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Fingerprints of hotphonon physics in timeresolved correlated quantum lattice dynamics
by E. Cappelluti, D. Novko
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Submission summary
Authors (as registered SciPost users):  Emmanuele Cappelluti · Dino Novko 
Submission information  

Preprint Link:  https://arxiv.org/abs/2110.13274v2 (pdf) 
Date accepted:  20220504 
Date submitted:  20220105 11:39 
Submitted by:  Cappelluti, Emmanuele 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approaches:  Theoretical, Computational 
Abstract
The time dynamics of the energy flow from electronic to lattice degrees of freedom in pumpprobe setups could be strongly affected by the presence of a hotphonon bottleneck, which can sustain longer coherence of the optically excited electronic states. Recently, hotphonon physics has been experimentally observed and theoretically described in MgB$_2$, the electronphonon based superconductor with $T_{\rm c}\approx 39$ K. By employing a combined abinitio and quantumfieldtheory approach and by taking MgB$_2$ as an example, here we propose a novel path for revealing the presence and characterizing the properties of hot phonons through a direct analysis of the information encoded in the lattice interatomic correlations. Such method exploits the underlying symmetry of the $E_{2g}$ hot modes characterized by a outofphase inplane motion of the two boron atoms. Since hot phonons occur typically at highsymmetry points of the Brillouin zone, with specific symmetries of the lattice displacements, the present analysis is quite general and it could aid in revealing the hotphonon physics in other promising materials, such as graphene, boron nitride, or black phosphorus.
Author comments upon resubmission
thank you very much for having sent us the Referee reports of our paper:
"Fingerprints of hotphonon physics in timeresolved correlated quantum lattice dynamics",
submitted for publication in SciPost Physics.
We thank also both of the Referees for their positive feedback and
for their valuable comments and recommendations, which help us in improving our manuscript.
We have revised the manuscript along the Referees suggestions
and we think that the revised version is now suitable for
publication in SciPost Physics.
We provide a detailed reply to the Referee comments
as well as a list of the modifications in the reply
to each of the Referee's questions (also attached below).
Thank you very much,
Emmanuele Cappelluti and Dino Novko
List of changes
*********************************************************
Reply to Referee 1
We are greatful to the Referee 1 for his/her valuable comments
and for the positive assessment of our manuscript.
Below, we provide an answer to each of his/her question.
> 1. Although the authors claim that their analysis is generic,
> but it is not obvious for me if the observed anomaly
> in the lattice displacement correlation factor is a unique fingerprint
> of the hot phonons. It seems to be just a signature of the "outofphase
> inplane motion of the two boron atoms" in MgB2. Would it be possible
> to observe similar features in any system with several atoms in the unit cell,
> if their optical modes are excited? Or would the same measure work
> in other systems with different symmetries of the hot phonon modes?
We thank the Referee for the comment.
Concerning the question whether
"the observed anomaly in the lattice
displacement correlation factor is a unique fingerprint of the hot phonons.
It seems to be just a signature of the "outofphase inplane motion
of the two boron atoms" in MgB2,
we definitively answer: yes.
"Outofphase inplane motion of the two boron atoms" is an extensive
way for defining the E_2g phonon branch,
and the sudden dominance of these modes with respect to other ones
is precisely the hotphonon physics we are describing.
We also confirm that, as stressed in the end of Sec. 1 of the manuscript,
the scenario investigated in this paper that is focused on MgB2, is quite
general and it can be applied to other systems, with several atoms in the
unit cell and with different symmetries of the hot phonon modes.
The clearest example could be underdoped cuprates [e.g., La_{2x}Sr_xCuO_4]
where hot phonons have been inferred from a threetemperature
modelling of timeresolved xray diffraction [Mansart et al;,
PRB 88, 054507 (2013)]. The symmetries of the relevant (hot) modes
are shown in Fig. 12 of that paper, involving modulation
of correlated motion of bond distances OO, LaLa, LaO etc.
