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Zero temperature momentum distribution of an impurity in a polaron state of one-dimensional Fermi and Tonks-Girardeau gases

by Oleksandr Gamayun, Oleg Lychkovskiy, Mikhail B. Zvonarev

Submission summary

As Contributors: Oleksandr Gamayun · Oleg Lychkovskiy · Mikhail Zvonarev
Arxiv Link: (pdf)
Date accepted: 2020-03-19
Date submitted: 2020-03-09 01:00
Submitted by: Zvonarev, Mikhail
Submitted to: SciPost Physics
Academic field: Physics
  • Condensed Matter Physics - Theory
  • Mathematical Physics
  • Quantum Physics
Approach: Theoretical


We investigate the momentum distribution function of a single distinguishable impurity particle which formed a polaron state in a gas of either free fermions or Tonks-Girardeau bosons in one spatial dimension. We obtain a Fredholm determinant representation of the distribution function for the Bethe ansatz solvable model of an impurity-gas δ-function interaction potential at zero temperature, in both repulsive and attractive regimes. We deduce from this representation the fourth power decay at a large momentum, and a weakly divergent (quasi-condensate) peak at a finite momentum. We also demonstrate that the momentum distribution function in the limiting case of infinitely strong interaction can be expressed through a correlation function of the one-dimensional impenetrable anyons.

Ontology / Topics

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Bethe Ansatz Fermi gases Tonks-Girardeau gas

Published as SciPost Phys. 8, 053 (2020)

Author comments upon resubmission

Response to Referee 1:

  1. Indeed, apart from the title and the abstract, we have mentioned the Tonks-Girardeau bosons only in the conclusions, without much explanation. Now, we are mentioning the Tonks-Girardeau bosons in the beginning of section 2, and give a reference to the section 2 of the work [25].

  2. Typos corrected

Response to Referee 2:

  1. We understand the logic of the proposed change. However, our point is that sections 3 to 7 are a natural continuation of the section 2 in that they altogether contain the result: the analytic formulas used to describe the momentum distribution function of the impurity. Hence, the reader who only wants the result about the momentum distribution may omit reading the last section 8. The above said, we would like to keep the order of the sections of the manuscript unchanged.

  2. We have explained vanishing of the integrals in Eq. (158) in more detail. Furthermore, section 8 contains now only formulas valid for any N. All calculations relevant for the large N limit are put into the appendix A. The title of this appendix has been changed accordingly.

Response to Referee 3:

We have added a paragraph to the "Conclusion" section discussing in brief the experimental findings in ultracold atomic gases related to the observable studied in our manuscript.

List of changes

Restored overall phase factor in Eq. (66), accidentally erased in the course of the editing of the manuscript. This does not affect any other equation or figure in the manuscript. Some notations are changed around Eqs. (66)-(71), and through the appendix A. Title change for the appendix A.

References [36] and [61] added

Beginning of the section 2 contains now an explanation of why our results are valid for the Tonks-Girardeau gas

Conclusion section contains now a paragraph discussing an experiment in ultracold atomic gases

An acknowledgement to Referees is added to the Acknowledgements section

Various minor stylistic/language changes, typos corrected

Reports on this Submission

Anonymous Report 4 on 2020-3-17 (Invited Report)






The authors improved the presentation of the section 8. As the suggestion to change the order of sections was optional, I recommend this new version for publication

  • validity: -
  • significance: -
  • originality: -
  • clarity: -
  • formatting: -
  • grammar: -

Anonymous Report 3 on 2020-3-10 (Invited Report)


After the revision, which incorporated the suggestions of all three referees,
I feel that the paper has improved significantly and, therefore, I warmly recommend its publication in SciPost Physics.

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

Anonymous Report 1 on 2020-3-10 (Invited Report)


I recommend that the revised version of this manuscript should be published in its present form.

  • validity: -
  • significance: -
  • originality: -
  • clarity: -
  • formatting: -
  • grammar: -

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