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Long time thermal asymptotics of nonlinear Luttinger liquid from inverse scattering

by Tom Price

Submission summary

As Contributors: Tom Price
Arxiv Link:
Date submitted: 2017-08-08
Submitted by: Price, Tom
Submitted to: SciPost Physics
Domain(s): Theoretical
Subject area: Condensed Matter Physics - Theory


I derive a Fredholm determinant for the thermal correlator of vertex operators with a conformally invariant density matrix and nonlinear time evolution. Using the method of nonlinear steepest descent I find the long time asymptotics at finite temperature. The asymptotics display exponentially small corrections in temperature to the finite temperature mobile impurity phenomenology.

Current status:

Editor-in-charge assigned, manuscript under review

Invited Reports on this Submission

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Anonymous Report 2 on 2017-10-15


1- New determinant representations for dynamical thermal correlators relevant to nonlinear Luttinger liquid physics
2- Asymptotic analysis


1- Calculations and logic not so easy to follow
2- Result limited to a simple form for the density matrix


In this manuscript the author considers dynamical thermal correlators in a non linear Luttinger liquid model. The main result is an exact Fredholm determinant formula for correlations of vertex operators in a conformally density matrix, generalizing previous results at zero temperature. This is done using the fact that a certain ratio of such correlators satisfies a nonlinear Schroedinger equation. The large time behavior is then obtained by mapping onto a Riemann-Hilbert problem, which can be solved asymptotically at both zero and finite temperature.

The paper is overall interesting, and some of the results are potentially quite nice. Several aspects can (and should) be improved however, especially regarding the writing of the manuscript. As it is several statements are cryptic, and should be supplemented by calculations. The logic behind some of the derivations could be better explained also, it takes the reader quite some effort to determine whether a given sentence is a side comment on the previous equation, a proof, or a general statement that will be useful next.

In section 3, the author should really make an effort to make the derivations easier to reproduce, especially bearing in mind that asymptotic analyses to Riemann-Hilbert problems are sometimes difficult to follow. Some more physical discussions of the final results would be desirable also.

The paper might be publishable, provided these concerns are addressed. Below is a list of more minor comments, some of which are still related to my points above.

1) The introduction is too short and narrow as it is. The author should spend more time introducing the problem, what motivates it, while discussing the literature.
2) Page 2, after equation (3), "The bosonization identity [...]". Add a reference.
3) Page 3. "It is easy to include them to any result [...]". How so?
4) Page 3. What is $m$ in equation (5)?
5) Page 4. "To this end it's" --> "To this end it is"
6) Page 5, around equation (5). Mention that $\sigma_3$ is a Pauli matrix.
7) Page 5, before equation (16). Write $\theta(k)$ instead of $\theta$ to avoid any ambiguity in the following.
8) Page 5, after equation (17). "The proof is to observe [...]". The proof of what?
9) Page 6. I do not understand what is meant exactly at the end of the first paragraph.
10) Page 6, "tends to the identity at infinity". When $z$ goes to infinity would be slightly more precise.
11) Page 7, section 3.2. The discussion at the beginning of the section could be made clearer I think. Also, some of the sentences in this paragraph read awkwardly.
12) Page 9, equation (33). The author should explain more precisely what are the contours i, II, III, IV, V and VI. Same comment in the caption to figure 1.
13) Page 11, equations (39), (40), (41). These results deserve a proper discussion, not only in the conclusion, especially given the fact that everything appears to be dominated by the zero temperature Fermi point even at finite temperature.
14) Appendix A. A few lines of introduction, explaining what is being computed in this appendix, would help. The sentence "As always we consider $x$ having a positive imaginary infinitesimal" reads awkwardly.

Requested changes

Improve the writing and the derivations, in particular in section 3.
Address my other minor comments.

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

Anonymous Report 1 on 2017-9-26


The paper contains interesting results.


The text is written very succinctly, therefore, it is difficult to follow the author's calculations.


The manuscript deals with large space-time asymptotic behavior of thermal correlation functions of vertex operators. The author obtains Fredholm determinant representations for these correlations and applies the Riemann--Hilbert approach to evaluating their asymptotics.

