Replica symmetry breaking in and around six dimensions

Critical point scaling in a field H applies for the limits t → 0, (where t = T /Tc − 1) and H → 0 but with the ratio R = t/H 2/∆ finite. ∆ is a critical exponent of the zero-field transition. We study the replicon correlation length ξ and from it the crossover scaling function f (R) defined via 1/(ξH 4/(d+2−η) ) ∼ f (R). We have calculated analytically f (R) for the mean-field limit of the Sherrington-Kirkpatrick model. In dimension d = 3 we have determined the exponents and the critical scaling function f (R) within two versions of the Migdal-Kadanoff (MK) renormalization group procedure. One of the MK versions gives results for f (R) in d = 3 in reasonable agreement with those of the Monte Carlo simulations at the values of R for which they can be compared. If there were a de Almeida-Thouless (AT) line for d ≤ 6 it would appear as a zero of the function f (R) at some negative value of R, but there is no evidence for such behavior. This is consistent with the arguments that there should be no AT line for d ≤ 6, which we review.

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“…which agrees with the expression for the AT line in six dimensions given by Parisi and Temesvári [23]. However, Eq.…”

Section: The At Line For D >supporting

confidence: 90%

“…[23] that it would be good to have an argument that at any fixed value of |t|, h AT went to zero in the limit d → 6, rather than just for the scaling limit |t| → 0. This is a non-perturbative task, but perhaps not impossible.…”

Section: The At Line For D >mentioning

confidence: 99%

Critical point scaling of Ising spin glasses in a magnetic field

Yeo

Moore

2015

Phys. Rev. B

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“…Some believe it should still apply without dramatic modifications [6,10]. Yet, a dissenting school of thought, the so-called droplet picture, predicts no phase transition in a field as soon as one goes below six spatial dimensions [11,12,13,14].…”

mentioning

confidence: 99%

The three-dimensional Ising spin glass in an external magnetic field: the role of the silent majority

Baity‐Jesi¹,

Baños²,

Cruz³

et al. 2014

J. Stat. Mech.

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Abstract. We perform equilibrium parallel-tempering simulations of the 3D Ising Edwards-Anderson spin glass in a field, using the Janus computer. A traditional analysis shows no signs of a phase transition. Yet, we encounter dramatic fluctuations in the behaviour of the model: Averages over all the data only describe the behaviour of a small fraction of it. Therefore we develop a new approach to study the equilibrium behaviour of the system, by classifying the measurements as a function of a conditioning variate. We propose a finite-size scaling analysis based on the probability distribution function of the conditioning variate, which may accelerate the convergence to the thermodynamic limit. In this way, we find a non-trivial spectrum of behaviours, where a part of the measurements behaves as the average, while the majority of them shows signs of scale invariance. As a result, we can estimate the temperature interval where the phase transition in a field ought to lie, if it exists. Although this would-be critical regime is unreachable with present resources, the numerical challenge is finally well posed.

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“…(3). The leading, linear contribution to g(x) is free from a singularity at d = 6, as it comes from an ultraviolet convergent one-loop graph 3 . Triangular insertions in the next, two-loop graphs, however, certainly produce singular terms like g(x) ∼ …”

Section: At and Above Six Dimensionsmentioning

confidence: 99%

“…[See also Fig. 2(b) and the discussion around it in 3 .] ǫ in (3) may be small, but must be kept fixed.…”

Section: At and Above Six Dimensionsmentioning

confidence: 99%

Comment on “Critical point scaling of Ising spin glasses in a magnetic field”

Temesvári

2016

Phys. Rev. B

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In a section of a recent publication, [J. Yeo and M.A. Moore, Phys. Rev. B 91, 104432 (2015)], the authors discuss some of the arguments in the paper by Parisi and Temesvári [Nuclear Physics B 858, 293 (2012)]. In this comment, it is shown how these arguments are misinterpreted, and the existence of the Almeida-Thouless transition in the upper critical dimension 6 reasserted.In a recent paper 1 by Yeo and Moore about the long debated existence of the Almeida-Thouless instability 2 in the short ranged Ising spin glass below the upper critical dimension six, the authors criticize in Sec. III. some of our statements and arguments in Ref.3 . In that paper we have demonstrated: firstly the incorrect reasoning of Ref.4 about the disappearence of the Almeida-Thouless (AT) transition line when approaching the upper critical dimension from above; secondly we have computed the AT line staying exactly in six dimensions (and not by a limiting process); and thirdly the ǫ-expansion was used to compute the AT line below six dimensions, and the relatively smooth behavior of it while crossing d = 6 (with fixed bare parameters) was exhibited. In what follows, we want to comment the discussion in Sec. III. of 1 . I. AT AND ABOVE SIX DIMENSIONSThe first order renormalization group (RG) equations for the six-dimensional model are worked out and solved in Sec. 3 of Ref.3 , the AT line follows from that calculation [see Eq. (37) in 3 ] 1 :where w 2 ≪ 1 was used. [Note that a minus sign in the denominator of Eq. (13) has been left out in 1 .] As it turns out from the discussion in Sec. 3 of Ref.3 , this approximation is valid if the scaling variable with zero scaling dimension (which is invariant under RG in d = 6) is small, i.e. and this condition is always satisfied whenever |r| ≪ 1 and w 2 ≪ 1; see also the middle part of Eq. (59) of that reference. Yeo and Moore 1 forget all about this derivation of the six-dimensional AT line; they deduce it from Eq. (11) of 1 by the limit ǫ → 0, and finally they argue that "Eq. (11) is not valid for this limit". We can absolutely agree with this last statement: the system at the upper critical dimension needs special care, physical quantities, like the critical magnetic field where replica symmetry breaking sets in, cannot be obtained by a limiting process of ǫ → 0. The point is that ǫ in Eq. (11) may be small, but fixed, while |r| ≪ 1, and the |r| ǫ/2 term in the denominator must be ignored. Taking account of this, the AT line above dimension six, Eq. (11) of 1 , must be written (consistently with the approximations used to derive it) as:This is just Eq. (28) of Ref.3 . This equation for the AT line above six dimensions must be supplemented by the range of its applicability, otherwise false conclusions like Eq. (12) in 1 [which is obviously incompatible with (1)] could be deduced. For this reason, we briefly repeat the two steps needed for the derivation of (3):1 We use here the notations of Ref. 1 . In fact |r| was called τ in 3 , whereas r had the role of the nonlinear scaling field associate...

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Replica symmetry breaking in and around six dimensions

Cited by 58 publications

References 24 publications

Critical point scaling of Ising spin glasses in a magnetic field

Critical point scaling of Ising spin glasses in a magnetic field

The three-dimensional Ising spin glass in an external magnetic field: the role of the silent majority

Comment on “Critical point scaling of Ising spin glasses in a magnetic field”

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