Kirkpatrick Spin-glass Model Research Articles

Unlearning regularization for Boltzmann machines

Boltzmann machines (BMs) are graphical models with interconnected binary units, employed for the unsupervised modeling of data distributions. When trained on real data, BMs show the tendency to behave like critical systems, displaying a high susceptibility of the model under a small rescaling of the inferred parameters. This behavior is not convenient for the purpose of generating data, because it slows down the sampling process, and induces the model to overfit the training-data. In this study, we introduce a regularization method for BMs to improve the robustness of the model under rescaling of the parameters. The new technique shares formal similarities with the unlearning algorithm, an iterative procedure used to improve memory associativity in Hopfield-like neural networks. We test our unlearning regularization on synthetic data generated by two simple models, the Curie–Weiss ferromagnetic model and the Sherrington–Kirkpatrick spin glass model. We show that it outperforms Lp -norm schemes and discuss the role of parameter initialization. Eventually, the method is applied to learn the activity of real neuronal cells, confirming its efficacy at shifting the inferred model away from criticality and coming out as a powerful candidate for actual scientific implementations.

Statistical mechanics of the maximum-average submatrix problem

We study the maximum-average submatrix problem, wherein given an N × N matrix J, one needs to find the k × k submatrix with the largest average number of entries. We investigate the problem for random matrices J, whose entries are i.i.d. random variables, by mapping it to a variant of the Sherrington–Kirkpatrick spin-glass model at fixed magnetisation. We analytically characterise the phase diagram of the model as a function of the submatrix average and the size of the submatrix k in the limit N→∞ . We consider submatrices of size k=mN with 0<m<1 . We find a rich phase diagram, including dynamical, static one-step replica symmetry breaking (1-RSB) and full-step RSB. In the limit of m → 0, we find a simpler phase diagram featuring a frozen 1-RSB phase, where the Gibbs measure comprises exponentially many pure states each with zero entropy. We discover an interesting phenomenon, reminiscent of the phenomenology of the binary perceptron: there are efficient algorithms that provably work in the frozen 1-RSB phase.

Role of quantum fluctuation ininducing ergodicity inthe spin glass phase and its effect inquantum annealing.

We review the numerical studies on the critical behaviour of the quantum Sherrington-Kirkpatrick (SK) spin glass model, which indicate that aquantum critical behaviour is observed up to a low but non-zero value of temperature. We revisit the numerical investigations on the spin glass order parameter distributions, which identify a low temperature along with high transverse field spin glass phase where the order parameter distribution becomes a delta function in the thermodynamic limit indicating the restoration of replica symmetry and ergodic nature of the system. In the remaining spin glass phase associated with high temperature and low transverse field, the observed distribution is broad akin to the Parisi order parameter distribution. This essentially indicates the non-ergodic behaviour of the system. We further discuss the annealing dynamics studies on the quantum SK model. Such investigations reveal the system size independence of annealing time when the annealing paths go through the ergodic spin glass region. Interestingly, when such dynamics are performed in the non-ergodic spin glass phase the annealing time becomes an increasing function of the system size. Spin autocorrelation shows faster relaxation in the ergodic spin glass region compared with that found in the non-ergodic spin glass region. This article is part of the theme issue 'Quantum annealing and computation: challenges and perspectives'.

Spherical Spin Glass Model with External Field

We analyze the free energy and the overlaps in the 2-spin spherical Sherrington Kirkpatrick spin glass model with an external field for the purpose of understanding the transition between this model and the one without an external field. We compute the limiting values and fluctuations of the free energy as well as three types of overlaps in the setting where the strength of the external field goes to zero as the dimension of the spin variable grows. In particular, we consider overlaps with the external field, the ground state, and a replica. Our methods involve a contour integral representation of the partition function along with random matrix techniques. We also provide computations for the matching between different scaling regimes. Finally, we discuss the implications of our results for susceptibility and for the geometry of the Gibbs measure. Some of the findings of this paper are confirmed rigorously by Landon and Sosoe in their recent paper which came out independently and simultaneously.

