Infinite-range Interactions Research Articles

Glassy systems under time-dependent driving forces: Application to slow granular rheology

We study the dynamics of a glassy model with infinite range interactions externally driven by an oscillatory force. We find a well-defined transition in the (temperature-amplitude-frequency) phase diagram between (i) a "glassy" state characterized by the slow relaxation of one-time quantities, aging in two-time quantities and a modification of the equilibrium fluctuation-dissipation relation; and (ii) a "liquid" state with a finite relaxation time. In the glassy phase, the degrees of freedom governing the slow relaxation are thermalized to an effective temperature. Using Monte Carlo simulations, we investigate the effect of trapping regions in phase space on the driven dynamics. We find that it alternates between periods of rapid motion and periods of trapping. These results confirm the strong analogies between the slow granular rheology and the dynamics of glasses. They also provide a theoretical underpinning to earlier attempts to present a thermodynamic description of moderately driven granular materials.

Hole-burning experiments within glassy models with infinite range interactions

We reproduce the results of nonresonant spectral hole-burning experiments with glassy models with infinite-range interactions that generalize the mode-coupling approach to nonequilibrium situations. We show that an ac field modifies the integrated linear response and the correlation in a way that depends on the amplitude and frequency of the pumping field. We study the effect of the waiting and recovery times and the number of oscillations applied. This calculation will help discriminate which results can and which cannot be attributed to spatial heterogeneities in real systems.

Nontrivial Phase Behaviour in the Infinite-Range Quantum Mattis Model

We have solved the quantum version of the Mattis model with infinite-range interactions. A variational approach gives the exact solution for the infinite-range system, in spite of the non-commutative nature of the quantum spin components; this implies that quantum effects are not predominant in determining the macroscopic properties of the system. Nevertheless, the model has a surprisingly rich phase behaviour, exhibiting phase diagrams with tricritical, three-phase and critical end points.

An infinite system of Brownian balls with infinite range interaction

We study an infinite system of Brownian hard balls, moving in R d and submitted to a smooth infinite range pair potential. It is represented by a diffusion process, which is constructed as the unique strong solution of an infinite-dimensional Skorohod equation. We also prove that canonical Gibbs states associated to the sum of the hard core potential and the pair potential are reversible measures for the dynamics.

(1 + ∞)-dimensional attractor neural networks

We solve a class of attractor neural network models with a mixture of 1D nearest-neighbour interactions and infinite-range interactions, which are both of a Hebbian-type form. Our solution is based on a combination of mean-field methods, transfer matrices, and 1D random-field techniques, and is obtained both for Boltzmann-type equilibrium (following sequential Glauber dynamics) and Peretto-type equilibrium (following parallel dynamics). Competition between the alignment forces mediated via short-range interactions, and those mediated via infinite-range ones, is found to generate novel phenomena, such as multiple locally stable `pure' states, first-order transitions between recall states, 2-cycles and non-recall states, and domain formation leading to extremely long relaxation times. We test our results against numerical simulations and simple benchmark cases, and find excellent agreement.

A Neural Network Model of an Ising Spin Glass

The behaviour of an Ising spin glass (S = 1/2) with infinite range interactions is modelled using a numerical simulation based on a neural network. Thermodynamic variables are defined on the network, and are found to obey the Thouless-Anderson—Palmer theory when the applied magnetic field is zero. When a magnetic field is applied along the spin direction, complex field-dependent behaviour appears, including a state in which the Edwards—Anderson order parameter is independent of temperature below the critical temperature.

Random field Ising chains with synchronous dynamics

We solve a class of attractor neural network models with a mixture of 1D nearest-neighbour and infinite-range interactions, which are of a Hebbian-type form. Our solution is based on a combination of mean-field methods, transfer matrices and 1D random-field techniques, and is obtained for Boltzmann-type equilibrium (following sequential Glauber dynamics) and Peretto-type equilibrium (following parallel dynamics). Competition between the alignment forces mediated via short-range interactions, and those mediated via infinite-range ones, is found to generate novel phenomena, such as multiple locally stable `pure' states, first-order transitions between recall states, 2-cycles and non-recall states, and domain formation leading to extremely long relaxation times. We test our results against numerical simulations and simple benchmark cases and find excellent agreement.