The possibility of probing the correlated motion of different pairs
provides more channels for actual detections of hot phonons.
Other materials (e.g., semiconducting systems such as GaAs) can be
also investigated.
In order to clarify in a better way this point, we have added
few useful references concerning hotphonon physics
in cuprates in the end of Sec. 1 and we added
a detailed discussion, and a new figure,
about the perspectives in cuprates and other materials
in a new Section (Sec. 4 of the revised manuscript).
> 2. I think the experimental feasibility of detecting the anomalous
> behavior in the correlation factor should be discussed.
We thank the Referee also about this point.
Considering MgB2, which is the template compound
considered in this manuscript, efficient probes of the interatomic
correlations has been provided in literature, at the steady level,
by means of neutron diffraction, which is however not particularly
suitable for femtosecond resolution.
The experimental landscape is however changing daybyday
and new experimental techniques are developed,
opening new promising pathways.
We discuss the experimental perspectives also
in the new Section 4.
*********************************************************
Reply to Referee 2
We thank also Referee 2 for her/his careful reading
and for the useful suggestions, which we have tried
to incorporate in the manuscript as best as we can.
We provide below a reply for each point with the corresponding changes.
> 1. I would ask the authors to write explicitly at the end of the Introduction
> that Sec. 2 is basically an extended explanation of part of Ref. [36],
> as well as to add explicitly in the captions of Figs. 1 and 2 which figures
> are adapted from Ref. [36].
We agree with the Referee.
Sec. 2 is mainly an overview (although with some novel elements,
like the direct comparison between 2T and 3T models) of former
Ref. [36] (current Ref. [37] in the revised manuscript).
As the Referee understand, this summary was needed and useful in order
to present the basic tools for the new analysis provided in the paper
and for making the manuscript selfcontained.
We however agree with the Referee that a most clear stress
that this Section has a strong overlap with Ref. [37] is due.
We have accordingly introduced a new paragraph at the end of the
Introduction, with the scheme of the present work,
clarifying that Sec. 2 presents theoretical modelling from Ref. [37].
We have stressed in the caption of Figs. 1 and 2 that
they are a readaptation from Ref. [37].
> 2. About Section 3  Timeresolved lattice dynamics. The authors introduce
> here the evaluation of the (projected) meansquare lattice displacements
> relative to the phonon modes following Refs. [68,69]. To this end,
> they explain Eq. (12) and show Fig. 3. Here, the phonon statistics
> enters via a BoseEinstein distribution evaluated at an effective
> lattice temperature: this approximates the non thermal phonon distribution
> that we expect out of equilibrium. Actually, a very similar analysis
> was carried out in Ref. [R1] (see below), which is not cited
> in the manuscript. Have a look in particular at Eqs. (3), (4) and (5)
> of that reference, as well as Fig. 4 and the animation in their
> supplementary material. The only difference is that the authors
> of Ref. [R1] use an approximated outof equilibrium and
> timedependent nonthermal distribution computed directly
> from firstprinciples. I would ask the authors to cite Ref. [R1],
> and  if they wish so  to add a sentence in order to clarify
> for the reader the advantages of their approach in the present context.
> [R1] Tong and Bernardi, Phys. Rev. Research 3, 023072 (2021)
We thank the Referee for poiting out to us such interesting reference
that, honestly speaking, we overlooked.
The approach described there, as well in other similar papers,
is very promising in a compelling modelling of outofequilibrium time
dynamics avoiding the assumption of effective temperature.
In our present manuscript this level of accuracy is somehow not
necessary, and, for sake of simplicity,
we employ the simplest (effectivetemperature based)
model that captures the relevant physics.
We comment on the perspectives of the approach of Ref. [R1]
(presently Ref. [34]) as well as of similar papers,
in a new paragraph in the Sect. 2.