The Free Energy of a Quantum Sherrington\u2013Kirkpatrick Spin-Glass Model for Weak Disorder

We extend two rigorous results of Aizenman, Lebowitz, and Ruelle in their pioneering paper of 1987 on the Sherrington–Kirkpatrick spin-glass model without external magnetic field to the quantum case with a “transverse field” of strength mathsf {b}. More precisely, if the Gaussian disorder is weak in the sense that its standard deviation mathsf {v}>0 is smaller than the temperature 1/beta , then the (random) free energy almost surely equals the annealed free energy in the macroscopic limit and there is no spin-glass phase for any mathsf {b}/mathsf {v}ge 0. The macroscopic annealed free energy turns out to be non-trivial and given, for any beta mathsf {v}>0, by the global minimum of a certain functional of square-integrable functions on the unit square according to a Varadhan large-deviation principle. For beta mathsf {v}<1 we determine this minimum up to the order (beta mathsf {v})^{4} with the Taylor coefficients explicitly given as functions of beta mathsf {b} and with a remainder not exceeding (beta mathsf {v})^{6}/16. As a by-product we prove that the so-called static approximation to the minimization problem yields the wrong beta mathsf {b}-dependence even to lowest order. Our main tool for dealing with the non-commutativity of the spin-operator components is a probabilistic representation of the Boltzmann–Gibbs operator by a Feynman–Kac (path-integral) formula based on an independent collection of Poisson processes in the positive half-line with common rate beta mathsf {b}. Its essence dates back to Kac in 1956, but the formula was published only in 1989 by Gaveau and Schulman.

Free energy of the quantum Sherrington–Kirkpatrick spin-glass model with transverse field

We consider the quantum Sherrington–Kirkpatrick (SK) spin-glass model with a transverse field and provide a formula for its free energy in the thermodynamic limit, valid for all inverse temperatures β &amp;gt; 0. To characterize the free energy, we use the path integral representation of the partition function and approximate the model by a sequence of finite-dimensional vector-spin glasses with Rd-valued spins. This enables us to use the results of Panchenko who generalized [Ann. Probab. 46(2), 829–864 (2018); ibid, 46(2), 865–896 (2018)] the Parisi formula to classical vector-spin glasses. As a consequence, we can express the thermodynamic limit of the free energy of the quantum SK model as the d → ∞ limit of the free energies of the d-dimensional approximations of the model.

Fundamental limits of detection in the spiked Wigner model

We study the fundamental limits of detecting the presence of an additive rank-one perturbation, or spike, to a Wigner matrix. When the spike comes from a prior that is i.i.d. across coordinates, we prove that the log-likelihood ratio of the spiked model against the nonspiked one is asymptotically normal below a certain reconstruction threshold which is not necessarily of a “spectral” nature, and that it is degenerate above. This establishes the maximal region of contiguity between the planted and null models. It is known that this threshold also marks a phase transition for estimating the spike: the latter task is possible above the threshold and impossible below. Therefore, both estimation and detection undergo the same transition in this random matrix model. Further information on the performance of the optimal test is also provided. Our proofs are based on Gaussian interpolation methods and a rigorous incarnation of the cavity method, as devised by Guerra and Talagrand in their study of the Sherrington–Kirkpatrick spin-glass model.

The effects of the one-step replica symmetry breaking on the Sherrington–Kirkpatrick spin glass model in the presence of random field with a joint Gaussian probability density function for the exchange interactions and random fields

The Sherrington–Kirkpatrick Ising spin glass model, in the presence of a random magnetic field, is investigated within the framework of the one-step replica symmetry breaking. The two random variables (exchange integral interaction Jij and random magnetic field hi) are drawn from a joint Gaussian probability density function characterized by a correlation coefficient ρ, assuming positive and negative values. The thermodynamic properties, the three different phase diagrams and system’s parameters are computed with respect to the natural parameters of the joint Gaussian probability density function at non-zero and zero temperatures. The low temperature negative entropy controversy, a result of the replica symmetry approach, has been partly remedied in the current study, leading to a less negative result. In addition, the present system possesses two successive spin glass phase transitions with characteristic temperatures.

Effects of smooth random surface on fluid monolayer thermodynamics

We consider the lattice gas approach to statistical mechanics of fluid adsorbed on random surfaces with fluid-fluid and fluid-surface potentials. It was shown that effective Hamiltonian contains quenched random interactions and random site fields. Their statistical features combine the properties of random geometry and fluid-fluid pair interaction potential. The high-temperature expansion leads to infinite-ranged random field model and Sherrington–Kirkpatrick spin-glass model. Thermodynamic properties are evaluated using replica theory procedure widely used to analyze quenched disorder systems. On the other hand we consider the random field model in random graph with finite connectivity instead of previous “infinite-ranged” approximations. This model has been investigated using finite connectivity technique. The replica symmetry ansatz for the order function is expressed in terms of an effective-field distribution. Analysis of random geometry effects on thermodynamic properties in such approach was done for the first time.

