In this chapter we present some results on numerical simulation in three-dimensional spin glasses. In fact, there are problems for which the analytical approach is out of reach. In such cases numerical simulations are a source of hints and suggestions. Their robustness is based on the fact that the asymptotic properties of the model for large volumes are identified. After describing the standard algorithm used (parallel tempering) we address the following problems: overlap equivalence among site and link overlaps, ultrametricity or hierarchical organization of the equilibrium states, decay of correlations, pure state identification, energy interfaces, and stiffness exponents related to the lower critical dimension. Introduction Numerical simulations have played a crucial role in the development of spin glass theory, especially in those cases in which exact results are unavailable due to formidable mathematical difficulties. Some of the first systematic works were by Bray and Moore (1984), Ogielski and Morgenstern (1985), and Bhatt and Young (1988). For a recent review, see Marinari et al . (1997) and the references therein. It is also interesting to check Newman and Stein (1996) and Marinari et al . (2000) for examples of the difficulties of reconciling theory with numerical simulations in finite-dimensional spin glasses. The general outcome of these (and other) numerical studies is that many features of themean-field theory are seen in the ever larger finite-volume systems accessible to numerical simulations. However, no definite conclusions can be reached by means of the sole use of computers.
In this chapter we present some results on numerical simulation in three-dimensional spin glasses. In fact, there are problems for which the analytical approach is out of reach. In such cases numerical simulations are a source of hints and suggestions. Their robustness is based on the fact that the asymptotic properties of the model for large volumes are identified. After describing the standard algorithm used (parallel tempering) we address the following problems: overlap equivalence among site and link overlaps, ultrametricity or hierarchical organization of the equilibrium states, decay of correlations, pure state identification, energy interfaces, and stiffness exponents related to the lower critical dimension. Introduction Numerical simulations have played a crucial role in the development of spin glass theory, especially in those cases in which exact results are unavailable due to formidable mathematical difficulties. Some of the first systematic works were by Bray and Moore (1984), Ogielski and Morgenstern (1985), and Bhatt and Young (1988). For a recent review, see Marinari et al . (1997) and the references therein. It is also interesting to check Newman and Stein (1996) and Marinari et al . (2000) for examples of the difficulties of reconciling theory with numerical simulations in finite-dimensional spin glasses. The general outcome of these (and other) numerical studies is that many features of themean-field theory are seen in the ever larger finite-volume systems accessible to numerical simulations. However, no definite conclusions can be reached by means of the sole use of computers.
Spin glass behavior was first seen in metallic alloys with magnetic impurities dispersed randomly in the main non-magnetic component. Due to the simultaneous presence of quenched disorder and competing ferromagnetic and antiferromagnetic interactions, the nonequilibrium behaviors of spin glasses are intricate and difficult to understand. Experimental research on spin glass materials indicate extremely slow dynamics, suggesting that the direct numerical simulation using local spin flips will take impractical time scales to replicate the nonequilibrium behaviors. This thesis introduces efficient algorithms and heuristics for the numerical simulation of simple models for spin glasses and discusses significant simulation results. The results provide insights into understanding the nonequilibrium behaviors, especially aging and memory, through studying microscopic constituent configurations. The methodology and conclusion presented in this thesis can be potentially extended to address general questions about disordered materials at nonequilibrium. Chapter 1 provides a thorough overview of spin glasses. A detailed introduction on complex nonequilibrium behaviors, including slow dynamics and memory effects, seen in typical experiments of spin glass materials, is presented. A general discussion about the characterization and categorization of memory effects found in many out-of-equilibrium systems is highlighted. Two prevalent theoretical models on spin glasses, the droplet and replica symmetry breaking theory, are described. Lastly, algorithm development and simulation techniques used to study spin glasses are covered. In Chapter 2, a patchwork heuristic is applied to accelerate the relaxation of spin glass models in dimensions d=1 and d=2 with the goal of investigating the dynamics of loss and recovery of spin configurations. A well-designed simulation protocol, consists of a aging and recovery stage, is used to mimic the temperature cycling experiment, where the temperature variations are replaced by the variations of the spin coupling realization. Eventual recovery of memory upon coarsening is seen in two-dimensional Ising spin glasses and one-dimensional clock models, while one-dimensional Ising spin systems do not exhibit configuration memory. A central result is that the coarsening recovery length scale sc grows faster than the aging length scale ℓ as a power law relation sc∼ℓ2.58. In Chapter 3, the microscopic dependence of spin glass equilibration on sample history, i.e., the initial configuration and the specific noise history of stochastic dynamics, is examined. The patchwork dynamics is used to simulate the glassy dynamics over a wide range of length scales. The dependence of the nearest-neighbor spin correlations at long time on initial spin configurations and on noise histories are compared. Most of the local spin correlations are independent of the details of evolution, due to the formation of rigid domains. The correlations on a fractal set of domain walls are found to have distinct dependence on the noise history and the initial state. The initial ``nature'' of the state is retained in the domain walls under coarsening more than the ``nurture'' of the noise, while the location of the domain walls at each scale is determined by the frozen disorder (``environment''). We also provide evidence that coarsening with local dynamics at finite temperature gives persistence of overlaps roughly consistent with coarsening at zero temperature, at least over short distances, supporting the use of zero-temperature heuristics to address finite-temperature dynamics and memory.
