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Dynamic correlation functions for integrable mobile impurities
by Andrew S. Campbell, Dimitri M. Gangardt
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Submission summary
As Contributors:  Dimitri Gangardt 
Arxiv Link:  http://arxiv.org/abs/1701.00810v1 (pdf) 
Date submitted:  20170105 01:00 
Submitted by:  Gangardt, Dimitri 
Submitted to:  SciPost Physics 
Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
We consider dynamical correlation functions near a spectral threshold in one dimensional quantum liquids. By using the phenomenological depleton model of mobile impurities we recover semiclassically the known leading power law behaviour of the correlation functions and express the corresponding edge exponents in terms of two phenomenological parameters: the number of depleted particles, $N$ and the superfluid phase drop $\pi J$. For integrable LiebLiniger and YangGaudin models we establish a rigorous relation between these parameters and the Bethe Ansatz shift functions of elementary excitations. This relation implies the absence of phonon back scattering from integrable impurities.
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Anonymous Report 4 on 2017215 (Invited Report)
 Cite as: Anonymous, Report on arXiv:1701.00810v1, delivered 20170215, doi: 10.21468/SciPost.Report.81
Strengths
The authors prove a relation between solutions of linear integral equations which allows one to make a connection between two apparently different expressions for edge exponents.
Weaknesses
In many cases, the authors present results as if they were discovered by them whereas these are common knowledge. (see report).
The paper is too long regarding to its original content.
Report
The paper "Dynamic correlation functions for integrable mobile impurities" by A.S. Campbell and D.M. Gangardt
deals with certain issues related to dynamical correlation functions in the Bose gas and the YangGaudin model.
More precisely, A. Kamenev and L. I. Glazman in "Dynamics of a onedimensional spinor Bose liquid: A phenomenological approach",
predicted a set of universal relations, valid for Galilei invariant models, expressing the edge exponents in terms of partial derivatives of the excitation's dispersion relations.
The work, A. Imambekov and L. I. Glazman "Exact Exponents of Edge Singularities in Dynamic Correlation Functions of 1D Bose Gas" and the work
M. B. Zvonarev, V. V. Cheianov and T. Giamarchi "Edge exponent in the dynamic spin structure factor of the YangGaudin model"
building on a mixture of nonlinear Luttinger liquid theory and Bethe Ansatz expressions for the spectrum argued other kinds of expressions for these
edge exponents, this time in terms of the shift function. The work N. Kitanine, K. K. Kozlowski, J. M. Maillet, N. A. Slavnov and V. Terras, "Form factor
approach to dynamical correlation functions in critical models" proved, on the basis of exact Bethe Ansatz calculations of correlation functions, the shift function based expressions
for the Bose gas.
The present work establishes an equivalence between these apparently different expressions for the edge exponents in the case of the two models.
M. Schecter, D. Gangardt and A. Kamenev in "Dynamics and Bloch oscillations of mobile impurities in onedimensional quantum liquids"
proposed a certain expression for a leading order photon backscattering amplitude that is responsible for the vanishing of a viscous
force acting on a moving impurity in a hydrodynamic approximation.
The present work establishes the vanishing of this leading order photon backscattering amplitude in the case of the two models.
The paper stars with a short introduction to the topic which misses the contribution of the integrable model community to the problem:
basing on the description of certain citations one may wonder whether the authors did take the time to read the papers they cite
(see a more precise discussion below). The introduction also allows the authors to present the problem of interest to the paper.
Section 2 applies the socalled depleton model to compute the exponents in the powerlaw behaviour of spectral function in terms of quantities they call the depleton charge N and
the kink size J. The purpose of this section is not understandable to me. The existence and expression for the edge exponents is well known on the basis of the nonlinear luttinger model approach ( Ref[2])
and even on the basis of exact, Bethe Ansatz based calculation (Ref [7]). There is thus low interest in rederiving these again in terms of some new
quantities (N and J). The derivation is rather obscure from the point of view of a novice to the field and useless for an expert. Furthermore, the effective model
provided by the authors leads to wrong conclusions in that it does not distinguishes the role played by the velocities of the deep excitations: the conclusions form formula
just above (11) are wrong since the spectral functions may also exhibit twosided singularities. Its usefullness is thus debatable.
