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arXiv:2102.06263 · Journal of Physics A Mathematical and Theoretical

Reply to comment on ‘Real-space renormalization-group methods for hierarchical spin glasses’

Michele Castellana

Published Apr 3, 2020• 0 citations
arXivHTMLPDF

Abstract

Abstract In their comment, Angelini et al object to the conclusion of (2019 J. Phys. A: Math. Theor. 52 445002), where we show that in (2013 Phys. Rev. B 87 134201) the exponent ν has been obtained by applying a mathematical relation in a regime where this relation is not valid. We observe that the criticism above on the mathematical validity of such relation has not been addressed in the comment. Our criticism thus remains valid, and disproves the conclusions of the comment. This constitutes the main point of this reply. In addition, we provide a point-by-point response and discussion of Angelini et al ’s claims. First, Angelini et al claim that the prediction 2 1/ ν = 1 of (2019 J. Phys. A: Math. Theor . 52 445002) is incorrect, because it results from the relation λ max = 2 1/ ν between the largest eigenvalue of the linearized renormalization-group (RG) transformation and ν , which cannot be applied to the ensemble renormalization group (ERG) method, given that for the ERG λ max = 1. However, the feature λ max = 1 is specific to the ERG transformation, and it does not give any grounds for questioning the validity of the general relation λ max = 2 1/ ν specifically for the ERG transformation. Second, Angelini et al claim that ν should be extracted from an early RG regime (A), as opposed to the asymptotic regime (B) used to estimate ν in (2019 J. Phys. A: Math. Theor . 52 445002), and that (B) is dominated by finite-size effects. However, (A) is a small-wavelength, non-critical regime, which cannot characterize the critical exponent ν related to the divergence of the correlation length. Also, the fact that (B) involves finite-size effects is a feature specific to the ERG, and gives no rationale for extracting ν from (A). Finally, we refute the remaining claims made by Angelini et al As a result of our analysis, we stand by our assertion that the ERG method yields a prediction given by 2 1/ ν = 1.

DOI
10.1088/1751-8121/ab8671

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Comment on ‘Real-space renormalization-group methods for hierarchical spin glasses’

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Abstract In the paper Angelini M C et al (2013 Phys. Rev. B 87 134201) we introduced a real-space renormalization group called ensemble renormalization group (ERG) and we applied it to the Edwards–Anderson model, obtaining estimates for the critical exponents in good agreement with those from Monte Carlo simulations. Recently the paper Castellana M (2019 J. Phys. A: Math. Theor. 52 445002) re-examined the ERG method from a different perspective, concluding that the previous results were wrong, and claiming that the ERG method predicts trivially wrong critical exponents. In this comment we explain why the conclusions reached by Castellana are wrong, as they are based on a misinterpretation of finite-size effects. We conclude that the ERG method remains a good RG method to obtain critical exponents in strongly disordered models (if properly used).

