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A localization transition underlies the modecoupling crossover of glasses
by Daniele Coslovich, Andrea Ninarello, Ludovic Berthier
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Authors (as registered SciPost users):  Ludovic Berthier · Daniele Coslovich · Andrea Ninarello 
Submission information  

Preprint Link:  https://arxiv.org/abs/1811.03171v2 (pdf) 
Date submitted:  20190510 02:00 
Submitted by:  Coslovich, Daniele 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Computational 
Abstract
We study the equilibrium statistical properties of the potential energy landscape of several glass models in a temperature regime so far inaccessible to computer simulations. We show that unstable modes of the stationary points undergo a localization transition in real space close to the modecoupling crossover temperature determined from the dynamics. The concentration of localized unstable modes found at low temperature is a nonuniversal, finite dimensional feature not captured by meanfield glass theory. Our analysis reconciles previous conflicting numerical results and provides a characteristic temperature for glassy dynamics that unambiguously locates the modecoupling crossover.
Current status:
Reports on this Submission
Anonymous Report 2 on 2019614 (Invited Report)
 Cite as: Anonymous, Report on arXiv:1811.03171v2, delivered 20190614, doi: 10.21468/SciPost.Report.1017
Report
The manuscript revisits the issue of the geometrical transition in the
potential energy landscape, which existence and possible link with the
modecoupling crossover of structural glasses was first discussed
about 20 years ago. It is a welcome new exploration the subject,
which now benefits from the existence of new models and increased
computing power which allows to explore temperatures which were out of
reach a decade ago.
The authors find saddles and "quasisaddles" of the potential energy
landscape and compute the fraction of unstable modes, $f_u$, and of
unstable *delocalized* modes, $f_{ud}$ as a function of temperature
for four model glassforming liquids. The main message of the
manuscript is this: from the numerical study of the four models, it is
clear that nothing happens to the $f_u(T)$ curves near the
modecoupling temperature $T_\text{MCT}$. However, if one considers
only *delocalized* unstable modes, one sees that the $f_{ud}(T)$
curves vanish near $T_\text{MCT}$ in a way compatible with a powerlaw
singularity.
The subject and approach are interesting, and a welcome attempt to
shed light on a difficult problem. Unfortunately, the manuscript has
issues serious enough that I cannot recommend publication, at least in
the present form.
1. The main issue with this study is the very definition of the
$f_u(T)$ and $f_{ud}(T)$ curves. While it is clear enough how to
compute $f_u$ once a saddle point has been found, it is not clear
how to associate a set of saddle points to a given temperature. In
practice, one starts from an instantaneous equilibrium
configuration and finds a saddle using some algorithm (in this case
lBFGS minimization of the squared force), and associates the saddle
to the temperature at which the instantaneous configuration was
produced. However, in ref. 40 it was shown that this procedure
leads to different $f_u(T)$ curves depending on the algorithm used
to find saddle points. The problem is aggravated by the fact that
with the algorithm used here, most of the time one does not find a
saddle, but a "quasisaddle" instead. Ref. 40 also showed that even
when a saddle is found, it is not necessarily that which is closest
to the instantaneous configuration. Although ref. 40 considered
only $f_u(T)$, it is reasonable to presume that the same holds for
$f_{ud}(T)$. Thus the curves of fig 1a and 1b are not very meaningful
unless a strong case can be made for the use of one algorithm over
another one. Of course it could be that different algorithms
produce curves are different but vanish at the same temperature,
but this remains to be shown. In any case, the exponent of eq. 9
is probaly meaningless.
2. On the other hand, the $f(e)$ curves (where $e$ is the energy *of
the saddle*) are much more robust (though ref. 40 found some small
algorithm bias). But the comparison in fig 1d and 1e is not very
meaningful: the threshold energy can be defined for $f_u(e)$ or for
$f_{ud}(e)$ as the energy at which either of the two curves vanish.
Since $f_{ud}(e)\le f_u(e)$ by definition, it is not surprising
that $f_{ud}(e)$ vanishes at a higher energy than $f_u(e)$. The
issue is which one of the two threshold energies can be linked to
the $T_\text{MCT}$ crossover. For the reasons stated above, the
$f(T)$ curves are suspect, so another approach is required. In
references 9 and 11, it was argued that one should compare the
energy of the saddles to the "bare" potential energy at
equilibrium, defined as $u_\text{eq}(T)3/2 k_B T$: the idea was to
find the bottom of the potential energy well within which the
system is moving. However, none of this is discussed in the
manuscript.
