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On mean field equations for spin glasses (1982)

Thompson, Colin J.

Springer

Journal of statistical physics 27 (1982), S. 457-472

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ISSN: 1572-9613

Schlagwort(e): Cayley tree ; iteration ; fixed point ; spin glass ; Gaussian distribution ; local mean-field theory ; SK equations ; TAP equations.

Quelle: Springer Online Journal Archives 1860-2000

Thema: Physik

Notizen: Abstract In this paper we study rigorously the random Ising model on a Cayley tree in the limit of infinite coordination numberz → 8. An iterative scheme is developed relating mean magnetizations and mean square magnetizations of successive shells far removed from the surface of the lattice. In this way we obtain local properties of the model in the (thermodynamic) limit of an infinite number of shells. When the coupling constants are independent Gaussian random variables the SK expressions emerge as stable fixed points of our scheme and provide a valid local mean-field theory of spin glasses in which negative local entropy (at low temperatures) while perfectly possible mathematically may still perhaps be physically undesirable. Finally we examine the TAP equations and show that if the average over bond disorder and the limitz → 8 are actually performed, one recovers our iterative scheme and hence the SK equations also in the thermodynamic limit.

Materialart: Digitale Medien

URL: http://dx.doi.org/10.1007/BF01011086

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Digitale Medien

Local properties of an Ising model on a Cayley tree (1982)

Thompson, Colin J.

Springer

Journal of statistical physics 27 (1982), S. 441-456

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ISSN: 1572-9613

Schlagwort(e): Ising model ; Cayley tree ; phase transition ; iteration ; fixed point ; bifurcation ; ferromagnetic ; antiferromagnetic.

Quelle: Springer Online Journal Archives 1860-2000

Thema: Physik

Notizen: Abstract The Ising model on a Cayley tree displays a peculiar (continuous order) phase transition with zero long-range order at all finite temperatures. When one studies expection values of spins far removed from the surface (which contains a finite fraction of the total number of spins in the thermodynamic limit), however, one obtains the so-called Bethe approximation. Here we study such a local description by setting up a simple recurrence relation for successive shell magnetizations far removed from the surface. In the ferromagnetic case the local magnetization is a fixed point of the iterative transformation, while in the antiferromagnetic case the fixed point bifurcates to a two-cycle of the transformation (for low temperatures and fields) giving rise to local sublattice magnetizations. In both cases, local thermodynamical properties are obtained by integration.

Materialart: Digitale Medien

URL: http://dx.doi.org/10.1007/BF01011085

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Ising model with competing interactions on a Cayley tree (1983)

Inawashiro, Sakari ; Thompson, Colin J. ; Honda, Goshi

Springer

Journal of statistical physics 33 (1983), S. 419-436

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ISSN: 1572-9613

Schlagwort(e): Cayley tree ; iteration ; fixed points ; cycles ; attractors ; chaos ; spin glass ; frequency locking ; devil's staircase

Quelle: Springer Online Journal Archives 1860-2000

Thema: Physik

Notizen: Abstract An iterative scheme is developed for a renormalized effective nearest-neighbor couplingK r and effective field per siteK r for spins in therth shell of a Cayley tree with nearest neighborJ, and next nearest neighborJ′, interactions between Ising spins on the lattice. In addition to the expected paramagnetic, ferromagnetic, and antiferromagnetic phases, we find an intermediate range ofJ'/J 〈 0 values whereX r, and Kr iterate to a continuous or quasicontinuous attractor in theX-K plane. In this range the local magnetization is mainly chaotic with oscillatory glasslike behavior. Embedded in the chaos, however, are regions of periodic and commensurate phases.

Materialart: Digitale Medien

URL: http://dx.doi.org/10.1007/BF01009804

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