# Mobile impurities in integrable models

### Submission summary

 As Contributors: Dimitri Gangardt Arxiv Link: http://arxiv.org/abs/1701.00810v2 (pdf) Date submitted: 2017-04-26 02:00 Submitted by: Gangardt, Dimitri Submitted to: SciPost Physics Academic field: Physics Specialties: Atomic, Molecular and Optical Physics - Theory Mathematical Physics Quantum Physics Approach: Theoretical

### Abstract

We use a mobile impurity or depleton model to study elementary excitations in one-dimensional integrable systems. For Lieb-Liniger and Yang-Gaudin models we express two phenomenological parameters characterising renormalised inter- actions of mobile impurities with superfluid background: the number of depleted particles, $N$ and the superfluid phase drop $\pi J$ in terms of the corresponding Bethe Ansatz solution and demonstrate, in the leading order, the absence of two-phonon scattering resulting in vanishing rates of inelastic processes such as viscosity experienced by the mobile impurities

### Ontology / Topics

See full Ontology or Topics database.

###### Current status:
Has been resubmitted

We have changed the overall scope of the paper, which is now focused on
calculations of effective interactions parameters (collective charges) of a
mobile impurity (depleton) model. We mention their relation to the edge
exponents in various correlation functions, but Section 2 in the previous
version, containing extended derivation of the correlation functions'
asymptotics, is moved now to Appendix A.

We now clearly state which results were obtained previously and give
appropriate credit. We still feel, however, that for the sake of completeness
and reader's convenience, these results should be included in the manuscript
and we state this in the text.

### List of changes

* Title and abstract changed

* Introduction rewritten

* Section 2 moved to Appendix A

* Credit to previously obtained results is given throughout the paper.

### Submission & Refereeing History

Resubmission 1701.00810v3 on 16 June 2017

Resubmission 1701.00810v2 on 26 April 2017
Submission 1701.00810v1 on 5 January 2017

## Reports on this Submission

### Anonymous Report 4 on 2017-5-31 (Invited Report)

• Cite as: Anonymous, Report on arXiv:1701.00810v2, delivered 2017-05-31, doi: 10.21468/SciPost.Report.153

### Strengths

As in the first report

### Weaknesses

Some citations are strange.

### Report

The authors did implement many of the referees' suggestions what has improved the paper. However, they also did made some quite queer amendments, especially in what concerns the bibliography and the citations.

In the present version, there is no mention -with the exception of reference 37- of all the efforts that where made within the integrable model community regarding to the calculation of correlation functions and extractions of the edge exponents, at least in some regimes.
The authors already did provide some citations in the first version of the paper and I have suggested some other literature, so as to provided a fairer picture. Since, in fine, the author's work relies directly on integrable techniques, I believe that making such an omission is unfair.

The authors inaccurately describe the content of certain papers. To be more precise:

In the middle of page 3, the authors discuss the goal of their paper. There they claim that their result "reproduces the conjectured identity of chiral linear combinations of the collective charges, so called chiral phase shifts, with BA shift functions".
This is not true. Paper 37 expresses the edge exponents in terms of shift functions by arguing directly that phase shifts in the unitary transform should be given by the same formula as for non-interacting fermions, and this is enough for their calculation.
The works 22,36 identify the coupling constants in their effective Hamiltonian by comparing their low-energy spectrum with the one issuing from the Bethe Ansatz. By the way, the authors do not prove the predictions issuing, per se, from the work 36 since it deals with the XXZ chain which is not considered in the author's paper.

I stress that, all these works directly provide an expression for the edge exponents in terms of the shift function. Thus, 37 and 38 provide the same expression for the edge exponents.

What the authors do is to prove the expressions for the phase shifts that were argued to hold in ref. 23 and in "Phenomenology of One-Dimensional Quantum Liquids Beyond the Low-Energy Limit" by A. Imambekov and L.I.Glazman. Curiously, that last paper is not cited in the present version of the paper while it was cited in the first one.

Page 4 The authors attribute to 14 the proposal of the relations (3)-(4). However that paper dates to 2012 while "Phenomenology of One-Dimensional Quantum Liquids Beyond the Low-Energy Limit" appeared in 2009, just as reference 23. Again, true, the expressions obtained there are written in another language but I believe that the correspondence between the two is not that hard.

Below (60). The identity was conjectured to hold much much earlier than [52]. However, the work [52] was the first to establish it.
See, e.g. "Conformal dimensions in Bethe ansatz solvable models" in what concerns the nested Bethe ansatz case.

### Requested changes

see report

• validity: ok
• significance: ok
• originality: ok
• clarity: ok
• formatting: acceptable
• grammar: excellent

### Author:  Dimitri Gangardt  on 2017-06-10  [id 142]

(in reply to Report 4 on 2017-05-31)

Response to Referee 153

(1) In the present version, there is no mention -with the exception of
reference 37- of all the efforts that where made within the integrable model
community regarding to the calculation of correlation functions and
extractions of the edge exponents, at least in some regimes. The authors
already did provide some citations in the first version of the paper and I
have suggested some other literature, so as to provided a fairer
picture. Since, in fine, the author's work relies directly on integrable
techniques, I believe that making such an omission is unfair.

Our response

In the newer version of the manuscript we tried to get the main message across
and skip unnecessary discussion irrelevant to our main results: the relation
of phenomenological parameters N, J and Bethe Ansatz shift fuctions. This was
done following Referees' suggestion to streamline the
discussion. Unfortunately, this resulted in vanishing of many citations from
the exactly integrable community not directly relevant for this manuscript.