Irreversible transitions in the exchange-striction model of spin-glass state

Based on the assumption of a negative volume dependence of random exchange integrals, it is possible to switch to a compressible Sherrington–Kirkpatrick spin-glass model. Within the proposed model, temperature–pressure phase diagrams were calculated and pressure- and magnetic-field-induced first-order phase transitions from the initial paramagnetic and spin-glass states to the ferromagnetic state were predicted. It was shown that the application of pressure in the spin-glass state not only increases and shifts magnetic susceptibility, but also reduces the critical magnetic fields of irreversible induced phase transitions from the spin-glass to the ferromagnetic state. The obtained results are used to describe the spin-glass state in (Sm1–xGdx)0.55Sr0.45MnO3.

Rare events analysis of temperature chaos in the Sherrington–Kirkpatrick model

We investigate the question of temperature chaos in the Sherrington–Kirkpatrick spin glass model, applying a recently proposed rare events based data analysis method to existing Monte Carlo data. Thanks to this new method, temperature chaos is now observable for this model, even with the limited size systems that can currently be simulated.

The Sherrington–Kirkpatrick spin glass model in the presence of a random field with a joint Gaussian probability density function for the exchange interactions and random fields

The magnetic systems with disorder form an important class of systems, which are under intensive studies, since they reflect real systems. Such a class of systems is the spin glass one, which combines randomness and frustration. The Sherrington–Kirkpatrick Ising spin glass with random couplings in the presence of a random magnetic field is investigated in detail within the framework of the replica method. The two random variables (exchange integral interaction and random magnetic field) are drawn from a joint Gaussian probability density function characterized by a correlation coefficient ρ. The thermodynamic properties and phase diagrams are studied with respect to the natural parameters of both random components of the system contained in the probability density. The de Almeida–Thouless line is explored as a function of temperature, ρ and other system parameters. The entropy for zero temperature as well as for non zero temperatures is partly negative or positive, acquiring positive branches as h0 increases.

Time evolution of the autocorrelation function in dynamical replica theory

Asynchronous dynamics given by the master equation in the Sherrington–Kirkpatrick (SK) spin-glass model is studied based on dynamical replica theory (DRT) with an extension to take into account the autocorrelation function. The dynamical behaviour of the system is approximately described by dynamical equations of the macroscopic quantities: magnetization, energy contributed by randomness and the autocorrelation function. The dynamical equations under the replica symmetry assumption are derived by introducing the subshell equipartitioning assumption and exploiting the replica method. The obtained dynamical equations are compared with Monte Carlo simulations, and it is demonstrated that the proposed formula describes well the time evolution of the autocorrelation function in some parameter regions. The study offers a reasonable description of the autocorrelation function in the SK spin-glass system.

Replica symmetry breaking in mean-field spin glasses through the Hamilton–Jacobitechnique

During the last few years, through the combined effort of the insight coming from physicalintuition and computer simulation, and the exploitation of rigorous mathematical methods,the main features of the mean-field Sherrington–Kirkpatrick spin glass model have beenfirmly established. In particular, it has been possible to prove the existence and uniquenessof the infinite-volume limit for the free energy, and its Parisi expression, in terms of avariational principle involving a functional order parameter. Even the expected property ofultrametricity, for the infinite-volume states, seems to be near to a complete proof.The main structural feature of this model, and related models, is the deep phenomenon ofspontaneous replica symmetry breaking (RSB), discovered by Parisi many years ago. Byexpanding on our previous work, the aim of this paper is to investigate a generalframework, where replica symmetry breaking is embedded in a kind of mechanical schemeof the Hamilton–Jacobi type. Here, the analog of the ‘time’ variable is a parametercharacterizing the strength of the interaction, while the ‘space’ variables rule outquantitatively the broken replica symmetry pattern. Starting from the simple cases, whereannealing is assumed, or replica symmetry, we build up a progression of dynamical systems,with an increasing number of space variables, which allow us to weaken the effect of thepotential in the Hamilton–Jacobi equation as the level of symmetry breaking is increased.This new machinery allows us to work out mechanically the generalK-step RSB solutions, in a different interpretation with respect to the replica trick, andeasily reveals their properties such as existence or uniqueness.

