Sherrington-Kirkpatrick Spin Glass Model Research Articles

On the relaxation of infinite-range spin glasses

The relaxation of the Sherrington-Kirkpatrick model for spin glasses is studied at low temperatures. A recently developed numerical method is used by which an infinitely large system is simulated, hence finite size effects are avoided. For parallel dynamics the decay of the energy as well as of the magnetization is investigated. For low temperatures we find evidence for a relaxation into a state which is characterized by non-equilibrium values of the energy and a non-vanishing remanent magnetization.

On the dynamics of a spherical spin-glass in a magnetic field

We carry out an analysis of the effect of a quenched magnetic field on the dynamics of the spherical Sherrington-Kirkpatrick spin-glass model. We show that there is a characteristic time introduced by the presence of the field. Firstly, for times sufficiently small the dynamic scenario of the zero field case - with aging effects - is reproduced. Secondly, for times larger than the characteristic time one sees equilibrium dynamics. This dynamical behaviour is reconciled with the geometry of the energy landscape of the model. We compare this behaviour with experimental observations at a finite applied field.

Quantum phase transitions in XY spin models.

The ground state of an $\mathrm{XY}$ model changes at a critical transverse field, above which there is a gap from the fully polarized ground state to the first excitation. We investigate the phase transition from this ``Mott insulator'' phase as the transverse field approaches its critical value. We calculate exactly some Green's functions and critical exponents for $d$-dimensional ferromagnetic lattices and for the Sherrington-Kirkpatrick spin-glass model. Finite-dimensional spin glasses are also discussed.

The Sherrington-Kirkpatrick model of spin glasses and stochastic calculus: The high temperature case

We study the fluctuations of free energy, energy and entropy in the high temperature regime for the Sherrington-Kirkpatrick model of spin glasses. We introduce here a new dynamical method with the help of brownian motions and continuous martingales indexed by the square root of the inverse temperature as parameter, thus formulating the thermodynamic formalism in terms of random processes. The well established technique of stochastic calculus leads us naturally to prove that these fluctuations are simple gaussian processes with independent increments, a generalization of a result proved by Aizenman, Lebowitz and Ruelle [1].

On the metastable states of the zero-temperature SK mode

We generalize the calculation of the avenge number of metastable states of the Sherrington-Kirkpatrick spin-glass model at zero temperature to the case where there is an external magnetic field and the case where there is an asymmetric bond distribution. The results are in agreement with the existence of the AT instability at zero temperature for all finite values of external field and average bond value. We conclude the paper by examining the question of dual metastability, i.e. when spin configurations are metastable both with and without the presence of a magnetic field.

First-order transition in a spin-glass model

We consider a generalization of the infinite-range Sherrington-Kirkpatrick spin-glass model with arbitrary spin S and the inclusion of crystal-field effects. For integer S, replica-symmetric calculations have shown the presence of both continuous and discontinuous transitions and a tricritical point. For S=1, we report a detailed numerical analysis of the replica-symmetric solutions. We locate the first-order boundary and clarify some inconsistencies of the previous analyses. Some analytic asymptotic expansions are used to support the numerical findings.

Experimental study and theoretical interpretation of hysteresis loops and Henkel plots in soft magnetic materials

In this paper we discuss an application of the moving Preisach model (MPM) and the zero-temperature Sherrington-Kirkpatrick spin-glass model (SKM) to the interpretation of hysteresis phenomena in soft materials, with particular attention to the analogies and differences between the models. Both models are compared with hysteresis measurements performed on strips of amorphous Fe-B-Si alloy. We describe experimental data using the same distribution of local coercivities and interactions fields in both of them. The results confirm that the MPM is actually able to reproduce the hysteresis behavior of soft magnetic materials and can be useful to discuss the complicated behavior of the SKM, where we explicitly know the form of interactions.

Several results on the finite-size corrections in the Sherrington-Kirkpatrick spin-glass model

The authors present several results on the finite-size corrections in the SK model within the spin-glass phase slightly below the transition temperature and along the de Almeida-Thouless (AT) line. In the first case, they study the finite-size corrections for the distribution of overlaps P(q) and for ultrametricity. Along the AT line, finite-size corrections to the free and internal energy are also obtained. Numerical results are in good agreement with their predictions.

Correlations in the Trotter dimension in the transverse Ising spin-glass model.

The nature of the paramagnetic phase of the transverse Sherrington-Kirkpatrick (SK) spin-glass model is not completely understood because of the quantum fluctuations. We use different approximations to describe these fluctuations. The resulting paramagnetic\char21{}spin-glass phase diagrams are compared to those obtained by numerical methods and by previous theories.

Cavity-fields approach to quantum XY spin-glass models in a transverse field.

The cavity-fields approach, recently developed for studying the Sherrington-Kirkpatrick model of spin glasses and its quantum version, is appropriately reformulated for the infinite-range XY spin-glass model in a transverse field. The equations for the phase diagram of the model are obtained and preliminary numerical results are also presented.

Study of hysteresis phenomena in Ising-like pseudospin systems with random local reversal fields and random spin-spin interactions

We study the hysteresis properties of a generalization of the Sherrington–Kirkpatrick spin-glass model, including, in addition to the random spin-spin coupling costants {Jij}, a random distribution {hci} of local spin coercive fields, hindering spin flipping. The system evolution is driven by the time-dependent external field H(t). We investigate numerically the behavior as a function of H of the average magnetization per spin, m(H), and of the spin internal field distribution P(h;H). A rich variety of m(H) hysteresis loops is obtained, whose properties are controlled by the parameters of the {Jij} and {hci} distribution. It is found that the contributions of {Jij} and {hci} to the coercive field of these loops simply add to each other. An approximate Fokker–Planck equation for P(h;H) is derived, which permits one to understand the main features of P(h;H). In particular, it leads to a natural interpretation of the fact, put in evidence by computer simulations, that P(h;H) is, for small h, a linear function of h, whose slope is independent of H and of the {hci} distribution.

