Edwards-Anderson Spin Glass Research Articles

Quantitative Disorder Effects in Low-Dimensional Spin Systems

The Imry–Ma phenomenon, predicted in 1975 by Imry and Ma and rigorously established in 1989 by Aizenman and Wehr, states that first-order phase transitions of low-dimensional spin systems are ‘rounded’ by the addition of a quenched random field coupled to the quantity undergoing the transition. The phenomenon applies to a wide class of spin systems in dimensions d≤2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d\\le 2$$\\end{document} and to spin systems possessing a continuous symmetry in dimensions d≤4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d\\le 4$$\\end{document}. This work provides quantitative estimates for the Imry–Ma phenomenon: in a cubic domain of side length L, we study the effect of the boundary conditions on the spatial and thermal average of the quantity coupled to the random field. We show that the boundary effect diminishes at least as fast as an inverse power of loglogL\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\log \\log L$$\\end{document} in general two-dimensional spin systems. For systems possessing a continuous symmetry, we show that the boundary effect diminishes at least as fast as an inverse power of L in two and three dimensions and at least as fast as an inverse power of loglogL\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\log \\log L$$\\end{document} in four dimensions. Finally, we establish a partial uniqueness results for translation-covariant Gibbs states, and prove that, for almost every realization of the random field, all such states must agree on the thermally-averaged value of the quantity coupled to the random field. Specific models of interest for the obtained results include the random-field q-state Potts model, the Edwards-Anderson spin glass model, and the random-field spin O(n) models.

Collective Monte Carlo updates through tensor network renormalization

We introduce a Metropolis-Hastings Markov chain for Boltzmann distributions of classical spin systems. It relies on approximate tensor network contractions to propose correlated collective updates at each step of the evolution. We present benchmark computations for a wide variety of instances of the two-dimensional Ising model, including ferromagnetic, antiferromagnetic, (fully) frustrated and Edwards-Anderson spin glass instances, and we show that, with modest computational effort, our Markov chain achieves sizeable acceptance rates, even in the vicinity of critical points. In each of the situations we have considered, the Markov chain compares well with other Monte Carlo schemes such as the Metropolis or Wolff’s algorithm: equilibration times appear to be reduced by a factor that varies between 4040 and 20002000, depending on the model and the observable being monitored. We also present an extension to three spatial dimensions, and demonstrate that it exhibits fast equilibration for finite ferro- and antiferromagnetic instances. Additionally, and although it is originally designed for a square lattice of finite degrees of freedom with open boundary conditions, the proposed scheme can be used as such, or with slight modifications, to study triangular lattices, systems with continuous degrees of freedom, matrix models, a confined gas of hard spheres, or to deal with arbitrary boundary conditions.

Canonical Monte Carlo multispin cluster method

We present a new Canonical Multispin-flip Cluster Monte Carlo algorithm for Ising model and describe efficient implementations for hybrid supercomputer. Our method takes advantage of the computing architecture for parallel and multi-threaded operations and uses a sequential cluster state update scheme. Due to the peculiarity of the implementation, the method is more effective for models with a restricted radius of interaction. It is based on combining a random selection of spin cluster by the Monte Carlo method with a complete enumeration of the all states of the selected cluster. To show how it works we applied our method to models of interacting magnetic Ising-moments: ferromagnetic Ising model, the Edwards–Anderson spin glass model, dipolar spin ice on hexagonal and pentagonal Cairo lattices.

Searching for spin glass ground states through deep reinforcement learning

Spin glasses are disordered magnets with random interactions that are, generally, in conflict with each other. Finding the ground states of spin glasses is not only essential for understanding the nature of disordered magnets and many other physical systems, but also useful to solve a broad array of hard combinatorial optimization problems across multiple disciplines. Despite decades-long efforts, an algorithm with both high accuracy and high efficiency is still lacking. Here we introduce DIRAC – a deep reinforcement learning framework, which can be trained purely on small-scale spin glass instances and then applied to arbitrarily large ones. DIRAC displays better scalability than other methods and can be leveraged to enhance any thermal annealing method. Extensive calculations on 2D, 3D and 4D Edwards-Anderson spin glass instances demonstrate the superior performance of DIRAC over existing methods. The presented framework will help us better understand the nature of the low-temperature spin-glass phase, which is a fundamental challenge in statistical physics. Moreover, the gauge transformation technique adopted in DIRAC builds a deep connection between physics and artificial intelligence. In particular, this opens up a promising avenue for reinforcement learning models to explore in the enormous configuration space, which would be extremely helpful to solve many other hard combinatorial optimization problems.

