Replica field theory for deterministic models. II. A non-random spin glass with glassy behaviour

Marinari, Enzo; Parisi, Giorgio; Ritort, Fèlix

doi:10.1088/0305-4470/27/23/011

Cited by 205 publications

(292 citation statements)

References 30 publications

(57 reference statements)

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“…In this direction it would be also interesting to investigate in a general way the type of phase transition for a generic eigenvalue distribution using analytical techniques such as those developed for the ROM. 17 It would also be interesting to investigate the behavior of the different terms in the TAP expansion to see how this change of behavior occurs and how this is related to the geometrical properties of the free energy landscape. Note that the information contained in the density of eigenvalues is closely related to the topological features of the energy landscape.…”

Section: Discussionmentioning

confidence: 99%

“…Setting mϭ1 in Eq. ͑51͒ gives 15 The precise way to locate both transitions (T d and T K ) was suggested in a series of papers by Kirkpatrick et al 16 and later on applied to several models such as the random orthogonal model, 17 Potts glasses, 18 and mean-field quantum spin glasses. 19 The static transition is located by expanding the corresponding free energy f (q,m) about mϭ1 and writing…”

Section: ͑50͒mentioning

confidence: 99%

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Squared interaction matrix Sherrington-Kirkpatrick model for a spin glass

Dean

Ritort

2002

Phys. Rev. B

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The mean-field theory of a spin glass with a specific form of nearest-and next-nearest-neighbor interactions is investigated. Depending on the sign of the interaction matrix chosen we find either the continuous replica symmetry breaking seen in the Sherrington-Kirkpartick model or a one-step solution similar to that found in structural glasses. Our results are confirmed by numerical simulations and the link between the type of spin-glass behavior and the density of eigenvalues of the interaction matrix is discussed.

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Section: Discussionmentioning

confidence: 99%

Section: ͑50͒mentioning

confidence: 99%

Squared interaction matrix Sherrington-Kirkpatrick model for a spin glass

Dean

Ritort

2002

Phys. Rev. B

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“…We identify two dangers: correlations between metastable states close to the ground state and too large degeneracy of the ground state. We apply the formalism to the periodic Ising spin model with orthogonal coupling matrix and find that it gives the same free energy as fiduciary Hamiltonian approach [1]. Further we show that it works in the intermediate temperature range but fails at low temperatures when metastable states become correlated.…”

mentioning

confidence: 87%

“…[1] although their properties at finite N are markedly different. Furthermore, this free energy is the same as obtained by fiduciary Hamiltonian approach [1]. Paramagnetic state.…”

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confidence: 99%

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Replica-symmetry breaking in long-range glass models without quenched disorder

Ioffe¹,

Lopatin²

2001

J. Phys.: Condens. Matter

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We discuss mean field theory of glasses without quenched disorder focusing on the justification of the replica approach to thermodynamics. We emphasize the assumptions implicit in this method and discuss how they can be verified. The formalism is applied to the long range Ising model with orthogonal coupling matrix. We find the one step replica-symmetry breaking solution and show that it is stable in the intermediate temperature range that includes the glass state but excludes very low temperatures. At very low temperatures this solution becomes unstable and this approach fails.The thermodynamics of glasses without quenched disorder is a long standing problem in statistical physics. The interest to this problem was renewed recently when it was understood that powerful methods developed for the glasses with quenched disorder can be often applied to this problem. [1][2][3][4][5][6]. In both systems the local magnetization in the ground state varies from site to site and different sites are typically non-equivalent. The qualitative reason why glasses without quenched disorder are more difficult to describe theoretically than spin glasses is the following. The mean field theory has to operate with the average magnetization (or its moments), not with quantities which depend on a realization and a particular state. The average quantities appear naturally in spin glasses after averaging over quenched disorder which makes all sites equivalent.A few methods were suggested to overcome this difficulty for the glasses without quenched disorder. First, a mapping of some glass models to the quenched disordered problems was suggested [1], this method has an obvious disadvantage that such mapping is difficult to guess. Second, it was noted that a typical dynamics in a glassy system leads not to a ground state but to one of many metastable states providing an effective averaging mechanism [7] which makes all sites equivalent even for glasses without quenched disorder. This method has a disadvantage that dynamical equations are much more difficult to solve than the statical ones. Very recently the cloning method was proposed that is based on the idea that even at low T a system of m clones might be distributed in its phase space over many low lying metastable states if m is chosen correctly and the properties of all these states are essentially equivalent to those of the ground state. [3][4][5][6] Generally, the partition sum of m weakly coupled clones is F e −N (mβF −S conf (F )) , where sum goes over free energies (per site) of metastable states, F , and S conf (F ) is their configurational entropy (S conf = 1 N ln(N states )). Assuming that dS conf (F )/dF is finite at the lowest F associated with the ground state one needs to chose m ∝ T at low T in order to avoid a complete dominance by a single (ground) state and the problems with site nonequivalence mentioned above. Distributing the system in the phase space provides the effective averaging mechanism in this approach. The main assumptions implicit in this approach are that...

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Nonequilibrium Fluctuations in Small Systems: From Physics to Biology

Ritort

2007

Advances in Chemical Physics

181

217

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In this paper I am presenting an overview on several topics related to nonequilibrium fluctuations in small systems. I start with a general discussion about fluctuation theorems and applications to physical examples extracted from physics and biology: a bead in an optical trap and single molecule force experiments. Next I present a general discussion on path thermodynamics and consider distributions of work/heat fluctuations as large deviation functions. Then I address the topic of glassy dynamics from the perspective of nonequilibrium fluctuations due to small cooperatively rearranging regions. Finally, I conclude with a brief digression on future perspectives.

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Replica field theory for deterministic models. II. A non-random spin glass with glassy behaviour

Cited by 205 publications

References 30 publications

Squared interaction matrix Sherrington-Kirkpatrick model for a spin glass

Squared interaction matrix Sherrington-Kirkpatrick model for a spin glass

Replica-symmetry breaking in long-range glass models without quenched disorder

Nonequilibrium Fluctuations in Small Systems: From Physics to Biology

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