Matching between typical fluctuations and large deviations in disordered systems:application to the statistics of the ground state energy in the SK spin-glass model

For the statistics of global observables in disordered systems, we discuss the matching betweentypical fluctuations and large deviations. We focus on the statistics of the ground state energyE0 in disorder models of two types: (i) for the directed polymer of lengthN in atwo-dimensional medium, for which many exact results exist; (ii) for the Sherrington–Kirkpatrick spin-glassmodel of N spins, for which various possibilities have been proposed. Here we stress thatit is very instructive to study, besides the behavior of the disorder averageE0av(N) and that of the standarddeviation ΔE0(N) ∼ Nωf that definesthe fluctuation exponent ωf, also the full probability distributionΠ(u) of the rescaled variableu = (E0(N) − E0av(N))/ΔE0(N): (a) numerically, theconvergence towards Π(u) is usually very rapid, so data for rather small sizes but with high statistics allow one to measure the two tailexponents η ± defined as ; in the generic case , this leads to explicit non-trivial terms in the asymptotic behaviors of the moments of the partition function when the combination[|n|Nωf] becomeslarge; (b) simple rare events arguments can usually be found for providing explicit relations betweenη ± andωf; these rare events usually correspond to ‘anomalous’ large deviation properties of thegeneralized form (the ‘usual’ large deviations formalism corresponds toκ ± = 1 = ρ ± ).

An eigenvalue method for computing the largest relaxation time of disorderedsystems

We consider the dynamics of finite size disordered systems as defined by a masterequation satisfying detailed balance. The master equation can be mapped ontoa Schrödinger equation in configuration space, where the quantum HamiltonianH has the generic form of an Anderson localization tight-binding model. The largest relaxation timeteq governingthe convergence towards Boltzmann equilibrium is determined by the lowest non-vanishing eigenvalueE1 = 1/teq ofH (the lowest eigenvaluebeing E0 = 0). So therelaxation time teq can be computed without simulating the dynamics by any eigenvalue method able to compute the first excitedenergy E1. Here we use the ‘conjugate gradient’ method to determineE1 for each disordered sample and present numerical results on the statistics of the relaxation timeteq over disordered samples of a given size for two models: (i) forthe random walk in a self-affine potential of Hurst exponentH on a two-dimensionalsquare of size L × L, we findthe activated scaling lnteq(L)∼Lψ with ψ = H as expected; (ii) for the dynamics of the Sherrington–Kirkpatrick spin glass model ofN spins, we findthe growth lnteq(N)∼Nψ with ψ = 1/3 inagreement with most previous Monte Carlo measures. In addition, we find that the rescaled distribution of(lnteq) decaysas e−uη forlarge u with a tail exponent of order . We give a rare-event interpretation of this value, that points towards a sample-to-samplefluctuation exponent of order for the barrier.

Compressible Sherrington–Kirkpatrick spin-glass model

We introduce a Sherrington–Kirkpatrick spin-glass model with the addition of elastic degrees of freedom. The problem is formulated in terms of an effective four-spin Hamiltonian in the pressure ensemble, which can be treated by the replica method. In the replica-symmetric approximation, we analyze the pressure–temperature phase diagram, and obtain expressions for the critical boundaries between the disordered and the ordered (spin-glass and ferromagnetic) phases. The second-order para–ferromagnetic border ends at a tricritical point, beyond which the transition becomes discontinuous. We use these results to make contact with the temperature–concentration phase diagrams of mixtures of hydrogen-bonded crystals.

Small-correlation expansions for the inverse Ising problem

We present a systematic small-correlation expansion to solve the inverse Ising problem and find a set of couplings and fields corresponding to a given set of correlations and magnetizations. Couplings are calculated up to the third order in the correlations for generic magnetizations and to the seventh order in the case of zero magnetizations; in addition, we show how to sum some useful classes of diagrams exactly. The resulting expansion outperforms existing algorithms on the Sherrington–Kirkpatrick spin-glass model.

Fluctuations of the free energy in the diluted SK-model

We derive limit theorems for the fluctuations of the free energy in a diluted version of the Sherrington–Kirkpatrick model. Our proofs are based on the classical approach of Aizenman et al. [M. Aizenman, J.L. Lebowitz, D. Ruelle, Some rigorous results on the Sherrington–Kirkpatrick spin glass model, Comm. Math. Phys. 112 (1987) 3–20].

Universality in Sherrington–Kirkpatrick's spin glass model

It was already clear to Sherrington and Kirkpatrick [Phys. Rev. B 17 (1978) 4384–4403] that the limiting free energy in Sherrington–Kirkpatrick's Spin Glass Model does not depend on the particular distribution of environments. We give here a mathematical proof of this fact.