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arXiv:cond-mat/9910333 · Physical Review Letters

Houdayer and Martin Reply:

J. Houdayer, Olivier Martin

Published Jan 31, 2000• 6 citations
arXivHTMLPDF

Abstract

The problem of the survival of a spin glass phase in the presence of a field has been a challenging one for a long time. To date, all attempts using equilibrium Monte Carlo methods have been unconclusive. In their comment to our paper, Marinari, Parisi and Zuliani use out-of-equilibrium measurements to test for an Almeida-Thouless line. In our view such a dynamic approach is not based on very solid foundations in finite dimensional systems and so cannot be as compelling as equilibrium approaches. Nevertheless, the results of those authors suggests that there is a critical field near B=0.4 at zero temperature. In view of this quite small value (compared to the mean field value), we have reanalyzed our data. We find that if finite size scaling is to distinguish between that small field and a zero field, we would need to go to lattice sizes of about 20x20x20.

DOI
10.1103/physrevlett.84.1057

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Reply to “Comment on ‘Critical point scaling of Ising spin glasses in a magnetic field’ ”

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In his Comment, Temesv\'ari objects to a remark in our paper [Phys. Rev. B 91, 104432 (2015)] that his result for the form of the Almeida-Thouless (AT) line obtained in an earlier paper by Parisi and Temev\'ari [Nucl. Phys. B 858, 293 (2012)] in six dimensions can be obtained by taking the limit of $d\ensuremath{\rightarrow}6$ in the equations valid for $d>6$ but that this violated one of the inequalities needed for their validity. He is just pointing out that they gave a derivation of the form of the AT line in six dimensions in Parisi and Temev\'ari [Nucl. Phys. B 858, 293 (2012)] which avoided this difficulty. However, it is still a perturbative approach and does not deal with the lack of a perturbative fixed point found by Bray and Roberts [J. Phys. C 13, 5405 (1980)] long ago.

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In a section of a recent paper [Phys. Rev. B 91, 104432 (2015)], the authors discuss some of the arguments in the paper by Parisi and Temesv\'ari [Nucl. Phys. B 858, 293 (2012)]. In this Comment, it is shown how these arguments are misinterpreted and the existence of the Almeida-Thouless transition $\mathit{in}$ the upper critical dimension six reasserted.

