Three-dimensional Edwards-Anderson Model Research Articles

Overlap renormalization group transformations for disordered systems

We establish a renormalization group approach which is implemented on the degrees of freedom defined by the overlap of two replicas to determine the critical fixed point and to extract four critical exponents for the phase transition of the three-dimensional Edwards-Anderson model. In addition, we couple the overlap order parameter to a fictitious field and introduce it within the two-replica Hamiltonian of the system to study its explicit symmetry-breaking with the renormalization group. Overlap transformations do not require a renormalization of the random couplings of a system to extract the critical exponents associated with the relevant variables of the renormalization group. We conclude by discussing the applicability of such transformations in the study of any phase transition which is fully characterized by an overlap order parameter.

Distribution of metastable states of spin glasses

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 high-dimensional 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 two-dimensional and the three-dimensional Edwards-Anderson model and compare with theoretical predictions.

Distribution of metastable states of Ising spin glasses

Local minima also known as inherent structures are expected to play an essential role for the behavior of spin glasses. Here, we propose techniques to efficiently sample these configurations in Monte Carlo simulations. For the Sherrington-Kirkpatrick and the three-dimensional Edwards-Anderson model their spectra are determined and compared to analytical results.

Nonequilibrium dynamics of the three-dimensional Edwards-Anderson spin-glass model with Gaussian couplings: strong heterogeneities and the backbone picture

We numerically study the three-dimensional Edwards-Anderson model with Gaussian couplings, focusing on the heterogeneities arising in its nonequilibrium dynamics. Results are analyzed in terms of the backbone picture, which links strong dynamical heterogeneities to spatial heterogeneities emerging from the correlation of local rigidity of the bond network. Different two-times quantities as the flipping time distribution and the correlation and response functions, are evaluated over the full system and over high- and low-rigidity regions. We find that the nonequilibrium dynamics of the model is highly correlated to spatial heterogeneities. Also, we observe a similar physical behavior to that previously found in the Edwards-Anderson model with a bimodal (discrete) bond distribution. Namely, the backbone behaves as the main structure that supports the spin-glass phase, within which a sort of domain-growth process develops, while the complement remains in a paramagnetic phase, even below the critical temperature.

Numerical study of the overlap Lee–Yang singularities in the three-dimensional Edwards–Anderson model

We have characterized numerically, using the Janus computer, the Lee–Yang complex singularities related to the overlap in the 3D Ising spin-glass with binary couplings over a wide range of temperatures (both in the critical and in the spin-glass phase). Studying the behavior of the zeros at the critical point, we have obtained an accurate measurement of the anomalous dimension in very good agreement with the values quoted in the literature. In addition, by studying the density of the zeros we have been able to characterize the phase transition and to investigate the Edwards–Anderson order parameter in the spin-glass phase, finding agreement with the values obtained using more conventional techniques.

Sampling the ground-state magnetization ofd-dimensionalp-body Ising models

We demonstrate that a recently introduced heuristic optimization algorithm [Phys. Rev. E 83, 046709 (2011)] that combines a local search with triadic crossover genetic updates is capable of sampling nearly uniformly among ground-state configurations in spin-glass-like Hamiltonians with $p$-spin interactions in $d$ space dimensions that have highly degenerate ground states. Using this algorithm we probe the zero-temperature ferromagnet to spin-glass transition point ${q}_{c}$ of two example models, the disordered version of the two-dimensional three-spin Baxter-Wu model [${q}_{c}=0.1072(1)$] and the three-dimensional Edwards-Anderson model [${q}_{c}=0.2253(7)$], by computing the Binder ratio of the ground-state magnetization.

Ultrametricity in the Edwards-Anderson Model

We test the property of ultrametricity for the spin-glass three-dimensional Edwards-Anderson model in zero magnetic field with numerical simulations up to 20(3) spins. We find an excellent agreement with the prediction of the mean field theory. Since ultrametricity is not compatible with a trivial structure of the overlap distribution, our result contradicts the droplet theory.

Factorization properties in the three-dimensional Edwards-Anderson model

Starting from the study of a linear combination of multi-overlaps which can be rigorously shown to vanish for large systems we numerically analyze the factorization properties of the link-overlaps multi-distribution for the 3D Gaussian Edward-Anderson spin-glass model. We find evidence of a pure factorization law for the multi-correlation functions. For instance the quantity [<Q_{12}^2> - <Q_{12}Q_{34}>]/<Q_{12}^2> tends to zero at increasing volumes. We also perform the same analysis for the standard overlap for which instead the lack of factorization persists increasing the size of the system. The necessity of a better understanding of the mutual relation between the two overlaps is pointed out.

“Glassy Dynamics” in Ising Spin Glasses –Experiment and Simulation–

The field-cooled magnetization (FCM) processes of Ising spin glasses under relatively small fields are investigated by experiment on Fe_{0.55}Mn_{0.45}TiO_3 and by numerical simulation on the three-dimensional Edwards-Anderson model. Both results are explained in a unified manner by means of the droplet picture. In particular, the cusp-like behavior of the FCM is interpreted as evidence, not for an equilibrium phase transition under a finite magnetic field, but for a dynamical (`blocking') transition frequently observed in glassy systems.

Measuring the fluctuation-dissipation ratio in glassy systems with no perturbing field.

A method is presented for measuring the integrated linear response in Ising spin system without applying any perturbing field. Large-scale simulations are performed in order to show how the method works. Very precise measurements of the fluctuation-dissipation ratio are presented for three different Ising models: the two-dimensional ferromagnetic model, the mean-field diluted three-spin model, and the three-dimensional Edwards-Anderson model.

Universal Finite-Size Scaling Functions in the 3D Ising Spin Glass

We study the three-dimensional Edwards-Anderson model with binary interactions by Monte Carlo simulations. Direct evidence of finite-size scaling is provided, and the universal finite-size scaling functions are determined. Monte Carlo data are extrapolated to infinite volume with an iterative procedure up to correlation lengths xi \approx 140. The infinite volume data are consistent with a conventional power law singularity at finite temperature Tc. Taking into account corrections to scaling, we find Tc = 1.156 +/- 0.015, nu = 1.8 +/- 0.2 and eta = -0.26 +/- 0.04. The data are also consistent with an exponential singularity at finite Tc, but not with an exponential singularity at zero temperature.