Infinite-range Spin-glass Model Research Articles

Effect of modularity on the Glauber dynamics of the dilute spin glass model

We study the Glauber dynamics of the dilute, infinite-ranged spin glass model, the so-called dilute Sherrington-Kirkpatrick (dSK) model. The dSK model has sparse couplings and can be classified by the modularity (M) of the coupling matrix. We investigate the effect of the modularity on the relaxation dynamics starting from a random initial state. By using the Glauber dynamics and the replica method, we derive the relaxation dynamics equations for the magnetization (m) and the energy per spin (r), in addition to the equation for the spin glass order parameter (q αβ ). In the replica symmetric (RS) analysis, we find that there are two solutions for the RS spin glass order parameter (q): q = 0which is stable for r 1/2 in the non-modular system and q = 0 which is stable for r 1/\(\sqrt 8 \) in the completely modular system. By substituting the proper q values into the equations for r, we find that the relaxation dynamics of r depends on the modularity, M. These results suggest that, in the context of evolutionary theory, the modularity may emerge spontaneously in the point-mutation-only framework (Glauber dynamics) under a changing environment.

Replica theory for Levy spin glasses**Dedicated to Professor Thomas Nattermann on the occasion of his 60th anniversary.

Infinite-range spin-glass models with Levy-distributed interactions show a spin-glasstransition with similarities to both the Sherrington–Kirkpatrick model and to disorderedspin systems on finite connectivity random graphs. Despite the diverging moments of thecoupling distribution the transition can be analyzed within the replica approach by workingat imaginary temperature. Within the replica symmetric approximation a self-consistentequation for the distribution of local fields is derived and from the instability of theparamagnetic solution to this equation the glass transition temperature is determined. Therole of the percolation of rare strong bonds for the transition is elucidated. The resultspartly agree and partly disagree with those obtained within the cavity approach. Numericalsimulations using parallel tempering are in agreement with the transition temperaturesfound.

Stability of a two-sublattice spin-glass model

We study the stability of the replica-symmetric solution of a two-sublattice infinite-range spin-glass model, which can describe the transition from an antiferromagnetic to a spin-glass state. The eigenvalues associated with replica-symmetric perturbations are in general complex. The natural generalization of the usual stability condition is to require the real part of these eigenvalues to be positive. The necessary and sufficient conditions for all the roots of the secular equation to have positive real parts is given by the Hurwitz criterion. The generalized stability condition allows a consistent analysis of the phase diagram within the replica-symmetric approximation.

Stability analysis of a two-dimensional uniaxial vortex glass.

The simplest model of a vortex glass is considered which is applicable for the description of a twodimensional uniaxial vortex crystal formed by the fluxon lines in a large area Josephson junction with inhomogeneous width. The analysis is performed in replica representation in terms of a free-energy functional which depends on the renormalized correlation function. The properties of different solutions of the Dyson equation are considered, the main attention being devoted to investigating the stability of these solutions. In particular the solution with the one-step replica symmetry breaking which corresponds to the absolute maximum of free energy is shown to be always stable ~when it exists at all!. The unimportance of higher-order corrections for the form of the asymptotic behavior of the correlation function in the phase with the one-step replica symmetry breaking is also demonstrated. I. INTRODUCTION The discovery of the high-Tc superconconducting materials have essentially increased the possibilities for the experimental observations of various phenomena related with the presence of vortices in superconductors ~vortex lattice melting, pinning, creep and so on!. This has led to the active development of theoretical investigation of these phenomena and the appearance of many new ideas ~for a recent review see Ref. 1!. In particular a suggestion has been made that at low temperatures a phase should exist in which the motion of the vortices is quenched by disorder and therefore the linear resistance is absent. 2,3 The properties of this phase ~including the multitude of metastable states separated by the diverging barriers! are expected to be more or less analogous to those of widely discussed infinite-range spin-glass models 4 and therefore it is usually called a vortex glass. It has been suggested 2,5 that the simplest model which allows one to analyze the large scale properties of a vortex crystal ~or charge-density wave! interacting with a random potential can be described by the Hamiltonian

Replica-symmetry breaking and quantum fluctuation effects in thep-spin interaction spin-glass model with a transverse field

The Ising infinite-range spin-glass model with p-spin interactions in the presence of a transverse field is studied for large but finite p using the Matsubara time representation and Parisi's scheme of replica-symmetry breaking. In the spin-glass phase, the corrections to the limit appear much more essential than in the classical counterpart. It is shown that the quantum fluctuations have the effect of destroying the spin-glass order and a lower temperature is required to stabilize the spin-glass phase. The spin autocorrelation function in the spin-glass phase is explicitly calculated as a function of p and the Matsubara time. The result is just complementary and consistent with that previously obtained for the paramagnetic state without using the replica method.

