Slow interaction dynamics in spin-glass models

Penney, R W; Sherrington, David C.

doi:10.1088/0305-4470/27/12/011

J. Phys. A: Math. Gen.

1994

DOI: 10.1088/0305-4470/27/12/011

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Slow interaction dynamics in spin-glass models

R W Penney¹,

David C. Sherrington²

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Cited by 15 publications

(52 citation statements)

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“…We solve our model upon making the replica-symmetric Ansatz, and calculate the de Almeida-Thouless (AT) lines [6] (of which here there are two types), where continuous transitions occur to phases with broken replica symmetry. In doing so we also improve the calculations of [3]. As §…”

mentioning

confidence: 99%

“…we take the adiabatic limit). For the dynamics of the couplings the following Langevin form is proposed (which is the natural adaptation to XY spins of the choices originally made in [1,3] for Ising spins):…”

mentioning

confidence: 99%

“…External biases K ij = µNB ij serve to steer the weights to some preferred values (note: this notation follows that of [3]). The B ij are choosen to be quenched random variables, drawn independently from a Gaussian probability distribution with mean B 0 /N and varianceB/N.…”

mentioning

confidence: 99%

“…Comparison with (3) shows that the order parameters m The final stage of the calculation is to make the replica symmetry (RS) Ansatz. Since observables with identical Roman indices refer to system copies with identical couplings, whereas observables with identical Roman indices and identical Greek indices refer to system copies with identical couplings and identical spins, in the present model the RS Ansatz takes the form q…”

mentioning

confidence: 99%

“…The stability of the RS solutions is, as always, expressed in terms of the eigenvalues of the matrix of second derivatives of quadratic fluctuations at the saddlepoint [6]. We have calculated all eigenvalues and their multiplicity following [3,6] (full details will be published elsewhere [9]). We find two replicon eigenvalues: λ G , associated with the Greek replicas and λ R , associated with the Roman replicas.…”

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confidence: 99%

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TheXYspin glass with slow dynamic couplings

Jongen

Bollé

Coolen

1998

J. Phys. A: Math. Gen.

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Abstract. We investigate an XY spin-glass model in which both spins and couplings evolve in time: the spins change rapidly according to Glauber-type rules, whereas the couplings evolve slowly with a dynamics involving spin correlations and Gaussian disorder. For large times the model can be solved using replica theory. In contrast to the XY-model with static disordered couplings, solving the present model requires two levels of replicas, one for the spins and one for the couplings. Relevant order parameters are defined and a phase diagram is obtained upon making the replica-symmetric Ansatz. The system exhibits two different spin-glass phases, with distinct de AlmeidaThouless lines, marking continuous replica-symmetry breaking: one describing freezing of the spins only, and one describing freezing of both spins and couplings.Recently various models with a coupled dynamics of fast Ising spins and slow couplings have been studied (see e.g. [1,5] and references therein). In addition to physical motivations, such as understanding the simultaneous learning and retrieval in recurrent neural networks or the influence of slow atomic diffusion processes in disordered magnetic systems, there is a more theoretical interest in such models in that they generate the replica formalism for a finite number of replicas n. Moreover, the replica number is found to have a physical meaning as the ratio of two temperatures (characterizing the stochasticity in the spin dynamics and the coupling dynamics, respectively). In this letter we extend the methods and results obtained for Ising spin models to a classical XY spin-glass with dynamic couplings, whose spin variables are physically more realistic than Ising ones. In addition, the XY model is closely related to neural network models of coupled oscillators, which provide a phenomenological description of neuronal firing synchronization in brain tissue. We solve our model upon making the replica-symmetric Ansatz, and calculate the de Almeida-Thouless (AT) lines [6] (of which here there are two types), where continuous transitions occur to phases with broken replica symmetry. In doing so we also improve the calculations of [3]. As §

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confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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TheXYspin glass with slow dynamic couplings

Jongen

Bollé

Coolen

1998

J. Phys. A: Math. Gen.

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Model glasses coupled to two different heat baths

2000

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In a $p$-spin interaction spherical spin-glass model both the spins and the couplings are allowed to change in the course of time. The spins are coupled to a heat bath with temperature $T$, while the coupling constants are coupled to a bath having temperature $T_{J}$. In an adiabatic limit (where relaxation time of the couplings is much larger that of the spins) we construct a generalized two-temperature thermodynamics. It involves entropies of the spins and the coupling constants. The application for spin-glass systems leads to a standard replica theory with a non-vanishing number of replicas, $n=T/T_J$. For $p>2$ there occur at low temperatures two different glassy phases, depending on the value of $n$. The obtained first-order transitions have positive latent heat, and positive discontinuity of the total entropy. This is the essentially non-equilibrium effect. The predictions of longtime dynamics and infinite-time statics differ only for $n<1$ and $p>2$. For $p=2$ correlation of the disorder (leading to a non-zero $n$) removes the known marginal stability of the spin glass phase. If the observation time is very large there occurs no finite-temperature spin glass phase. In this case there are analogies with the broken-ergodicity dynamics in the usual spin-glass models and non-equilibrium (aging) dynamics. A generalized fluctuation-dissipation relation is derived.Comment: revtex, 28 pages, 7 figures, 3 table

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Partially Annealed Disorder and Collapse of Like-Charged Macroions

2008

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Slow interaction dynamics in spin-glass models

Cited by 15 publications

References 16 publications

TheXYspin glass with slow dynamic couplings

TheXYspin glass with slow dynamic couplings

Model glasses coupled to two different heat baths

Partially Annealed Disorder and Collapse of Like-Charged Macroions

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