Uniqueness of Ground State in the Edwards–Anderson Spin Glass Model

It is proven rigorously that the ground state in the Edwards-Anderson spin glass model is unique in any dimension for almost all continuous random exchange interactions under a condition that a single spin breaks the global ${\mathbb Z}_2$ symmetry. This theorem implies that replica symmetry breaking does not occur at zero temperature. The site- and bond-overlap are concentrated at their maximal values. It is argued that behaviors of short range spin glass models are much different from those of mean field spin glass models near zero temperature. Errata have been attached at the final page.

Changing the universality class of the three-dimensional Edwards-Anderson spin-glass model by selective bond dilution

The three-dimensional Edwards-Anderson spin-glass model present strong spatial heterogeneities well characterized by the so-called backbone, a magnetic structure that arises as a consequence of the properties of the ground state and the low-excitation levels of such a frustrated Ising system. Using extensive Monte Carlo simulations and finite size scaling, we study how these heterogeneities affect the phase transition of the model. Although, we do not detect any significant difference between the critical behavior displayed by the whole system and that observed inside and outside the backbone, surprisingly, a selective bond dilution of the complement of this magnetic structure induces a change of the universality class, whereas no change is noted when the backbone is fully diluted. This finding suggests that the region surrounding the backbone plays a more relevant role in determining the physical properties of the Edwards-Anderson spin-glass model than previously thought. Furthermore, we show that when a selective bond dilution changes the universality class of the phase transition, the ground state of the model does not undergo any change. The opposite case is also valid, i. e., a dilution that does not change the critical behavior significantly affects the fundamental level.

Nonflat histogram techniques for spin glasses.

We study the bimodal Edwards-Anderson spin glass comparing established methods, namely the multicanonical method, the 1/k ensemble, and parallel tempering, to an approach where the ensemble is modified by simulating power-law-shaped histograms in energy instead of flat histograms as in the standard multicanonical case. We show that by this modification a significant speed-up in terms of mean round-trip times can be achieved for all lattice sizes taken into consideration.

Record-dynamics description of spin-glass thermoremanent magnetization: A numerical and analytical study.

Thermoremanent magnetization data for the three-dimensional Edwards-Anderson (EA) spin glass are generated using the waiting time method as a simulational tool and interpreted using record dynamics. We verify that clusters of contiguous spins are overturned by quakes, nonequilibrium events linked to record-sized energy fluctuations, and we show that quaking is a log-Poisson process, i.e., a Poisson process whose average depends on the logarithm of the system age, counted from the initial quench. Our findings compare favorably with experimental thermoremanent magnetization findings and with the spontaneous fluctuation dynamics of the EA model. The logarithmic growth of the size of overturned clusters is related to similar experimental results and to the growing length scale of the spin-spin spatial correlation function. The analysis buttresses the applicability of the waiting time method as a simulational tool, and of record dynamics as a coarse-graining method for aging dynamics.

High-precision studies of domain-wall properties in the 2D Gaussian Ising spin glass

In two dimensions, short-range spin glasses order only at zero temperature, where efficient combinatorial optimization techniques can be used to study these systems with high precision. The use of large system sizes and high statistics in disorder averages allows for reliable finite-size extrapolations to the thermodynamic limit. Here, we use a recently introduced mapping of the Ising spin-glass ground-state problem to a minimum-weight perfect matching problem on a sparse auxiliary graph to study square-lattice samples of up to 10 000 × 10 000 spins. We propose a windowing technique that allows to extend this method, that is formally restricted to planar graphs, to the case of systems with fully periodic boundary conditions. These methods enable highly accurate estimates of the spin-stiffness exponent and domain-wall fractal dimension of the 2D Edwards-Anderson spin-glass with Gaussian couplings. Studying the compatibility of domain walls in this system with traces of stochastic Loewner evolution (SLE), we find a strong dependence on boundary conditions and compatibility with SLE only for one out of several setups.

