Sherrington-Kirkpatrick Model Research Articles

Griffiths-Type Theorems for Short-Range Spin Glass Models

We establish relations between different characterizations of order in spin glass models. We first prove that the broadening of the replica overlap distribution indicated by a nonzero standard deviation of the replica overlap R1,2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$R^{1,2}$$\\end{document} implies the non-differentiability of the two-replica free energy with respect to the replica coupling parameter λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda $$\\end{document}. In Z2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {Z}}_2$$\\end{document} invariant models such as the standard Edwards–Anderson model, the non-differentiability is equivalent to the spin glass order characterized by a nonzero Edwards–Anderson order parameter. This generalization of Griffiths’ theorem is proved for any short-range spin glass models with classical bounded spins. We also prove that the non-differentiability of the two-replica free energy mentioned above implies replica symmetry breaking in the literal sense, i.e., a spontaneous breakdown of the permutation symmetry in the model with three replicas. This is a general result that applies to a large class of random spin models, including long-range models such as the Sherrington-Kirkpatrick model and the random energy model. There is a 25-minute video that explains the main results of the present work:https://youtu.be/BF3hJiY1xvI

Complete replica solution for the transverse field Sherrington-Kirkpatrick spin glass model with continuous-time quantum Monte Carlo method

Complete replica solution for the transverse field Sherrington-Kirkpatrick spin glass model with continuous-time quantum Monte Carlo method

Maximum-Likelihood Detection with QAOA for Massive MIMO and Sherrington-Kirkpatrick Model with Local Field at Infinite Size

Maximum-Likelihood Detection with QAOA for Massive MIMO and Sherrington-Kirkpatrick Model with Local Field at Infinite Size

Bounds on the covariance matrix of the Sherrington–Kirkpatrick model

Bounds on the covariance matrix of the Sherrington–Kirkpatrick model

Wavelength-division multiplexing optical Ising simulator enabling fully programmable spin couplings and external magnetic fields.

Recently various physical systems have been proposed for modeling Ising spin Hamiltonians appealing to solve combinatorial optimization problems with remarkable performance. However, how to implement arbitrary spin-spin interactions is a critical and challenging problem in unconventional Ising machines. Here, we propose a general gauge transformation scheme to enable arbitrary spin-spin interactions and external magnetic fields as well, by decomposing an Ising Hamiltonian into multiple Mattis-type interactions. With this scheme, a wavelength-division multiplexing spatial photonic Ising machine (SPIM) is developed to show the programmable capability of general spin coupling interactions. We exploit the wavelength-division multiplexing SPIM to simulate three spin systems: ±J models, Sherrington-Kirkpatrick models, and only locally connected J1-J2 models and observe the phase transitions. We also demonstrate the ground-state search for solving Max-Cut problem with the wavelength-division multiplexing SPIM. These results promise the realization of ultrafast-speed and high-power efficiency Boltzmann sampling to a generalized large-scale Ising model.

Bound on annealing performance from stochastic thermodynamics, with application to simulated annealing.

Annealing is the process of gradually lowering the temperature of a system to guide it towards its lowest energy states. In an accompanying paper [Y. Luo etal., Phys. Rev. E 108, L052105 (2023)10.1103/PhysRevE.108.L052105], we derived a general bound on annealing performance by connecting annealing with stochastic thermodynamics tools, including a speed limit on state transformation from entropy production. We here describe the derivation of the general bound in detail. In addition, we analyze the case of simulated annealing with Glauber dynamics in depth. We show how to bound the two case-specific quantities appearing in the bound, namely the activity, a measure of the number of microstate jumps, and the change in relative entropy between the state and the instantaneous thermal state, which is due to temperature variation. We exemplify the arguments by numerical simulations on the Sherrington-Kirkpatrick (SK) model of spin glasses.

