The Blume-Emery-Griffiths spin glass is studied by renormalization-group theory in d=3. The boundary between the ferromagnetic and paramagnetic phases has first-order and two types of second-order segments. This topology includes an inverted tricritical point, first-order transitions replacing second-order transitions as temperature is lowered. The phase diagrams show disconnected spin-glass regions, spin-glass and paramagnetic reentrances, and complete reentrance, where the spin-glass phase replaces the ferromagnet as temperature is lowered for all chemical potentials.
The time evolution of the extremely diluted Blume-Emery-Griffiths neural network model is studied, and a detailed equilibrium phase diagram is obtained exhibiting pattern retrieval, fluctuation retrieval and self-sustained activity phases. It is shown that saddle-point solutions associated with fluctuation overlaps slow down considerably the flow of the network states towards the retrieval fixed points. A comparison of the performance with other three-state networks is also presented.
We study the Blume-Emery-Griffiths spin glass model in presence of an attractive coupling between real replicas, and evaluate the effective potential as a function of the density overlap. We find that there is a region, above the first order transition of the model, where metastable states with a large density overlap exist. The line where these metastable states appear should correspond to a purely dynamical transition, with a breaking of ergodicity. Differently from what happens in p-spin glasses, in this model the dynamical transition would not be the precursor of a 1-step RSB transition, b…
We study the Blume-Emery-Griffiths model in a random crystal field in two and three dimensions, through a real-space renormalization-group approach and a mean-field approximation, respectively. According to the two-dimensional renormalization-group calculation, non-symmetry-breaking first-order phase transitions are eliminated and symmetry-breaking discontinuous transitions are replaced by continuous ones, when disorder is introduced. On the other hand, the mean-field calculation predicts that first-order transitions are not eliminated by disorder, although some changes are introduced in the p…
We introduce a spherical version of the frustrated Blume-Emery-Griffiths model and solve exactly the statics and the Langevin dynamics for zero particle-particle coupling (K=0). In this case the model exhibits an equilibrium transition from a disordered to a spin glass phase which is always continuous for nonzero temperature. The same phase diagram results from the study of the dynamics. Furthermore, we notice the existence of a nonequilibrium time regime in a region of the disordered phase, characterized by aging as occurs in the spin glass phase. Due to a finite equilibration time, the syste…
The optimal capacity of a diluted Blume-Emery-Griffiths neural network is studied as a function of the pattern activity and the embedding stability using the Gardner entropy approach. Annealed dilution is considered, cutting some of the couplings referring to the ternary patterns themselves and some of the couplings related to the active patterns, both simultaneously (synchronous dilution) or independently (asynchronous dilution). Through the de Almeida-Thouless criterion it is found that the replica-symmetric solution is locally unstable as soon as there is dilution. The distribution of the c…
Renormalization group on hierarchical lattices is often considered a valuable tool to understand the critical behavior of more complicated statistical mechanical models. In presence of quenched disorder, however, in many model cases predictions obtained with the Migdal-Kadanoff bond removal approach fail to quantitatively and qualitatively reproduce critical properties obtained in the mean-field approximation or by numerical simulations in finite dimensions. In order to critically review this limitation we analyze the behavior of Ising and Blume-Emery-Griffiths models on more complicated hiera…
The parallel dynamics of the fully connected Blume-Emery-Griffiths neural network model is studied for arbitrary temperature. By employing a probabilistic signal-to-noise approach, a recursive scheme is found determining the time evolution of the distribution of the local fields and, hence, the evolution of the order parameters. A comparison of this approach is made with the generating functional method, allowing to calculate any physical relevant quantity as a function of time. Explicit analytic formula are given in both methods for the first few time steps of the dynamics. Up to the third ti…
We study the equilibrium properties of the Blume-Emery-Griffiths model with bilinear quenched disorder in the case of attractive as well as repulsive biquadratic interactions. The global phase diagram of the system is calculated in the context of the replica symmetric mean field approximation.
