Dynamic correlation functions for integrable mobile impurities

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.

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.

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 Yang-Gaudin model.

More precisely, A. Kamenev and L. I. Glazman in "Dynamics of a one-dimensional 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 Yang-Gaudin model"
building on a mixture of non-linear 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 one-dimensional 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 so-called depleton model to compute the exponents in the power-law 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 non-linear luttinger model approach ( Ref[2])
and even on the basis of exact, Bethe Ansatz based calculation (Ref [7]). There is thus low interest in re-deriving 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 two-sided 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 Gaudin-Yang 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 so-called phonon backscattering amplitude vanishes for the Bose gas and the Yang-Gaudin 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 two-sided singularities.


- "relation between these parameters and the Bethe
Ansatz shift functions of elementary excitations". This is a miss-leading 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 one-body 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 Tracy-Vadiya.

- "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 ad-hoc procedure to get the result. The conclusions of the method are not correct since
one misses the existence of two-sided 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 non-linear 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 non-linear Luttinger liquid theory in [30], in what concerns the Bose gas, and predictions in [33]
by similar means for the Gaudin-Yang 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 Gaudin-Yang 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 Lieb-Liniger and Yang-Gaudin models in terms of shift functions F (ν|λ) and F̃ (ν|λ) 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 non-exhaustive 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.

see report

- Nice presentation

- Claims about novelty are exaggerated

The authors study dynamical correlation functions in one-dimensional
quantum systems. Using a semiclassical analysis, they relate the edge
exponents of power-laws in the response functions to the
phenomenological parameters of their mobile-impurity 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 liquid-impurity 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
mobile-impurity 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.

- Make claims about the existence of analytical exponents of Bethe ansatz solvable models more precise.

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.

1. The volume of the manuscript does not correspond to the essentially new content.

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 re-derive 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 Lieb-Liniger and Yang-Gaudin 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 Lieb-Liniger 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 Lieb-Liniger and Yang-Gaudin 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.

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.