> 3. In Ref. [69], the interatomic correlation factors BB and BMg
> are found to have a value around or below 0.1 in a temperature
> range from 0 to 600 K (Fig. 5c and d of that reference). In Fig. 4b
> of the present manuscript, the values corresponding to the same
> quantities are sensibly higher (0.225 0.275) both at negative
> time delay and at 0.4 ps, where according to Fig. 2a the
> temperatures
> should be below 500 K. Can the authors clarify this?
We thank the Referee for this comment that allows us to clarify
a subtle point.
The Refeee is right in noticing such discrepancy in the absolute value.
On this regard one should keep in mind that other scattering sources
different than the electronphonon coupling (more particularly:
disorder) might play a role in the experimental analysis,
affecting the quantitative estimate of the correlation functions
under steady conditions.
The presence of disorder is expected to lead to an overall
offshifts of the value of the correlation factors, not altering
the sign and the presence of hotphononinduced anomaly
on which we focus in this paper.
In order not to affect the readibility of the manuscript,
we have discussed in detail these aspects in a long note (Ref. [83]).
> 4. I think the authors may be more specific about the generalization
> of their approach (in this, I agree with point 1. of the first Referee).
> In my opinion, a generalization of the proposed approach may be
> schematized in this way: (1) we know or suspect some specific
> symmetry constraints on the hotphonon modes of some system
> (in this case, MgB2 hotphonon modes are characterized
> by outofphase boron planar oscillations);
> (2) we devise a suitable correlation factor between atomic displacements
> which is sensitive to such symmetries (in the present case,
> atomic pair projection of BB relative displacements);
> (3) we check that at equilibrium / twotemperature level there is
> no sensible effect; (4) finally, we investigate the changes at the
> outofequilibrium / threetemperature level: if there is
> a marked change in correlation factor (in the MgB2 case, the BB factor drops),
> then we may conclude that a hot phonon scenario is likely.
> Do the authors agree with this stepbystep summary?
We confirm that the scheme of the Referee is essentially the scheme
we have in mind.
We thank the Referee for such feedback.
Often something that appears obvious to the authors, is not to obvious
for the reader.
Following the suggestion of Referee 1, we have thus gladly
explicitly summarized such scheme in the second paragraph
of new Sect. 4.
>5. If the authors agree with the scheme in point 3: can they make
> another example of a correlation factor not involving outofphase
> oscillations of a specific atom but some other symmetry?
The Referee touches an interesting point.
Although from the theoretical point of view, hot phonons not related
to outofphase lattice displacements are certainly possible,
from the point of view of material science the conditions
to observe them might be more difficult.
On the one hand, favourable conditions for hot phonons are usually
realized for the zonecenter optical modes, which intrinsically involve
opposite displacement for inner atoms in a unit cell.
However, it is still possible in principle to devise systems
with many atoms in a unit cell where the outofphase motion
of a subset of atoms is accompanied by an inphase motion
of another subset of atoms.
Other possible conditions can be encountered when hot phonons
are stored in the edgezone modes. Finally, one should consider the
possibility where a large electronphonon coupling, along
with possible nesting conditions, induces a strong Kohn anomaly
on the acoustic branches, where also nonthermal populations
can be sustained.
Although such complex investigation goes beyond the aims
of the present work, the search for real systems where hot phonons
are associated with an inphase lattice motion is thus an interesting subject
that we hope can be further developed in the future.
We have briefly commented about these aspects in Sect. 4.
> 6. I think that the authors should discuss a bit more
> the possible limitations of this approach. For example,
> the case of a correlation factor failing to "decrease", while
> hotphonon behavior is clearly observed by the threetemperature model
> or experimentally. Or, the correlation factor "decreasing" without
> hotphonon behavior being observed or while the two and
> threetemperature models give similar results.
We have followed the Referee's advise, smoothing some strong statements, e.g.
"we suggest that a smoking gun evidence" > "we suggest that a suitable evidence";
"it provides a striking tool" > "it provides a useful tool".
We agree that a theoretical prediction is always subject to a possible
failure, because of additional and/or spurious effects,
but it is hard at this point to comment about possible failures
before they occur.