Strong phase separation in a model of sedimenting lattices

We study the steady state resulting from instabilities in crystals driven through a dissipative medium, for instance, a colloidal crystal which is steadily sedimenting through a viscous fluid. The problem involves two coupled fields, the density and the tilt; the latter describes the orientation of the mass tensor with respect to the driving field. We map the problem to a one-dimensional lattice model with two coupled species of spins evolving through conserved dynamics. In the steady state of this model each of the two species shows macroscopic phase separation. This phase separation is robust and survives at all temperatures or noise levels- hence the term strong phase separation. This sort of phase separation can be understood in terms of barriers to remixing which grow with system size and result in a logarithmically slow approach to the steady state. In a particular symmetric limit, it is shown that the condition of detailed balance holds with a Hamiltonian which has infinite-ranged interactions, even though the initial model has only local dynamics. The long-ranged character of the interactions is responsible for phase separation, and for the fact that it persists at all temperatures. Possible experimental tests of the phenomenon are discussed.

An Exactly Solvable Model of Disordered System

An exactly solvable model of disordered system with infinite-ranged interaction is solved. We find that almost every eigenvector is localized.

M-Vector spin glasses in the presence of uniaxial anisotropies

The effects of the two lowest-order uniaxial anisotropy fields on the phase diagrams of the classical m-vector spin glass are analyzed. The model is defined in terms of infinite–range interactions and the replica approach is used to study the system. The replica-symmetric phase diagrams present qualitative modifications with respect to those with no uniaxial anisotropies; new features, in particular, reentrance effects arise. In some cases, the reentrant critical frontiers are totally inside the region of instability of the replica-symmetric solution and may disappear within more general parametrizations, whereas in other cases, they coincide with the limit of stability of such a solution and should persist under replica-symmetry breaking.

Nonequilibrium dynamics of infinite particle systems with infinite range interactions

We discuss the existence and uniqueness of nonequilibrium dynamics of infinitely many particles interacting via superstable pair interactions in one and two dimensions. The interaction is allowed to be of infinite range and singular at the origin. Under suitable regularity conditions on the interaction potential, we show that if the potential decreases polynomially as the distance between interacting two particles increases, then the tempered solution to the system of Hamiltonian equations exists. Moreover, if the potential satisfies further that either it has a subexponential decreasing rate or it is everywhere two-times continuously differentiable, then we show that the tempered solution is unique. The results extend those of Dobrushin and Fritz obtained for finite range interactions.

Random anisotropy in lattice gas model

We consider a lattice-gas model with infinite-range interaction with site dependent random anisotropy distributed with a Gaussian distribution. The random anisotropy lattice-gas analogous of the random field Ising model is solved exactly using a replica theory. We show that, at finite temperature, the introduction of disorder eliminates completely the phase transition, and destroy the equivalence between real gases and Ising magnets. Whereas at T = 0, the density of occupied sites has a step-like behavior as function of the random anisotropy.

Addendum to “Solitons for Discrete Systems with Infinite Interaction Range”

Addendum to “Solitons for Discrete Systems with Infinite Interaction Range”

The infinite-range two-real-Ising replicas spin glass

A spin-glass model, with two Ising coupled systems (real replicas), is investigated in the limit of infinite-range interactions, through the replica method. Besides the Ising representation, the model is also considered as a discrete two-component four-state clock spin glass. The phase diagram of the model is obtained, within the replica-symmetry approximation, exhibiting a collinear spin-glass phase, i.e., no transverse four-state clock ordering. The order parameter conjugated to the coupling field presents a discontinuity along the spin-glass phase, when the field is reversed. The validity limits of the replica-symmetry solution are analysed.

On the applicability of mode coupling theory to a ϕ 4-model with first order phase transition

A ϕ 4-model with symmetric double-well-like on-site potential and anharmonic, infinite range interactions is investigated. This model exhibits a first order phase transition at a temperature T c. The time-dependent displacement correlation function is studied in the framework of the mode coupling theory (MCT). Depending on the choice of slow modes, MCT makes qualitatively different predictions which are compared with MD-results. These numerical results suggest that only the order parameter mode {ie1-1} should be considered as slow. In that case it is shown that MCT yields a dynamical transition in the supercooled high-temperature phase {ie1-2} at a temperature T* which coincides with the spinodal temperature T s (T s = 0 for our model) where the metastable supercooled phase becomes instable.