Section 3 establishes a link between the expressions obtained by the authors for the edge exponents in section 2 and those obtained earlier in the literature.
Its presence in the paper is only justified by the presence of section 2 which introduces different parameters ($N$ and $J$) than those
used earlier in the literature. However, as I stated the presence of section 2 does not seem that useful.
Section 4 presents various Bethe Ansatz issued expressions for the energies and momenta of the excitations in the Bose gas and the GaudinYang model.
It leaves one with the wrong impression that some of these expressions where discovered by the authors whereas, in fact, they are known for easily more than
30 years (see below for a deeper explanation).
Section 5 contains the first original result of the paper, namely the proof of relations (12). It is based on establishing certain functional relations satisfied by solutions
to linear integral equations. True, some of the identities that are obtained are undoubtedly new. However, this section again leaves one with the wrong impression
that most of the expressions where discovered by the authors whereas these have been established a long time ago.
Section 6 applies the results of section 5 to the proof that the socalled phonon backscattering amplitude vanishes for the Bose gas and the YangGaudin model.
This is the second original input of this work.
However, one should keep in mind that the expression obtained in [40] for this amplitude is only a leading order one. Hence, higher order processes could, in principle, still lead to backscattering/existence of a
viscous force. Thus, the conclusions of this section should be softened in that it only provides a first order check.
The paper contains several appendices which contain technical details relative to manipulations of solutions of linear integral equations. Most of the handlings
in these appendices are well known. Yet, by the way these are discussed, one gets impression that
these were invented by the authors. In fact, most of the proofs presented by the authors can be found in Ref. [3] which is cited at other instance in the paper.
It is however not cited as the source for these proofs.
To summarise, the two original results obtained in the paper are interesting and worth publishing. However, the huge "dressing" up of these discoveries
by well known content is not necessary, especially that, the way it is written, it produces the wrong impression that this content was also discovered by the authors.
I would not object if the paper reproduced known proofs, for the reader's convenience, while making a clear statement that
"this is not the original part of the work". However, even then some parts could be shorten.
I believe that the original result deserve publication if the content is significantly reduced and
the presentation simplified. If these efforts are done, I would recommend the paper for publication.
Below, I list in more details the various problems encountered in this work.
Abstract
"we recover semiclassically" is a wrong statement since the author's phenomenological approach does not reproduce the existence of twosided singularities.
 "relation between these parameters and the Bethe
Ansatz shift functions of elementary excitations". This is a missleading statement (see discussion in section 3)
Thus it should be corrected to something of the sort " We establish a rigorous relation between Galilei invariance based predictions for the edge exponents and exact Bethe Ansatz calculations.''Which is closer to the truth.
 "implies the absence of phonon back scattering". This is an overstatement in that it is only proved to leading order.
Introduction
 "Progress in this direction has been recently reported in Refs.[7, 8, 9, 10, 11]." The statement is wrong and shows the lack of knowledge of the content of the
cited works. Regarding to the progress in the calculation of correlation functions and matrix elements of local operators, one could cite the important works of the Kyoto
group on elementary blocks or further developments on qKZ equations and fermionic bases, the resolution of the inverse problem by the Lyon group or the calculation of thermal correlators by the Wüppertal group to name a few.
The cited works [7, 8, 9, 10, 11] characterise, on the basis of exact calculations, various critical behaviours
of the correlation functions in the XXZ spin chain or the Bose gas. In particular, the work [7] characterises various dynamical properties of the correlators in the Bose
gas the edge singular behaviour of spectral functions in particular.
 Again "Another approach
was undertaken in Refs. [12, 13, 14] where the matrix elements where calculated numerically." is an huge understatement of the content of these works.
Not only the matrix elements where computed there but, in fact, a full numeric characterisation of the spectral functions was achieved, in particular by showing
the critical excitation tresholds.
 Reference [15] is correct but I trust that the original credit should be given to Luther and Peschel and Haldane.
 "This description is sufficient to calculate reliably the static correlation functions
such as the onebody density matrix and the corresponding momentum distribution [17]." The use of reliably for an approximate method sounds strange to me.
In any case, this was achieved on the basis of exact rigorous calculations much earlier: see the works of the Kyoto group and of TracyVadiya.
