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Impurities in a onedimensional Bose gas: the flow equation approach
by F. Brauneis, H.W. Hammer, M. Lemeshko, and A. G. Volosniev
This is not the current version.
Submission summary
As Contributors:  Artem Volosniev 
Arxiv Link:  https://arxiv.org/abs/2101.10958v3 (pdf) 
Date submitted:  20210528 13:57 
Submitted by:  Volosniev, Artem 
Submitted to:  SciPost Physics 
Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
A few years ago, flow equations were introduced as a technique for calculating the groundstate 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 onedimensional Bose gases with one and two impurities. For a single mobile impurity, we use flow equations to validate the meanfield results obtained upon the LeeLowPines transformation. We show that the meanfield approximation is accurate for all values of the bosonimpurity 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 impurityimpurity interactions induced by the Bose gas. We find that leading order perturbation theory fails when bosonimpurity interactions are stronger than bosonboson interactions. The meanfield approximation reproduces the flow equation results for all values of the bosonimpurity interaction strength as long as bosonboson interactions are weak.
Current status:
Author comments upon resubmission
We thank the Referees for taking the time to review our paper. We are happy to see the overall positive evaluation of our paper. The comments in the reports helped us to significantly improve our manuscript. We hope that the revised version is ready for publication in SciPost Physics.
Below, we provide a pointbypoint reply to the comments in the reports. For convenience, we quote the comments of the Referees. The main changes in the manuscript are shown in blue. Minor changes are not highlighted in the manuscript.
Reply to Anonymous Report 1:
Major comments:
Referee: "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."
Our reply: We changed all figures which were not readable in black and white. We modified the captions accordingly.
Referee: "I think that notation f_1 and f_2 is not as obvious as it would be with f_{2b} and f_{GP} denoting twobody and GrossPitaevskii solutions used as input for generating the correlations."
Our reply: We thank the Referee for this useful suggestion. We have modified the text and figures accordingly. Note that we chose to use f_{1b} instead of the suggested f_{2b} to emphasize that the reference state is the solution of a oneboson Hamiltonian.
Referee: "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."
 Our reply: Please note that we are using the system of units in which hbar=M=1. This implies that g/rho and c/rho are dimensionless.
We added footnote 4 to clarify this point:
"In general, the dimensionless LiebLiniger parameter is defined as gamma=Mg/(hbar^2rho), which leads to gamma=g/rho in our units (hbar=M=1)."
Referee: "“One can show that the pair correlation function of the LiebLiniger 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."
Our reply: We thank the Referee for alerting us of this potentially confusing statement. Please note that the density of bosons is measured in the frame comoving with the impurity. We have modified the text to stress this point and to clarify the relation between pair correlation function and density in the comoving frame, see the discussion around Eqs. (1719) of the revised manuscript. We have also identified a typo in our formula. The revised text presents the correct expression.
Referee: "“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”"
Our reply: We thank the Referee for identifying this imprecise statement. To clarify it, we have rewritten the discussion, which now reads as:
“In our studies, we noticed that the reference state f_{GP} is generally a better choice than f_{1b}. In comparison to IMSRG(f_{1b}), the scheme IMSRG(f_{GP}) allows us to obtain converged results for a larger range of parameters. In particular, IMSRG(f_{GP}) is more reliable for large systems, and large bosonboson interactions.”
Referee: "Fig 4c, there is a nonmonotonic dependence that looks rather spurious. Is it possible to reduce the errorbars?"
Our reply: Unfortunately, it is not possible to reduce the errorbars using out truncation scheme. Note that the density of bosons at the position of the impurity is small. Therefore, large errorbars in Fig. 4c is a result of working with small numbers, which cannot be avoided, unless we significantly modify our approach.
In the revised version, we have added a sentence to clarify this statement. See footnote 8: “Note that we expect that the exact curve for c/ρ= 0.5 in Fig. 4 c) is monotonous. Our calculations of thiscurve have large errorbars, which allow for an apparently nonmonotonous behavior.”
Referee: "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"
Our reply: We thank the referee for this comment. We have modified the discussion accordingly: “ For c >0, the agreement between MonteCarlo and the meanfield approximation is reasonable for all available data points. For attractive interactions, the difference between the results is more noticeable, which implies that the MFA leads to less accurate results for c <0, see also Appendix B, where we present some additional data for the case with attractive interactions”
Referee: "Please comment on the values of C3 in the fit to the energy. Is that a linear or quadratic behavior?"
