Ising Spin Glass Model Research Articles

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.

Cluster percolation in the two-dimensional Ising spin glass.

Suitable cluster definitions have allowed researchers to describe many ordering transitions in spin systems as geometric phenomena related to percolation. For spin glasses and some other systems with quenched disorder, however, such a connection has not been fully established, and the numerical evidence remains incomplete. Here we use Monte Carlo simulations to study the percolation properties of several classes of clusters occurring in the Edwards-Anderson Ising spin-glass model in two dimensions. The Fortuin-Kasteleyn-Coniglio-Klein clusters originally defined for the ferromagnetic problem do percolate at a temperature that remains nonzero in the thermodynamic limit. On the Nishimori line, this location is accurately predicted by an argument due to Yamaguchi. More relevant for the spin-glass transition are clusters defined on the basis of the overlap of several replicas. We show that various such cluster types have percolation thresholds that shift to lower temperatures by increasing the system size, in agreement with the zero-temperature spin-glass transition in two dimensions. The overlap is linked to the difference in density of the two largest clusters, thus supporting a picture where the spin-glass transition corresponds to an emergent density difference of the two largest clusters inside the percolating phase.

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.

Asymmetric phase diagrams, algebraically ordered Berezinskii-Kosterlitz-Thouless phase, and peninsular Potts flow structure in long-range spin glasses.

The Ising spin-glass model on the three-dimensional (d=3) hierarchical lattice with long-range ferromagnetic or spin-glass interactions is studied by the exact renormalization-group solution of the hierarchical lattice. The chaotic characteristics of the spin-glass phases are extracted in the form of our calculated, in this case continuously varying, Lyapunov exponents. Ferromagnetic long-range interactions break the usual symmetry of the spin-glass phase diagram. This phase-diagram symmetry breaking is dramatic, as it is underpinned by renormalization-group peninsular flows of the Potts multicritical type. A Berezinskii-Kosterlitz-Thouless (BKT) phase with algebraic order and a BKT-spin-glass phase transition with continuously varying critical exponents are seen. Similarly, for spin-glass long-range interactions, the Potts mechanism is also seen, by the mutual annihilation of stable and unstable fixed distributions causing the abrupt change of the phase diagram. On one side of this abrupt change, two distinct spin-glass phases, with finite (chaotic) and infinite (chaotic) coupling asymptotic behaviors are seen with a spin-glass to spin-glass phase transition.

Phase transition dynamics in the three-dimensional ±J Ising model.

By using frustration-preserving hard-spin mean-field theory, we investigated the phase-transition dynamics in the three-dimensional field-free ±J Ising spin-glass model. As the temperature T is decreased from paramagnetic phase at high temperatures, with a rate ω=-dT/dt in time t, the critical temperature depends on the cooling rate through a clear power law ω^{a}. With increasing antiferromagnetic bond fraction p, the exponent a increases for the transition into the ferromagnetic case for p<p_{c} and decreases for the transition into the spin-glass phase for p>p_{c}, signaling the ferromagnetic - spin-glass phase transition at p_{c}≈0.22. The relaxation time is also investigated in the adiabatic case ω=0 and the dynamic exponent zν is found to increase with increasing p.

Nontrivial maturation metastate-average state in a one-dimensional long-range Ising spin glass: Above and below the upper critical range.

Understanding the low-temperature pure state structure of spin glasses remains an open problem in the field of statistical mechanics of disordered systems. Here we study Monte Carlo dynamics, performing simulations of the growth of correlations following a quench from infinite temperature to a temperature well below the spin-glass transition temperature T_{c} for a one-dimensional Ising spin-glass model with diluted long-range interactions. In this model, the probability P_{ij} that an edge {i,j} has nonvanishing interaction falls as a power law with chord distance, P_{ij}∝1/R_{ij}^{2σ}, and we study a range of values of σ with 1/2<σ<1. We consider a correlation function C_{4}(r,t). A dynamic correlation length that shows power-law growth with time ξ(t)∝t^{1/z} can be identified in the data and, for large time t,C_{4}(r,t) decays as a power law r^{-α_{d}} with distance r when r≪ξ(t). The calculation can be interpreted in terms of the maturation metastate averaged Gibbs state, or MMAS, and the decay exponent α_{d} differentiates between a trivial MMAS (α_{d}=0), as expected in the droplet picture of spin glasses, and a nontrivial MMAS (α_{d}≠0), as in the replica-symmetry-breaking (RSB) or chaotic pairs pictures. We find nonzero α_{d} even in the regime σ>2/3 which corresponds to short-range systems below six dimensions. For σ<2/3, the decay exponent α_{d} follows the RSB prediction for the decay exponent α_{s}=3-4σ of the static metastate, consistent with a conjectured statics-dynamics relation, while it approaches α_{d}=1-σ in the regime 2/3<σ<1; however, it deviates from both lines in the vicinity of σ=2/3.