3. The significance of "quasisaddles" is not discussed enough. While
the present study confirms earlier results in the sense that the
properties of the fraction of unstable modes behaves similarly in
both saddles and "quasisaddles", the dynamical relevance of the
latter is not clear. It was argued in ref. 18 that "quasisaddles"
are not "quasi" at all, i.e. they are simply nonstationary points
where the force is an eigenvector of the hessian with zero
eigenvalue. While the fact that they behave similar to true
stationary points is intriguing, the conceptual differences between
the two call for an analysis of the $f(T)$ curves of true saddles
in all the models considered.
4. In p. 6 the authors write "These deviations were previously
attributed to finitesize effects [40], but they actually stem from
localized modes" (referring to deviations from linearity in the
$f_u(e)$ curve near $e_\text{th}$. This statement is wrong: the
comment in ref. 40 concerns the saddle point approximation, and
applies both to $f_u(e)$ and $f_{ud}(e)$. If one defines the
threshold energy as the point where the complexity $\Sigma(f,e)$
(where $f$ can be $f_u$ or $f_{ud}$) first has a maxium at $f=0$,
then finite systems will actually have a small but nonzero average
values of $f$ even below $e_\text{th}$ (because $f$ is a positive
quantity, and saddle points continue to exist below the threshold,
only that they become subdominant with respect to minima [or
delocalized saddles]).
5. Also in p. 6, "Such a representation [plotting $f$ vs. the energy
of the saddle] reveals an intrinsic property of the landscape and
does not depend on the way in which the latter is sampled". This
is not quite true: ref. 40 found a small algorithm bias even in the
$f(e)$ representation.
Anonymous Report 1 on 201966 (Invited Report)
 Cite as: Anonymous, Report on arXiv:1811.03171v2, delivered 20190606, doi: 10.21468/SciPost.Report.1000
Strengths
1New approach to an old long standing debate (taking advantage of recent algorithmic and conceptual progresses)
2Accurate analysis of the results under multiple perspectives
Weaknesses
1The wording in the text is at times too terse, something could have been explained better.
Report
I am definitely in favour of publication after the following comments are taken into account and corresponding minor revisions are implemented in the text.
Requested changes
1I do not see any reason to put Table 1 on second page. I found reference to that in the text only at pg.6, unless I am mistaken. It should be put in correspondence of the text referring to it or add a first introductory comment to the Table earlier on if felt important that it appears in first pages.
2A few more words are to be devoted to explain what are quasi stationary points (it is not really clear as it is) and it should be clearly stated what it is the issue with them and how it is solved for the current data analysis (the current explanation is not well structure and a bit confusing).
3The distribution of radii for the Polydisperse particles n=12 is not specified.
4It is said that "the fraction of delocalized unstable modes [...] goes strictly to zero when the temperature drops below T_MCT". This seems an accurate description of data for 5050 and ternary mixture, not quite so for the two polydisperse systems (n=12 and 18) and the network liquid for which the fraction of delocalized unstable modes goes to zero but somewhat below T_MCT. Nothing is said about this discrepancy. The Authors should comment more about that. Do they relate this to a small error in the estimation of T_MCT? Some finite size effects? Inflection points? Other reasons?
5The sentence "Finally the trend in Fig. 1(c) superficially suggests a correlation between glass forming ability [27] and the concentration of localized unstable modes." must be expanded. In which way the suggested correlation goes? Can the Author spend a few more words on glass forming ability and what does this correlation, if any, imply?
6When INM are mentioned on pg.7, it would be nice to read a brief explanation of why INM where not successful and in which way in comparison the adopted method is expected to be more reliable.
7On pg.9 it would be beneficial to have some more words to introduce soft stable modes and to speculate about the proposed connection between their scaling with the one observed for the delocalized portion of the unstable spectrum.
8Fig4 why data about the unstable part of the spectrum of stationary points are only presented for ternary mixtures? I did not find anywhere in text where it is commented if this result is robust across the different models studied. This should be clarified.