Distribution of interevent avalanche times in disordered and frustrated spin systems

Hysteresis loops and the associated avalanche statistics of spin systems, such as the random-field Ising and Edwards-Anderson spin-glass models, have been extensively studied. A particular focus has been on self-organized criticality, manifest in power-law distributions of avalanche sizes. Considerably less work has been done on the statistics of the times between avalanches. This paper considers this issue, generalizing the work of Nampoothiri et al. [Phys. Rev. E 96, 032107 (2017)] in one space dimension to higher space dimensions. In addition to the interevent statistics of all avalanches, we also consider what happens when events are restricted to those exceeding a certain threshold size. Doing so raises the possibility of altering the definition of time to count the number of small events between the large ones, which provides for an analog to the concept of natural time introduced by the geophysics community with the goal of predicting patterns in seismic events. We analyze the distribution of time and natural time intervals both in the case of models that include only nearest-neighbor interactions, as well as models with (sparse) long-range couplings.

Mesoscopic real-space structures in spin-glass aging: The Edwards-Anderson model

Isothermal simulational data for the 3D Edwards-Anderson spin glass are collected at several temperatures below $T_{\rm c}$ and, in analogy with a recent model of dense colloidal suspensions,interpreted in terms of clusters of contiguous spins overturned by quakes, non-equilibrium events linked to record sized energy fluctuations. We show numerically that, to a good approximation, these quakes are statistically independent and constitute a Poisson process whose average grows logarithmically in time. The overturned clusters are local projections on one of the two ground states of the model, and grow likewise logarithmically in time. Data collected at different temperatures $T$ can be collapsed by scaling them with $T^{1.75}$, a hitherto unnoticed feature of the E-A model, which we relate on the one hand to the geometry of configuration space and on the other to experimental memory and rejuvenation effects. The rate at which a cluster flips is shown to decrease exponentially with the size of the cluster, as recently assumed in a coarse grained model of dense colloidal dynamics. The evolving structure of clusters in real space is finally sssociated to the decay of the thermo-remanent magnetization. Our analysis provides an unconventional coarse-grained description of spin glass aging as statistically subordinated to a Poisson quaking process and highlights record dynamics as a viable common theoretical framework for aging in different systems.

Fractal dimension of interfaces in Edwards-Anderson spin glasses for up to six space dimensions.

The fractal dimension of domain walls produced by changing the boundary conditions from periodic to antiperiodic in one spatial direction is studied using both the strong-disorder renormalization group algorithm and the greedy algorithm for the Edwards-Anderson Ising spin-glass model for up to six space dimensions. We find that for five or fewer space dimensions, the fractal dimension is lower than the space dimension. This means that interfaces are not space filling, thus implying that replica symmetry breaking is absent in space dimensions fewer than six. However, the fractal dimension approaches the space dimension in six dimensions, indicating that replica symmetry breaking occurs above six dimensions. In two space dimensions, the strong-disorder renormalization group results for the fractal dimension are in good agreement with essentially exact numerical results, but the small difference is significant. We discuss the origin of this close agreement. For the greedy algorithm there is analytical expectation that the fractal dimension is equal to the space dimension in six dimensions and our numerical results are consistent with this expectation.

Analysis and optimization of population annealing.

Population annealing is an easily parallelizable sequential Monte Carlo algorithm that is well suited for simulating the equilibrium properties of systems with rough free-energy landscapes. In this work we seek to understand and improve the performance of population annealing. We derive several useful relations between quantities that describe the performance of population annealing and use these relations to suggest methods to optimize the algorithm. These optimization methods were tested by performing large-scale simulations of the three-dimensional (3D) Edwards-Anderson (Ising) spin glass and measuring several observables. The optimization methods were found to substantially decrease the amount of computational work necessary as compared to previously used, unoptimized versions of population annealing. We also obtain more accurate values of several important observables for the 3D Edwards-Anderson model.

Number of thermodynamic states in the three-dimensional Edwards-Anderson spin glass

The question of the number of thermodynamic states present in the low-temperature phase of the three-dimensional Edwards-Anderson Ising spin glass is addressed by studying spin and link overlap distributions using population annealing Monte Carlo simulations. We consider overlaps between systems with the same boundary condition-which are the usual quantities measured-and also overlaps between systems with different boundary conditions, both for the full systems and also within a smaller window within the system. Our results appear to be fully compatible with a single pair of pure states such as in the droplet/scaling picture. However, our results for whether or not domain walls induced by changing boundary conditions are space filling or not are also compatible with scenarios having many thermodynamic states, such as the chaotic pairs picture and the replica symmetry breaking picture. The differing results for spin overlaps in same and different boundary conditions suggest that finite-size effects are very large for the system sizes currently accessible in low-temperature simulations.