On Absence of disorder chaos for spin glasses on $\mathbb{Z} ^{d}$

We identify simple mechanisms that prevent a strong form of disorder chaos for edge overlap of the Ising spin glass model on $\mathbb{Z} ^{d}$. This was first shown by Chatterjee in the case of Gaussian couplings. We present three proofs of the theorem for general couplings with continuous distribution based on the presence in the coupling realization of stabilizing features of positive density.

GPU-based Ising computing for solving max-cut combinatorial optimization problems

In VLSI physical design, many algorithms require the solution of difficult combinatorial optimization problems such as max/min-cut, max-flow problems etc. Due to the vast number of elements typically found in this problem domain, these problems are computationally intractable leading to the use of approximate solutions. In this work, we explore the Ising spin glass model as a solution methodology for hard combinatorial optimization problems using the general purpose GPU (GPGPU). The Ising model is a mathematical model of ferromagnetism in statistical mechanics. Ising computing finds a minimum energy state for the Ising model which essentially corresponds to the expected optimal solution of the original problem. Many combinatorial optimization problems can be mapped into the Ising model. In our work, we focus on the max-cut problem as it is relevant to many VLSI design automation problems. Our method is inspired by the observation that Ising annealing process is very amenable to fine-grain massive parallel GPU computing. We will illustrate how the natural randomness of GPU thread scheduling can be exploited during the annealing process to create random update patterns and allow better GPU resource utilization. Furthermore, the proposed GPU-based Ising computing can handle any general Ising graph with arbitrary connections, which was shown to be difficult for existing FPGA and other hardware based implementation methods. Numerical results show that the proposed GPU Ising max-cut solver can deliver more than 2000X speedup over the CPU version of the algorithm on some large examples, which shows huge performance improvement for addressing many hard optimization algorithms for solving practical VLSI design automation problems.

Two-Photon Driven Kerr Resonator for Quantum Annealing with Three-Dimensional Circuit QED

We propose a realizable circuit QED architecture for engineering states of a superconducting resonator off-resonantly coupled to an ancillary superconducting qubit. The qubit-resonator dispersive interaction together with a microwave drive applied to the qubit gives rise to a Kerr resonator with two-photon driving that enables us to efficiently engineer the quantum state of the resonator such as generation of the Schrodinger cat states for resonator-based universal quantum computation. Moreover, the presented architecture is easily scalable for solving optimization problem mapped into the Ising spin glass model, and thus served as a platform for quantum annealing. Although various scalable architecture with superconducting qubits have been proposed for realizing quantum annealer, the existing annealers are currently limited to the coherent time of the qubits. Here, based on the protocol for realizing two-photon driven Kerr resonator in three-dimensional circuit QED (3D cQED), we propose a flexible and scalable hardware for implementing quantum annealer that combines the advantage of the long coherence times attainable in 3D cQED and the recently proposed resonator based Lechner-Hauke-Zoller (LHZ) scheme. In the proposed resonator based LHZ annealer, each spin is encoded in the subspace formed by two coherent state of 3D microwave superconducting resonator with opposite phase, and thus the fully-connected Ising model is mapped onto the network of the resonator with local tunable three-resonator interaction. This hardware architecture provides a promising physical platform for realizing quantum annealer with improved coherence.

Many-body localization-delocalization transition in the quantum Sherrington-Kirkpatrick model

We analyze many-body localization (MBL) to delocalization transition in Sherrington-Kirkpatrick (SK) model of Ising spin glass (SG) in the presence of a transverse field $\Gamma$. Based on energy resolved analysis, which is of relevance for a closed quantum system, we show that the quantum SK model has many-body mobility edges separating MBL phase which is non-ergodic and non-thermal from the delocalized phase which is ergodic and thermal. The range of the delocalized regime increases with increase in the strength of $\Gamma$ and eventually for $\Gamma$ larger than $\Gamma_{CP}$ the entire many-body spectrum is delocalized. We show that the Renyi entropy is almost independent of the system size in the MBL phase, hinting towards an area law in this infinite range model while the delocalized phase shows volume law scaling of Renyi entropy. We further obtain spin glass transition curve in energy density $\epsilon$-$\Gamma$ plane from the collapse of eigenstate spin susceptibility. We demonstrate that in most of the parameter regime SG transition occurs close to the MBL transition indicating that the SG phase is non-ergodic and non-thermal while the paramagnetic phase is delocalized and thermal.