Potts models on Feynman diagrams

We investigate numerically and analytically Potts models on `thin' random graphs - generic Feynman diagrams, using the idea that such models may be expressed as the limit of a matrix model. The thin random graphs in this limit are locally tree-like, in distinction to the `fat' random graphs that appear in the planar Feynman diagram limit, , more familiar from discretized models of two-dimensional gravity. The interest of the thin graphs is that they give mean-field theory behaviour for spin models living on them without infinite range interactions or the boundary problems of genuine tree-like structures such as the Bethe lattice. q-state Potts models display a first-order transition in the mean field for q>2, so the thin-graph Potts models provide a useful test case for exploring discontinuous transitions in mean-field theories in which many quantities can be calculated explicitly in the saddle-point approximation. Such discontinuous transitions also appear in multiple Ising models on thin graphs and may have implications for the use of the replica trick in spin-glass models on random graphs.

Central peak for a scalar ϕ 4-model: mode coupling approach revisited

We reinvestigate the mode coupling approach to the central peak which occurs in the vicinity of a structural phase transition at T c. For a scalar ϕ 4-model it is shown that the use of renormalized vertices leads to quite different results compared to recent calculations with bare vertices. Particularly, we prove that the latter are obtained in leading order of the anharmonicity constant of the on-site potential from a perturbational treatment of the renormalized vertices. Again, this mode coupling approach may yield a dynamical transition at a temperature T c'(≥ T c) at which the dynamics becomes nonergodic, i.e. a central peak occurs. For a ϕ 4- model with infinite range interactions our theoretical predictions are consistent with numerical results. Furthermore, if the fluctuations in the vicinity of Tc are Gaussian, no dynamical transition occurs above Tc. Therefore the temperature T 0'obtained from the Ginzburg criterion sets an upper bound for T c'. If a dynamical transition occurs, it is shown that the nonergodicity parameter as function of wave vector q and temperature T follows from an universal master function.

Fermion condensation and non Fermi liquid behavior in a model with long range forces

The phenomenon of the so called Fermion condensation, a phase transition analogous to Bose condensation but for Fermions, postulated in the past to occur in systems with strong momentum dependent forces, is reanalysed in a model with infinite range interactions. The strongly non Fermi Liquid behavior of this system is demonstrated analytically at $T=0$ and at $T\neq 0$ in the superconducting and normal phases. The validity of the quasiparticle picture is investigated and seems to hold true for temperatures less than the characteristic temperature $T_f$ of the Fermion condensation.

Newtonian versus Langevin dynamics: Example of ϕ4-model with infinite range interactions

The functional integral representation for the generating functional (GF) of the canonically averaged ensemble with an underlying Newtonian dynamics is obtained. It is shown that for this representation the non-linear fluctuation-dissipation theorem (NFDT) has the same form as for the Langevin dynamics case. This GF-representation is used for the investigation of the dynamics of the ϕ 4-model with infinite range interactions at T > T c. It is shown that the kinetic equation for the complete correlation function has the same form as for the Langevin dynamics case, which was considered before. All peculiarities of Newtonian dynamics are absorbed by one-particle (2-point and 4-point) correlator and response functions. The analysis of this equation shows that the 1/ N-fluctuations (where N is the number of particles) restore the ergodicity of the system with the characteristicsrate τ −1 ∼ μ 2/ N, where μ is a coupling constant.

Tricritical Behavior of Ising Spin Glasses with Charge Fluctuations.

We show that tricritical points displaying unusal behaviour exist in phase diagrams of fermionic Ising spin glasses as the chemical potential or the filling assumes characteristic values. Exact results for infinite range interaction and a one loop renormalization group analysis of thermal tricritical fluctuations for finite range models are presented. Surprising similarities with zero temperature transitions and a new $T=0$ tricritical point of metallic quantum spin glasses are derived.