magnetization as a function of the interaction mean value Jo.
A New Method to Compute the Configurational Entropy in Spin Glasses
2001
We propose a new method to compute the configurational entropy of glassy systems as a function of the free energy of valleys at a given temperature, in the framework of the Stillinger and Weber approach. In this method, which we call free-energy inherent structures (FEIS) approach, valleys are represented by inherent structures that are statistically grouped according to their free-energy rather than the energy as is commonly done in the standard procedure. The FEIS method provides a further step toward a description of the relaxational behavior of glassy systems in terms of a free energy measure. It can be used to determine the character of the glass transition as well as the mode coupling and the Kauzmann temperatures. We illustrate the usefulness of the method by applying it to simple models of glasses and spin glasses.
Distribution of metastable states of spin glasses
Journal of Physics: Conference Series
The complex behavior of systems like spin glasses, proteins or neural networks is typically explained in terms of a rugged energy or fitness landscape. Within the highdimensional conformation space of these systems one finds features like barriers, saddle points, and metastable states whose number-at least in the case of spin glasses-grows exponentially with the size of the system. We propose a novel Monte Carlo sampling algorithm that employs an ensemble of short Markovian chains in order to visit all metastable states with equal probability. We apply this algorithm in order to measure the number of metastable states for the twodimensional and the three-dimensional Edwards-Anderson model and compare with theoretical predictions.
Metastable states and space-time phase transitions in a spin-glass model
Physical Review E, 2010
We study large deviations of the dynamical activity in the random orthogonal model (ROM). This is a fully connected spin-glass model with one-step replica symmetry breaking behaviour, consistent with the random first-order transition scenario for structural glasses. We show that this model displays dynamical (space-time) phase-transitions between active and inactive phases, as demonstrated by singularities in large deviation functions. We argue that such transitions are generic in systems with long-lived metastable states.
Stability properties and probability distribution of multi-overlaps in dilute spin glasses
2007
We prove that the Aizenman-Contucci relations, well known for fully connected spin glasses, hold in diluted spin glasses as well. We also prove more general constraints in the same spirit for multi-overlaps, systematically confirming and expanding previous results. The strategy we employ makes no use of self-averaging, and allows us to generate hierarchically all such relations within the framework of Random Multi-Overlap Structures. The basic idea is to study, for these structures, the consequences of the closely related concepts of stochastic stability, quasi-stationarity under random shifts, factorization of the trial free energy. The very simple technique allows us to prove also the phase transition for the overlap: it remains strictly positive (in average) below the critical temperature if a suitable external field is first applied and then removed in the thermodynamic limit. We also deduce, from a cavity approach, the general form of the constraints on the distribution of mult...