Spin glasses on thin graphs

In a recent paper [C. Baillie, D.A. Johnston and J.-P. Kownacki, Nucl. Phys. B 432 (1994) 551] we found strong evidence from simulations that the Ising antiferromagnet on “thin” random graphs — Feynman diagrams — displayed a mean-field spin-glass transition. The intrinsic interest of considering such random graphs is that they give mean-field theory results without long-range interactions or the drawbacks, arising from boundary problems, of the Bethe lattice. In this paper we reprise the saddle-point calculations for the Ising and Potts ferromagnet, antiferromagnet and spin glass on Feynman diagrams. We use standard results from bifurcation theory that enable us to treat an arbitrary number of replicas and any quenched bond distribution. We note the agreement between the ferromagnetic and spin-glass transition temperatures thus calculated and those derived by analogy with the Bethe lattice or in previous replica calculations. We then investigate numerically spin glasses with a ± J bond distribution for the Ising and Q = 3, 4, 10, 50 state Potts models, paying particular attention to the independence of the spin-glass transition from the fraction of positive and negative bonds in the Ising case and the qualitative form of the overlap distribution P( q) for all of the models. The parallels with infinite-range spin-glass models in both the analytical calculations and simulations are pointed out.

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.

Numerical study of the TAP equations for infinite-range spin glass models

Abstract The nature of spin glass phase for infinite-range model is investigated by an approach without relying on the replica method. Mean field equations are solved numerically and order parameters such as q(x) are obtained. First, to study a kind of quantum effects, the SK model in a transverse field is investigated within the pair approximation. The nature of the spin glass phase is shown to be unchanged by the presence of a transverse field. Secondly, a generalized SK model with a crystal field is studied to clarify the nature of the first order transition. The first order spin glass transition is described consistently. The distribution of pi = 〈S2i〉 is doubly peaked in the spin glass phase on the first order transition line.

First-order transitions in an infinite-range spin-glass model

The spin-glass phase and the first-order spin-glass transitions are investigated for a generalized Sherrington-Kirkpatrick Ising spin-glass model with S=1 in a crystal field. Mean field equations for the model are derived and they are solved numerically. The double peak structure in the distribution of pi=(Si2) is shown to be important near the first order transitions. The transitions are qualitatively similar to the ground state transition.

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.

Distribution of barrier heights in infinite-range spin glass models

The spherical model of a spin glass with infinite-range interactions is studied to obtain the distribution of barrier heights between time-reversed states. The problem is shown to be equivalent at leading order to that of finding the distribution of the difference between the largest two eigenvalues of the exchange matrix. For the case where the exchange matrix is Gaussian orthogonal an approximate distribution is quoted and verified numerically. The barriers between time-reversed states scale with system size, N, as (1-T/Tc)N1/3. The barrier heights in the Sherrington and Kirkpatrick model of a spin glass are also studied. Using the mean-field equations of Thouless, Anderson and Palmer (1977) a perturbation analysis in terms of the eigenvectors of the Hessian is performed. In the scaling limit T to Tc, N to infinity with (1-T/Tc)N1/3 fixed, macroscopic condensation into the largest eigenvector occurs and the barrier between time-reversed states scales with system size as (1-T/Tc)N1/3.

A heuristic approach to the infinite-ranged spin-glass model

The local minima of the Sherrington-Kirkpatrick model (1978) are studied numerically by introducing a heuristic method. The author's approach naturally leads to their ultrametricity. For finite systems some observations on this method suggest the relation d1-d2 approximately d3 among three distances of three low-energy local minima d1>or=d2>or=d3. Overlap functions are studied numerically, giving consistent results with the above relation and the results of the replica symmetry breaking theory.

