Sherrington-Kirkpatrick Spin Glass Model Research Articles

Supersymmetric complexity in the Sherrington-Kirkpatrick model.

By using a supersymmetric approach we compute the complexity of the metastable states in the Sherrington-Kirkpatrick spin-glass model. We prove that the supersymmetric complexity is exactly equal to the Legendre transform of the thermodynamic free energy, thus providing a recipe to find the complexity once the free energy is known. Our results suggest that the supersymmetry may be a useful tool for the calculation of the entropy of metastable states in generic glassy systems.

Extended variational principle for the Sherrington-Kirkpatrick spin-glass model

The recent proof by F. Guerra that the Parisi ansatz provides a lower bound on the free energy of the SK spin-glass model could have been taken as offering some support to the validity of the purported solution. In this work we present a broader variational principle, in which the lower bound, as well as the actual value, are obtained through an optimization procedure for which ultrametic/hierarchal structures form only a subset of the variational class. The validity of Parisi's ansatz for the SK model is still in question. The new variational principle may be of help in critical review of the issue.

On the Thermodynamics of the Hopfield Neural Network with External Field

In the paper thermodynamic properties of an artificial neural network are analyzed in a way analogous to spin glasses theory. Synaptic connections are calculated numerically according to the Hebb rule and their distribution is obtained for different characteristics of stored patterns. The phase diagrams and magnetization are established in dependence on the temperature of the network and the external field (threshold). It was showed that changing control parameters typical of artificial neural network (i.e. number of stored patterns and pattern bias level) one obtains the results similar to the Sherrington-Kirkpatrick model of spin glass.

Central limit theorem for fluctuations in the high temperature region of the Sherrington–Kirkpatrick spin glass model

In a region above the Almeida–Thouless line, where we are able to control the thermodynamic limit of the Sherrington–Kirkpatrick model and to prove replica symmetry, we show that the fluctuations of the overlaps and of the free energy are Gaussian, on the scale 1/N, for large N. The method we employ is based on the idea we recently developed of introducing quadratic coupling between two replicas. The proof makes use of the cavity equations and of concentration of measure inequalities for the free energy.

Correlation timescales in the Sherrington-Kirkpatrick model

We investigate the dynamical behavior of the Sherrington-Kirkpatrick mean field model of spin glasses by numerical simulation. All the time scales tau we have measured behave like ln(tau) propto N^epsilon, where N is the number of spins and epsilon \simeq 1/3. This is true whether the autocorrelation function used to define tau is sensitive to the full reversal of the system or not.

Out-of-equilibrium dynamics of the hopfield model in its spin-glass phase

In this paper, we study numerically the out-of-equilibrium dynamics of the Hopfield model for associative memory inside its spin-glass phase. Aside from its interest as a neural network model, it can also be considered as a prototype of a fully connected magnetic system with randomness and frustration. By adjusting the ratio between the number of stored configurations p and the total number of neurons N, one can control the phase-space structure, whose complexity can vary between the simple mean-field ferromagnet (when p=1) and that of the Sherrington Kirkpatrick spin-glass model (for a properly taken limit of an infinite number of patterns). In particular, little attention has been devoted to the spin-glass phase of this model. In this paper, we analyze the two-time autocorrelation function, the decay of the magnetization and the distribution of overlaps between states. The results show that within the spin-glass phase of the model, the dynamics exhibits aging phenomena and presents features that suggest a non trivial breaking of replica symmetry.

Spreading of damage in a two-dimensional Ising model with dipolar interactions

In this work we use the spreading of damage technique to study the dynamical properties of a two-dimensional Ising model with competition between ferromagnetic exchange (J0) and antiferromagnetic dipolar (Jd) interactions. In recent works it was shown that, depending on the ratio ? = J0/Jd, the ground state of the model corresponds to ferromagnetic stripes whose width increases with ?. In this striped phase the system displays a slow dynamics, characterized by the formation and growth of magnetic domains. We found that, within these phases, the spreading of damage technique allows one to identify two dynamical phases, with a behaviour similar to that observed in the Sherrington-Kirkpatrick model of spin glasses.

