Hierarchical solutions of the Sherrington-Kirkpatrick model: Exact asymptotic behavior near the critical temperature

Janiš, V.; Klic, A.

doi:10.1103/physrevb.74.054410

Cited by 10 publications

(28 citation statements)

References 14 publications

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“…Its ratio χ 1 /α at the AT line as a function of temperature is plotted in Figure 2. The ratio diverges in zero magnetic field where m = 0 and both χ 1 and m are linearly proportional to θ = (T c − T )/T c , while α ∼ 2θ 2 [11]. It is interesting to notice that the asymptotic solution near the AT line reduces to 1RSB in non-zero magnetic fields.…”

Section: Asymptotic Solution Near De Almeida-thouless Instability Linementioning

confidence: 93%

Replica-symmetry breaking: discrete and continuous schemes in the Sherrington–Kirkpatrick model

Janiš¹,

Klic²,

Ringel

2008

J. Phys. A: Math. Theor.

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Abstract. We study hierarchies of replica-symmetry-breaking solutions of the Sherrington-Kirkpatrick model. Stationarity equations for order parameters of solutions with an arbitrary number of hierarchies are set and the limit to infinite number of hierarchical levels is discussed. In particular, we demonstrate how the continuous replica-symmetry breaking scheme of Parisi emerges and how the limit to infinite-many hierarchies leads to equations for the order-parameter function of the continuous solution. The general analysis is accompanied by an explicit asymptotic solution near the de Almeida-Thouless instability line in the nonzero magnetic field.

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Section: Asymptotic Solution Near De Almeida-thouless Instability Linementioning

confidence: 93%

Replica-symmetry breaking: discrete and continuous schemes in the Sherrington–Kirkpatrick model

Janiš¹,

Klic²,

Ringel

2008

J. Phys. A: Math. Theor.

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“…It seems that at least for the SK model, continuous measures of the Parisi solution form a complete space and the Parisi free energy determines the exact, marginally stable solution. We demonstrated it explicitly in the asymptotic region below the critical temperature of the spin-glass phase in zero magnetic field 9 and recently also in the nonzero magnetic field. 10 On the other hand, there are models, such as the Potts spin glass, 11 where a discrete onestep RSB appears to be stable on a finite temperature interval.…”

Section: Free-energy Functional For the Parisi Solutionmentioning

confidence: 95%

Free-energy functional for the Sherrington-Kirkpatrick model: The Parisi formula completed

Janiš

2008

Phys. Rev. B

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The Parisi formula for the free energy of the Sherrington-Kirkpatrick model is completed to a closed-form generating functional. We first find an integral representation for a solution of the Parisi differential equation and represent the free energy as a functional of order parameters. Then, we set stationarity equations for local maxima of the free energy determining the order-parameter function on interval ͓0, 1͔. Finally, we show without resorting to the replica trick that the solution of the stationarity equations leads to a marginally stable thermodynamic state.

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“…, m K ) for arbitrary K via the asymptotic expansion below the transition temperature in a small parameter θ = 1 − T /T c . 21 Only this asymptotic solution was able to resolve the question of the structure of the equilibrium state. We found the leading order of the order parameters…”

Section: A Ising Spin Glassmentioning

confidence: 99%

Ergodicity breaking in frustrated disordered systems: replicas in mean-field spin-glass models

2015

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We discuss ergodicity breaking in frustrated disordered systems with no apparent broken symmetry of the Hamiltonian and present a way how to amend it in the lowtemperature phase. We demonstrate this phenomenon on mean-field models of spin glasses. We use replicas of the spin variables to test thermodynamic homogeneity of ergodic equilibrium systems. We show that replica-symmetry breaking reflects ergodicity breaking and is used to restore an ergodic state. We then present explicit asymptotic solutions for the Ising, Potts and p-spin glasses. Each of the models shows a different low-temperature behavior and the way the replica symmetry and ergodicity are broken.Keywords: ergodicity and thermodynamic homogeneity; replications of phase space; mean-field theory of spin glasses; discrete and continuous replica-symmetry breaking; asymptotic solutions with broken replica symmetry

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Hierarchical solutions of the Sherrington-Kirkpatrick model: Exact asymptotic behavior near the critical temperature

Cited by 10 publications

References 14 publications

Replica-symmetry breaking: discrete and continuous schemes in the Sherrington–Kirkpatrick model

Replica-symmetry breaking: discrete and continuous schemes in the Sherrington–Kirkpatrick model

Free-energy functional for the Sherrington-Kirkpatrick model: The Parisi formula completed

Ergodicity breaking in frustrated disordered systems: replicas in mean-field spin-glass models

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