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

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The present paper discusses the applicability of a renormalization group approach to the polaron and bi-polaron problem in weakly interacting one-dimensional Bose gases. The method developed in nuclear physics has recently been successfully extended by some of the authors to weakly interacting Bose gases and to calculate the self energy of the Bose-polaron, a single impurity embedded in the gas. In the present paper the authors develop the approach further and discuss extensions to observables other than the energy as well as to more involved problems, in particular the bi-polaron problem, i.e. the interaction of two impurities mediated by the Bose gas. The paper has two major messages. The first is the discussion of a specific problem, the polaron and bi-polaron in a 1D Bose gas. The second is of methodological nature and concerns the applicability and limitation of the flow-equation approach. While with respect to the first aspect the paper is well written and contains interesting results that deserve publication, I do have some problems with the second. Here considerable revision is needed, both in terms of presentation and in terms of substantiating claims.

Concerning the physics part of the paper

The bi-polaron problem in one dimensional Bose gases has recently regained considerable attention and the authors calculate the interaction energy and interaction potential, as well as the density and phase fluctuations of the quasi-condensate around two impurities. The impurity positions are fixed, i.e. the Born-Oppenheimer limit is considered. The authors find sizable deviations from previous perturbative results when the impurity-boson interaction becomes strong. They also discuss the system-size dependence of observables and point out their relevance for the analysis of cold gas experiments. The results obtained with the flow equation approach agree very well with those obtained from a mean field approximation.

Some comments:

(1) The flow equation results are used to benchmark the mean-field results. There is however no proof or at least some convincing arguments why the flow equation approach should be superior to the mean-field approximation. The authors compare the flow-equation results along with mean-field results with exact data obtained from Bethe ansatz in the integrable case. It would be much more convincing if a parameter case could be found, where there is a sizable difference between flow equations and mean field. Perhaps the author can comment on this.

(2) The authors have calculated the impurity-impurity interaction potential. Ref.[23] predicts an exponential behavior at short distances and a power-law at large distances. The large-distance scaling is due to Casimir-like forces induced by phonon exchange and cannot be captured in the mean field approach. It would be interesting to see if the flow equation approach follows the mean field results or is actually able to reproduce the Casimir forces. Furthermore other recent work in Ref.[76] seems to suggest a linear interaction potential at short distances. Fig. 9 seems to show something different. Can the authors comment on this?

(3) What is the reason that the case of attractive impurity-boson interaction is not included. While one could expect that the mean-field predictions are less accurate here, in contrast to the statement just before eq.(29), the mean-field solutions can be obtained similarly to the repulsive case and it would be interesting to see if the flow equations predict different results than the mean-field approach.

(4) In the extrapolation of the bi-polaron energies to the thermodynamic limit obtained either from flow-equations or mean-field in Fig. 9c and d it appears that the flow equations predict an oscillatory correction on top of the mean field result. This should be discussed.

Concerning the methodological aspect of the paper

The authors use the example of a bose polaron and bi-polaron to illustrate the power of their renormalization approach to weakly interacting Bose gases and use the results as benchmarks for a mean-field theory. While it is certainly very desirable to have methods at hand that allow to tackle these type of problems beyond the mean field level, the authors have not convinced me that this is actually the case for their flow-equation approach. As stated in point (2), the paper would substantially gain if at least some arguments could be provided that the flow equations capture effects beyond the mean-field level. This could be achieved e.g. by comparing to exact solutions, e.g. from Bethe ansatz or from Luttinger liquid theory (see e.g. Ref.[29]), or to QMC simulations, e.g. from Ref.[14] for larger values of the Lieb-Liniger parameter of the Bose gas.

Some comments on the presentation:

The introduction of the renormalization scheme in Sec. 2 is not self contained and thus incomprehensible for a reader unfamiliar with the renormalization scheme. Here the presentation should be improved.

(5) It would be beneficial to explain in a few words what the goal of the unitary transformations governed by the flow equation (1) is. What is meant if the authors say: "to choose the operatore eta(s) .... to steer the flow in the desired direction."? What desired direction?

(6) Right after eq.(2) the reference state is mentioned without any introduction. The latter is only given in the following subsection.

(7) Normal ordering with respect to the reference state is mentioned but only defined in the Appendix.

(8) The flow equation results show error bars from "relative truncation error". Reference to Appendix A.4 should be made here to explain how the truncation error was obtained.

In summary the manuscript contains interesting and novel results warranting in principle publication but needs revision.

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1 - the problem is interesting in relation to current experiments with polarons in ultracold atoms
2 - the proposed method allows calculation of the spatial properties
3 - a good agreement is found to the known results which serves as a verification of the method

1 - I find that the Authors did not pay enough attention to make the Manuscript clearly written. The quality of the text and figures can be improved.