The autoregressive neural network architecture of the Boltzmann distribution of pairwise interacting spins systems

Autoregressive Neural Networks (ARNNs) have shown exceptional results in generation tasks across image, language, and scientific domains. Despite their success, ARNN architectures often operate as black boxes without a clear connection to underlying physics or statistical models. This research derives an exact mapping of the Boltzmann distribution of binary pairwise interacting systems in autoregressive form. The parameters of the ARNN are directly related to the Hamiltonian’s couplings and external fields, and commonly used structures like residual connections and recurrent architecture emerge from the derivation. This explicit formulation leverages statistical physics techniques to derive ARNNs for specific systems. Using the Curie–Weiss and Sherrington–Kirkpatrick models as examples, the proposed architectures show superior performance in replicating the associated Boltzmann distributions compared to commonly used designs. The findings foster a deeper connection between physical systems and neural network design, paving the way for tailored architectures and providing a physical lens to interpret existing ones.

Universality of superconcentration in the Sherrington–Kirkpatrick model

AbstractWe study the universality of superconcentration for the free energy in the Sherrington–Kirkpatrick model. In [10], Chatterjee showed that when the system consists of spins and Gaussian disorders, the variance of this quantity is superconcentrated by establishing an upper bound of order , in contrast to the bound obtained from the Gaussian–Poincaré inequality. In this paper, we show that superconcentration indeed holds for any choice of centered disorders with finite third moment, where the upper bound is expressed in terms of an auxiliary nondecreasing function that arises in the representation of the disorder as for standard normal. Under an additional regularity assumption on , we further show that the variance is of order at most .

Mean-field theory is exact for Ising spin glass models with Kac potential in non-additive limit on Nishimori line

Recently, Mori (2011 Phys. Rev. E 84 031128) has conjectured that the free energy of Ising spin glass models with the Kac potential in the non-additive limit, such as the power-law potential in the non-additive regime, is exactly equal to that of the Sherrington–Kirkpatrick model in the thermodynamic limit. In this study, we prove that his conjecture is true on the Nishimori line at any temperature in any dimension. One of the key ingredients of the proof is the use of the Gibbs–Bogoliubov inequality on the Nishimori line. We also consider the case in which the probability distribution of the interaction is symmetric, where his conjecture is true at any temperature in one dimension but is an open problem in the low-temperature regime in two or more dimensions.

Meta-learning digitized-counterdiabatic quantum optimization

The use of variational quantum algorithms for optimization tasks has emerged as a crucial application for the current noisy intermediate-scale quantum computers. However, these algorithms face significant difficulties in finding suitable ansatz and appropriate initial parameters. In this paper, we employ meta-learning using recurrent neural networks to address these issues for the recently proposed digitized-counterdiabatic quantum approximate optimization algorithm (QAOA). By combining meta-learning and counterdiabaticity, we find suitable variational parameters and reduce the number of optimization iterations required. We demonstrate the effectiveness of our approach by applying it to the MaxCut problem and the Sherrington–Kirkpatrick model. Our method offers a short-depth circuit ansatz with optimal initial parameters, thus improving the performance of the state-of-the-art QAOA.

Extended Series of Correlation Inequalities in Quantum Systems

A systematic derivation provides extended series of correlation inequalities in quantum systems. Each order in truncated Taylor expansion of the spectral representation for the Duhamel correlation function gives its lower and upper bounds. The obtained bound on the Duhamel function and the square root interpolation method enable us to derive a variational solution of specific free energy in the transverse field Sherrington-Kirkpatrick model.

On the concavity of the TAP free energy in the SK model

We analyze the Hessian of the Thouless–Anderson–Palmer (TAP) free energy for the Sherrington–Kirkpatrick model, below the de Almeida–Thouless line, evaluated in Bolthausen’s approximate solutions of the TAP equations. We show that the empirical spectral distribution weakly converges to a measure with negative support below the AT line, and that the support includes zero on the AT line. In this “macroscopic” sense, we show that TAP free energy is concave in the order parameter of the theory, i.e. the random spin-magnetizations. This proves a spectral interpretation of the AT line. We also find different magnetizations than Bolthausen’s approximate solutions at which the Hessian of the TAP free energy has positive outlier eigenvalues. In particular, when the magnetizations are assumed to be independent of the disorder, we prove that Plefka’s second condition is equivalent to all eigenvalues being negative. On this occasion, we extend the convergence result of Capitaine et al. (Electron. J. Probab. 16, no. 64, 2011) for the largest eigenvalue of perturbed complex Wigner matrices to the GOE.

Biases in inverse Ising estimates of near-critical behavior.