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.

Hyperscaling breakdown and Ising spin glasses: The Binder cumulant

Among the Renormalization Group Theory scaling rules relating critical exponents, there are hyperscaling rules involving the dimension of the system. It is well known that in Ising models hyperscaling breaks down above the upper critical dimension. It was shown by Schwartz (1991) that the standard Josephson hyperscaling rule can also break down in Ising systems with quenched random interactions. A related Renormalization Group Theory hyperscaling rule links the critical exponents for the normalized Binder cumulant and the correlation length in the thermodynamic limit. An appropriate scaling approach for analyzing measurements from criticality to infinite temperature is first outlined. Numerical data on the scaling of the normalized correlation length and the normalized Binder cumulant are shown for the canonical Ising ferromagnet model in dimension three where hyperscaling holds, for the Ising ferromagnet in dimension five (so above the upper critical dimension) where hyperscaling breaks down, and then for Ising spin glass models in dimension three where the quenched interactions are random. For the Ising spin glasses there is a breakdown of the normalized Binder cumulant hyperscaling relation in the thermodynamic limit regime, with a return to size independent Binder cumulant values in the finite-size scaling regime around the critical region.

Critical and Griffiths-McCoy singularities in quantum Ising spin glasses on d-dimensional hypercubic lattices: A series expansion study.

We study the ±J transverse-field Ising spin-glass model at zero temperature on d-dimensional hypercubic lattices and in the Sherrington-Kirkpatrick (SK) model, by series expansions around the strong-field limit. In the SK model and in high dimensions our calculated critical properties are in excellent agreement with the exact mean-field results, surprisingly even down to dimension d=6, which is below the upper critical dimension of d=8. In contrast, at lower dimensions we find a rich singular behavior consisting of critical and Griffiths-McCoy singularities. The divergence of the equal-time structure factor allows us to locate the critical coupling where the correlation length diverges, implying the onset of a thermodynamic phase transition. We find that the spin-glass susceptibility as well as various power moments of the local susceptibility become singular in the paramagnetic phase before the critical point. Griffiths-McCoy singularities are very strong in two dimensions but decrease rapidly as the dimension increases. We present evidence that high enough powers of the local susceptibility may become singular at the pure-system critical point.

Inverse freezing in a van Hemmen Fermionic Ising Spin Glass model under transverse and random magnetic fields

The van Hemmen Fermionic Ising Spin Glass (vH FISG) model in the presence of a transverse and a random magnetic field is adopted to study the inverse freezing (IF) transition, without using the replica method to treat the disorder. In this model, the spin interactions are given by a combination of random variables that follow Gaussian distribution. The random field (RF) also follows a Gaussian distribution. The introduction of allow us to investigate the IF under the effects of a disorder which is not a source of frustration. A particularity of this fermionic formalism is that the chemical potential and the provide a magnetic dilution and quantum spin flip mechanism, respectively. The results show a reentrant transition from the spin glass (SG) to the paramagnetic (PM) phase in the absence of and . This reentrance appears for a certain range of , in which is found a PM phase (at low temperatures) with lower entropy than the SG state, characterising the IF. However, the IF is gradually suppressed when the effects are intensified. In addition, the IF is completely destroyed by the quantum fluctuations provided by . Therefore, nontrivial disorder combined with dilution can bring a scenario favourable to the IF occurrence, while random fields and quantum fluctuations are against the IF.

Nonstoquastic Hamiltonians and quantum annealing of an Ising spin glass

We study the role of Hamiltonian complexity in the performance of quantum annealers. We consider two general classes of annealing Hamiltonians: stoquastic ones, which can be simulated efficiently using the quantum Monte Carlo algorithm, and nonstoquastic ones, which cannot be treated efficiently. We implement the latter by adding antiferromagnetically coupled two-spin driver terms to the traditionally studied transverse-field Ising model, and compare their performance to that of similar stoquastic Hamiltonians with ferromagnetically coupled additional terms. We focus on a model of long-range Ising spin glass as our problem Hamiltonian and carry out the comparison between the annealers by numerically calculating their success probabilities in solving random instances of the problem Hamiltonian in systems of up to 17 spins. We find that, for a small percentage of mostly harder instances, nonstoquastic Hamiltonians greatly outperform their stoquastic counterparts and their superiority persists as the system size grows. We conjecture that the observed improved performance is closely related to the frustrated nature of nonstoquastic Hamiltonians.