 "It is known that the LL theory fails to describe correctly the dynamical correlations even
in the low energy limit [2]." The reference to [2] is very strange. This is known from much much earlier works on dynamical correlators
in the XX chain.
 paragraph 2 on page 3. The authors do not cite ref [7] where it was shown that the edge exponents can be computed exactly for integrable models.
Section 2
 "It is well known (see Ref.[2]" parenthesis missing.
 The arguments of the section are unclear and look like an adhoc procedure to get the result. The conclusions of the method are not correct since
one misses the existence of twosided singularities (check [2]).
 The sentence "In the next section we show that this result is in complete agreement with the results of
Refs. [18, 30, 46]." omits the contributions of the integrable model community.
Section 3
 The authors introduce the parameters $N$ and $J$ in their depleton model. These parametrised the edge exponents as computed within their method.
By requiring consistence of their depleton model's conclusion with the predictions for the edge exponents issuing from the reasonings based on the nonlinear Luttinger
liquid model and arguments of Galilean invariance, they identify J and N
with partial derivatives of the excitation's dispersion relation.
Then, after making this identification, they claim to recover the various predictions, in terms of the phase shifts, that appeared in the literature.
However, from the very start, their parameters where tuned in a way to correspond to the phase shift predictions in the first place.
Thus, this section does not really contain any derivation.
 For all these reasons, I believe that it would be reasonable to start the results of the paper from this section and simply state the expressions for the pĥase shifts (12)
as predicted by Galilean invariance. And then to discuss the exact results obtained in [7],
and earlier predicted on the basis of a mixture of Bethe Ansatz calculation and nonlinear Luttinger liquid theory in [30], in what concerns the Bose gas, and predictions in [33]
by similar means for the GaudinYang model.
Section 4
The denomination "shift function" for the solution to (22) is unusual, this object is traditionally called "dressed phase" or scattering phase, see ref [3] equation (4.39).
The matter is that F used by the authors only identifies with the shift functions in very specific cases, as actually discussed in ref [3].
 All integral representations present in this section are well known. They are all listed and proven in, say ref [3]. For instance (23)(24) is equation (4.6)(4.7) in [3], while (25) is equation
(4.28) and (4.19) in [3], (26) is (4.9). In particular the equivalence between these representations is proven in [3]. For this reason, the statement "The proof of equivalence of (25), (26) and (23), (24) can be found in Appendix C.1"
is very missleading. I repeat that I have no problem if the authors recall the proofs (although these are of not much use to the "original" part of the research presented in the paper)
but they have to very clearly state they they only recall the proof of a known result.
 I have no immediate reference in mind for the similar types of representations in the GaudinYang model, but these are also very well known results (and the proof is similar). The authors should thus give the relevant credit.
Section 5
"The results of previous section allows us to obtain the dispersion relation ε(k) of excitation
in LiebLiniger and YangGaudin models in terms of shift functions F (νλ) and F̃ (νλ) correspondingly". The sentence is missleading since it insinuates that there were some new results
in section 4 whereas it is just a reminder of well known facts.
 "Indeed, the previous studies [30, 33] suggested" this disregards that relation (30) was established through exact calculations for the Bose gas.
 Most relations proposed in section 5.1 and 5.2 are well known. What makes the original part is only their combination to check the nice prediction (12).
However, the authors should not give themselves the credit of having established some specific differential relations.
 In the paragraph below (35) the authors reintroduce a mass $m$, eventhough they stated taking $m=1/2$ at some earlier point. They should make up their mind once for all.
 The parenthesis in (46), (47), (48) are too large.
Section 6
 One should keep in mind that the results of [40] stem form a linearisation procedure, hence an approximation of some more complex equation. As such, $\Gamma_{+}$ only represents
the leading contributions to the viscous force. Higher order scattering contributions may also contribute to the effect. Hence, showing that $\Gamma_{+}=0$ only shows
that to the leading order such effects do not take place. It is still an interesting, but weaker, result.
Section 7
 "respect to the Bethe Ansatz parameters mentioned above ." too much space
 "We expect that due to the general structure of Bethe Ansatz equations our results can be
generalised to other models soluble by nested BA, such as , the fermionic Hubbard model and
integrable spin chains." I disagree with the conclusions. The authors use a trick that heavily depends on the bare energy being quadratic, which fails for
more complex models.
