Edwards-Anderson Research Articles

Haake–Lewenstein–Wilkens approach to spin-glasses revisited

We revisit the Haake–Lewenstein–Wilkens approach to Edwards–Anderson (EA) model of Ising spin glass (SG) (Haake et al 1985 Phys. Rev. Lett. 55 2606). This approach consists in evaluation and analysis of the probability distribution of configurations of two replicas of the system, averaged over quenched disorder. This probability distribution generates squares of thermal copies of spin variables from the two copies of the systems, averaged over disorder, that is the terms that enter the standard definition of the original EA order parameter, . We use saddle point/steepest descent (SPSD) method to calculate the average of the Gaussian disorder in higher dimensions. This approximate result suggest that at in 3D and 4D. The case of 2D seems to be a little more subtle, since in the present approach energy increase for a domain wall competes with boundary/edge effects more strongly in 2D; still our approach predicts SG order at sufficiently low temperature. We speculate, how these predictions confirm/contradict widely spread opinions that: (i) There exist only one (up to the spin flip) ground state in EA model in 2D, 3D and 4D; (ii) there is (no) SG transition in 3D and 4D (2D). This paper is dedicated to the memories of Fritz Haake and Marek Cieplak.

Spin glass in the bond-diluted J1−J2 Ising model on the square lattice

We use Monte Carlo (MC) methods to simulate a two-dimensional (2D) bond-diluted Ising model on the square lattice which has frustration between the nearest-neighbor interaction J1 and the next-nearest-neighbor interaction J2. In this paper, we use the parallel tempering algorithm to study the thermodynamics for different diluted ratio x and give the phase diagram. The presence of both frustration and disorder results in a spin-glass phase, which exists between the stripe antiferromagnetic phase and the Neel phase. We present the ground-state energy of T -> 0 and the size-dependence of Edwards-Anderson (EA) order parameter for the spin glass phase. By scaling the mean energy and the EA order parameter from the simulated annealing with the Kibble-Zurek (KZ) mechanism, we obtain two different dynamic exponents z_E and z_q for the spin glass phase. Experimentally, this model has close implication with the FeAs plane of the iron-based superconductor BaFe2(As(1-x)P(x))2, where a spin-glass like phase was found.

Zero-Temperature Fluctuations in Short-Range Spin Glasses

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 dimension greater than or equal to two. 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.

Universality of Chaos and Ultrametricity in Mixed p‐Spin Models

AbstractWe prove disorder universality of chaos phenomena and ultrametricity in the mixed p‐spin model under mild moment assumptions on the environment. This establishes the longstanding belief among physicists that the solution of mean‐field models with Gaussian disorder also holds for different environments. Our results extend to the mixed p‐spin model as well as to different spin glass models. These include universality of quenched disorder chaos in the Edwards‐Anderson (EA) model and quenched concentration for the magnetization in both EA and mixed p‐spin models under non‐Gaussian environments. In addition, we show quenched self‐averaging for the overlap in the random field Ising model under small perturbation of the external field.© 2015 Wiley Periodicals, Inc.

On the number of ground states of the Edwards–Anderson spin glass model

Les états fondamentaux du modèle de verre de spins de Edwards–Anderson (EA) sont étudiés sur des graphes infinis de degré fini. Les états fondamentaux sont les configurations de spins qui minimisent de manière locale l’Hamiltonien pour chaque ensemble fini de sommets. Un problème avec des implications importantes en physique et en mathématique est de déterminer le nombre d’états fondamentaux pour le modèle sur $\mathbb{Z}^{d}$ pour un $d>1$ donné. Ce problème est la version équivalente pour les modèles de verre de spins du problème du nombre de géodésiques infinies en percolation de premier passage et du nombre d’états fondamentaux du modèle d’Ising ferromagnétique désordonné. Il a été montré récemment par Newman, Stein et les deux auteurs que sur le demi-plan $\mathbb{Z}\times\mathbb{N}$, il existe un unique état fondamental (modulo un flip global des spins) produit par la limite faible des états fondamentaux des volumes finis pour un choix spécifique des conditions frontières. Dans cet article, nous étudions l’ensemble de tous les états fondamentaux sur le graphe infini $\mathbb{Z}\times\mathbb{N}$. Nous montrons que le nombre d’états fondamentaux est deux (correspondant à un flip global des spins) ou infini. Ceci est le premier résultat sur l’ensemble de tous les états fondamentaux pour une dimension non-triviale. Dans la première partie, nous développons des outils qui sont pertinents à la résolution du problème analogue sur $\mathbb{Z}^{d}$.