9Why is the average over all modes used to find particles involved in the rearrangements? It is natural to ask whether different pictures emerge for modes with lambda>lambda_0 or lambda>lambda_0. May be a distribution of displacements for what is now labelled as a 'mobile' particle could show a non trivial (bimodal?) behaviour? I find the choice of focusing on the average a little coarse and simplistic.
Author: Daniele Coslovich on 20191008 [id 625]
(in reply to Report 1 on 20190606)
We thank the referee for his/her careful and constructive report. He/she provided a number of suggestions to improve the manuscript, which we mostly followed.
1I do not see any reason to put Table 1 on second page. I found reference to that in the text only at pg.6, unless I am mistaken. It should be put in correspondence of the text referring to it or add a first introductory comment to the Table earlier on if felt important that it appears in first pages.
We agree with the referee, the table now appears in the Results section.
2A few more words are to be devoted to explain what are quasi stationary points (it is not really clear as it is) and it should be clearly stated what it is the issue with them and how it is solved for the current data analysis (the current explanation is not well structure and a bit confusing).
Prompted by the remarks of both referees, we have signifincantly expanded our discussion and analysis of the similarities and differences between stationary and quasistationary points. We now include an additional set of optimizations obtained with the eigenvectorfollowing method, see the Methods section as well as the reponse to the other referee for more details. These new data are presented and discussed along with the previous ones in the Results section. Additional comments on the distinction between quasistationary and stationary points have been added in the Introduction and in the Methods section.
3The distribution of radii for the Polydisperse particles n=12 is not specified.
We have specified that the distribution is the same as for polydisperse particles with n=18.
4It is said that "the fraction of delocalized unstable modes [...] goes strictly to zero when the temperature drops below T_MCT". This seems an accurate description of data for 5050 and ternary mixture, not quite so for the two polydisperse systems (n=12 and 18) and the network liquid for which the fraction of delocalized unstable modes goes to zero but somewhat below T_MCT. Nothing is said about this discrepancy. The Authors should comment more about that. Do they relate this to a small error in the estimation of T_MCT? Some finite size effects? Inflection points? Other reasons?
We see two separate issues here. First, the fraction of delocalized modes for the network liquid does not go to zero within the explored range of temperature. This may be an intrinsic feature of this kind of liquids or a feature specific to quasistationary points. In fact, we could barely locate any stationary point for this model, no matter the optimization algorithm. We have added a related comment on this on page 8.
Concerning polydisperse particles, the fractions go to zero but at a temperature lower than T_MCT by about 10% (n=18) and 20% (n=12). These discrepancies are likely due to the wellknown inherent uncertainty in fitting the MCT crossover from dynamic data. Or they might perhaps reflect a worse overall consistency with the geometric transition scenario. We have added a brief comment about this on page 7, pointing out the difficulties in determining the MCT crossover.
5The sentence "Finally the trend in Fig. 1(c) superficially suggests a correlation between glass forming ability [27] and the concentration of localized unstable modes." must be expanded. In which way the suggested correlation goes? Can the Author spend a few more words on glass forming ability and what does this correlation, if any, imply?
We have expanded the corresponding discussion on page 7 and pointed to Ref. 27 for the discussion about the glassforming ability of the studied models. We do not have a clear picture of what this correlation may imply and we prefer to leave this specific point open.
6When INM are mentioned on pg.7, it would be nice to read a brief explanation of why INM where not successful and in which way in comparison the adopted method is expected to be more reliable.
To address this point, we have added a comment on page 10 about INM, pointing out which INM modes need to be filtered in order to establish contact with the MCT crossover, see Ref. 49.
7On pg.9 it would be beneficial to have some more words to introduce soft stable modes and to speculate about the proposed connection between their scaling with the one observed for the delocalized portion of the unstable spectrum.
Prompted by the referee's remark, we have added a comment on page 11 about the quasilocalized modes observed in the stable, low frequency portion of the the spectrum. The possible connection with the unstable spectrum becomes even more intriguing in the light of the new data for stationary points, which shows a depletion close to lambda=0. Please see the new Fig. 6 and its corresponding discussion. Work to test this connection is currently under way.