Appendices
 Most of the relations established in the appendices have already been established and a major part thereof is even extremely well known.
The authors give proofs without making it explicit in the text that it is only a reminder of known facts is missleading.
 Some nonexhaustive examples:
 The authors cite [50] to credit it for (55). Then they use it to prove (58). Yet (58) was proven in 1998 in the paper of Korepin and Slavonv "The New Identity for the Scattering Matrix
of Exactly Solvable Models".
 Above (74), the fact that the resolvent kernel is symmetric is not proven in [50]. It is a simple consequence of the theory of linear integral operators.
 (75) the proof that can be found in [3].
 The authors should thus make a clear identification of what is new and what is only a copying of known results.
Requested changes
see report
Anonymous Report 3 on 2017214 (Invited Report)
 Cite as: Anonymous, Report on arXiv:1701.00810v1, delivered 20170214, doi: 10.21468/SciPost.Report.78
Strengths
 Nice presentation
Weaknesses
 Claims about novelty are exaggerated
Report
The authors study dynamical correlation functions in onedimensional
quantum systems. Using a semiclassical analysis, they relate the edge
exponents of powerlaws in the response functions to the
phenomenological parameters of their mobileimpurity model. Moreover,
they determine these phenomenological parameters exactly for two
integrable models in terms of the shift functions of their Bethe ansatz
solutions. Finally, they show that the phonon backscattering rate due to
impurities vanishes for integrable models.
The paper is very nicely written. The semiclassical derivation of the
edge exponents in terms of the parameters J and N, and their relation to
the phase shifts obtained from the conventional calculation based on
gauging away the liquidimpurity interaction are well explained and
useful. However, I think the authors need to be more careful about their
claims of novelty. In fact, as far as I know, the predictions of the
mobileimpurity models have been checked analytically for some models.
The claim "The equivalence of the edge exponents obtained from the Bethe
Ansatz solution and from variation of the dispersion relation was so far
established only numerically." is in this generality not correct. But I
think a careful rewording of these claims should suffice to make this
paper acceptable for publication.
Requested changes
 Make claims about the existence of analytical exponents of Bethe ansatz solvable models more precise.
Author: Dimitri Gangardt on 20170426 [id 121]
(in reply to Report 3 on 20170214)
We are grateful to the Referee who judges that our paper "is very nicely
written" and its results are "useful". We followed his/her suggestion and made
our claims more precise. In particular, we do not claim to be the first to
obtain edge exponents exactly via Bethe Ansatz solution but rather demonstrate
directly the relation Eq. 22 (new version of the manuscript) between
phenomenological parameter of depleton (mobile impurity model and Bethe Ansatz
Bethe function.
Anonymous Report 1 on 2017122 (Invited Report)
 Cite as: Anonymous, Report on arXiv:1701.00810v1, delivered 20170122, doi: 10.21468/SciPost.Report.67
Strengths
1. The statement
'The equivalence of the edge exponents obtained from the Bethe Ansatz solution and from variation of the dispersion relation was so far established only numerically'
is not correct.
Weaknesses
1. The volume of the manuscript does not correspond to the essentially new content.
Report
The Authors study dynamical correlation functions near a spectral threshold
in one dimensional quantum models. The paper can be divided in three parts.
The first part deals with a phenomenological approach based on the method of mobile impurities. Here the Authors rederive the edge exponents obtained previously by this method in a series of works.
The most significant part of the manuscript (including appendices) is devoted to integrable systems. Here LiebLiniger and YangGaudin models are considered. The elementary excitations in these models are treated as mobile impurities.
This allows the Authors to express the edge exponents in terms of the shift function. The Authors claim that 'The equivalence of the edge exponents obtained from the Bethe Ansatz solution and from variation of the dispersion relation was so far established only numerically'. I do not agree with this statement. I would like to draw the attention of the Authors to the paper [7] where the edge exponents were derived within the Bethe ansatz approach via pure analytical technique. Thus, at least for the LiebLiniger model the aforementioned equivalence was already proved, and hence, it cannot be
considered as a new result.
Finally, in the third part of the paper the Authors prove that the phonon backscattering amplitude vanishes for LiebLiniger and YangGaudin models. This result confirms a conjecture that in integrable systems this amplitude vanishes due to the existence of infinitely many integrals of motion.