Comment on “Evidence of Non-Mean-Field-Like Low-Temperature Behavior in the Edwards-Anderson Spin-Glass Model”

A recent interesting paper [Yucesoy et al. Phys. Rev. Lett. 109, 177204 (2012), arXiv:1206:0783] compares the low-temperature phase of the 3D Edwards-Anderson (EA) model to its mean-field counterpart, the Sherrington-Kirkpatrick (SK) model. The authors study the overlap distributions P_J(q) and conclude that the two models behave differently. Here we notice that a similar analysis using state-of-the-art, larger data sets for the EA model (generated with the Janus computer) leads to a very clear interpretation of the results of Yucesoy et al., showing that the EA model behaves as predicted by the replica symmetry breaking (RSB) theory.

Numerical results for the Edwards-Anderson spin-glass model at low temperature

We have simulated Edwards-Anderson (EA) as well as Sherrington-Kirkpatrick systems of ${L}^{3}$ spins. After averaging over large sets of EA system samples of $3\ensuremath{\le}L\ensuremath{\le}10$, we obtain accurate numbers for distributions $p(q)$ of the overlap parameter $q$ at very low-temperature $T$. We find $p(0)/T\ensuremath{\rightarrow}0.233(4)$ as $T\ensuremath{\rightarrow}0$. This is in contrast with the droplet scenario of spin glasses. We also study the number of mismatched links---between replica pairs---that come with large scale excitations. Contributions from small scale excitations are discarded. We thus obtain for the fractal dimension of outer surfaces of $q\ensuremath{\sim}0$ excitations in the EA model ${d}_{s}\ensuremath{\rightarrow}2.59(3)$ as $T\ensuremath{\rightarrow}0$. This is in contrast with ${d}_{s}\ensuremath{\rightarrow}3$ as $T\ensuremath{\rightarrow}0$ that is predicted by mean-field theory for the macroscopic limit.

Progress in the mathematical theory of quantum disordered systems

We review recent progress in the mathematical theory of quantum disordered systems: the Anderson transition, including some joint work with Marchetti, the (quantum and classical) Edwards-Anderson (EA) spin-glass model and return to equilibrium for a class of spin-glass models, which includes the EA model initially in a very large transverse magnetic field.

Extreme Value Statistics Distributions in Spin Glasses

We study the probability distribution of the pseudocritical temperature in a mean-field and in a short-range spin-glass model: the Sherrington-Kirkpatrick and the Edwards-Anderson (EA) model. In both cases, we put in evidence the underlying connection between the fluctuations of the pseudocritical point and the extreme value statistics of random variables. For the Sherrington-Kirkpatrick model, both with Gaussian and binary couplings, the distribution of the pseudocritical temperature is found to be the Tracy-Widom distribution. For the EA model, the distribution is found to be the Gumbel distribution. Being the EA model representative of uniaxial magnetic materials with quenched disorder like Fe(0.5)Mn)0.5)TiO(3) or Eu(0.5)Ba(0.5)MnO(3), its pseudocritical point distribution should be a priori experimentally accessible.

Inference algorithm for finite-dimensional spin glasses: Belief propagation on the dual lattice

Starting from a cluster variational method, and inspired by the correctness of the paramagnetic ansatz [at high temperatures in general, and at any temperature in the two-dimensional (2D) Edwards-Anderson (EA) model] we propose a message-passing algorithm--the dual algorithm--to estimate the marginal probabilities of spin glasses on finite-dimensional lattices. We use the EA models in 2D and 3D as benchmarks. The dual algorithm improves the Bethe approximation, and we show that in a wide range of temperatures (compared to the Bethe critical temperature) our algorithm compares very well with Monte Carlo simulations, with the double-loop algorithm, and with exact calculation of the ground state of 2D systems with bimodal and Gaussian interactions. Moreover, it is usually 100 times faster than other provably convergent methods, as the double-loop algorithm. In 2D and 3D the quality of the inference deteriorates only where the correlation length becomes very large, i.e., at low temperatures in 2D and close to the critical temperature in 3D.

