On classical spin-glass models

Grensing, D.; Kühn, Reimer

doi:10.1051/jphys:01987004805071300

Cited by 11 publications

(5 citation statements)

References 4 publications

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“…We have shown that the distribution of spin-spin interaction constants can be found directly by way of calculations of classical Equations (5) and analysis of statistical data of simulation. It is theoretically shown (see inequality (16)) that at least for the 1D spin-glass problem the distribution of the spin-spin interaction constants can not be Gauss-Edwards-Anderson type. In particular, the analysis of numerical data of simulation shows that they obey to Lévys alpha-stable distribution law.…”

Section: Discussionmentioning

confidence: 99%

“…The algorithm works as follows. Randomly M sets of initial parameters are generated and parallel calculations of Equation (17) for unknown variables i x and i y transact taking into account conditions (16) and (18). However only specifying initial conditions is not enough for the solution of these equations.…”

Section: Numerical Experimentsmentioning

confidence: 99%

“…In the models of second type, the bondrandomness is expressed in terms of some underlining hidden site-randomness and is thus of a superficial nature. However, it has been pointed out in the works [14][15][16] that this feature retains an important physical aspect of true spin-glasses, viz. that they are random with respect to the positions of magnetic impurities.…”

Section: Introductionmentioning

confidence: 99%

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A New Parallel Algorithm for Simulation of Spin-Glass Systems on Scales of Space-Time Periods of an External Field

Gevorkyan¹,

Abajyan²,

Sukiasyan³

2011

JMP

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We study the statistical properties of an ensemble of disordered 1D spatial spin-chains (SSCs) of certain length in the external field. On nodes of spin-chain lattice the recurrent equations and corresponding inequality conditions are obtained for calculation of local minimum of a classical Hamiltonian. Using these equations for simulation of a model of 1D spin-glass an original high-performance parallel algorithm is developed. Distributions of different parameters of unperturbed spin-glass are calculated. It is analytically proved and shown by numerical calculations that the distribution of the spin-spin interaction constant in the Heisenberg nearest-neighboring Hamiltonian model as opposed to the widely used Gauss-Edwards-Anderson distribution satisfies the Lévy alpha-stable distribution law which does not have variance. We have studied critical properties of spin-glass depending on the external field amplitude and have shown that even at weak external fields in the system strong frustrations arise. It is shown that frustrations have a fractal character, they are self-similar and do not disappear at decreasing of calculations area scale. After averaging over the fractal structures the mean values of polarizations of the spin-glass on the scales of external field's space-time periods are obtained. Similarly, Edwards-Anderson's ordering parameter depending on the external field amplitude is calculated. It is shown that the mean values of polarizations and the ordering parameter depending on the external field demonstrate phase transitions of first-order.

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Section: Discussionmentioning

confidence: 99%

Section: Numerical Experimentsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A New Parallel Algorithm for Simulation of Spin-Glass Systems on Scales of Space-Time Periods of an External Field

Gevorkyan¹,

Abajyan²,

Sukiasyan³

2011

JMP

View full text Add to dashboard Cite

show abstract

“…In the models of second type the bond-randomness is expressed in terms of some underlining hidden site-randomness and is thus of a superficial nature. It has been pointed out in the works [14][15][16], however, this feature retains an important physical aspect of true spin glasses, viz. they are random with respect to the positions of magnetic impurities.…”

Section: Introductionmentioning

confidence: 97%

On Modeling of Statistical Properties of Classical 3D Spin Glasses

Gevorkyan,

Abajyan,

Ayryan

2011

Preprint

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We study statistical properties of 3D classical spin glass layer of certain width and infinite length. The 3D spin glass is represented as an ensemble of disordered 1D spatial spin-chains (SSC) where interactions are random between spin-chains (nonideal ensemble of 1D SSCs). It is proved that at the limit of Birkhoff's ergodic hypothesis performance 3D spin glasses can be generated by Hamiltonian of disordered 1D SSC with random environment. Disordered 1D SSC is defined on a regular lattice where one randomly oriented spin is put on each node of lattice. Also it is supposed that each spin randomly interacts with six nearest-neighboring spins (two spins on lattice and four in the environment). The recurrent transcendental equations are obtained on the nodes of spin-chain lattice. These equations combined with the Silvester conditions allow step by step construct spin-chain in the ground state of energy where all spins are in minimal energy of classical Hamiltonian. On the basis of these equations an original high-performance parallel algorithm is developed for 3D spin glasses simulation. Distributions of different parameters of unperturbed spin glass are calculated. In particular, it is analytically proved and by numerical calculations shown that the distribution of spin-spin interaction constant in Heisenberg nearest-neighboring Hamiltonian model as opposed to widely used Gauss-Edwards-Anderson distribution satisfies Lévy alpha-stable distribution law which does not have variance. A new formula is proposed for construction of partition function in kind of one-dimensional integral on energy distribution of 1D SSCs.

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“…The case where M is is strictly bounded could be termed "standard disordered mean field", and it is this type of models that were studied by Pastur and Figotin in 1977, the case of two patterns having been introduced by Luttinger [Lut] shortly before that. Such "site-disorder" models were studied again intensely some years later by a number of people, emphasizing applications of large deviation methods [vHvEC,vH1,GK,vHGHK,vH2,AGS2,JK,vEvHP]. A general large deviation theory for such systems was obtained by Comets [Co] somewhat later.…”

Section: Introductionmentioning

confidence: 99%

Hopfield Models as Generalized Random Mean Field Models

Bovier

Gayrard

1998

Mathematical Aspects of Spin Glasses and Neural Networks

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We give a comprehensive self-contained review on the rigorous analysis of the thermodynamics of a class of random spin systems of mean field type whose most prominent example is the Hopfield model. We focus on the low temperature phase and the analysis of the Gibbs measures with large deviation techniques. There is a very detailed and complete picture in the regime of "small α"; a particularly satisfactory result concerns a non-trivial regime of parameters in which we prove 1) the convergence of the local "mean fields" to gaussian random variables with constant variance and random mean; the random means are from site to site independent gaussians themselves; 2) "propagation of chaos", i.e. factorization of the extremal infinite volume Gibbs measures, and 3) the correctness of the "replica symmetric solution" of Amit, Gutfreund and Sompolinsky [AGS]. This last result was first proven by M. Talagrand [T4], using different techniques. # Work partially supported by the Commission of the European Union under contract CHRX-CT93-0411† To appear in "Mathematics of spin glasses and neural networks", A.

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On classical spin-glass models

Cited by 11 publications

References 4 publications

A New Parallel Algorithm for Simulation of Spin-Glass Systems on Scales of Space-Time Periods of an External Field

A New Parallel Algorithm for Simulation of Spin-Glass Systems on Scales of Space-Time Periods of an External Field

On Modeling of Statistical Properties of Classical 3D Spin Glasses

Hopfield Models as Generalized Random Mean Field Models

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