Dynamics of the finite SK spin-glass

The dynamical behavior of a Sherrington-Kirkpatrick spin-glass model consisting of a large but finite number of Ising spins with a time evolution given by Glauber dynamics is investigated. Starting from the resummation of a diagrammatic expansion we derive a differential equation for the response function which allows us to handle nonperturbative effects. This enables us to find explicit expressions for the dynamical behavior of response and correlation function on time scales related to those free energy barriers which diverge with system sizeN. For the largest of these barriers we find a behavior proportional toN λ with λ=1/3.

Spectral moments for systems described by an ensemble of hamiltonians

We generalize Van Vleck's method of moments for disordered solid-state that can be described by an ensemble of hamiltonians. Special reference is made to N.M.R. spectroscopy. Inhomogeneous broadening for a sample of free spins and the more complex problem of interacting spins are discussed in detail. The cumulant expansion and the Liouville space formalism are introduced as tools to simplify the expressions for the moments and to obtain physical approximations. As a particular application the Sherrington-Kirkpatrick model for spin glasses is derived and interpreted physically.

Ultrametricity in the infinite-range Ising spin glass

Monte Carlo simulations have been used to study the development of ultrametricity in the low-temperature phase of the infinite-range Sherrington-Kirkpatrick model of the Ising spin glass. Equilibrium configuration overlap data have been obtained for sizes N between 16 and 512 spins at the transition temperature T=Tc and in the spin-glass phase at T=0.6Tc. At the lower temperature, the data are consistent with the emergence, in the thermodynamic limit, of isosceles triangles formed by the mutual overlaps of three configurations, as predicted by Mezard and co-workers (1984, 1987). Evidence for equilateral triangles is not clear and may require larger sizes. This behaviour is contrasted with the data at T=Tc, where all overlap distributions are found to scale with size in terms of a single exponent.

Some rigorous results on the Sherrington-Kirkpatrick spin glass model

We prove that in the high temperature regime (T/J>1) the deviation of the total free energy of the Sherrington-Kirkpatrick (S-K) spin glass model from the easily computed log Av(Z N ({βJ})) converges in distribution, asN → ∞, to a (shifted) Gaussian variable. Some weak results about the low temperature regime are also obtained.

Some comments on the Sherrington-Kirkpatrick model of spin glasses

In this paper the high-temperature phase of general mean-field spin glass models, including the Sherrington-Kirkpatrick (SK) model, is analyzed. The free energy in zero magnetic field is calculated explicitly for the SK model, and uniform bounds on quenched susceptibilities are established. It is also shown that, at high temperatures, mean-field spin glasses are limits of short-range spin glasses, as the range of the interactions tends to infinity.

Static allocation in massively parallel computers

In this paper we present the statistical mechanical treatment of an optimization problem related to static allocation in massively parallel computers. We show that there is a strong relation between this problem and the Sherrington-Kirkpatrick model for spin glasses. This analogy makes it likely that the space of solutions shows an ultrametric structure. The significance of the results to parallel processing is discussed, and particular emphasis is given to the complexity of finding the optimal allocation. We discuss directions for further research in this novel area.

A mathematical reformulation of Derrida's REM and GREM

The large system limit of the Random Energy Model (REM) and generalized Random Energy Model (GREM) of Derrida is investigated, and found to be universal. This permits systematic calculations of relevance in particular to Parisi's solution of the Sherrington-Kirkpatrick spin-glass model.

A statistical mechanical study of Boltzmann machines

Hinton and Sejnowski (1983) have described recently a novel statistical mechanical system which they named the Boltzmann machine. The interesting property of Boltzmann machines is that they can learn to recognise the structure in a set of patterns simply by being shown an example subset of patterns. In this paper some numerical simulations of Boltzmann machines are reported. It is found that the annealing schedule proposed by Ackley, Hinton and Sejnowski (1985) is adequate to obtain a Boltzmann distribution of states, on which the key part of the algorithm depends, but it is clear that the algorithm will require massive computations for large networks. It is also found that there is a window of annealing temperatures at which learning is possible, and the sensitivity of the learning rate to temperature can be understood in terms of the density of states at low energies. Direct calculations of the partition function in small instances of Boltzmann machines are used to characterise the number of states which are thermally accessible for particular annealing schedules. Finally, since Boltzmann machines bear some resemblance to models of disordered magnetic systems, a comparison is made with results for the Sherrington-Kirkpatrick spin-glass model. Both systems support multiple metastable states (i.e. stable with respect to single spin flips), but, in contrast to the SK spin glass, Boltzmann machines exhibit a random distribution of low-energy states in terms of Hamming distance.

Spin-wave gap in uniaxial spin glasses.

The spectrum of elementary excitations (``spin waves'') is computed within the Sherrington-Kirkpatrick model for spin glasses with single-ion uniaxial anisotropy. It is shown that in the longitudinal spin-glass phase (when only spin components along the ``easy axis'' are frozen) the spin-wave spectrum has a gap which vanishes, as the anisotropy is reduced, at a nonzero value of the anisotropy, namely at the critical value for the onset of transverse spin-glass freezing. The gap is computed as a function of the anisotropy.