Short-Range Spin Glasses and Random Overlap Structures

Properties of Random Overlap Structures (ROSt)’s constructed from the Edwards-Anderson (EA) Spin Glass model on ℤd with periodic boundary conditions are studied. ROSt’s are ℕ×ℕ random matrices whose entries are the overlaps of spin configurations sampled from the Gibbs measure. Since the ROSt construction is the same for mean-field models (like the Sherrington-Kirkpatrick model) as for short-range ones (like the EA model), the setup is a good common ground to study the effect of dimensionality on the properties of the Gibbs measure. In this spirit, it is shown, using translation invariance, that the ROSt of the EA model possesses a local stability that is stronger than stochastic stability, a property known to hold at almost all temperatures in many spin glass models with Gaussian couplings. This fact is used to prove stochastic stability for the EA spin glass at all temperatures and for a wide range of coupling distributions. On the way, a theorem of Newman and Stein about the pure state decomposition of the EA model is recovered and extended.

Spin order and lattice frustration in optimally doped manganites: A high-temperature NMR study

The physics underlying the complex glassy phenomena, which accompany the formation of polarons in optimally doped manganites (ODM) is a cumbersome issue with many unexplained aspects. In this article we present $^{139}$La and $^{55}$Mn NMR in the temperature range 80K - 900K of ODM La$_{0.67}$Ca$_{0.33}$MnO$_3$. We show that local lattice distortions, established in the Paramagnetic (PM) phase for $T<700$K, induce a genuine spin-glass state, which for $T<T_c$ consolidates with the Ferromagnetic (FM) state into a single thermodynamic phase. Comparative NMR experiments on La$_{0.77}$Ca$_{0.23}$MnO$_3$, La$_{0.59}$Ca$_{0.41}$MnO$_3$, and La$_{0.70}$Sr$_{0.30}$MnO$_3$ demonstrate the dominant role of lattice distortions, which appear to control (i) the stability of the spin glass phase component and (ii) the kind (1st or 2nd order) of the PM-FM phase transition. The experimental results are in agreement with the predictions of the compressible random bond - random field Ising model, where consideration of a strain field induced by lattice distortions, is shown to invoke at $T_c$ a discontinuous (1st order like) change of both the FM and the "glassy" Edwards-Anderson (EA) order parameters.

DISTRIBUTION OF PURE STATES IN SHORT-RANGE SPIN GLASSES

We review the structure of the spin glass phase in the infinite-range Sherrington–Kirkpatrick model and the short-range Edwards–Anderson (EA) model. While the former is now believed to be understood, the nature of the latter remains unresolved. However, considerable insight can be gained through the use of the metastate, a mathematical construct that provides a probability measure on the space of all thermodynamic states. Using tools provided by the metastate construct, possibilities for the nature of the organization of pure states in short-range spin glasses can be considerably narrowed. We review the concept of the "ordinary" metastate, and also newer ideas on the excitation metastate, which has been recently used to prove existence of only a single pair of ground states in the EA Ising model in the half-plane. We close by presenting a new result, using metastate methods, on the number of mixed states allowed in the EA model.

Domain growth within the backbone of the three-dimensional±JEdwards-Anderson spin glass

The goal of this work is to show that a ferromagneticlike domain-growth process takes place within the backbone of the three-dimensional $\ifmmode\pm\else\textpm\fi{}J$ Edwards-Anderson (EA) spin glass model. To sustain this affirmation we study the heterogeneities displayed in the out-of-equilibrium dynamics of the model. We show that both correlation function and mean flipping time distribution present features that have a direct relation with spatial heterogeneities, and that they can be characterized by the backbone structure. In order to gain intuition we analyze the pure ferromagnetic Ising model, where we show the presence of dynamical heterogeneities in the mean flipping time distribution that are directly associated to ferromagnetic growing domains. We extend a method devised to detect domain walls in the Ising model to carry out a similar analysis in the three-dimensional EA spin-glass model. This allows us to show that there exists a domain-growth process within the backbone of this model.

