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A localization transition underlies the mode-coupling crossover of glasses

by Daniele Coslovich, Andrea Ninarello, Ludovic Berthier

Submission summary

As Contributors: Daniele Coslovich · Andrea Ninarello
Arxiv Link:
Date submitted: 2019-05-10
Submitted by: Coslovich, Daniele
Submitted to: SciPost Physics
Domain(s): Computational
Subject area: Statistical and Soft Matter Physics


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 mode-coupling crossover temperature determined from the dynamics. The concentration of localized unstable modes found at low temperature is a non-universal, finite dimensional feature not captured by mean-field glass theory. Our analysis reconciles previous conflicting numerical results and provides a characteristic temperature for glassy dynamics that unambiguously locates the mode-coupling crossover.

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Reports on this Submission

Anonymous Report 2 on 2019-6-14 Invited Report


The manuscript revisits the issue of the geometrical transition in the
potential energy landscape, which existence and possible link with the
mode-coupling 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 "quasi-saddles" 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 glass-forming 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
mode-coupling 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 power-law

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 "quasi-saddle" 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

3. The significance of "quasi-saddles" 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 "quasi-saddles", the dynamical relevance of the
latter is not clear. It was argued in ref. 18 that "quasi-saddles"
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.

  • validity: low
  • significance: good
  • originality: good
  • clarity: high
  • formatting: good
  • grammar: excellent

Anonymous Report 1 on 2019-6-6 Invited Report


1-New approach to an old long standing debate (taking advantage of recent algorithmic and conceptual progresses)
2-Accurate analysis of the results under multiple perspectives


1-The wording in the text is at times too terse, something could have been explained better.


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

1-I 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.

2-A 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).

3-The distribution of radii for the Polydisperse particles n=12 is not specified.

4-It 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 50-50 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?

5-The 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?

6-When 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.

7-On 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.

8-Fig4 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.

9-Why 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.

  • validity: high
  • significance: high
  • originality: good
  • clarity: ok
  • formatting: good
  • grammar: good

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