As we have seen on several occasions, a great many of the physical phenomena of interest concern collective behaviour or strongly correlated particles. While it is sometimes possible to obtain useful insight using approximation methods, such as perturbation theory, reliable controlled approximations do not exist. In such situations, numerical simulations have become an indispensable tool. We shall not attempt in this chapter to discuss computer programming nor shall we discuss many of the powerful methods, both algorithmic and in data analysis, that have been developed since numerical simulations have become an important research tool. Instead, we shall assume that the reader already has enough familiarity with computer programming to apply the notions which we discuss here. Our goal is to explain enough of the basics of the Monte Carlo method (classical and quantum) to allow the reader to apply it readily to interesting equilibrium statistical physics problems of the type discussed in this book. The problems in this chapter have been tailored with this in mind. The programs needed are relatively simple and the physics illustrates many of the phenomena already seen: phase transitions with and without spontaneous symmetry breaking, critical exponents, diverging susceptibilities, scaling, vortices, superfluids etc. Consequently, some of the problems are somewhat long and may be better considered as mini-projects. Markov chains, convergence and detailed balance Even relatively innocent looking models are impossible to solve exactly. The exact solution of the two-dimensional Ising model is known but not in the presence of an external magnetic field. In three dimensions, even in the absence of a magnetic field, the exact solution is not known. In fact, relatively few models can be solved exactly; see Reference [14] for an excellent review of the subject.
We present a massive equilibrium simulation of the three-dimensional Ising spin glass at low temperatures. The Janus special-purpose computer has allowed us to equilibrate, using parallel tempering, L = 32 lattices down to T ≈ 0.64 T c . We demonstrate the relevance of equilibrium finite size simulations to understanding experimental non-equilibrium spin glasses in the thermodynamical limit by establishing a time-length dictionary. We conclude that non-equilibrium experiments performed on a timescale of 1 h can be matched with equilibrium results on L ≈ 110 lattices. A detailed investigation of the probability distribution functions of the spin and link overlap, as well as of their correlation functions, shows that Replica Symmetry Breaking is the appropriate theoretical framework for the physically relevant length scales. Besides, we improve over existing methodologies in ensuring equilibration in parallel tempering simulations.