Comment on “Ultrametricity in the Edwards-Anderson Model”

In a recent interesting Letter Contucci {\it et al.} have investigated several properties of the three-dimensional (3d) Edwards-Anderson (EA) Ising spin glass. They claim to have found strong numerical evidence for the presence of a complex ultrametric structure similar to the one described by the replica symmetry breaking solution of the mean field model. We illustrate by numerical simulations that the relations used by Contucci {\it et al.} as evidence for an ultrametric structure in the 3d EA model are fulfilled to similar accuracy in the two-dimensional EA model, which is well-described by the droplet picture and has no spin glass phase at finite temperature. We conclude that the data presented in the Contucci {\it et al.} Letter is not sufficient to dismiss the possibility that, e.g., the droplet model might describe the behavior of the 3d EA model.

On the application of the Edwards–Anderson order parameter to the Coulomb glass

AbstractIt has been suggested in the literature to use a kind of a modified Edwards–Anderson (EA) order parameter borrowed from the field of spin glasses in order to characterize the correlated distribution of electrons governed by the long‐range many‐body interactions in a disordered system with localized states (the Coulomb glass). So far this idea has been applied to the lattice model of the disordered media. We show by straight‐forward Monte Carlo computer simulations that such an order parameter is of little use for interacting charge systems with spatial disorder. It turns out that the EA parameter for the spatially disordered system tends to zero at much higher temperatures than in the case of the lattice. Furthermore, the value of the modified EA parameter appears dependent on the system size, which makes this parameter rather useless. (© 2008 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Strong Dynamical Heterogeneities in the Violation of the Fluctuation-Dissipation Theorem in Spin Glasses

We analyze numerically the violation of the fluctuation-dissipation theorem (FDT) in the +/-J Edwards-Anderson (EA) spin-glass model. Using single spin probability densities we reveal the presence of strong dynamical heterogeneities, which correlate with ground-state information. The physical interpretation of the results shows that the spins can be divided into two sets. In 3D, one set forms a compact structure which presents a coarseninglike behavior with its characteristic violation of the FDT, while the other asymptotically follows the FDT. Finally, we compare the dynamical behavior observed in 3D with 2D.

Critical behavior of the three-dimensional bond-diluted Ising spin glass: Finite-size scaling functions and universality

We study the three-dimensional (3D) bond-diluted Edwards-Anderson (EA) model with binary interactions at a bond occupation of 45% by Monte Carlo (MC) simulations. Using an efficient cluster MC algorithm we are able to determine the universal finite-size scaling (FSS) functions and the critical exponents with high statistical accuracy. We observe small corrections to scaling for the measured observables. The critical quantities and the FSS functions indicate clearly that the bond-diluted model for dilutions above the critical dilution ${p}^{*}$, at which a spin glass (SG) phase appears, lies in the same universality class as the 3D undiluted EA model with binary interactions. A comparison with the FSS functions of the 3D site-diluted EA model with Gaussian interactions at a site occupation of 62.5% gives very strong evidence for the universality of the SG transition in the 3D EA model.

Temperature Chaos and Bond Chaos in Edwards-Anderson Ising Spin Glasses: Domain-Wall Free-Energy Measurements

Domain-wall free energy sigmaF, entropy sigmaS, and the correlation function C(temp) of sigmaF are measured independently in the four-dimensional +/-J Edwards-Anderson (EA) Ising spin glass. The stiffness exponent theta, the fractal dimension of domain walls d(s), and the chaos exponent zeta are extracted from the finite-size scaling analysis of sigmaF, sigmaS, and C(temp), respectively, well inside the spin-glass phase. The three exponents are confirmed to satisfy the scaling relation zeta = d(s)/2 - theta derived by the droplet theory within our numerical accuracy. We also study bond chaos induced by random variation of bonds, and find that the bond and temperature perturbations yield the universal chaos effects described by a common scaling function and the chaos exponent. These results strongly support the appropriateness of the droplet theory for the description of chaos effect in the EA Ising spin glasses.

Cluster Monte Carlo Algorithms for Diluted Spin Glasses

Recently a cluster Monte Carlo algorithm has been used very successfully in the two-dimensional Edwards-Anderson (EA) model. We show that this algorithm and a variant thereof can also be used successfully in models with a non-zero spin glass transition temperature. The application of such algorithms to the site-diluted EA model in three dimensions is discussed and the efficiency of the two algorithms is compared among each other and to parallel tempering. Finally, we give evidence for a spin glass transition in the three-dimensional site-diluted EA model with Gaussian couplings at a site occupation of p = 62.5 %.