Modeling of polar clusters in disordered perovskites: The S-K model with tunneling

The polar clusters generated by random non-central impurities in quantum paraelectrics determine to a great extent the complex dielectric behaviour observed in systems like SCT, KTN or KTL. The competition between a quantum paraelectric phase and an impurity-driven ferroelectric or glass phase depends on the concentration and on the nature of the interaction between clusters. This work presents a simple model in which each cluster is represented by a quantum two level system involving an effective Ising dipolar moment η and a tunneling energy A. The interactions between clusters are taken into account by following the guide-lines of the Sherrington-Kirkpatrick model for spin glasses. General expressions for the polarization (P) and the Edwards-Anderson order parameter (q) are given and the phase diagram involving temperature, the normalized mean interaction energy Jo/J and tunneling energy A/Jo is built.

On the Distribution of Overlaps in the Sherrington–Kirkpatrick Spin Glass Model

This paper describes some of the analytic tools developed recently by Ghirlanda and Guerra in the investigation of the distribution of overlaps in the Sherrington–Kirkpatrick spin glass model and of Parisi's ultrametricity. In particular, we introduce to this task a simplified (but also generalized) model on which the Gaussian analysis is made easier. Moments of the Hamiltonian and derivatives of the free energy are expressed as polynomials of the overlaps. Under the essential tool of self-averaging, we describe with full rigour, various overlap identities and replica independence that actually hold in a rather large generality. The results are presented in a language accessible to probabilists and analysts.

Coalescents With Multiple Collisions

For each finite measure $\Lambda$ on [0,1] a coalescent Markov process, with state space the compact set of all partitions of the set $\mathbb{N}$ of positive integers, is constructed so the restriction of the partition to each finite subset of $\mathbb{N}$ is a Markov chain with the following transition rates: when the partition has b blocks, each $k$-tuple of blocks is merging to form a single block at rate $\int_0^1x^{k-2}(1-x)^{b-k}\Lambda(dx)$. Call this process a $\Lambda$-coalescent. Discrete measure-valued processes derived from the $\Lambda$-coalescent model a system of masses undergoing coalescent collisions. Kingman’s coalescent, which has numerous applications in population genetics, is the $\delta_0$-coalescent for $\delta_0$ a unit mass at 0. The coalescent recently derived by Bolthausen and Sznitman from Ruelle’s probability cascades, in the context of the Sherrington–Kirkpatrick spin glass model in mathematical physics, is the $U$-coalescent for $U$ uniform on [0,1]. For $\Lambda=U$, and whenever an infinite number of masses are present, each collision in $\Lambda$-coalescent involves an infinite number of masses almost surely, and the proportion of masses involved exists as a limit almost surely and is distributed proportionally to $\Lambda$. The two-parameter Poisson–Dirichlet family of random discrete distributions derived from a stable subordinator, and corresponding exchangeable random partitions of $\mathbb{N}$ governed by a generalization of the Ewens sampling formula, are applied to describe transition mechanisms for processes of coalescence and fragmentation, including the $U$-coalescent and its time reversal.

A new simple version of the replica method

The moments of the partition function of a disordered system are used to determine the probability distribution of the free energy. The relation between them is essentially given by a Legendre transformation. A cut-off must be introduced to ensure the normalization of the distribution function. The method yields reasonable approximations or even rigorous results for the mean free energy without taking into account replica symmetry breaking (RSB). It therefore represents a conceptually and technically simple alternative to Parisi's RSB-scheme. Investigated examples are the random-energy model and the Sherrington-Kirkpatrick model of spin glasses.

Fermionic Thouless-Anderson-Palmer equations

We derive the Thouless-Anderson-Palmer (TAP) equations for the fermionic Ising spin glass. It is found that, just as in the classical Sherrington-Kirkpatrick spin-glass model, the conditions for stability and for validity of the free energy are equivalent. We determine the breakdown of the paramagnetic phase. Numerical solutions of the fermionic TAP equations at T = 0 allowed us to localize a first-order transition between the spin-glass phase and the paramagnetic phase at µ≈0.8. We computed at zero temperature the filling factor ν(µ) and the distribution of the internal fields. The saddle-point equations resulting from the calculation of the number of solutions to the TAP equations were found to be much more complicated, as in the classical case.