The manuscript contains important and interesting results provided they are correct. However, the text is written so succinctly that it is difficult for the reader to follow the author's calculations. This remark especially concerns Section 3 on the RHP asymptotic analysis. In particular, I did not find an answer to the question of why the deformation of the jump contour is the same for the cases of zero and finite temperatures. This situation is very atypical. As a rule, these deformations are very different, which leads to different asymptotic behavior of the solution to the RHP. I would like to have an explanation of this atypical effect. Besides, I would like to understand why a vicinity of the Fermi point contributes to the asymptotics at finite temperature. This is quite natural in the case of zero temperature,
because the Fermi--Dirac distributions turn into the step-functions, leading to the non-analyticity in the jump-matrix in the Fermi point. However, for finite temperature, the jump-matrix is analytic in this point, thus, it is not obvious that it gives a contribution to the asymptotics of the solution to the RHP. Finally, turning back to the case of zero temperature, I should notice that strictly speaking, one has to describe the behavior of the RHP solution in the points of non-analyticity, otherwise the solution to the RHP is not unique. If this is not the case for the given RHP, it would be desirable to have some explanation.

I would like to have answers to the formulated questions before making a final decision on the manuscript. Concluding, I want to make few remarks concerning the notation and the citations.

I think that the author should give definitions of ALL the symbols used in equations, even if the meaning of some of them (like m, T, \beta) is intuitively clear for readers. Moreover, m's in Eq. (5) and Eq. (21) have different meaning.

To the best of my knowledge, Ref. [13] has nothing to do with the mobile impurity method, and it should rather be attributed to the QISM.

Ref. [16] deals with the low temperature limit, while there exists a paper arXiv:1011.3149 by the same authors where the case of a finite temperature is considered. Ref. [20] also should be attributed to the list of citations on the QUANTUM inverse scattering. Thus, the number of results obtained in this field becomes rather big, therefore, the first part of the sentence on line 9 of Introduction ( ‘...even for integrable systems the situation is rather less clear cut’) contradicts to the second part (‘ in spite of remarkable progress from quantum inverse scattering’).

Requested changes

1. To answer the questions formulated in the report.
2. To define all the symbols used in equations.
3. To make corrections in the citation list.

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

Author Tom Price on 2017-09-30

(in reply to Report 1 on 2017-09-26)

I thank the referee for their careful reading of the paper, comments on the literature, and thoughtful suggestions and questions. Here I answer their questions.

Deformation of the Riemann--Hilbert problem for arbitrary T occurs via the piecewise analytic transformation Eqn~(33), with regions of the complex plane labeled in Fig. 1. This includes zero temperature as a special, simpler case that is also treated separately in Section 3.2. The reflection coefficients $R, \bar{R}$ contain Fermi-Dirac factors and therefore have poles along the imaginary axis at nonzero T that merge into a cut when T=0. For $T < vk_*$ the slotted contour cannot be closed over the imaginary axis without encountering these poles, or the cut at T=0. This justifies the contributions from the Fermi point to the asymptotics even when T is nonzero, as is expected on physical grounds in finite temperature extensions of the mobile impurity model. Personally I find it encouraging that the asymptotics at low $T \ll uk_*$ approach those of zero temperature and would be interested to know of situations where this is not the case.

I chose to separate the zero and nonzero temperature calculations because at zero temperature the only RHP that needs to be solved, Eqns 26--27, is elementary. At nonzero temperatures the model RHP is not triangular and calculation is extremely lengthy, although a standard in the literature as it is just the asymptotics of the nonlinear Schrodinger equation when no nonanalyticities are encountered in the reflection coefficients. Consequently I only give the result we need in Eqn. 35 and refer the reader to Ref.~30 for details.

Regarding the comment "the case of zero temperature, I should notice that strictly speaking, one has to describe the behavior of the RHP solution in the points of non-analyticity, otherwise the solution to the RHP is not unique. If this is not the case for the given RHP, it would be desirable to have some explanation." The behaviour of the RHP near the points of non-analyticity for zero temperature are given in Eqns. 25, 27 and 29.

I thank the referee again for their comments and hope they find my reply helpful.

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