Short-Range Spin Glasses: The Metastate Approach (original) (raw)
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.
A probabilistic approach to the models of spin glasses
Journal of Statistical Physics, 1983
Introducing the notions of quenched and annealed probability measures, a systematic study of some problems in the description of spin glasses is attempted. Inequalities and variational principles for the free energies are derived. The absence of spontaneous breakdown of the gauge symmetry is discussed and some high-temperature properties are studied. Examples of annealed models with more than one phase transition are shown.
Random energy levels and low-temperature expansions for spin glasses
2001
In a previous paper (cond-mat/0106554) we showed the existence of two new zero-temperature exponents (λ and θ ′) in two dimensional Gaussian spin glasses. Here we introduce a novel lowtemperature expansion for spin glasses expressed in terms of the gap probability distributions for successive energy levels. After presenting the numerical evidence in favor of a random-energy levels scenario, we analyze the main consequences on the low-temperature equilibrium behavior. We find that the specific heat is anomalous at low-temperatures c ∼ T α with α = −d/θ ′ which turns out to be linear for the case θ ′ = −d.
Zero-Temperature Fluctuations in Short-Range Spin Glasses
Journal of Statistical Physics, 2016
We consider the energy difference restricted to a finite volume for certain pairs of incongruent ground states (if they exist) in the d-dimensional Edwards-Anderson (EA) Ising spin glass at zero temperature. We prove that the variance of this quantity with respect to the couplings grows at least proportionally to the volume in any d ≥ 2. An essential aspect of our result is the use of the excitation metastate. As an illustration of potential applications, we use this result to restrict the possible structure of spin glass ground states in two dimensions.
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.
Zero-Temperature Dynamics of ±J Spin Glasses¶and Related Models
Communications in Mathematical Physics, 2000
We study zero-temperature, stochastic Ising models σ t on Z d with (disordered) nearest-neighbor couplings independently chosen from a distribution µ on R and an initial spin configuration chosen uniformly at random. Given d, call µ type I (resp., type F) if, for every x in Z d , σ t x flips infinitely (resp., only finitely) many times as t → ∞ (with probability one) -or else mixed type M. Models of type I and M exhibit a zero-temperature version of "local non-equilibration". For d = 1, all types occur and the type of any µ is easy to determine. The main result of this paper is a proof that for d = 2, ±J models (where µ = αδ J + (1 − α)δ −J ) are type M, unlike homogeneous models (type I) or continuous (finite mean) µ's (type F). We also prove that all other noncontinuous disordered systems are type M for any d ≥ 2. The ±J proof is noteworthy in that it is much less "local" than the other (simpler) proof. Homogeneous and ±J models for d ≥ 3 remain an open problem.
Fragile-glass behavior of a short-range p-spin model
Physical Review B, 1996
We propose a short-range generalization of the p-spin interaction spin-glass model. The model is well suited to test the idea that an entropy collapse is at the bottom line of the dynamical singularity encountered in structural glasses. The model is studied in three dimensions through Monte Carlo simulations, which put in evidence fragile glass behavior with stretched exponential relaxation and super-Arrhenius behavior of the relaxation time. Our data are in favor of a Vogel-Fulcher behavior of the relaxation time, related to an entropy collapse at the Kauzmann temperature. We, however, encounter difficulties analogous to those found in experimental systems when extrapolating thermodynamical data at low temperatures. We study the spin-glass susceptibility, investigating the behavior of the correlation length in the system. We find that the increase of the relaxation time is accompanied by a very slow growth of the correlation length. We discuss the scaling properties of off-equilibrium dynamics in the glassy regime, finding qualitative agreement with the mean-field theory.
Frustration and ground-state degeneracy in spin glasses
Physical Review B, 1977
The problem of an Ising model with random nearest-neighbor interactions is reformulated to make manifest Toulouse's recent suggestion that a broken "lattice gauge symmetry" is responsible for the unusual properties of spin glasses. Exact upper and lower bounds on the ground-state energy for models in which the interactions are of constant magnitude but fluctuating sign are obtained, and used to place restrictions on possible geometries of the unsatisfied interactions which must be present in the ground state. Proposed analogies between the ferromagnetspin-glass phase boundary at zero temperature and a percolation threshold for the "strings" of unsatisfied bonds are reviewed in the light of this analysis. Monte Carlo simulations show that the upper bound resulting from a "one-dimensional approximation" to the spin-glass ground-state energy is reasonably close to the actual result. The transition between spin glass and ferromagnet at 0 K appears to be weakly first order in these models. The entropy of the ground state is obtained from the temperature dependence of the internal energy, and compared with the density of free spins at very low temperatures. For a two-dimensional spin glass in which half the bonds are antiferromagnetic, S(0)-0.099 k~; for the analogous three-dimensional spin glass the result is S(0)-0.062 k~. Monte Carlo kinetic simulations are reported which demonstrate the existence and stability of a fieldcooled moment in the spin-glass ground state.
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.
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.