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arXiv:1610.04747 · Physical review. B./Physical review. B

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

T. Temesvári

Published Nov 1, 2016• 3 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.94.176401

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

Joonhyun Yeo, M A Moore

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.

arXiv:1610.05764•Physical review. B./Physical review. B•Nov 1, 2016•1 citation

Critical point scaling of Ising spin glasses in a magnetic field

Joonhyun Yeo, M A Moore

Critical point scaling in a field $H$ 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 $f(R)$ defined via $1/(\ensuremath{\xi}{H}^{4/(d+2\ensuremath{-}\ensuremath{\eta})})\ensuremath{\sim}f(R)$. 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 $d\ensuremath{\le}6$, 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 $d\ensuremath{\le}6$, which we review.

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Some recent results concerning phase transitions in spin glasses

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I give an introduction to spin glasses, including a short discussion of experiments and theoretical concepts. I then discuss recent results from Monte Carlo simulation about two topics (i) the transition in a Heisenberg spin glass, and (ii) the transition in a magnetic field (Almeida-Thouless line) for an Ising spin glass.

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The very existence of a phase transition for spin glasses in an external magnetic field is controversial, even in high dimensions. We carry out massive simulations of the Ising spin-glass in a field, in six dimensions (which, according to classical-but not generally accepted-field-theoretical studies, is the upper critical dimension). We obtain results compatible with a second-order phase transition and estimate its critical exponents for the simulated lattice sizes. The detailed analysis performed by other authors of the replica symmetric Hamiltonian, under the hypothesis of critical behavior, predicts that the ratio of the renormalized coupling constants remain bounded as the correlation length grows. Our numerical results are in agreement with this expectation.

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Evidence of de Almeida–Thouless line below six dimensions

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Abstract We study the critical behavior of the Ising spin glass in five spatial dimensions through large-scale Monte Carlo simulations and finite-size scaling analysis. Numerical evidence for a phase transition is found both with and without an externally applied magnetic field. The critical exponents are computed in both cases. We compute with a 10% accuracy the lower critical dimension at zero magnetic field, finding a result consistent with estimates obtained with entirely different methods, by combining our estimates of critical exponents in five dimensions with previous results for other spatial dimensions. When the results in a magnetic field are compared with previous results in six spatial dimensions, qualitative differences emerge in the scaling behavior of the correlation functions at zero external momentum. This anomalous scaling does not extend to other wavevectors. We do not find indications of a quasi first-order phase transition in a magnetic field.

arXiv:2510.14446•Journal of Statistical Mechanics Theory and Experiment•Nov 1, 2025•1 citation
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.
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.
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.
We show how the recent computer simulation data reported by Caracciolo et al. for the three-dimensional ±\pm± JJJ Ising spin glass in a uniform magnetic field are, in contrast to their claims, quite consistent with the simple scaling or droplet theory developed by McMillan, Fisher and Huse, and Bray and Moore, and do notnotnot provide evidence for a de Almeida-Thouless transition.
The very existence of a phase transition for spin glasses in an external magnetic field is controversial, even in high dimensions. We carry out massive simulations of the Ising spin-glass in a field, in six dimensions (which, according to classical-but not generally accepted-field-theoretical studies, is the upper critical dimension). We obtain results compatible with a second-order phase transition and estimate its critical exponents for the simulated lattice sizes. The detailed analysis performed by other authors of the replica symmetric Hamiltonian, under the hypothesis of critical behavior, predicts that the ratio of the renormalized coupling constants remain bounded as the correlation length grows. Our numerical results are in agreement with this expectation.
Abstract We study the critical behavior of the Ising spin glass in five spatial dimensions through large-scale Monte Carlo simulations and finite-size scaling analysis. Numerical evidence for a phase transition is found both with and without an externally applied magnetic field. The critical exponents are computed in both cases. We compute with a 10% accuracy the lower critical dimension at zero magnetic field, finding a result consistent with estimates obtained with entirely different methods, by combining our estimates of critical exponents in five dimensions with previous results for other spatial dimensions. When the results in a magnetic field are compared with previous results in six spatial dimensions, qualitative differences emerge in the scaling behavior of the correlation functions at zero external momentum. This anomalous scaling does not extend to other wavevectors. We do not find indications of a quasi first-order phase transition in a magnetic field.