Our reply: We thank the Referee for this suggestion. The revised version now states that the parameter C3 is in between 1 and 2, see footnote 11. To be more precise, we find that for c_2/rho=0.02 C3 is between 1.25 and 1.7 depending on d. For c_2/rho=0.1, C3 is between 1.1 and 2. We also checked that these windows are consistent with the value obtained by fitting to MF results. We noticed that large values of d usually imply smaller values of C3.
Minor Comments: We thank the Referee for providing us with minor comments [The revised version addresses all of them.]. They helped us to significantly improve our manuscript.
Reply to Anonymous Report 2
Referee Report 2:
Referee: "The flow equation results are used to benchmark the meanfield results. There is however no proof or at least some convincing arguments why the flow equation approach should be superior to the meanfield approximation. The authors compare the flowequation results along with meanfield 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."
Our reply: We thank the Referee for alerting us of this weakness in interpreting our results. Indeed, the flow equations are more accurate than the meanfield approximation. We had demonstrated this in our previous works, for example, by calculating the energy of the LiebLiniger model, see [1] in the revised manuscript. In that paper, for gamma=1 and Nsim 10, the flow equations yield the exact energy with 10% accuracy, the meanfield energy for the same parameters is about 50% larger.
In the revised version, we address this issue in the Introduction: “IMSRG was recently extended to cold Bose gases [1, 2]. It was tested by calculating the groundstate energies of the LiebLiniger model and a onedimensional (1D) Bose gas with an impurity atom (‘Bose polaron’) [1, 2]. Those works demonstrate that flow equations allow one to go beyond meanfield approximation without relying on manybody perturbation theory. In the present work, we use IMSRG to calculate the density and phase fluctuations of the Bose gas."
In addition, following the recommendation of the Referee, we illustrate a parameter regime for which there is a sizable difference between flow equations and mean field, see new Appendix B in the revised manuscript. In particular, we demonstrate that for an attractive impurity the flow equations predict significant phase fluctuations, which are beyond the meanfield approximation.
Referee: "The authors have calculated the impurityimpurity interaction potential. Ref. [23] predicts an exponential behavior at short distances and a powerlaw at large distances. The largedistance scaling is due to Casimirlike 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?"
Our reply: We absolutely agree, a numerical validation of the longrange Casimirlike force would be an extremely interesting result. Unfortunately, that force is very weak. The Casimirlike force is important only at distances of the order of 510 xi (xi for the healing length), which are larger than the typical sizes we consider. We have added a corresponding remark to the manuscript, see also very recent works on the topic in Refs. [76,77], where a similar conclusion is reached.
To the best of our knowledge, currently, there are no numerical techniques capable of calculating the Casimirlike force. In particular, it is also outofreach of the stateoftheart Quantum Monte Carlo results [76].
Please note that 2101.11997 suggests linear interaction only for etatoinfty, i.e., not for the case presented in Fig. 9. We also expect a linear behavior when c_2 becomes very large, since our equations are identical to those of 2101.11997. The case of large c_2 is however not discussed in our work.
Referee: "What is the reason that the case of attractive impurityboson interaction is not included. While one could expect that the meanfield predictions are less accurate here, in contrast to the statement just before eq.(29), the meanfield 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 meanfield approach."
Our reply: We thank the Referee for this comment. Our plan is to consider the system with attractive bosonimpurity interactions in another publication. For finite systems, the attractive case is fundamentally different from the repulsive case. In particular, there is a transition from a manybody bound state to a state in which the bosons may occupy scattering states. This transition requires a separate discussion, which is beyond the scope of the present paper. Following the recommendation of the Referee, we have added Appendix B to the revised manuscript. The Appendix shows some data for a Bose gas with a single attractive impurity. In particular, it illustrates significant phase fluctuations that happen close to the aforementioned transition.
Referee: "In the extrapolation of the bipolaron energies to the thermodynamic limit obtained either from flowequations or meanfield 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."