(2) In the middle of page 3, the authors discuss the goal of their
paper. There they claim that their result "reproduces the conjectured identity
of chiral linear combinations of the collective charges, so called chiral
phase shifts, with BA shift functions". This is not true. Paper 37 expresses
the edge exponents in terms of shift functions by arguing directly that phase
shifts in the unitary transform should be given by the same formula as for
non-interacting fermions, and this is enough for their calculation. The works
22,36 identify the coupling constants in their effective Hamiltonian by
comparing their low-energy spectrum with the one issuing from the Bethe
Ansatz.

Our response

The argument of paper by Imambekov and Glazman assumes implicitely that
parameters of the effective mobile impurity are given in terms of BA shift
functions. It is indeed done by evoking free fermions and their effective
phase shifts, but no mathematical proof is given. Papers by Cheianov and
Pustilnik as well as Zvonarev, Cheianov, Giamarchi do not provide proof
either. The 2012 review [1] states that the equivalence between chiral phase
shifts given in terms of the impurity dispersion and BA shift functions
(dressed phases) is an open problem. Referee is right by saying that the form
(6) of the chiral phase shifts first appeared in in the publication
"Phenomenology of One-Dimensional Quantum Liquids Beyond the Low-Energy Limit"
by A. Imambekov and L.I.Glazman, but as it is explained in this paper
parameters of the effective mobile impurity model used in all the cited
references can be obtained from the impurity dispersion. Here we demonstrate
the missing relation of the parameters (in the form of N, J) of effective
impurities to the BA shift functions. In a sense this is nothing but the
"comparing their low-energy spectrum with the one issuing from the Bethe
Ansatz".

We are grateful to Referee for pointing out the omission of the above paper by
Imambekov and Glazman, which is now Ref. 32 in the new version of the
manuscript.

(4) Page 4 The authors attribute to 14 the proposal of the relations
(3)-(4). However that paper dates to 2012 while "Phenomenology of
One-Dimensional Quantum Liquids Beyond the Low-Energy Limit" appeared in 2009,
just as reference 23. Again, true, the expressions obtained there are written
in another language but I believe that the correspondence between the two is
not that hard.

Our response

By no means we claim that Ref [14] (co-authored by one of us)
was the first to establish relations between effective impurity parameters
and its dispersion. Indeed, the relations (3)-(4) are equivalent to Eq. (6)
under linear transformation (7) from N,J to the chiral phases as we point out
in our work citing the appropriate references. However, the depleton model,
(1), (2) was formulated in Ref. [14] in terms of collective charges N,J using
a different approach and we prefer to start with (3)-(4) rather than
deriving them from (6) and (7).

(5) Below (60). The identity was conjectured to hold much much earlier than
[52]. However, the work [52] was the first to establish it.

Our response

We are grateful to Referee for pointing it out and have modified the sentence
around Eq. (60) correspondingly.

### Anonymous Report 3 on 2017-5-25 (Invited Report)

• Cite as: Anonymous, Report on arXiv:1701.00810v2, delivered 2017-05-25, doi: 10.21468/SciPost.Report.149

-

-

### Report

In their work, Campbell and Gangardt re-derive some important results of the theory of Nonlinear Luttinger liquids. Most notably, they re-affirm the absence of viscosity for excitations describing the edge of the spectral continuum in two specific integrable models: the Lieb-Liniger model and the Yang-Gaudin model for BOSONS with spin 1/2.

While there is little new physics in their work, I believe it is methodically useful and should be published in SciPost. The previous Referees already fought heavily for assigning the proper credit to various groups, so I spare the Authors from some minor grievances.

### Requested changes

I have two specific requests to the Authors:

(1) In the Introduction (p. 3), please substantiate of modify the phrase: "The equivalence
of these results to those obtained in Ref. [37] using phenomenological mobile impurity model
relies on the conjectured relation between the collective charges and BA shift functions which
we demonstrate here." It is not clear how the notion of "collective charges" was involved in demonstrating the equivalence.

(2) It seems to me that the Authors consider only the bosonic Yang-Gaudin model. If this is the case, that must be clearly stated in the beginning of the manuscript and repeated in the main text in the sections devoted to that model.

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

### Author:  Dimitri Gangardt  on 2017-06-10  [id 141]

(in reply to Report 3 on 2017-05-25)

(1) In the Introduction (p. 3), please substantiate of modify the phrase:
"The equivalence of these results to those obtained in Ref. [37] using
phenomenological mobile impurity model relies on the conjectured relation
between the collective charges and BA shift functions which we demonstrate
here." It is not clear how the notion of "collective charges" was involved in
demonstrating the equivalence.

Our response:

We have removed this phrase and slightly modified the preceding sentence.

(2) It seems to me that the Authors consider only the bosonic Yang-Gaudin
model. If this is the case, that must be clearly stated in the beginning of
the manuscript and repeated in the main text in the sections devoted to
that model.

Our response:

The referee is absolutely right and we have replaced Yang-Gaudin model by
bosonic Yang-Gaudin model throughout the manuscript.

### Anonymous Report 2 on 2017-5-23 (Invited Report)

• Cite as: Anonymous, Report on arXiv:1701.00810v2, delivered 2017-05-23, doi: 10.21468/SciPost.Report.146

### Strengths

Nicely written paper containing important results.

### Weaknesses

No obvious weaknesses.

### Report

The authors have responded adequately to the questions raised by the referees in the first round. I recommend the paper for publication.

### Requested changes

No changes required.

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

### Anonymous Report 1 on 2017-4-27 (Invited Report)

• Cite as: Anonymous, Report on arXiv:1701.00810v2, delivered 2017-04-27, doi: 10.21468/SciPost.Report.123

### Strengths

1. The paper contains new important results.

### Weaknesses

1. I do not see evident weaknesses.

### Report

I appreciate the changes made by the authors and recommend the manuscript for publication.

### Requested changes

1. No changes required.

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