Inverse Ising inference allows pairwise interactions of complex binary systems to be reconstructed from empirical correlations. Typical estimators used for this inference, such as pseudo-likelihood maximization (PLM), are biased. Using the Sherrington-Kirkpatrick model as a benchmark, we show that these biases are large in critical regimes close to phase boundaries, and they may alter the qualitative interpretation of the inferred model. In particular, we show that the small-sample bias causes models inferred through PLM to appear closer to criticality than one would expect from the data. Data-driven methods to correct this bias are explored and applied to a functional magnetic resonance imaging data set from neuroscience. Our results indicate that additional care should be taken when attributing criticality to real-world data sets.

Nonequilibrium thermodynamics of the asymmetric Sherrington-Kirkpatrick model

Most natural systems operate far from equilibrium, displaying time-asymmetric, irreversible dynamics characterized by a positive entropy production while exchanging energy and matter with the environment. Although stochastic thermodynamics underpins the irreversible dynamics of small systems, the nonequilibrium thermodynamics of larger, more complex systems remains unexplored. Here, we investigate the asymmetric Sherrington-Kirkpatrick model with synchronous and asynchronous updates as a prototypical example of large-scale nonequilibrium processes. Using a path integral method, we calculate a generating functional over trajectories, obtaining exact solutions of the order parameters, path entropy, and steady-state entropy production of infinitely large networks. Entropy production peaks at critical order-disorder phase transitions, but is significantly larger for quasi-deterministic disordered dynamics. Consequently, entropy production can increase under distinct scenarios, requiring multiple thermodynamic quantities to describe the system accurately. These results contribute to developing an exact analytical theory of the nonequilibrium thermodynamics of large-scale physical and biological systems and their phase transitions.

Finite-Size Relaxational Dynamics of a Spike Random Matrix Spherical Model.

We present a thorough numerical analysis of the relaxational dynamics of the Sherrington-Kirkpatrick spherical model with an additive non-disordered perturbation for large but finite sizes N. In the thermodynamic limit and at low temperatures, the perturbation is responsible for a phase transition from a spin glass to a ferromagnetic phase. We show that finite-size effects induce the appearance of a distinctive slow regime in the relaxation dynamics, the extension of which depends on the size of the system and also on the strength of the non-disordered perturbation. The long time dynamics are characterized by the two largest eigenvalues of a spike random matrix which defines the model, and particularly by the statistics concerning the gap between them. We characterize the finite-size statistics of the two largest eigenvalues of the spike random matrices in the different regimes, sub-critical, critical, and super-critical, confirming some known results and anticipating others, even in the less studied critical regime. We also numerically characterize the finite-size statistics of the gap, which we hope may encourage analytical work which is lacking. Finally, we compute the finite-size scaling of the long time relaxation of the energy, showing the existence of power laws with exponents that depend on the strength of the non-disordered perturbation in a way that is governed by the finite-size statistics of the gap.

Model templates: transdisciplinary application and entanglement

The omnipresence of the same basic equations, function forms, algorithms, and quantitative methods is one of the most spectacular characteristics of contemporary modeling practice. Recently, the emergence of the discussion of templates and template transfer has addressed this striking cross-disciplinary reach of certain mathematical forms and computational algorithms. In this paper, we develop a notion of a model template, consisting of its mathematical structure, ontology, prototypical properties and behaviors, focal conceptualizations, and the paradigmatic questions it addresses. We apply this notion to three widely disseminated and powerful model templates: the Sherrington-Kirkpatrick model of spin glasses, scale-free networks, and the Kuramoto model of synchronization. We argue that what appears to be an interdisciplinary model transfer between different domains turns out, from a broader perspective, to be the application of transdisciplinary model templates across a multitude of domains. We also point out a further feature of template-based modeling that so far has not been discussed: template entanglement. Such entanglement enhances and makes manifest the conceptual side of model templates.