8Fig4 why data about the unstable part of the spectrum of stationary points are only presented for ternary mixtures? I did not find anywhere in text where it is commented if this result is robust across the different models studied. This should be clarified.
To address the point raised by the referee, we performed the following modifications:
 We pointed out on page 11 which features are "universal" across models and/or type of points and which features are not.
 We have added the full N and Tdependence of the spectra in the Supplement section of the accompanying project document.
9Why is the average over all modes used to find particles involved in the rearrangements? It is natural to ask whether different pictures emerge for modes with lambda>lambda_0 or lambda>lambda_0. May be a distribution of displacements for what is now labelled as a 'mobile' particle could show a non trivial (bimodal?) behaviour? I find the choice of focusing on the average a little coarse and simplistic.
We chose to show the average over all unstable modes because this quantity was found to correlate well with the propensity of motion in Ref. 58. We found that filtering the distribution P(e_u^i) according to the localization of the modes, as suggested by the referee, does not lead to interesting insight, even though his/her point is definitely valid. Promising alternative approaches to quantify the spatial correlations in the modes are the methods of Ref. 37 and Ref. 57, and we have added a related comment on page 12. Work in this direction is under way and will be presented in a separate publication.
Author: Daniele Coslovich on 20191008 [id 624]
(in reply to Report 2 on 20190614)We thank the referee for his/her critical but constructive report, which prompted us to extend and improve our analysis. The referee raises some concerns about the minimization algorithm, but also suggests in point 2. a possible strategy to check the internal consistency of our results. We have thus addressed these concerns by (i) perfoming additional optimizations using a different method locating true saddle points and (ii) exploiting his/her suggestion in point 2. We also fully took into account the additional remarks raised in the report.
The main issue raised by the referee is not specific to our work and indeed has plagued previous research on this topic since the early studies of the Rome group in the early 2000's. The problem is quite intricate and somewhat technical, and we did our best to disentangle several subissues that the referee has condensed in the paragraph above. We also point out that a simple strategy to tackle the problem is offered by the referee himself/herself in point 2. and we address it below.
The first subissue concerns the algorithmdependence of the curves f_u(T). The results presented in the previous version of the manuscript were based on force minimizations using the lBFGS algorithm (Wminimizations). To address the referee's concerns, we considered the hybrid eigenvectorfollowing (EF) method. In a first stage, we implemented the algorithm following Refs. 18, 20, but we encountered serious convergence issues. Similar issues were reported in the above papers (especially at high temperature) and can be alleviated by "fixing" the number of target unstable modes after a prescribed number, M, of iterations, as done already in Refs. 18, 20. This is the approach used in Ref. 32 (formerly Ref. 40) and which produced "algorithmdependent" f_u(T) curves, as mentioned by the referee. We realized, however, that this dependence is inherent in the hybrid EF method: larger values of M leads to smaller values of f_u, but also decrease the success rate of the optimizations. On top of that, given M, the final value of f_u depends on the choice of the initial trust radius, which is a free parameter of the algorithm. Thus, the algorithmdependence of the f_u(T) curves observed in Ref. 32 is partly artificial: the curves may be shifted upwards and downwards by choosing different values of M or of the initial trust radii. We also found a fundamental issue with the hybrid EF method: it displays an erratic behavior whenever the system passes through regions of zero curvature. This issue deteriorates the success rate of the optimizations (unless M is very small) and is not specific to the complex highdimensional landscape of supercooled liquids: it can be reproduced under some circumstances even on the simple MullerBrown surface. An account of these difficulties is given in the Methods section of the revised manuscript.
Based on these preliminary explorations and from a survey of the current literature (including a recent but unfortunately unpublished study by C. Ruscher, PhD thesis at the University of Strasbourg), we got to the unsatisfactory conclusion that the hybrid EF method does not provide a robust, parameterfree mapping between instantaneous configuration and stationary points. However, by fixing f_u from the outset, the original EF method can at least be used to address the second subissue mentioned by the referee: Wminimizations, while providing a welldefined mapping from initial configurations, only rarely locate true stationary points. Prompted by both referees, we decided to investigate this issue more carefully and we performed extensive searches of stationary points for the ternary mixture model using the EF method. Given the above problems, the value of f_u was fixed from the outset to the one of the point found by Wminimization starting from the same instantaneous configuration. This approach does not provide an independent mapping, of course, but it enables a straightforward and more extensive comparison between quasistationary and stationary points. Through this new analysis we evidenced a small qualitative difference between these two types of points which had so far passed undetected and which we further discuss below. The main point, after sorting out all these technical issues not mentioned in the referee's comment, is that the bulk of the conclusions at the core of the paper (localization transition of the unstable modes) remain unaffected. We include and discuss these new results throughout the revised manuscript.