Dynamic scaling in the two-dimensional Ising spin glass with normal-distributed couplings

We carry out simulated annealing and employ a generalized Kibble-Zurek scaling hypothesis to study the two-dimensional Ising spin glass with normal-distributed couplings. The system has an equilibrium glass transition at temperature T=0. From a scaling analysis when T→0 at different annealing velocities v, we find power-law scaling in the system size for the velocity required in order to relax toward the ground state, v∼L^{-(z+1/ν)}, the Kibble-Zurek ansatz where z is the dynamic critical exponent and ν the previously known correlation-length exponent, ν≈3.6. We find z≈13.6 for both the Edwards-Anderson spin-glass order parameter and the excess energy. This is different from a previous study of the system with bimodal couplings [Rubin etal., Phys. Rev. E 95, 052133 (2017)2470-004510.1103/PhysRevE.95.052133] where the dynamics is faster (z is smaller) and the above two quantities relax with different dynamic exponents (with that of the energy being larger). We argue that the different behaviors arise as a consequence of the different low-energy landscapes: for normal-distributed couplings the ground state is unique (up to a spin reflection), while the system with bimodal couplings is massively degenerate. Our results reinforce the conclusion of anomalous entropy-driven relaxation behavior in the bimodal Ising glass. In the case of a continuous coupling distribution, our results presented here also indicate that, although Kibble-Zurek scaling holds, the perturbative behavior normally applying in the slow limit breaks down, likely due to quasidegenerate states, and the scaling function takes a different form.

Jump events in a 3D Edwards–Anderson spin glass

The statistical properties of infrequent particle displacements, greater than a certain distance, are known as jump dynamics in the context of structural glass formers. We generalize the concept of a jump to the case of a spin glass, by dividing the system into small boxes, and considering the infrequent cooperative spin flips in each box. Jumps defined this way share similarities with jumps in structural glasses. We perform numerical simulations for the 3D Edwards–Anderson model, and study how the properties of these jumps depend on the waiting time after a quench. Similar to the results for structural glasses, we find that while jump frequency depends strongly on time, the jump duration and jump length are roughly stationary. At odds with some results reported on studies of structural glass formers, at long enough times, the rest time between jumps varies as the inverse of jump frequency. We give a possible explanation for this discrepancy. We also find that our results are qualitatively reproduced by a fully-connected trap model.

Numerical simulations of Ising spin glasses with free boundary conditions: The role of droplet excitations and domain walls.

The relative importance of the contributions of droplet excitations and domain walls on the ordering of short-range Edwards-Anderson spin glasses in three and four dimensions is studied. We compare the spin overlap distribution functions of periodic and free boundary conditions using population annealing Monte Carlo. For system sizes up to about 1000 spins, spin glasses show nontrivial spin overlap distributions. Periodic boundary conditions may trap diffusive domain walls which can contribute to small spin overlaps, and the other contribution is the existence of low-energy droplet excitations within the system. We use free boundary conditions to minimize domain-wall effects, and show that low-energy droplet excitations are the major contribution to small overlaps in numerical simulations. Free boundary conditions has stronger finite-size effects, and is likely to have the same thermodynamic limit with periodic boundary conditions.

No percolation in low temperature spin glass

We consider the Edwards-Anderson Ising Spin Glass model for temperatures $T\geq 0.$ We define notions of Boltzmann-Gibbs measure for the Edwards-Anderson spin glass at a given temperature, and of unsatisfied (frustrated) edges, and recall the notion of ground states. We prove that for low positive temperatures, in almost every spin configuration the graph formed by the unsatisfied edges is made of finite connected components. Similarly, for zero temperature, we show that in almost every ground state the graph of unsatisfied edges is a forest all of whose components are finite. In other words, for low enough temperatures the unsatisfied edges do not percolate.

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.

Disordered XYZ spin chain simulations using the spectrum bifurcation renormalization group

We study the disordered XYZ spin chain using the recently developed Spectrum Bifurcation Renormalization Group (SBRG) numerical method. With strong disorder, the phase diagram consists of three many body localized (MBL) spin glass phases. We argue that, with sufficiently strong disorder, these spin glass phases are separated by marginally many-body localized (MBL) critical lines. We examine the critical lines of this model by measuring the entanglement entropy and Edwards-Anderson spin glass order parameter, and find that the critical lines are characterized by an effective central charge c'=ln2. Our data also suggests continuously varying critical exponents along the critical lines. We also demonstrate how long-range mutual information can distinguish these phases.