Ising spin glasses in two dimensions: Universality and nonuniversality.

Following numerous earlier studies, extensive simulations and analyses were made on the continuous interaction distribution Gaussian model and the discrete bimodal interaction distribution Ising spin glass (ISG) models in two dimensions [Lundow and Campbell, Phys. Rev. E 93, 022119 (2016)1539-375510.1103/PhysRevE.93.022119]. Here we further analyze the bimodal and Gaussian data together with data on two other continuous interaction distribution two-dimensional ISG models, the uniform and the Laplacian models, and three other discrete interaction distribution models, a diluted bimodal model, an "antidiluted" model, and a more exotic symmetric Poisson model. Comparisons between the three continuous distribution models show that not only do they share the same exponent η≡0 but that to within the present numerical precision they share the same critical exponent ν also, and so lie in a single universality class. On the other hand the critical exponents of the four discrete distribution models are not the same as those of the continuous distributions, and the present data strongly indicate that they differ from one discrete distribution model to another. This is evidence that discrete distribution ISG models in two dimensions have nonzero values of the critical exponent η and do not lie in a single universality class.

Ising spin glasses in dimension five.

Ising spin-glass models with bimodal, Gaussian, uniform, and Laplacian interaction distributions in dimension five are studied through detailed numerical simulations. The data are analyzed in both the finite-size scaling regime and the thermodynamic limit regime. It is shown that the values of critical exponents and of dimensionless observables at criticality are model dependent. Models in a single universality class have identical values for each of these critical parameters, so Ising spin-glass models in dimension five with different interaction distributions each lie in different universality classes. This result confirms conclusions drawn from measurements in dimension four and dimension two.

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.

Variational perturbation and extended Plefka approaches to dynamics on random networks: the case of the kinetic Ising model

We describe and analyze some novel approaches for studying the dynamics of Ising spin glass models. We first briefly consider the variational approach based on minimizing the Kullback–Leibler divergence between independent trajectories and the real ones and note that this approach only coincides with the mean field equations from the saddle point approximation to the generating functional when the dynamics is defined through a logistic link function, which is the case for the kinetic Ising model with parallel update. We then spend the rest of the paper developing two ways of going beyond the saddle point approximation to the generating functional. In the first one, we develop a variational perturbative approximation to the generating functional by expanding the action around a quadratic function of the local fields and conjugate local fields whose parameters are optimized. We derive analytical expressions for the optimal parameters and show that when the optimization is suitably restricted, we recover the mean field equations that are exact for the fully asymmetric random couplings (Mézard and Sakellariou 2011 J. Stat. Mech. 2011 L07001). However, without this restriction the results are different. We also describe an extended Plefka expansion in which in addition to the magnetization, we also fix the correlation and response functions. Finally, we numerically study the performance of these approximations for Sherrington–Kirkpatrick type couplings for various coupling strengths and the degrees of coupling symmetry, for both temporally constant but random, as well as time varying external fields. We show that the dynamical equations derived from the extended Plefka expansion outperform the others in all regimes, although it is computationally more demanding. The unconstrained variational approach does not perform well in the small coupling regime, while it approaches dynamical TAP equations of (Roudi and Hertz 2011 J. Stat. Mech. 2011 P03031) for strong couplings.

Interface free-energy exponent in the one-dimensional Ising spin glass with long-range interactions in both the droplet and broken replica symmetry regions.

The one-dimensional Ising spin-glass model with power-law long-range interactions is a useful proxy model for studying spin glasses in higher space dimensions and for finding the dimension at which the spin-glass state changes from having broken replica symmetry to that of droplet behavior. To this end we have calculated the exponent that describes the difference in free energy between periodic and antiperiodic boundary conditions. Numerical work is done to support some of the assumptions made in the calculations and to determine the behavior of the interface free-energy exponent of the power law of the interactions. Our numerical results for the interface free-energy exponent are badly affected by finite-size problems.