The infinite-range clock spin glass model: an investigation of the relevance of reflection symmetry

The infinite-range S-state clock spin glass model is investigated by means of a Landau expansion. Within the replica symmetric theory a second-order phase transition is found for all values of S. By applying replica symmetry-breaking theory it is shown that although the S=3 case has the peculiar behaviour already observed for Potts models, the absence of reflection symmetry is qualitatively irrelevant for all other odd-state clock glasses. For S<or=5 this model is in the same class as the xy spin glass.

Universality of the Spin-Glass Transition in the Cd1−xMnXTe System

AbstractUniversality of the spin-glass (SG) transition in bulk Cd1−xMnxTe was investigated by determining the scaling behavior of the spin-glass order parameter q as a function of the reduced temperature t = (Tg-T)/Tg, where Tg is the transition temperature. q(T) was determined from the SQUID magnetometric data both above and below the transition. It can be shown that, q(T) = 1 + T[│θ│ − C/χ(T)]−1. C and θ, the Curie and Curie-Weiss parameters, were obtained from a non-linear regression of the dc low field susceptibility χ(T) above Tg. q(T) thus obtained exhibits the cannonical order parameter criteria, viz. q(T&gt;Tg) = 0, q(T→Tg)= 0, and q(T→Tg)→0, and q(T→0)→1. In range of Mn concentration studied (0.3 ≦ x ≦ 0.55), Tg ranged between 13 K and 23.5 K. Thus q as a function of absolute temperature behaves differently for different x. However, universality is clearly evidenced when the dependence on t is determined. q(t) exhibits a universal scaling law. We observe q ˜ tβ with β ≅ 0.95. Overall good agreement was noted with the Sherrington-Kirkpatrick (SK) infinite range SG model. Observed value of β is in excellent agreement with the model prediction β −1. But in the Cd1−xMnxTe system we find θ&lt;0, indicating a net antiferromagnetic interaction and │θ│/Tg &gt; 1 as opposed to the SK model. We believe this to be additional evidence that the spin interactions are not distributed as a Gaussian function.

Self-averaging correlation functions in the mean field theory of spin glasses

In the infinite range spin glass model, we consider the staggered spin 03C303BB associated with a given eigenvector of the interaction matrix. We show that the thermal average of 03C3203BB is a self-averaging quantity and we compute it.

Stability conditions for the infinite-range vector spin glass

Using the recently formulated method of Plefka (1971), the authors derive the TAP free-energy functional for the infinite-range vector spin glass model. The stability condition, in the case of zero magnetic field, has a simple form: the average of the squares of the single-site susceptibilities is less than or equal to the inverse variance of the random interaction. For an applied magnetic field, stability conditions are also obtained.

Stability of Parisi's solution of a spin glass model

The condition of stability of Parisi's solution of the infinite-range spin glass model is obtained and examined near the transition temperature. The matrix of second derivatives delta 2F/ delta Qalpha beta delta Qgamma nu has non-negative eigenvalues only. There are two continuous branches of the eigenvalues and zero is an accumulation point of the eigenvalues.

Staggered-magnetization approach to spin-glasses

A method is developed to solve the Sherrington-Kirkpatrick infinite-ranged spin-glass model, considering the staggered magnetizations associated with the eigenvalue spectrum of the random interaction matrix as the spin-glass order parameters. It is shown that the spin-glass ordering is not associated with a macroscopic condensation of eigenstates, but instead each of the $N$ eigenstates acquires, below ${T}_{c}$, nonzero magnetization of $O(1)$.

Mean field theory for Heisenberg spin glasses

The infinite-ranged spin glass model is studied in the general case of m-component spins. Results for the magnetization laws, the shape of the phase diagram, the nature of the transitions are presented.

Stability and susceptibility in Parisi's solution of a spin glass model

The stability of Parisi's solution (1979-80) of the infinite-range spin glass model is examined near the transition temperature. It is found that the quadratic terms in the variation of the free energy are all non-positive. The free energy is only a local maximum with respect to q(x) if q is constrained to be a monotonic function of x. The connection between replica symmetry breaking and failure of linear response theory for magnetic susceptibility is shown in more detail. It is shown that the order parameter susceptibility is divergent, as expected, in the spin glass phase.