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.
Quasi-one-dimensional spin systems described by an Ising-like Hamiltonian with a strong space anisotropy (s=1/2) are investigated. Magnetic properties of this model are examined in the approximation including Gaussian fluctuations of molecular field. This paper reports an attempt at obtaining more accurate results for Gaussian fluctuation of molecular field by an exact formula for mean fluctuations of a spin.
A simple spin system is constructed to simulate dynamics of asset prices and studied numerically. The outcome for the distribution of prices is shown to depend both on the dimension of the system and the introduction of price into the link measure. For dimensions below 2, the associated risk is high and the price distribution is bimodal. For higher dimensions, the price distribution is Gaussian and the associated risk is much lower. It is suggested that the results are relevant to rare assets or situations where few players are involved in the deal making process.
The dynamics of a kicked, anisotropic, damped spin is reduced to a two-dimensional map. This map exhibits such features as bifurcation diagrams, regular or chaotic attractors/repellors and intermittent-like transitions between two strange attractors. With increase of damping a transition from chaos to the fixed point attractor occurs. On the contrary to the Hamiltonian case the type of magnetic anisotropy plays a crucial role for damped models.
An Ising (s=1) model of ferromagnetic nanoparticles and ultrathin films of the sc structure deposited on a non-magnetic substrate is considered. The substrate was assumed to affect the crystal field around the atoms lying closest to it. Consequently, the one-ion anisotropy constant of spin moments of these atoms becomes dependent on the appropriate component of the tensor of the magnetic structure deformation. This dependence was assumed to be linear. To obtain approximations of the Gaussian fluctuations of molecular field, the generalised equilibrium reduced density operator, along with the Feynman diagram technique were used. As a result, temperature dependences of the spatial distributions of mean fluctuations of the magnetic field versus the changes in the one-ion anisotropy constant induced by the non-magnetic substrate were obtained.
In this work we investigate a mixed spin-1/2 and spin-1 Ising model on a decorated square lattice. In addition to standard pair interactions between nearest neighbors, we also consider in our calculations three-site four-spin interactions and the effect of single ion anisotropy. Applying the well-known decoration-iteration transformation, we derive a simple relation between the partition function of the model under investigation and the Onsager's partition function for the simple square lattice. With the help of this mapping relation we are able to calculate exact expressions for all relevant physical quantities of the system (for example, the magnetization, internal energy, Gibbs free energy, entropy and specific heat). The most interesting results have been found in the model without pair interactions. In this case, besides the standard ordered and disordered phases, there also appears an interesting phase which is ordered only partially. Analyzing relevant physical quantities of the system, we have found that the partial ordering is closely related to very strong frustrations that lead to non-zero entropy at T=0.
We present a study of the magnetic properties of a mixed ferro-ferrimag-netic ternary alloy of the type AB_p C_{1-p} on a cubic lattice consisting of three different Ising spins S_A =3/2, S_B = 2, and S_C = 5/2. We employ the mean-field approximation and Monte Carlo simulation to find the compensation temperatures of the system for selected values of the parameters in the model Hamiltonian. In particular, the relation between considered mixed ferro-ferrimagnetic model and magnetic properties of the ternary metal Prussian blue analog such as (Fe_p^{II}Mn_{1-p}^{II})_1.5 [Cr^{III}(CN)_6] · nH_2O is discussed.
Geometrically frustrated quantum spin systems, with competing antiferromagnetic couplings, show the Kahn degenerate frustration for some specific values of Heisenberg Hamiltonian parameters. It has been recently shown for rings with a defect bond and centered rings. In the case of classical counterparts of these systems, degenerated configurations with the lowest energy are present for the energy function parameter greater than a certain threshold. In these domains such configurations are planar but non-collinear with continuous changes of the net magnetic moment with respect to the Hamiltonian parameter. Outside these domains there is unique collinear ground state configuration (neglecting choice of the net magnetic moment direction). However, these collinear configurations are the same in both non-frustrated and geometrically frustrated domains. Numerically exact calculations for quantum systems strongly confirm that determined properties of their classical counterparts realize the classical limit s→∞.
The S=1 pseudospin formalism was recently proposed to describe the charge degree of freedom in a model high-T_{c} cuprate with the on-site Hilbert space reduced to the three effective valence centers, nominally Cu^{1+;2+;3+}. With small corrections the model becomes equivalent to a strongly anisotropic S=1 quantum magnet in an external magnetic field. We have applied a generalized mean-field approach and quantum Monte-Carlo technique for the model 2D S=1 system to find the ground state phase with its evolution under deviation from half-filling and different correlation functions. Special attention is given to the role played by the on-site correlation ("single-ion anisotropy").