The main subjects of this dissertation are spin glass applications in other disciplines and spin glass algorithms. Spin glasses are magnetic systems with disorder and frustration, and the essential physics of spin glasses lies not in the details of their microscopic interactions but rather in the competition between quenched ferromagnetic and antiferromagnetic interactions. Concepts that arose in the study of spin glasses have led to applications in areas as diverse as computer science, biology, and finance, as well as a variety of others.\n\nIn the first part of this dissertation I study the equilibrium and non-equilibrium properties of Boolean decision problems with competing interactions on scale-free networks in an external bias (a magnetic field). First, I perform finite-temperature Monte Carlo simulations in a field to test the robustness of the spin-glass phase and I show that the system has a spin-glass phase in a field, i.e., it exhibits a de Almeida–Thouless line. Then I study avalanche distributions when the system is driven by a field at zero temperature to test whether the system displays self-organized criticality. The numerical results suggest that avalanches (damage) can spread across the entire system with nonzero probability when the decay exponent of the interaction degree is less than or equal to 2, i.e., that Boolean decision problems on scale-free networks with competing interactions can be fragile when the system is not in thermal equilibrium.\n\nIn the second part of this dissertation I discuss the best-case performance of quantum annealers on native spin-glass benchmarks, i.e., how chaos can affect success probabilities. We perform classical parallel-tempering Monte Carlo simulations of the archetypal benchmark problem, an Ising spin glass, on the native chip topology. Using realistic uncorrelated noise models for the D-Wave Two quantum annealer, I study the best-case resilience, or the probability that the ground-state configuration is not affected by random fields and random-bond fluctuations found on the chip. We compute the upper-bound success probabilities for different instance classes based on these simple error models, and I present strategies for developing robust and hard benchmark instances.\n\nIn the third part of this dissertation I present a cluster algorithm for Ising spin glasses that works in any space dimension and speeds up thermalization by several orders of magnitude at temperatures where thermalization is typically difficult. Our isoenergetic cluster moves are based on the Houdayer cluster algorithm for two-dimensional spin glasses and lead to a speedup over conventional state-of-the-art methods that increases with the system size. We illustrate the benefits (improved thermalization and achievement of more equiprobable sampling of ground states) of the isoenergetic cluster moves in two and three space dimensions, as well as in the nonplanar Chimera topology found in the D-Wave quantum annealing machine.\n\nFinally, I study the thermodynamic properties of the two-dimensional Edwards-Anderson Ising spin-glass model on a square lattice using the tensor renormalization group method based on a higher-order singular-value decomposition. Our estimates of the partition function without a high precision data type lead to negative values at very low temperatures, thus illustrating that the method can not be applied to frustrated magnetic systems.
The statics-dynamics correspondence in spin glasses relate non-equilibrium results on large samples (the experimental realm) with equilibrium quantities computed on small systems (the typical arena for theoretical computations). Here we employ statics-dynamics equivalence to study the Ising spin-glass critical behavior in three dimensions. By means of Monte Carlo simulation, we follow the growth of the coherence length (the size of the glassy domains), on lattices too large to be thermalized. Thanks to the large coherence lengths we reach, we are able to obtain accurate results in excellent agreement with the best available equilibrium computations. To do so, we need to clarify the several physical meanings of the dynamic exponent close to the critical temperature.
Spin glasses have been the subject of intense study and considerable controversy for decades, and the low-temperature phase of short-range spin glasses is still poorly understood. Our main goal is to improve our understanding in this area and find an answer to the following question: Are there only a single pair or a countable infinity of pure states in the low temperature phase of the EA spin glass? To that aim we first start by introducing spin glasses and provide a brief history of their research, then proceed to describe our method of simulation, the parallel tempering Monte Carlo algorithm. Next, we present the results of a large-scale numerical study of the equilibrium three-dimensional Edwards-Anderson Ising spin glass with Gaussian disorder. In order to understand how the parallel tempering algorithm works, we measure various static, as well as dynamical quantities, such as the autocorrelation times and round-trip times for the parallel tempering Monte Carlo method. We examine the correlation between static and dynamic observables for _ 5000 disorder realizations and up to 1000 spins down to temperatures at 20% of the critical temperature, and our results show that autocorrelation times are directly correlated with the roughness of the free-energy landscape. In the following chapters, the three- and four-dimensional Edwards-Anderson and mean-field Sherrington-Kirkpatrick Ising spin glasses are studied again via large-scale Monte Carlo simulations at low temperatures, deep within the spinglass phase. Performing a careful statistical analysis of several thousand independent disorder realizations and using an observable that detects peaks in the overlap distribution, we show that the Sherrington-Kirkpatrick and Edwards-Anderson models have a distinctly different low-temperature behavior. We arrive to the following conclusion: The structure of the spin-glass overlap distribution for the Edwards-Anderson model suggests that its low-temperature phase has only a single pair of pure states. Finally we present results for several new observables, along with a few preliminary studies and suggestions for future research.