The referee mentioned in passing a third subissue related to Wminimizations: Ref. 32 provides some evidence that the stationary points located by Wminimizations are not necessarily the "closest ones" to the corresponding instantaneous configurations. We performed additional analysis and qualitatively confirmed this behavior, however we also observed that: (i) the difference between the distances of the Wminimized point and the closest stationary point is on average rather small (of the order of 10%, consistent with Ref. 32) and crucially, (ii) the average fraction f_u of the closest points is statistically indistinguishable from the one of the bulk of the points. Thus, even though Wminimizations do not always locate the closest points, the mapping that defines f_u(T) is unaffected and remains rather welldefined.
Our take on this is that there exists, to date, no satisfactory mapping between instantaneous configurations and stationary points for highdimensional glassy landscapes (methods like Newton homotopy still have to prove their relevance for this field). Nonetheless, the conclusion we got after 3 months of exploration of this intricate problem is that, despite the known issue about quasistationary points, Wminimizations provide a rather welldefined mapping, i.e. f_u(T), which is more robust than the one of the alternative hybrid EF. Some evidence that this mapping is not irrelevant to the dynamics was given long ago in Ref. 58.
We may agree that the agreement with the exponent found in meanfield pspin models is a bit surprising (rather than "meaningless"), but it is also clear from the text that we do not make a strong case for it: we only state that the data can be "described approximately" by the power law in Eq. 10 with exponent 3/2. Actually, the agreement with the new set of stationary points is quite good.
Finally, a simple strategy to circumvent the above issues comes from a suggestion of the referee in point 2, to which we now arrive.
We thank the referee for this suggestion. We determined the temperature at which the bare energy, which is a proxy to the inherent structure energy, crosses the threshold energy determined from the vanishing of delocalized modes f_ud. We found that it matches well the MCT crossover temperatures for all models except the polydisperse n=12. A similar discrepancy is observed in Fig. 1(b) and is likely due to the inherent uncertainties and ambiguities in fitting the dynamic data (choice of the temperature range). Indeed, the temperature at which the crossing occurs agrees well with the localization temperature T_lambda. This new analysis is included in the new Fig.2, which now gathers all the plots on the energydependence of the data. The fact that the bare energy reaches the threshold energy around the localization transition is a nice confirmation of our previous analysis. It also corroborates the validity of the mapping f_u(T) determined from Wminimizations, since it provides quantitatively similar localization temperatures, thus solving the intricate "algorithmdependence" issue mentioned in point 1. above.
We would also like to emphasize that, while it is clear that f_ud(e_s) must vanish at a T higher or equal than f_u(e_s), our central result is that f_ud(e) vanishes for all fragile liquids, while f_u(e_s) does not (not even in the thermodynamic limit, see point 4. below), except perhaps for the 5050 mixture. This finding is another strong indication that f_ud is much more relevant than f_u to establish the connection with the MCT crossover. We trust that these results will fully address the referee's concerns.
Prompted by both referees, we have addressed this issue in greater detail. Our main conclusions on this point are:
Throughout the revised manuscript we discuss in much more detail the similarities and differences between the two types of points, now that we have excellent statistics for both. In particular, we point out on page 11 that "the difference between stationary and quasistationary points does not boil down simply to the presence of an inflection mode" but also that these discrepancies do not affect the localization properties.
Prompted by the referee's final remark, we have also added a panel to Fig.11 in the Appendix showing the curves f_u(T) for true stationary points for all models and system sizes for which these points could be sampled. The comparison of the available data for f_u and f_ud for true stationary points support our conclusions about the unstable modes localization.
We removed that specific sentence about finite size effects from page 7. However please note the following:
We have revised this sentence, which now reads "an intrinsic property of the landscape and it is fairly insensitive to the way in which the latter is sampled."