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Impurities in a one-dimensional Bose gas: the flow equation approach

by F. Brauneis, H.-W. Hammer, M. Lemeshko, and A. G. Volosniev

Submission summary

As Contributors: Artem Volosniev
Arxiv Link: https://arxiv.org/abs/2101.10958v4 (pdf)
Date accepted: 2021-06-29
Date submitted: 2021-06-23 10:24
Submitted by: Volosniev, Artem
Submitted to: SciPost Physics
Academic field: Physics
Specialties:
  • Atomic, Molecular and Optical Physics - Theory
  • Condensed Matter Physics - Computational
Approach: Theoretical

Abstract

A few years ago, flow equations were introduced as a technique for calculating the ground-state energies of cold Bose gases with and without impurities. In this paper, we extend this approach to compute observables other than the energy. As an example, we calculate the densities, and phase fluctuations of one-dimensional Bose gases with one and two impurities. For a single mobile impurity, we use flow equations to validate the mean-field results obtained upon the Lee-Low-Pines transformation. We show that the mean-field approximation is accurate for all values of the boson-impurity interaction strength as long as the phase coherence length is much larger than the healing length of the condensate. For two static impurities, we calculate impurity-impurity interactions induced by the Bose gas. We find that leading order perturbation theory fails when boson-impurity interactions are stronger than boson-boson interactions. The mean-field approximation reproduces the flow equation results for all values of the boson-impurity interaction strength as long as boson-boson interactions are weak.

Published as SciPost Phys. 11, 008 (2021)



Author comments upon resubmission

Dear Editor,

thank you for communicating to us the editorial decision on our manuscript on June 14th.

We have revised the manuscript in accord with the second reports of the Referees.
We hope that the revised version of our manuscript is ready for publication in SciPost Physics.

Best wishes,

Artem Volosniev, on behalf of the authors

List of changes

We thank the Referees for reviewing our work. The revised version of the manuscript addresses all comments of the Referees.
For convenience of the reader, we highlight major changes in the manuscript. In addition, we provide a list of changes below.

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List of Changes in Response to Anonymous Report 3

1. The Referee "Oscillations in the impurity-impurity potential in Fig.9b: The authors state at the end of page 19 that (i) they expect a monotonous increase and (ii) that it is difficult for them to estimate the error bars for the curve in the thdyn. limit. All curves in Fig. 9a and 9b are shown with error bars, except the curves in the thdn. limit. On first glance the crosses suggest however that there are error bars plotted as well. To avoid confusion I suggest to explicitly mention in the figure caption that no error bars are given here since their estimation is difficult."

Our reply: We thank the Referee for this remark. The revised captions of Figs. 9 and 12 state explicitly that we do not present error bars for the curves in the thermodynamic limit. To avoid any confusion, we have also used another symbol to present our data in the thermodynamic limit.

2. The Referee: "The arguments given by the authors in the text and in Appendix B for why attractive impurity-boson interactions are not treated in detail and referred to a future publication are not convincing. The comparision of mean-field contact parameter in Fig.5a for c <0 does not appear much worse than in Fig.5b for c>0. The same applies to Fig.15 a-d. The most convincing argument for me is the plot of the phase fluctuations in Fig. 15e. I suggest to refer to this plot when arguing that the attractive case requires more careful analysis."

Our reply: To address this comment of the Referee, we have modified the discussion in Sec. 3.4. In particular we added a sentence: "Note in particular Fig. 15, which indicates large phase fluctuations for moderate impurity-boson interactions, in contrast to the repulsive case."

3. The Referee: "The definition of normal ordering with respect to the reference state, now explained a bit more in detail in Appendix A.1, is simple enough and yet of sufficient importance for a non-specialist reader to understand the idea of the approach, that it should be put into th main text of the paper."

Our reply: To address this comment of the Referee, we moved our discussion on normal ordering to the main text, see Sec. 2.2.

4. The Referee: "In the introduction the authors say at the beginning of page 3 that it is of particular interest to compare their flow equation approach with Wilson-type RG techniques and that the IM-SRG complements that technique. I did not quite understand this comment. It seems that the Wilson-type RG approach gives results which deviate substantially already from the mean-field result of the authors, see paragraph before Sec. 3.3."

Our reply: We thank the Referee for pointing out this potentially confusing statement. The Wilson-type RG technique is formulated in momentum space. Our work is formulated in real space. The difference between our result and the Wilson-type RG indicates that the Bose polaron problem can be studied more accurately in the real-space formulation. To avoid any confusion, we moved this discussion from the introduction to footnote 6.

5. The Referee: "Finally it would be helpful to add to the legend (not the figure caption) in Fig.3 that the reference state in the left figure is f_GP and in the right figure it is f_1b."

Our reply: We have modified the figures accordingly.

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List of changes in Response to Anonymous Report 1

The Referee: "As a final remark, I strongly suggest using the full units in the figures and equations. Although the choice hbar=1 and m=1 is often used, it does not allow to use the criteria of proper units for checking the correctness of a certain expression."

Our reply: We thank the Referee for this suggestion. To address this comment, we have introduced proper dimensionless quantities at the beginning of Sec. 3, and have modified all equations and figures accordingly.

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