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Abstract In their comment, Angelini et al object to the conclusion of (2019 J. Phys. A: Math. Theor. 52 445002), where we show that in (2013 Phys. Rev. B 87 134201) the exponent ν has been obtained by applying a mathematical relation in a regime where this relation is not valid. We observe that the criticism above on the mathematical validity of such relation has not been addressed in the comment. Our criticism thus remains valid, and disproves the conclusions of the comment. This constitutes the main point of this reply. In addition, we provide a point-by-point response and discussion of Angelini et al ’s claims. First, Angelini et al claim that the prediction 2 1/ ν = 1 of (2019 J. Phys. A: Math. Theor . 52 445002) is incorrect, because it results from the relation λ max = 2 1/ ν between the largest eigenvalue of the linearized renormalization-group (RG) transformation and ν , which cannot be applied to the ensemble renormalization group (ERG) method, given that for the ERG λ max = 1. However, the feature λ max = 1 is specific to the ERG transformation, and it does not give any grounds for questioning the validity of the general relation λ max = 2 1/ ν specifically for the ERG transformation. Second, Angelini et al claim that ν should be extracted from an early RG regime (A), as opposed to the asymptotic regime (B) used to estimate ν in (2019 J. Phys. A: Math. Theor . 52 445002), and that (B) is dominated by finite-size effects. However, (A) is a small-wavelength, non-critical regime, which cannot characterize the critical exponent ν related to the divergence of the correlation length. Also, the fact that (B) involves finite-size effects is a feature specific to the ERG, and gives no rationale for extracting ν from (A). Finally, we refute the remaining claims made by Angelini et al As a result of our analysis, we stand by our assertion that the ERG method yields a prediction given by 2 1/ ν = 1.
Abstract In the paper Angelini M C et al (2013 Phys. Rev. B 87 134201) we introduced a real-space renormalization group called ensemble renormalization group (ERG) and we applied it to the Edwards–Anderson model, obtaining estimates for the critical exponents in good agreement with those from Monte Carlo simulations. Recently the paper Castellana M (2019 J. Phys. A: Math. Theor. 52 445002) re-examined the ERG method from a different perspective, concluding that the previous results were wrong, and claiming that the ERG method predicts trivially wrong critical exponents. In this comment we explain why the conclusions reached by Castellana are wrong, as they are based on a misinterpretation of finite-size effects. We conclude that the ERG method remains a good RG method to obtain critical exponents in strongly disordered models (if properly used).
Abstract We focus on two real-space renormalization-group (RG) methods recently proposed for a hierarchical model of a spin glass: a sample-by-sample method, in which the RG transformation is performed separately on each disorder sample, and an ensemble RG (ERG) method (Angelini M C, Parisi G and Ricci-Tersenghi F 2013 Ensemble renormalization group for disordered systems Phys. Rev . B 87 134201) in which the transformation is based on an average over samples. Above the upper critical dimension, the sample-by-sample method yields the correct mean-field value for the critical exponent related to the divergence of the correlation length, while it does not predict the correct qualitative behavior of below the upper critical dimension. On the other hand, the ERG procedure has been claimed to predict the correct behavior of both above and below the upper critical dimension. Here, we straighten out the reasons for the discrepancy between the two methods above, by demonstrating that the ERG method predicts a marginally stable critical fixed point, thus implying a prediction for the critical exponent given by . This prediction disagrees, on a qualitative and quantitative level, both with the mean-field value of above the critical dimension, and with numerical estimates of below the upper critical dimension. Therefore, our results show that finding a real-space RG method for spin glasses which yields the correct prediction for universal quantities below the upper critical dimension is still an open problem, for which our analysis may provide some general guidance for future studies.
In a recent work [M. Castellana and G. Parisi, Phys. Rev. E 82, 040105(R) (2010)], the large-scale behavior of the simplest non-mean-field spin-glass system has been analyzed, and the critical exponent related to the divergence of the correlation length has been computed at two loops within the ε-expansion technique by two independent methods. By performing the explicit calculation of the critical exponents at two loops, one obtains that the two methods yield the same result. This shows that the underlying renormalization group ideas apply consistently in this disordered model, in such a way that an ε-expansion can be set up. The question of the extension to high orders of this ε-expansion is particularly interesting from the physical point of view. Indeed, once high orders of the series in ε for the critical exponents are known, one could check the convergence properties of the series, and find out if the ordinary series resummation techniques, yielding very accurate predictions for the Ising model, work also for this model. If this is the case, a consistent and predictive non-mean-field theory for such a disordered system could be established. In that regard, in this work we expose the underlying techniques of such a two-loop computation. We show with an explicit example that such a computation could be quite easily automatized, i.e., performed by a computer program, in order to compute high orders of the ε-expansion, and so eventually make this theory physically predictive. Moreover, all the underlying renormalization group ideas implemented in such a computation are widely discussed and exposed.
We have recently shown that in nonequilibrium spin systems at criticality the limit ${X}^{\ensuremath{\infty}}$ of the fluctuation-dissipation ratio X(t,tw)X(t,{t}_{w})X(t,tw​) for t⪢tw⪢1t⪢{t}_{w}⪢1t⪢tw​⪢1 can be measured using observables such as magnetization or energy [Phys. Rev. E 68, 016116 (2003)]. Pleimling argues in a Comment [preceding paper, Phys. Rev. 70, 018101 (2004)] on our paper that for such observables correlation and response functions are dominated by one-time quantities dependent only on ttt, and are therefore not suitable for a determination of ${X}^{\ensuremath{\infty}}$. Using standard scaling forms of correlation and response functions, as used by Pleimling, we show that our data do have a genuine two-time dependence and allow X(t,tw)X(t,{t}_{w})X(t,tw​) and ${X}^{\ensuremath{\infty}}$ to be measured, so that Pleimling's criticisms are easily refuted. We also compare with predictions from renormalization-group calculations, which are consistent with our numerical observation of a fluctuation-dissipation plot for the magnetization that is very close to a straight line. A key point remains that coherent observables make measurements of ${X}^{\ensuremath{\infty}}$ easier than the traditionally used incoherent ones, producing fluctuation-dissipation plots whose slope is close to ${X}^{\ensuremath{\infty}}$ over a much larger range.
The large scale behavior of the simplest non-mean-field spin-glass system is analyzed, and the critical exponent related to the divergence of the correlation length is computed at two loops within the ε-expansion technique with two independent methods. The techniques presented show how the underlying ideas of the renormalization group apply also in this disordered model, in such a way that an ε-expansion can be consistently set up. By pushing such calculation to high orders in ε, a consistent non-mean-field theory for such disordered system could be established, giving a substantial contribution the development of a predictive theory for real spin glasses.