Reflection symmetry in mean-field replica-symmetric spin glasses

Gribova, N. V.; Ryzhov, V. N.; Schelkacheva, T. I.; Tareyeva, E. E.

doi:10.1016/s0375-9601(03)01059-4

Cited by 15 publications

(12 citation statements)

References 20 publications

(21 reference statements)

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“…2, 3, and 4 have much in common. All of them are models with cubic terms in the free energy of the regular nonrandom problem, and their behavior undoubtedly differs from the behavior of models without cubic terms [13]. It is convenient to trace several of their distinguishing characteristics with the following model.…”

Section: Role Of the Reflection Symmetrymentioning

confidence: 99%

“…The presence of cubic terms in the free-energy expansion leads to a first-order phase transition, and their absence results in a second-order phase transition. In several papers where systems with random interaction were studied in the mean-field approximation, it was also noted that the cubic terms play an important role, and attempts were made to formulate the universality properties (see, e.g., [9]- [11] and [12], [13] and the references therein).…”

Section: Introductionmentioning

confidence: 99%

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Spin-glass-type state in complex nonmagnetic systems

2009

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Section: Role Of the Reflection Symmetrymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Spin-glass-type state in complex nonmagnetic systems

2009

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“…отличны от нуля, мы имеем ψ 3 -ную модель и в результате получаем сосущество-вание ориентационного стекла и дальнего ориентационного порядка, как показано в работах [18], [19], [28]. Наше дальнейшее рассмотрение наиболее близко следует работе [28].…”

Section: ориентационное стеклоunclassified

“…. [28]. Одновре-менное сосуществование ориентационного упорядочения (параметр порядка x ̸ = 0) и стекла (параметр порядка q ̸ = x 2 ) соответствует экспериментальным данным из работ [1], [2].…”

Section: ориентационное стеклоunclassified

Ориентационное упорядочение и переход в состояние ориентационного стекла в фуллерите $\mathrm C_{60}$

Tareeva¹,

Shchelkacheva²,

Shchelkacheva³

et al. 2008

ТМФ

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“…This is because using continuous variables in the case of twoparticle interactions effectively restores the reflection symmetry. We can easily see this by constructing the Ginzburg-Landau Hamiltonian using the Hubbard-Stratonovich transformation similarly to what was done in [13]. Indeed, the series for Tr A e φA contain odd powers of the quantity φ in the discrete case if Tr A 2n+1 = 0, while the corresponding integrals always vanish in the continuous case, and these terms are absent.…”

mentioning

confidence: 91%

Spherical analogues of some non-Ising spin glass models: Exact solutions

Tareyeva

2011

Theor Math Phys

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We consider several models of non-Ising spin glass for which we can naturally formulate an analogue of spherical conditions well known for the Ising model. We calculate the partition functions of the obtained models exactly. The random matrix method and the replica method yield identical results.Here, we consider continuous analogues of some known discrete non-Ising models of spin glass, namely, models of the spin-1 spin glass, quadruple glass, and the Potts spin glass with an arbitrary number of states. In these models, we can naturally introduce a condition analogous to the well-known spherical condition for the continuous analogue of the Ising model [1], which consequently allows solving these new "spherical" models exactly. The essential result is that exact results for these models obtained with the random matrix theory coincide with the results obtained using the replica method here.Although it is well known that spherical models are quite nonrealistic from the physical standpoint, they are rather important because they are rare examples of multiparticle systems for which partition functions can be calculated explicitly.The spherical approximation was first formulated by Berlin and Kac in [1], where discrete Ising spins with S 2 i = 1 were replaced with continuous spins satisfying the global condition i S 2 i = N , where the summation ranges all N sites of a lattice. The obtained problem was solved explicitly for the nearestneighbor interaction on various three-dimensional lattices.Soon after the appearance of the first theoretical papers on spin glasses, a spherical variant of the Sherrington-Kirkpatrick model [2], i.e., a model with an infinite radius of random interactions, was proposed, and its exact solution was found [3]. Models of spin glass with soft spins are in general quite popular in the kinetic approach to spin glasses because the presence of one-site terms allows describing the dynamical behavior of the glass. The spherical model of p-spin glass of quasi-Ising spins (see, e.g., [4], [5]), whose behavior provides a scenario of the liquid-glass transition, is the best-known model.In [6], we proposed an exactly solvable Potts spin glass model with three states and with additional conditions generalizing the spherical conditions for Ising spins. We there obtained an exact solution using random matrix theory. We also showed that the replica method gives the same solution. The obtained solution is replica-symmetric and stable, and we therefore do not need to break the replica symmetry. This distinguishes the spherical variant from just the Potts model with three states in which the replica-symmetric solution is stable only in a finite temperature range near the phase transition to the glass state [7], [8].This paper is a straightforward continuation of [6] in two directions. First, we consider spherical models of a Potts spin glass with an arbitrary number of states. Second, we break the Potts symmetry of the model in [6], which provides the possibility to study spherical analogues of the axial quad...

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Reflection symmetry in mean-field replica-symmetric spin glasses

Cited by 15 publications

References 20 publications

Spin-glass-type state in complex nonmagnetic systems

Spin-glass-type state in complex nonmagnetic systems

Ориентационное упорядочение и переход в состояние ориентационного стекла в фуллерите $\mathrm C_{60}$

Spherical analogues of some non-Ising spin glass models: Exact solutions

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