Dyson's equations for the (Ising) spin-glass

Dominicis, C. De; Kondor, Imre; Temesvári, T.

doi:10.1051/jp1:1994189

Cited by 20 publications

(43 citation statements)

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“…As discussed above, there are actually many phenomenological similarities between the spin-glass phase and the 2D XY model, provided length scales, rather than time scales, are compared 19,47 . The power-law decay of C 4 (r, t) suggests that the whole spin-glass phase is in a certain sense critical, at least in the 'zero-overlap' sector which was indeed found to be mass-less in replica field analysis 60 . However, this is not in contradiction with the existence of a finite correlation length ξ(T ) separating critical from activated dynamics within a single 'state'.…”

Section: F the 'Kovacs Effect'mentioning

confidence: 88%

Geometrical aspects of aging and rejuvenation in the Ising spin glass: A numerical study

2002

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We present a comprehensive study of non-equilibrium phenomena in the low temperature phase of the Edwards-Anderson Gaussian spin glass in 3 and 4 spatial dimensions. Many effects can be understood in terms of a time dependent coherence length, ℓT , such that length scales smaller that ℓT are equilibrated, whereas larger length scales are essentially frozen. The time and temperature dependence of ℓT is found to be compatible with critical power-law dynamical scaling for small times/high temperatures, crossing over to an activated logarithmic growth for longer times/lower temperatures, in agreement with recent experimental results. The activated regime is governed by a 'barrier exponent' ψ which we estimate to be ψ ∼ 1.0 and ψ ∼ 2.3 in 3 and 4 dimensions, respectively. We observe for the first time the rejuvenation and memory effects in the four dimensional sample, which, we argue, is unrelated to 'temperature chaos'. Our discussion in terms of length scales allows us to address several experimentally relevant issues, such as super-aging versus sub-aging effects, the role of a finite cooling rate, or the so-called Kovacs effect.

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Section: F the 'Kovacs Effect'mentioning

confidence: 88%

Geometrical aspects of aging and rejuvenation in the Ising spin glass: A numerical study

2002

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Section: Introductionmentioning

confidence: 91%

“…Thus our results will be applicable in situations where only a single RSB step is needed, as well as in the case of full-fledged RSB with R → ∞. The results for this "continuous" case (R → ∞, n → 0), derived by a completely different method, will be published in [12].Although almost trivial in principle, our analysis will, inevitably, be very complicated in actual details. It is clearly impossible to reproduce the often very lenghty calculations here, and we shall have to use the phrase "it can be shown" frequently.…”

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confidence: 97%

Block diagonalizing ultrametric matrices

Temesvári

Dominicis

Kondor

1994

J. Phys. A: Math. Gen.

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The problem of diagonalizing a class of complicated matrices, to be called ultrametric matrices, is investigated. These matrices appear at various stages in the description of disordered systems with many equilibrium phases by the technique of replica symmetry breaking. The residual symmetry, remaining after the breaking of permutation symmetry between replicas, allows us to bring all ultrametric matrices to a block diagonal form by a common similarity transformation. A large number of these blocks are, in fact, of size 1 × 1, i.e. in a vast sector the transformation actually diagonalizes the matrix. In the other sectors we end up with blocks of size (R + 1) × (R + 1) where R is the number of replica symmetry breaking steps. These blocks cannot be further reduced without giving more information, in addition to ultrametric symmetry, about the matrix. Similar results for the inverse of a generic ultrametric matrix are also derived. PACS classification numbers: 75.10.Nr, 05.50.+q. * E-mail:temtam@hal9000.elte.hu IntroductionLow temperature disordered systems often possess many equilibrium phases. The technique of replica symmetry breaking (RSB) provides a theoretical framework in which these systems can be described analytically , starting from a microscopic basis. Discovered and developed in the theory of spin glasses [1], RSB has recently penetrated into a number of other problems, including the theory of random manifolds [2][3][4], random field problems [5,6], protein folding [7][8][9], vortex pinning [10], etc. In each of these theories randomness is handled via the replica trick, and the multitude of equilibrium phases is captured by breaking the permutation symmetry between the replicas. As always, symmetry breaking means that the low temperature solutions realize a particular subgroup of the underlying symmetry group of the theory, here of the permutation group. The proper choice of the subgroup proved to be a highly nontrivial task in the case of RSB. The succesful Ansatz for the symmetry breaking pattern, proposed by Parisi originally in the context of spin glasses, turned out to embody a particular, hierarchical organization of the equilibrium phases, usually referred to as ultrametricity [1].The corresponding subgroup determines the structure not only of the order parameter, but also of all other quantities in the theory, like self-energies, propagators, etc. The structure imposed by this residual symmetry on quantities depending on two replica indices is by now widely known. The algebra of such quantities has been worked out by Parisi [11] with further results, most notably on the inversion problem, added by Mézard and Parisi [2]. At a certain stage of the development of RSB theories, however, one has to face also more complicated objects, depending on three or four replica indices. The structure of these is much harder to grasp and their algebra is much more involved than that of the two-index quantities. Our purpose here is to analyse and exploit the structure imposed by ultametricity on fou...

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“…We consider the case, relevant for spin glasses, where the function r of (15) is non-zero and q m ≤q ≤ q M (i.e.q is in the support of P (q)). In the usual Ansatz for replica symmetry one finds that [27][28][29][30] …”

Section: B the Correlation Functionsmentioning

confidence: 99%

Untitled

Marinari

Parisi

Ricci-Tersenghi

et al. 2000

Journal of Statistical Physics

222

278

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We discuss replica symmetry breaking (RSB) in spin glasses. We update work in this area, from both the analytical and numerical points of view. We give particular attention to the difficulties stressed by Newman and Stein concerning the problem of constructing pure states in spin glass systems. We mainly discuss what happens in finite-dimensional, realistic spin glasses. Together with a detailed review of some of the most important features, facts, data, and phenomena, we present some new theoretical ideas and numerical results. We discuss among others the basic idea of the RSB theory, correlation functions, interfaces, overlaps, pure states, random field, and the dynamical approach. We present new numerical results for the behaviors of coupled replicas and about the numerical verification of sum rules, and we review some of the available numerical results that we consider of larger importance (for example, the determination of the phase transition point, the correlation functions, the window overlaps, and the dynamical behavior of the system).

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Dyson's equations for the (Ising) spin-glass

Cited by 20 publications

References 0 publications

Geometrical aspects of aging and rejuvenation in the Ising spin glass: A numerical study

Geometrical aspects of aging and rejuvenation in the Ising spin glass: A numerical study

Block diagonalizing ultrametric matrices

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