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The problem of the survival of a spin glass phase in the presence of a field has been a challenging one for a long time. To date, all attempts using equilibrium Monte Carlo methods have been unconclusive. In their comment to our paper, Marinari, Parisi and Zuliani use out-of-equilibrium measurements to test for an Almeida-Thouless line. In our view such a dynamic approach is not based on very solid foundations in finite dimensional systems and so cannot be as compelling as equilibrium approaches. Nevertheless, the results of those authors suggests that there is a critical field near B=0.4 at zero temperature. In view of this quite small value (compared to the mean field value), we have reanalyzed our data. We find that if finite size scaling is to distinguish between that small field and a zero field, we would need to go to lattice sizes of about 20x20x20.
Critical point scaling in a field HHH applies for the limits $t\ensuremath{\rightarrow}0$ (where $t=T/{T}_{c}\ensuremath{-}1)$ and $H\ensuremath{\rightarrow}0$ but with the ratio $R=t/{H}^{2/\ensuremath{\Delta}}$ finite. $\ensuremath{\Delta}$ is a critical exponent of the zero-field transition. We study the replicon correlation length $\ensuremath{\xi}$ and from it the crossover scaling function f(R)f(R)f(R) defined via $1/(\ensuremath{\xi}{H}^{4/(d+2\ensuremath{-}\ensuremath{\eta})})\ensuremath{\sim}f(R)$. We have calculated analytically f(R)f(R)f(R) for the mean-field limit of the Sherrington-Kirkpatrick model. In dimension d=3d=3d=3, we have determined the exponents and the critical scaling function f(R)f(R)f(R) within two versions of the Migdal-Kadanoff (MK) renormalization group procedure. One of the MK versions gives results for f(R)f(R)f(R) in d=3d=3d=3 in reasonable agreement with those of the Monte Carlo simulations at the values of RRR for which they can be compared. If there were a de Almeida-Thouless (AT) line for $d\ensuremath{\le}6$, it would appear as a zero of the function f(R)f(R)f(R) at some negative value of RRR, but there is no evidence for such behavior. This is consistent with the arguments that there should be no AT line for $d\ensuremath{\le}6$, which we review.
In ref. cond-mat/9811419 Houdayer and Martin analyze the T=0 3d EA spin glass with a magnetic field BBB. By using a new, powerful method, they determine an effective critical field BcB_cBc​ as a function of the lattice size LLL. They use their results to deduce that the model is behaving like in the droplet approach and not like the mean-field theory. We show here, by using some unpublished data, that this very interesting method and numerical results are completely compatible with the behavior implied by the Replica Symmetry Breaking theory.
In his Comment, Temesv\'ari objects to a remark in our paper [Phys. Rev. B 91, 104432 (2015)] that his result for the form of the Almeida-Thouless (AT) line obtained in an earlier paper by Parisi and Temev\'ari [Nucl. Phys. B 858, 293 (2012)] in six dimensions can be obtained by taking the limit of $d\ensuremath{\rightarrow}6$ in the equations valid for d>6d>6d>6 but that this violated one of the inequalities needed for their validity. He is just pointing out that they gave a derivation of the form of the AT line in six dimensions in Parisi and Temev\'ari [Nucl. Phys. B 858, 293 (2012)] which avoided this difficulty. However, it is still a perturbative approach and does not deal with the lack of a perturbative fixed point found by Bray and Roberts [J. Phys. C 13, 5405 (1980)] long ago.
In a section of a recent paper [Phys. Rev. B 91, 104432 (2015)], the authors discuss some of the arguments in the paper by Parisi and Temesv\'ari [Nucl. Phys. B 858, 293 (2012)]. In this Comment, it is shown how these arguments are misinterpreted and the existence of the Almeida-Thouless transition in\mathit{in}in the upper critical dimension six reasserted.
We study in Ising spin glasses the finite-size effects near the spin-glass transition in zero field and at the de Almeida-Thouless transition in a field by Monte Carlo methods and by analytical approximations. In zero field, the finite-size scaling function associated with the spin-glass susceptibility of the Sherrington-Kirkpatrick mean-field spin-glass model is of the same form as that of one-dimensional spin-glass models with power-law long-range interactions in the regime where they can be a proxy for the Edwards-Anderson short-range spin-glass model above the upper critical dimension. We also calculate a simple analytical approximation for the spin-glass susceptibility crossover function. The behavior of the spin-glass susceptibility near the de Almeida-Thouless transition line has also been studied, but here we have only been able to obtain analytically its behavior in the asymptotic limit above and below the transition. We have also simulated the one-dimensional system in a field in the non-mean-field regime to illustrate that when the Imry-Ma droplet length scale exceeds the system size one can then be erroneously lead to conclude that there is a de Almeida-Thouless transition even though it is absent.
We study the de Almeida-Thouless (AT) line in the one-dimensional power-law diluted XY spin-glass model, in which the probability that two spins separated by a distance r interact with each other, decays as 1/r^{2σ}. Tuning the exponent σ is equivalent to changing the space dimension of a short-range model. We develop a heat bath algorithm to equilibrate XY spins; using this in conjunction with the standard parallel tempering and overrelaxation sweeps, we carry out large-scale Monte Carlo simulations. For σ=0.6, which is in the mean-field regime above six dimensions-it is similar to being in 10 dimensions-we find clear evidence for an AT line. For σ=0.75 and σ=0.85, which are in the non-mean-field regime and similar to four and three dimensions, respectively, our data is like that found in previous studies of the Ising and Heisenberg spin glasses when reducing the temperature at fixed field. For σ=0.75, there is evidence from finite-size-scaling studies for an AT transition but for σ=0.85, the evidence for a transition is nonexistent. We have also studied these systems at fixed temperature varying the field and discovered that at both σ=0.75 and at σ=0.85 there is evidence of an AT transition! Confusingly, the correlation length and spin-glass susceptibility as a function of the field are both entirely consistent with the predictions of the droplet picture and hence the nonexistence of an AT line. In the usual finite-size critical point scaling studies used to provide evidence for an AT transition, there is seemingly good evidence for an AT line at σ=0.75 for small values of the system size N, which is strengthening as N is increased, but for N>2048 the trend changes and the evidence then weakens as N is further increased. We have also studied with fewer bond realizations the system at σ=0.70, which is the analog of a system with short-range interactions just below six dimensions, and found that it is similar in its behavior to the system at σ=0.75 but with larger finite-size corrections. The evidence from our simulations points to the complete absence of the AT line in dimensions outside the mean-field region and to the correctness of the droplet picture. Previous simulations which suggested there was an AT line can be attributed to the consequences of studying systems which are just too small. The collapse of our data to the droplet scaling form is poor for σ=0.75 and to some extent also for σ=0.85, when the correlation length becomes of the order of the length of the system, due to the existence of excitations which only cost a free energy of O(1), just as envisaged in the TNT picture of the ordered state of spin glasses. However, for the case of σ=0.85 we can provide evidence that for larger system sizes, droplet scaling will prevail even when the correlation length is comparable to the system size.