A generalised integer S Ising spin glass model is analysed using the replica formalism. The bilinear couplings are assumed to have a Gaussian distribution with ferromagnetic mean <J_ij> = Jo. Incorporation of a quadrupolar interaction term and a chemical potential leads to a richer phase diagram with transitions of first and second order. The first order transition may be interpreted as a phase separation, and contrary to what has been argued previously, it persists in the presence of disorder. Finally, the stability of the replica symmetric solution with respect to fluctuations in replica spa…
A Blume-Emery-Griffiths perceptron model is introduced and its optimal capacity is calculated within the replica-symmetric Gardner approach, as a function of the pattern activity and the imbedding stability parameter. The stability of the replica-symmetric approximation is studied via the analogue of the Almeida-Thouless line. A comparison is made with other three-state perceptrons.
The study of the mean-field static solution of the Random Blume-Emery-Griffiths-Capel model, an Ising-spin lattice gas with quenched random magnetic interaction, is performed. The model exhibits a paramagnetic phase, described by a stable Replica Symmetric solution. When the temperature is decreased or the density increases, the system undergoes a phase transition to a Full Replica Symmetry Breaking spin-glass phase. The nature of the transition can be either of the second order (like in the Sherrington-Kirkpatrick model) or, at temperature below a given critical value, of the first order in t…
The effect of the random quantum transverse field $Ω$ on the tricritical behavior of the spin-1 Blume- Emery- Griffiths (BEG) model is studied using effective field theory. It is found, that the tricritical behavior depends on both the biquadratic interaction $K$, single- ion anisotropy $Δ$, and the concentration $p$ of the disorder of $Ω$. Indeed, there exist a special value $p_{1}$ of the probability $p$ below which the tricritical behavior disappears.
The Blume-Emery-Griffiths model with a random crystal field is studied in a random graph architecture, in which the average connectivity is a controllable parameter. The disordered average over the graph realizations is treated by replica symmetry formalism of order parameter functions. A self-consistent equation for the distribution of local fields is derived and numerically solved by a population dynamics algorithm. The results show that the average connectivity amounts to changes in the topology of the phase diagrams. Phase diagrams for representative values of the model parameters are comp…
The thermodynamic and retrieval properties of fully connected Blume-Emery-Griffiths networks, storing ternary patterns, are studied using replica mean-field theory. Capacity-temperature phase diagrams are derived for several values of the pattern activity. It is found that the retrieval phase is the largest in comparison with other three-state neuron models. Furthermore, the meaning and stability of the so-called quadrupolar phase is discussed as a function of both the temperature and the pattern activity. Where appropriate, the results are compared with the diluted version of the model.
The parallel dynamics of the fully connected Blume-Emery-Griffiths neural network model is studied at zero temperature for arbitrary using a probabilistic approach. A recursive scheme is found determining the complete time evolution of the order parameters, taking into account all feedback correlations. It is based upon the evolution of the distribution of the local field, the structure of which is determined in detail. As an illustrative example, explicit analytic formula are given for the first few time steps of the dynamics. Furthermore, equilibrium fixed-point equations are derived and com…
The short- and long-time dynamics of model systems undergoing a glass transition with apparent inversion of Kauzmann and dynamical arrest glass transition lines is investigated. These models belong to the class of the spherical mean-field approximation of a spin-$1$ model with $p$-body quenched disordered interaction, with $p>2$, termed spherical Blume-Emery-Griffiths models. Depending on temperature and chemical potential the system is found in a paramagnetic or in a glassy phase and the transition between these phases can be of a different nature. In specific regions of the phase diagram coe…
We obtain the phase diagram for the Blume-Capel model with bimodal distribution for random crystal fields, in the space of three fields: temperature, crystal field and magnetic field. We find that three critical lines meet at a tricritical point, but only for weak disorder. As disorder strength increases there is no tricritical point in the phase diagram. We instead find a bicritical end point, where only two of the critical lines meet on a first order surface in the H=0 plane. For intermediate strengths of disorder, the phase diagram has critical end points along with the bicritical end point…