Glauber dynamics of the SK model: theory and simulations in the high-temperature phase

The low-temperature Glauber dynamics of the Sherrington-Kirkpatrick spin-glass model is studied using a combination of theory and computer simulations. The theoretical approach follows the spirit of Mori's continuous fraction expansion. The predicted amplitude of the long-time decay of the time-dependent equilibrium spin-spin correlation function agrees well with the Glauber dynamics simulation results. The theory predicts a universal dynamic exponent of in the low-temperature phase. The simulations cannot distinguish between a universal exponent and a temperature-dependent one.

Glauber dynamics of the SK model: theory and simulations in the low-temperature phase

The low-temperature Glauber dynamics of the Sherrington-Kirkpatrick spin-glass model is studied using a combination of theory and computer simulations. The theoretical approach follows the spirit of Mori's continuous fraction expansion. The predicted amplitude of the long-time decay of the time-dependent equilibrium spin-spin correlation function agrees well with the Glauber dynamics simulation results. The theory predicts a universal dynamic exponent of in the low-temperature phase. The simulations cannot distinguish between a universal exponent and a temperature-dependent one.

Exchange Monte Carlo Dynamics in the SK Model

The equilibration process of the Sherrington–Kirkpatrick spin-glass model is investigated using the exchange Monte Carlo (MC) method. It is found that a MC time scale for equilibration, τ eq , is nearly proportional to the system size and weakly depends on temperature, in contrast to the thermal activation behavior observed in a conventional MC simulation. Fluctuation of the overlap distribution over samples is confirmed to follow the non trivial probability law predicted by the replica-symmetry-breaking theory.

Kinetic theory approach to the SK spin glass model with Glauber dynamics

I present a new method to analyze Glauber dynamics of the Sherrington-Kirkpatrick (SK) spin glass model. The method is based on ideas used in the classical kinetic theory of fluids. I apply it to study spin correlations in the high temperature phase ($T\ge T_c$) of the SK model at zero external field. The zeroth order theory is equivalent to a disorder dependent local equilibrium approximation. Its predictions agree well with computer simulation results. The first order theory involves coupled evolution equations for the spin correlations and the dynamic (excess) parts of the local field distributions. It accounts qualitatively for the error made in the zeroth approximation.

Equilibrium Properties of the Ising Frustrated Lattice Gas

We study the equilibrium properties of an Ising frustrated lattice gas with a mean field replica approach. This model bridges usual {\em Spin Glasses} and a version of {\em Frustrated Percolation} model, and has proven relevant to describe the glass transition. It shows a rich phase diagram which in a definite limit reduces to the known Sherrington-Kirkpatrick spin glass model.

The Legendre transform of two replicas of the Sherrington-Kirkpatrick spin glass model

We prove a variational inequality linking the values of the free energy per site at different temperatures. This inequality is based on the Legendre transform of the free energy of two replicas of the system. We prove that equality holds whenβ≤1/\(\sqrt 2 \) and fails when 1/\(\sqrt 2 \) 1/\(\sqrt 2 \). We also use our inequality numerically within the low temperature region to improve (slightly) the best previously known lower bounds for the free energy and the ground state energy per site.

The Legendre transform of two replicas of the Sherrington-Kirkpatrick spin glass model

The Legendre transform of two replicas of the Sherrington-Kirkpatrick spin glass model

Dynamical Probability Distribution Function of the SK Model at High Temperatures

The microscopic probability distribution function of the Sherrington-Kirkpatrick (SK) model of spin glasses is calculated explicitly as a function of time by a high-temperature expansion. The resulting formula to the third order of the inverse temperature shows that an assumption made by Coolen, Laughton and Sherrington in their recent theory of dynamics is violated. Deviations of their theory from exact results are estimated quantitatively. Our formula also yields explicit expressions of the time dependence of various macroscopic physical quantities when the temperature is suddenly changed within the high-temperature region.