We hope that our manuscript can provide a useful guide to
experimental groups for a direct check, and we are at the moment
confident that the anomalies in the correlated motion
here predicted can be revealed.
> 7. I am curious whether strong nonadiabatic effects,
> as in the case of MgB2, would be detrimental to the quantitative accuracy
> of the proposed method, since the DFPT phonon normal modes become
> illdefined in the presence of these effects, probably even more
> so out of equilibrium. What do the authors think?
> 8. Also, the analysis of the interatomic correlations rests on the
> harmonic approximation for the phonon normal modes, and on the fact
> that they remain harmonic outofequilibrium. The authors may want to
> comment on this.
We thank the Referee for arising these points.
As the Referee might have noticed from the list of our previous publications,
the role of nonadiabaticity is a crucial topic that is within the core
of the main interests of the both authors.
MgB2 presents an undoubtful degree of nonadiabaticity which
is intrinsically entangled with anharmoncity.
We therefore completely agree with the Referee that investigating
the interplay between nonadiabaticity and the quantum lattice
dynamics (including anharmoncity), also within the context of hot phonons,
is a crucial task both for a quantitative description of the effects
of the electronphonon coupling as well as a suitable path
for revealing nonadiabatic effects.
This is however a formidable and delicate task where no welldefined
procedure is assessed in the scientific community so that
different approches have been advanced according different aims,
leading sometimes to controversial debate.
For these reasons, and since nonadiabaticity and anharmonicity
are not fundamental ingredients of the analysis here presented,
in this work we didn't address explicitly the inclusion
of nonadiabatic/anharmonic effects.
A proper inclusion of nonadiabatic/anharmonic effects is however a topic
that we (and hopefully even other researchers)
will certainly pursue in future works.
> 9. Fig. 2(c) introduces the electronphonon coupling strength
> lambda but this quantity is never defined.
We thank the Referee for pointing out this lack.
We have added the definition of lambda in the present Eq. (1)
> 10. In Eq. (12) the phonon distribution is denoted as n. I just
> assumed that this is the BoseEinstein distribution,
> which however appears as b in Eqs. (56).
> Can the authors solve this discrepancy?
It was a typo. We thank the Referee.
We have used the unique notation b(x) for the BoseEinstein factor
all through the paper.
> 11. At the bottom of page 8, the authors write
> "... with a sharp increase of rho_{BB} ...".
> I assume they mean "decrease", considering Fig. 4(b)?
We thank the Referee for having signalized us also
this typo, which we have properly corrected.
> 12. Both in the abstract and in the conclusion, the authors qualify
> their approach as a 'quantumfieldtheory approach'.
> I do not understand why. I would qualify this approach a
> 'semiclassical' instead. First, the threetemperature model seems
> to me a semiclassical approach: Boltzmanntype equations
> using DFTcomputed parameters (therefore at an assumed
> effective thermodynamic equilibrium), and in the
> relaxation time approximation.
> Second, the correlation functions between DFPTcomputed
> atomic displacements are also expectation values on quantum
> harmonic oscillators in effective equilibrium, which may certainly
> correspond to the diagonal elements of an approximated equilibrium
> phonon Green's function, but still I wouldn't call this quantum field theory.
We agree with the Referee that the expression
"quantumfieldtheoryapproach" was somehow misleading
and, following the Referee's advice,
we have corrected it as "semiclassical approach"
Published as SciPost Phys. 12, 173 (2022)
Reports on this Submission
Report
The authors have revised their manuscript following the suggestions of both referees. All the raised questions are answered, and the comments are taken into account.
I feel that the revised version of the manuscript is well written, addresses an interesting subject, and in particular its claims now sound more realistic.
Report
I wish to extend my gratitude to the authors for thoroughly engaging with my comments.
I know mine was a somewhat longer report than authors may wish to receive, yet I was quite interested in the work.
I found their replies quite informative and their edits to the manuscript excellent.