Diabatic quantum and classical annealing of the Sherrington-Kirkpatrick model

Quantum annealing is a contender to solve combinatorial optimization problems based on quantum dynamics. While significant efforts have been undertaken to investigate the quality of the solutions and the required run times, much less attention has been paid to understanding the dynamics of quantum annealing and the process leading to the solution during the sweep itself. In this comprehensive study, we investigate various aspects of the quantum annealing dynamics using different approaches. We perform quantum annealing, simulated quantum annealing, and classical annealing on several hundred instances of the Sherrington-Kirkpatrick model with intermediate system sizes up to 22 spins using numerical simulations. We observe qualitative differences between the quantum and classical methods, in particular at intermediate times, where a peak in the fidelity, also known as diabatic bump, appears for hard instances. Furthermore, we investigate the two-point correlation functions, which feature differences at intermediate times as well. At short times, however, the methods are similar again, which can be explained by relating the short-time expansion of quantum annealing to a high-temperature expansion, thus allowing one in principle to find the classical solution already at short times, albeit at prohibitive sampling cost.

Mean Field Spin Glass Models Under Weak External Field

We study the fluctuation and limiting distribution of free energy in mean-field spin glass models with Ising spins under weak external fields. We prove that at high temperature, there are three sub-regimes concerning the strength of external field $h \approx \rho N^{-\alpha}$ with $\rho,\alpha\in (0,\infty)$. In the super-critical regime $\alpha < 1/4$, the variance of the log-partition function is $\approx N^{1-4\alpha}$. In the critical regime $\alpha = 1/4$, the fluctuation is of constant order but depends on $\rho$. Whereas, in the sub-critical regime $\alpha>1/4$, the variance is $\Theta(1)$ and does not depend on $\rho$. We explicitly express the asymptotic mean and variance in all three regimes and prove Gaussian central limit theorems. Our proofs mainly follow two approaches. One utilizes quadratic coupling and Guerra's interpolation scheme for Gaussian disorder, extending to many other spin glass models. However, this approach can prove the CLT only at very high temperatures. The other one is a cluster-based approach for general symmetric disorders, first used in the seminal work of Aizenman, Lebowitz, and Ruelle~(Comm.~Math.~Phys.~112 (1987), no.~1, 3--20) for the zero external field case. It was believed that this approach does not work if the external field is present. We show that if the external field is present but not too strong, it still works with a new cluster structure. In particular, we prove the CLT up to the critical temperature in the Sherrington-Kirkpatrick (SK) model when $\alpha \ge 1/4$. We further address the generality of this cluster-based approach. Specifically, we give similar results for the multi-species SK model and diluted SK model.

Magnetic proximity in a coupled ferromagnet–spin glass system

We study the competition between ferromagnetic and spin glass phases using a system of coupled infinite-range Ising and Sherrington-Kirkpatrick models. We obtain the replica-symmetric solution for the free energy of this system in terms of magnetization and Edwards-Anderson order parameters in both subsystems. Using these order parameters we are able to identify different phases in the system and determine which phase is dominant for different strengths of the coupling between two subsystems. We observe that both subsystems are in the same phase although with different order parameters. The phase boundary between paramagnetic-ferromagnetic and paramagnetic--spin glass phases is more or less similar to that in the Sherrington-Kirkpatrick phase diagram. But the boundary between ferromagnetic--spin glass phases becomes qualitatively different as the coupling between the two subsystems changes. Remarkably, for intermediate values of the coupling, this phase boundary is such that there could be a reentrant transition between spin glass and ferromagnetic phases by increasing the temperature. We found that for some range of the coupling, the second-order transition between these phases turns into first order at a tricritical point. Further, we carry out the stability analysis by considering deviations from the replica-symmetric solution.

Inference of interactions between players based on asynchronously updated evolutionary game data

The interactions between players of the prisoner’s dilemma game are inferred using observed game data. All participants play the game with their counterparts and gain corresponding rewards during each round of the game. The strategies of each player are updated asynchronously during the game. Two inference methods of the interactions between players are derived with naïve mean-field (nMF) approximation and maximum log-likelihood estimation (MLE), respectively. Two methods are tested numerically also for fully connected asymmetric Sherrington–Kirkpatrick models, varying the data length, asymmetric degree, payoff, and system noise (coupling strength). We find that the mean square error of reconstruction for the MLE method is inversely proportional to the data length and typically half (benefit from the extra information of update times) of that by nMF. Both methods are robust to the asymmetric degree but work better for large payoffs. Compared with MLE, nMF is more sensitive to the strength of couplings and prefers weak couplings.