The multicritical behavior of the Blume-Capel model with infinite-range interactions is investigated by introducing quenched disorder in the crystal field $Δ_{i}$, which is represented by a superposition of two Gaussian distributions with the same width $σ$, centered at $Δ_{i} = Δ$ and $Δ_{i} = 0$, with probabilities $p$ and $(1-p)$, respectively. A rich variety of phase diagrams is presented, and their distinct topologies are shown for different values of $σ$ and $p$. The tricritical behavior is analyzed through the existence of fourth-order critical points as well as how the complexity of th…
We study the effect of different symmetric random field distributions: trimodal and Gaussian on the phase diagram of the infinite range Blume-Capel model. For the trimodal random field, the model has a very rich phase diagram. We find three new ordered phases, multicritical points like tricritical point (TCP), bicritical end point (BEP), critical end point (CEP) along with some multi-phase coexistence points. We also find re-entrance at low temperatures for some values of the parameters. On the other hand for the Gaussian distribution the phase diagram consists of a continuous line of transiti…
We investigate the scaling of the interfacial adsorption of the two-dimensional Blume-Capel model using Monte Carlo simulations. In particular, we study the finite-size scaling behavior of the interfacial adsorption of the pure model at both its first- and second-order transition regimes, as well as at the vicinity of the tricritical point. Our analysis benefits from the currently existing quite accurate estimates of the relevant (tri)critical-point locations. In all studied cases, the numerical results verify to a level of high accuracy the expected scenarios derived from analytic free-energy…
We study the mean-field static solution of the Blume-Emery-Griffiths-Capel model with quenched disorder, an Ising-spin lattice gas with quenched random magnetic interaction. The thermodynamics is worked out in the Full Replica Symmetry Breaking scheme. The model exhibits a high temperature/low density paramagnetic phase. When the temperature is decreased or the density increased, the system undergoes a phase transition to a Full Replica Symmetry Breaking spin-glass phase. The nature of the transition can be either of the second order (like in the Sherrington-Kirkpatrick model) or, at temperatu…
Serial vs parallel recall in the Blume-Every-Griffiths neural networks
Istituto Nazionale di Alta Matematica Francesco Severi（意大利高等数学研究所弗朗切斯科·塞韦里）；Università del Salento（萨伦托大学）；Dipartimento di Matematica e Fisica, Università del Salento（萨伦托大学数学与物理系）；Istituto Nazionale di Fisica Nucleare, Sezione di Lecce（意大利国家核物理研究院莱切分部）；Dipartimento di Informatica, Università di Pisa（比萨大学计算机科学系）；Sapienza Università di Roma（罗马智慧大学）；Dipartimento di Scienze di Base e Applicazioni all’Ingegneria, Sapienza Università di Roma（罗马智慧大学基础科学与工程应用科学系）；Istituto Nazionale di Fisica Nucleare, Sezione di Roma（意大利国家核物理研究院罗马分部）；CNR Nanotec, Sezione di Lecce（意大利国家研究委员会纳米技术研究所莱切分部）
Hamiltonians for general multi-state spin-glass systems with Ising symmetry are derived for both sequential and synchronous updating of the spins. The possibly different behaviour caused by the way of updating is studied in detail for the (anti)-ferromagnetic version of the models, which can be solved analytically without any approximation, both thermodynamically via a free-energy calculation and dynamically using the generating functional approach. Phase diagrams are discussed and the appearance of two-cycles in the case of synchronous updating is examined. A comparative study is made for the…
We analyze the Blume-Emery-Griffiths model with disordered magnetic interaction displaying the inverse freezing phenomenon. The behaviour of this spin-1 model in crystal field is studied throughout the phase diagram and the transition and spinodal lines for the model are computed using the Full Replica Symmetry Breaking Ansatz that always yelds a thermodynamically stable phase. We compare the results both with the quenched disordered model with Ising spins on lattice gas - where no reentrance takes place - and with the model with generalized spin variables recently introduced by Schupper and S…
We analyze the Blume-Emery-Griffiths model with disordered magnetic interaction that displays the inverse freezing phenomenon. The behavior of this spin-1 model in crystal field is studied throughout the phase diagram and the transition and spinodal lines for the model are computed using the Full Replica Symmetry Breaking Ansatz that always yields a thermodynamically stable phase. We compare the results both with the formulation of the same model in terms of Ising spins on lattice gas, where no reentrance takes place, and with the model with generalized spin variables recently introduced by Sc…
