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On weak ergodicity breaking in meanfield spin glasses
by Giampaolo Folena, Francesco Zamponi
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Submission summary
Authors (as registered SciPost users):  Giampaolo Folena 
Submission information  

Preprint Link:  https://arxiv.org/abs/2303.00026v2 (pdf) 
Date submitted:  20230509 14:39 
Submitted by:  Folena, Giampaolo 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
The weak ergodicity breaking hypothesis postulates that outofequilibrium glassy systems lose memory of their initial state despite being unable to reach an equilibrium stationary state. It is a milestone of glass physics, and has provided a lot of insight on the physical properties of glass aging. Despite its undoubted usefulness as a guiding principle, its general validity remains a subject of debate. Here, we present evidence that this hypothesis does not hold for a class of meanfield spin glass models. While most of the qualitative physical picture of aging remains unaffected, our results suggest that some important technical aspects should be revisited.
Author comments upon resubmission
The two reports are positive and only a few small changes and precisations have been requested.
We have carefully considered all recommendations and we have modified the text accordingly.
We think that our manuscript is ready for publication.
List of changes
Report 1:
 properly introduced the “overlap”, the “characteristic polynomial” and the “complexity”.
 added footnote to explain that $\lambda$ is chosen to maximize the “discrepacy”
 added comment on T_onset crossover versus transition.
Report 3:
 added a footnote to comment about previous long time numerical solutions
 rephrased the sentence to answer to the misquotation of [22]
 clarified that in [54] the considered transition is of continuous type
Current status:
Reports on this Submission
Report
I thank the authors for their response. I agree with reviewer 1 that a plot showing the impact of the time step dt on the extrapolation is important to support their finding. I believe that with this modification the paper is ready for publication.
Report #1 by Anonymous (Referee 4) on 2023524 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2303.00026v2, delivered 20230524, doi: 10.21468/SciPost.Report.7242
Report
As I said in the previous report the paper deserves publication. However I ask the authors to make a small additional effort to address an important technical issue. The key ingredient of the paper is the numerical solution of the offequilibrium dynamical equations by time discretization. Therefore the outcome depends on the time spacing dt. The correct procedure is then to extrapolate the solution to dt=0 considering different values of dt's. It seems that instead the authors report only data at fixed dt=.05 and do not mention the issue at all.
Take for instance fig. 13 (a), it suggests that the numerical curve with dt=.05 does not have large corrections at least up to time t=100 due to the agreement with the results from the series expansion, BUT this is not enough to argue that the agreement will continue to larger times. In order to do so one should compare the result with smaller dt to be sure that the results can be trusted.
Now this is not a minor issue, take for instance the problem I raised in the previous report to which the authors replied:
"We have tried the suggested procedure, i.e. plotting the excess energy versus C(t,0) and performing a 3parameters fit (see attached figure CorrEn.pdf: (a) for 2spin and (b) for 3spin). However, the results remain more consistent with the strong ergodicity breaking scenario, without adding further insights. We thus prefer to not add them to the paper, in order to avoid overcharging it."
Let me note that the interest of such a plot is not to argue if favor or strong ergodicity breaking or not.
Indeed any claim on strong ergodicity breaking is based on extrapolations to infinite time and therefore prone to endless discussions, given that there is no analytical prediction on how the correlation with the initial condition should decay to zero. The aim of the parametric plot is essentially different, the question is: **assuming** that there is weak ergodicity breaking, does the asymptotic energy go to the threshold one? Looking from the figure attached to the reply one would say that this is not the case and this is a very interesting statement.
Therefore I urge the authors to put the figure in the manuscript (there are already 16 figures in the paper, I do not think that one more will change much in terms of readability).
However this is clearly an instance in which one would like to be sure that the data do not suffer from a systematic discretization error. Therefore I would like to see the same parametric curves with a different dt so that the data plotted can be taken confidently to describe the continuum dt=0 limit. The reader must know if the data they are seeing are reliable and this could be done in an appendix.
Requested changes
1) put the correlationenergy parametric figure in the text
2) display the effect of the dicretization dt on the data of the aforementioned figure in a separate figure, to be put in the appendix
Author: Giampaolo Folena on 20230610 [id 3722]
(in reply to Report 1 on 20230524)
Referee
As I said in the previous report the paper deserves publication. However I ask the authors to make a small additional effort to address an important technical issue. The key ingredient of the paper is the numerical solution of the offequilibrium dynamical equations by time discretization. Therefore the outcome depends on the time spacing dt. The correct procedure is then to extrapolate the solution to dt=0 considering different values of dt's. It seems that instead the authors report only data at fixed dt=.05 and do not mention the issue at all. Take for instance fig. 13 (a), it suggests that the numerical curve with dt=.05 does not have large corrections at least up to time t=100 due to the agreement with the results from the series expansion, BUT this is not enough to argue that the agreement will continue to larger times. In order to do so one should compare the result with smaller dt to be sure that the results can be trusted.
Response We agree with the referee that it is of fundamental importance to show that the chosen dt=0.05 gives a small error with respect to the exact dt > 0 limit. Indeed this is the case, and we substantiated it in an additional appendix A, which provides explicit plots (Fig.13) of the error as a function of the time, showing that the error for dt=0.05 is small and decreases with time, and therefore it does not corrupt the analysis. We remark that that an analogous analysis of the error was provided in a previous article by one of the authors.
Referee
Now this is not a minor issue, take for instance the problem I raised in the previous report to which the authors replied:
"We have tried the suggested procedure, i.e. plotting the excess energy versus C(t,0) and performing a 3parameters fit (see attached figure CorrEn.pdf: (a) for 2spin and (b) for 3spin). However, the results remain more consistent with the strong ergodicity breaking scenario, without adding further insights. We thus prefer to not add them to the paper, in order to avoid overcharging it."
Let me note that the interest of such a plot is not to argue if favor or strong ergodicity breaking or not. Indeed any claim on strong ergodicity breaking is based on extrapolations to infinite time and therefore prone to endless discussions, given that there is no analytical prediction on how the correlation with the initial condition should decay to zero. The aim of the parametric plot is essentially different, the question is: assuming that there is weak ergodicity breaking, does the asymptotic energy go to the threshold one? Looking from the figure attached to the reply one would say that this is not the case and this is a very interesting statement. Therefore I urge the authors to put the figure in the manuscript (there are already 16 figures in the paper, I do not think that one more will change much in terms of readability). However this is clearly an instance in which one would like to be sure that the data do not suffer from a systematic discretization error. Therefore I would like to see the same parametric curves with a different dt so that the data plotted can be taken confidently to describe the continuum dt=0 limit. The reader must know if the data they are seeing are reliable and this could be done in an appendix.
Response As suggested by the referee we have added a new plot of the energy vs the Correlation (Fig.8) and commented accordingly. We remark that the analysis of the error in dt provide evidence that the results shown in Fig.8 should have a 'very small' difference with the exact results in the dt>0 limit. For further confirmation we have added the extroplated (dt>0) line in Fig.8 for the (3+11)spin model (dashed dotted red line) for comparison.
Author: Giampaolo Folena on 20230610 [id 3723]
(in reply to Report 2 on 20230529)We have followed the suggestion of reviewer 1 and added an appendix which shows the impact of the time step dt .