Authors study the problem of one or two impurities in a one-dimensional Bose gas. It is proposed to use the renormalization group method in real space for the calculation of the density profile and induced interactions. The validity of the method is demonstrated in comparison with known results. The accuracy of perturbation theories is analyzed, both for repulsive and attractive impurities. The obtained results are relevant to the present experiments with ultracold dilute gases.

The considered problem is important in the context of recent experiments with polarons. The Authors contribute to the progress in the field and the proposed method can be applied to different systems. I find that the obtained results deserve an eventual publication in Sci. Post. journal. But before that a number of issues have to be settled down.

Major comments
1) The manuscript contains a large number of figures and many of them (Figs 1, 2, 6, etc) are not readable in black and white version. I suggest the Authors modify the figures and check if they are easy to read in a printed version.
2) I think that notation f_1 and f_2 is not as obvious as it would be with f_{2b} and f_{GP} denoting two-body and Gross-Pitaevskii solutions used as input for generating the correlations.
3) The ratios “g/ρ” and “c/ρ” which are used to quantify the interaction strength are used as dimensionless parameters, while this is not the case. Here “g” has units of the coupling constant and “ρ” of the density.
4) “One can show that the pair correlation function of the Lieb-Liniger model, g2, is identical to the density of the bosons…” I disagree, for example, for c= ∞ and γ=∞, the pair correlation function g2 = 1 - sin(πρz) ²/ (πρz) ² has oscillations which are always below the asymptotic density while the density of bosons has oscillations which exceed the bulk density.
5) “In our studies, we have noticed that the reference state f2 allows us to investigate a larger range of parameters…” Please specify what exactly is meant by “allowing to investigate”
6) Fig 4c, there is a non-monotonic dependence that looks rather spurious. Is it possible to reduce the errorbars ?
7) Discussion below Fig. 5. “For attractive interactions, the agreement is only quantitative for large values of |c|”, this statement contradicts what is shown in Fig. 5. The agreement is not perfect, but reasonable for repulsive interactions. Exactly the same can be said about the attractive case. In order to keep the claim, I guess, one has to go to a stronger attraction
8) Please comment on the values of C3 in the fit to the energy. Is that a linear or quadratic behavior?
As an overall observation, the text is ambiguous in a number of places, is often repetitive “results agree with … results”

Minor comments:
9) Abstract: “observables other than the energy. As an example, we calculate the energies”. Either remove “the energies” or rephrase.
10) Abstract “for two static impurities, we calculate .. correlations”. “Correlation functions” would sound more appropriate, in any case, expand which correlation functions are actually calculated.
11) Abstract does not contain references to renormalization group methods, this has to be stated explicitly
12) Introduction “when solving a Bose-polaron problem on is interested only in local properties …”. And if I am still interested in non-local properties, then what?
13) “The IM-SRG was previously applied to study Bose gases with large condensate fraction”. Is that a 3D system? As the article talks about 1D gas, this should be specified.
14) Approximations are implemented -> assumed
15) I am not sure that it is important to say that Python was used to perform the simulations, the text was written in Latex, etc. For me, mentioning that Runge-Kutta method was used is sufficient.
16) “function which, we discuss” I am not sure you can discuss a function, please rephrase.
17) “the Lieb-Liniger parameter γ and the healing length ξ = 1/ (γ^{1/2} ρ)”. The provided expression applies only for small values of γ, this has to be specified.
18) “The function g2 is known for mesoscopic systems” rephrase
19) “We estimate that ξρ = 7.1”, not clear what “estimate” means here, xi is known exactly
20) “The condensate fraction is a global quantity.” Please rephrase, I am not sure I understand what exactly is meant and why this sentence is here.
21) “we do not present phase fluctuations”, rephrase
22) Figure 5. I suggest putting “c<0” and “c>0” labels inside of the figures
23) Figure 7. I suggest putting “c_2 = …” labels inside of the figures
24) Figure 7, it seems that the errorbars are overestimated.
25) Sections 4.2 and 4.4 “Approach to the thermodynamic limit”. The title is ambiguous. What is “approach” here? Does it refer to a method that describes the thermodynamic or rather to approaching the thermodynamic limit?
26) “We compare the potential … with calculations”, rephrase
27) Figure 9, use the same vertical axis limits in (a) and (c), and (b) and (d). Add “c_2=…” to the legend
28) “each boson is attached to the impurity”. Please explain in more detail.
29)“to understand the induced interactions”, rephrase
30) “The full mean-field solution seems to be cumbersome”. Please specify what is actually meant by that.

I suggest to carefully read the Manuscript, checking what is actually written and if this is what was really intended. The ill-formed phrases and ambiguities do not help to an easier comprehension. As well, it might worth it to give it to read to someone else.