On infrared divergences in spin glasses

Ferrero, Marco; Parisi, Giorgio

doi:10.1088/0305-4470/29/14/008

Cited by 7 publications

(13 citation statements)

References 19 publications

(38 reference statements)

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“…There is a feeling that this result may be exact, and that it corresponds to some kind of Goldstone theorem (see [27] and references therein, and [31]). …”

Section: B the Correlation Functionsmentioning

confidence: 93%

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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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“…There is a feeling that this result may be exact, and that it corresponds to some kind of Goldstone theorem (see [27] and references therein, and [31]). …”

Section: B the Correlation Functionsmentioning

confidence: 93%

Untitled

Marinari

Parisi

Ricci-Tersenghi

et al. 2000

Journal of Statistical Physics

222

278

View full text Add to dashboard Cite

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“…Remark 4 : Several works in the past [43,44,45] have discussed the necessity of calculating the fluctuations around the saddle point to obtain the leading finite size corrections, using in particular linear response theory to determine the fluctuations of the Parisi function. The main difference between these works and our present approach is that our starting point to calculate finite size corrections is not based on Parisi's ansatz.…”

Section: Remarkmentioning

confidence: 99%

Finite size corrections in the random energy model and the replica approach

Derrida¹,

Mottishaw²

2015

J. Stat. Mech.

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We present a systematict and exact way of computing finite size corrections for the random energy model, in its low temperature phase. We obtain explicit (though complicated) expressions for the finite size corrections of the overlap functions. In its low temperature phase, the random energy model is known to exhibit Parisi's broken symmetry of replicas. The finite size corrections given by our exact calculation can be reproduced using replicas if we make specific assumptions about the fluctuations (with negative variances!) of the number and sizes of the blocks when replica symmetry is broken. As an alternative we show that the exact expression for the non-integer moments of the partition function can be written in terms of coupled contour integrals over what can be thought of as "complex replica numbers". Parisi's one step replica symmetry breaking arises naturally from the saddle point of these integrals without making any ansatz or using the replica method. The fluctuations of the "complex replica numbers" near the saddle point in the imaginary direction correspond to the negative variances we observed in the replica calculation. Finally our approach allows one to see why some apparently diverging series or integrals are harmless.

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“…Distribution-like components were first identified among the spin glass propagators by two of us [48], see also [49]. Independently, Ferrero and Parisi put them to use in their recent analysis of the IR divergences in spin glasses [24].…”

Section: The Far Infrared Regionmentioning

confidence: 99%

Beyond the Sherrington-Kirkpatrick Model

Dominicis¹,

Kondor²,

Temesvári³

1997

Series on Directions in Condensed Matter Physics

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The state of art in spin glass field theory is reviewed. We start from an Edwards-Anderson-type model in finite dimensions, with finite but long range forces, construct the effective field theory that allows one to extract the long wavelength behaviour of the model, and set up an expansion scheme (the loop expansion) in the inverse range of the interaction. At the zeroth order we recover mean field theory. We evaluate systematic corrections to this around Parisi's replica symmetry broken solution. At the level of quadratic fluctuations we derive a set of coupled integral equations for the free propagators of the theory and show how they can be solved for short, intermediate and extreme long distances. To reveal the physical meaning of these results, we relate the various propagator components to overlaps of spin-spin correlation functions inside a single phase space valley resp. between different valleys. Next we calculate the first loop corrections to the theory above 8 dimensions, where we find that it maps back onto mean field theory, with basically temperature independent renormalization of the coupling constants, thereby demonstrating that Parisi's mean field theory is, at least perturbatively, stable against finite range corrections. In the range between six and eight dimensions various physical quantities pick up nontrivial temperature dependences which can, however, still be determined exactly. Upon approaching the upper critical dimension (d = 6) of the model, scaling which is badly violated in Parisi's mean field theory is gradually restored. Below 6 dimensions one should apply renormalization group methods. Unfortunately, the structure of RG is not completely understood in spin glass theory. Nevertheless, the first corrections in 6 − d to e.g. the exponent of the order parameter can still be calculated, moreover exponentiation to this power can be checked at the next order. The theory is, however, plagued by infrared divergences due to the presence of zero modes and soft modes. Systematic methods (like those developed in the O(n) model) to handle these infrared singularities are not yet available in spin glass theory.

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On infrared divergences in spin glasses

Abstract: By studying the structure of infrared divergences in a toy propagator in the replica approach to the Ising spin glass below T c , we suggest a possible cancellation mechanism which could decrease the degree of singularity in the loop expansion.

Cited by 7 publications

References 19 publications

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Finite size corrections in the random energy model and the replica approach

Beyond the Sherrington-Kirkpatrick Model

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