I believe that essentially new results contains only in Section 6 (Phonon backscattering amplitude) and apparently in Eqs. (40) and (49). In all the preceding sections the Authors actually reproduce the results already known. I think that the volume of the manuscript does not correspond to the essentially new content. Therefore, it can be significantly reduced. I also believe that the Authors should provide a more complete description of the results of
[7] and correspondingly adjust some of their statements.
Requested changes
1. I suggest to reduce the volume of the manuscript. I believe that the equivalence of the edge exponents obtained from the Bethe Ansatz solution and from variation of the dispersion relation was already proved. Therefore, there is no need to prove it again.
Author: Dimitri Gangardt on 20170426 [id 120]
(in reply to Report 1 on 20170122)
We followed the referee's suggestion and removed the misleading statement
mentioned in his/her report. We have completely rewritten the manuscript,
including its title and abstract. In its new version the manuscript does not
pretend to calculate correlations functions but focuses on mobile impurities
and their interactions with background liquid. Phenomenological parameters
entering theoretical description of the interactions were obtained in Ref. 14,
irrespectively of the edge exponents of correlation functions. Here we only
show that they coincide with chiral phase shifts used to calculate powerlaw
edge exponents.
The aim of the manuscript is not to calculate edge exponents as in the paper
by Kitanine et al. (Ref. 38) but to express the phenomenological interaction
parameters (collective charges N,J) via Bethe Ansatz solutions of certain
integrable models. Using this result in the expression 8 for the edge
exponents we indeed recover the results of Ref. 38 and we state it explicitly
(references, section and equation numbers are given as they are in the new
version).
We stress, that the goal of our work is to demonstrate the absence of
twophonon scattering and dissipation. The results of Ref. 38 cannot be used
for this and this necessitates our derivation of expressions 22,33, which
might be lengthy but we cannot think of any shortcuts for the moment. However
we think that the new structure of the manuscript with the discussion of
correlation functions delegated to Appendix is more straightforward and clear.
We hope that the introduction in the rewritten manuscript
reflects this ideas and provides enough motivation for our calculations. In
addition we give full reference to previous results and state, where
necessary, that this is not part of the original research.
Author: Dimitri Gangardt on 20170426 [id 122]
(in reply to Report 4 on 20170215)We are thankful to the Referee for reading carefully our manuscript and providing us with such a detailed guidance for its improvement. Before we turn to the specific questions raised by the referee we would like to stress that the manuscript has been completely reorganised and its main parts were substantially rewritten. The title and the abstract were modified accordingly and do not anymore suggest that we pretend to calculate known dynamical correlation functions.
Here are the main points:
The Referee is right by presenting the first result of our manuscript as a proof of equivalence between different approaches to edge exponents in onedimensional quantum liquids. We state it explicitly in the introduction of the revised version.
We agree that the inelastic processes are considered to the leading, twophonon, order. We now state it explicitly throughout the text.
We took into account the Referee's unsatisfaction with Section 2 and followed his advice to start the manuscript with Section 3. Section 2 was revised and its content became Appendix A as it is indeed not crucial for the main message of the manuscript. We feel, however, that our approach based on path integral provides a new insight on the mobile impurity model and connects to our previous work in which dynamics of mobile impurities was obtained using this technique. Our method shows that the powerlaw singularities are nothing but a semiclassical approximation to the path integral justified by the logarithmically large action of phonons and we recover all known results this way.
In the first version of the manuscript it was implicitly assumed that the velocity of the depleton in Section 2 (Appendix A in the new version) is smaller than sound velocity leading to onesided singularity only. The missing twoside singularities alluded to by the Referee do appear in our approach under a careful analysis of the time integration contour which we provide in the revised version.
We now address the Referee's remarks in more detail. References, section and equation numbers are given as they are in the new version unless stated otherwise.
Abstract
Introduction
Appendix A (Section 2 in the old version)
Section 2 (3 in the old version)
Section 3 (4 in the old version)
Section 4 (5 in the old version)
Section 5 (6 in the new version)
Section 6 (7 in the old version)
Appendices
Appendix B
Appendix C
Appendices D.1 We give full credit to Ref. 32 for the prove of equivalence of different expressions for momentum and energy of excitations.
Appendices D.2 and D.3