The phase diagram of the Askhin-Teller model in two dimensions is determined. Numerical calculations are performed for the simple square L × L lattice using transfer matrix technique. Exploiting finite-size scaling all unknown critical lines were obtained with good accuracy. An extended version of the Ashkin-Teller model is also considered within the molecular field renormalization group method and the critical surface for three-parameter odd-parity Hamiltonian is calculated.
The motion of domain walls in thin garnet films was investigated numerically using Slonczewski's equations of wall motion for the case of periodic drive field. The type of the wall motion was analyzed by observation of phase trajectories and spatio-temporal diagrams. It was found that depending on the period and amplitude of the drive field the motion of the wall is periodic or chaotic, reflecting the character of the dynamical processes connected with horizontal Bloch lines in the wall.
The route to chaos of domain wall in thin magnetic film, which is described by Słonczewski's equations of motion, is analyzed numerically. Hagedorn's model of surface stray field is applied. Ranges of periodic and chaotic wall motion as a function of constant in time, drive field are found. Comparison of results with those obtained for Hubert's model of the stray field is made.
The critical coupling and spontaneous magnetization curve for the 2D-Ising model are calculated using the effective-field method with correlations. An application of the method to the quantum S=1/2 1D-Heisenberg model is presented and reliable low-temperature estimates of the specific heat are evaluated.
Ground-state phase diagram of the mixed spin-1/2 and spin-1 Ising-Heisenberg model on doubly decorated planar lattices is examined using the generalized decoration-iteration transformation. The main attention is devoted to the comparison of the ground-state properties of the quantum Ising-Heisenberg model and its semi-classical Ising analogue.
The temperature driven reorientation of the magnetization observed in thin ferromagnetic films and multilayer systems is studied theoretically. We take into account layer dependent magnetic anisotropies. The free energy and the direction of magnetization are calculated by using a perturbational approach as well as a nonperturbational treatment. It is shown that in most cases a continuous magnetic reorientation is obtained. An increasing anisotropy energy as compared to the exchange interaction leads to an increasing width of the magnetic reorientation.
Within the framework of the effective-field theory with correlations we investigate effects of an external magnetic field and random site dilution on basic thermodynamic quantities, such as the magnetization and the magnetic susceptibility, on the geometrically frustrated triangular lattice Ising antiferromagnet. Behavior of these quantities is presented in the temperature-field parameter space for selected mild degrees of dilution. It is found that, besides the anomalies associated with phase transitions from the ferrimagnetic to the paramagnetic state, in certain regions of the parameter space these functions display some more anomalies and peculiarities, as a result of joint effects of the geometrical frustration, magnetic dilution, thermal fluctuations and the applied magnetic field.
In this paper, we consider an Ising model with three competing interactions (nearest neighbor, next-nearest neighbor, and ternary prolonged neighbor) on the Cayley tree of order two, investigated by Ganikhodjaev et al. We study translation-invariant Gibbs measures of the Ising model with these competing interactions. Also, we investigate the set of the extreme Gibbs measures called Markov random fields with memory 2 of the model.
The Blume-Emery-Griffiths model for spins S=1 in a bilayer with z=5 nearest neighbours is studied with the use of Gaussian fluctuations approximation. The fluctuations of two molecular fields, connected with two order parameters, are introduced. Their influence on phase diagrams for non-negative values of the biquadratic coupling constant is taken under consideration. The results are compared with those obtained by the mean-field approximation and discussed.
The universal sequence of the ground states for antiferromagnetic frustrated rings with the odd number of the local spins s and a single bond defect α described by the isotropic Heisenberg Hamiltonian is discussed. The Lieb-Mattis energy level ordering in a pentanuclear ring is revealed and the arising magnetisation steps are demonstrated.
We study effects of the next-next-nearest-neighbour antiferromagnetic (J₃ < 0) interaction on critical properties (or phase diagram) of the frustrated spin-½ J₁-J₂-J₃ Ising antiferromagnet on the honeycomb lattice by using the effective-field theory with correlations. Beside the ground-state energy, we find that there is a region of J₃ < 0 in which the frustrated honeycomb lattice antiferromagnet exhibits a tricritical point, at which the phase transition changes from the second order to the first one on the line between Néel antiferromagnetic and paramagnetic phases.
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