Our reply: We thank the Referee for this remark. We believe that the oscillating behavior is due to the numerical accuracy of our results. In particular, the amplitude of oscillations is within the errorbars for N=70. Unfortunately, it is not easy for us to increase the accuracy for these large systems.
We have added a clarifying discussion to the revised manuscript (see page 19): “The truncation error in the IMSRG method grows rapidly with the number of particles.This rapid growth rules out a reliable extrapolation of the errorbars to the thermodynamic limit. Therefore, we give no estimate for the accuracy of C1, which leads to an apparently oscillating character of the potential in the thermodynamic limit. We expect that the exact potential is a monotonically increasing function of the distance between the impurities,d.”
Referee: "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?"
Our reply: Following the recommendation of the Referee, we have added a corresponding discussion to the revised manuscript. See the discussion around Eqs. (1) and (2):
“[…]which transforms the Hamiltonian matrix into a blockdiagonal form, i.e., it decouples the“groundstate” matrix element from all excitations (see Fig. 1).”
“ In our work, eta(s) is chosen from the matrix elements that describe the couplings between the ‘condensate’ and its excitations such that these couplingsbecome weaker as the flow progresses, see Fig. 1. A detailed construction of eta(s) is presented in Appendix A.”
Referee: "Right after eq.(2) the reference state is mentioned without any introduction. The latter is only given in the following subsection."
Our reply: We thank the Referee for this comment. We have modified the text to make the discussion more clear. In particular, we no longer mention the reference state in Chap. 2.1 “Flow equations”. Instead, we moved the discussion of our truncation scheme (in which we mentioned the reference state) to Chap. 2.2 “Reference state”.
Referee: "Normal ordering with respect to the reference state is mentioned but only defined in the Appendix."
Our reply: Following the recommendation of the Referee, in the revised version, we make a more clear reference to the definition of normal ordering in the Appendix.
Referee: "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."
Our reply: We thank the Referee for this omission in the original version of our manuscript. We have added a discussion which clarifies the calculation of errorbars, and makes a more explicit reference to the Appendix.
Chapter 2.2 now starts as: “In general, it is impossible to solve Eq. (1) for a manyparticle system without approximations. The complexity is due to the commutator [η,H]: It leads to manybody terms, which are not present in the initial Hamiltonian H(s= 0). To solve Eq. (1), the manybody terms must be truncated at some order. To define a truncation hierarchy, we write the Hamiltonian in second quantization using normal ordering with respect to a reference state (for a definition of normal ordering see Appendix A.1). The reference state, Ψref, should approximate an eigenstate (here the ground state) of the Hamiltonian well, otherwise the IMSRG transformation cannot map Ψref onto the exact state. Upon normal ordering, we truncate threebody excitations and beyond, see Fig. 1. To estimate the introduced truncation error, we use the threebody elements and second order perturbation theory for matrices, see Fig. 1 and Appendix A.2.”
Submission & Refereeing History
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Reports on this Submission
Anonymous Report 3 on 2021614 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2101.10958v3, delivered 20210614, doi: 10.21468/SciPost.Report.3061
Strengths
1) benchmark of meanfield results for Bose polaron and bipolaron in 1D gas
2) extends previous work of authors to different observables such as density profile of Bose gas and interaction potential
Weaknesses
1) bipolaron problem discussed only for asymmetric case of one impurity interacting infinitely strongly and the second with variable strength
2) benefit of the flow equation approach becomes only apparent after consulting earlier work of the authors
Report
The authors have essentially addressed all the points of my previous report and I recommend publication after they consider the following optional comments.
Requested changes
1) Oscillations in the impurityimpurity 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.
2) The arguments given by the authors in the text and in Appendix B for why attractive impurityboson interactions are not treated in detail and referred to a future publication are not convincing. The comparision of meanfield 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 ad. 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.
3) 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 nonspecialist reader to understand the idea of the approach, that it should be put into th main text of the paper.
4) 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 Wilsontype RG techniques and that the IMSRG complements that technique. I did not quite understand this comment. It seems that the Wilsontype RG approach gives results which deviate substantially already from the meanfield result of the authors, see paragraph before Sec. 3.3.
5) 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.
Anonymous Report 1 on 2021611 (Invited Report)
Report
I find that most of my comments were adequately taken into account and I recommend a publication once my last comment is answered.
Requested changes
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.