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Polarons and bipolarons in a two-dimensional square lattice
by Shanshan Ding, G. A. Domínguez-Castro, Aleksi Julku, Arturo Camacho-Guardian, Georg M. Bruun
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Submission summary
| Authors (as registered SciPost users): | Shanshan Ding · Aleksi Julku |
| Submission information | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/2212.00890v1 (pdf) |
| Date submitted: | 2022-12-05 08:46 |
| Submitted by: | Ding, Shanshan |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approaches: | Theoretical, Computational |
Abstract
Quasiparticles and their interactions are a key part of our understanding of quantum many-body systems. Quantum simulation experiments with cold atoms have in recent years advanced our understanding of isolated quasiparticles, but so far they have provided limited information regarding their interactions and possible bound states. Here, we show how exploring mobile impurities immersed in a Bose-Einstein condensate (BEC) in a two-dimensional lattice can address this problem. First, the spectral properties of individual impurities are examined, and in addition to the attractive and repulsive polarons known from continuum gases, we identify a new kind of quasiparticle stable for repulsive boson-impurity interactions. The spatial properties of polarons are calculated showing that there is an increased density of bosons at the site of the impurity both for repulsive and attractive interactions. We then derive an effective Schr\"odinger equation describing two polarons interacting via the exchange of density oscillations in the BEC, which takes into account strong impurity-boson two-body correlations. Using this, we show that the attractive nature of the effective interaction between two polarons combined with the two-dimensionality of the lattice leads to the formation of bound states - i.e. bipolarons. The wave functions of the bipolarons are examined showing that the ground state is symmetric under particle exchange and therefore relevant for bosonic impurities, whereas the first excited state is doubly degenerate and odd under particle exchange making it relevant for fermionic impurities. Our results show that quantum gas microscopy in optical lattices is a promising platform to explore the spatial properties of polarons as well as to finally observe the elusive bipolarons.
Current status:
Reports on this Submission
Strengths
1 - The considered topic is interesting
2 - The paper is written well
Weaknesses
1 - some figures can be improved
Report
The Authors perform an analytical study of the problem of impurities in a 2D square lattice. The polaron problem is related to the current experiments and the obtained results are relevant in that context. I find the article to be well-written and I recommend its acceptance once my comments are addressed.
Requested changes
The following comments should be addressed
1 - unit filling $n_0 = 1$ is used "for concreteness". It would be useful to have a comment if important differences are expected for different values of $n_0$
2 - Induced interactions: explain better the region of applicability of the results
3 - Induced interactions: is it possible to comment on what is the behavior of the long-range tail (power law, exponential, etc)?
4- I advise adding Fig 3b with $k_y=0$ data
5 - Make figures readable in black and white version (Fig. 3, etc)
6- "Here and in the rest of the paper we take the value $U_B/t_B = 0.07$." Provide motivation for using this specific value.
7- Below Eq. (4), "We have defined $z =$ ...". Explain the physical meaning of $z$.
8- In the used notation indices "Bk", "Ik" in the energy spectrum should be deciphered as index + argument, which might be confusing. Instead, double arguments are given with parenthesis $(k,\omega)$. It would be less confusing use $(k)$ as an argument and B/I as an index.
9- "... scattering of an impurity atom and a boson ...". I think "atom" can be omitted in similar phrases, or atom should be added also to "boson", as the scattering occurs between two atoms, instead one is called an atom and the other is not.
10- "we assume that the interaction $U_B$ is weak so that it is accurately described by Bogoliubov theory". What should be accurately described by the Bogoliubov theory? Interaction?
11- "states with center of mass momentum zero", rephrase.
12- "We denote ... as the upper polaron", I would advise to emphasize the name "{\em upper polaron}"
13- "Equation (8) shows ... residue" -> "Equation (8a)"
14- The placement of the figures is weird, the first one is inside the text while the rest of the figures are at the end of the paper. Arrange figures properly.
15 - Introduction, "BECs" abbreviation is not introduced in the text of the article
16- below Eq .(1), "Here, $\hat b_i$..." -> "Here, operators $\hat b_i$..."
17- In the Abstract, "... the attractive nature of the effective interactions between two polarons combined with the two-dimensionality ...". It is a general feature of the second-order perturbative theories to provide a negative correction so that attractive interactions appear in all dimensionalities. Is the low dimensionality a keypoint here? Or should it be necessarily 2D?
18- Fig. 2, I advise to a add a vertical black line at $x=0$ position (no additional caption is needed)
19- Fig. 2, the horizontal dashed line seems to correspond to 1/2 and it is not clear if this is a coincidence as typically 1/2 is not a special value for the residue
20- Fig. 4, Figures are too small, 0 and 1 values are not shown. The projection angle in (b) panel is not optimal. Alternatively, top view as in Fig. 5 can be tried. Try to improve the figures.
Report #1 by Anonymous (Referee 1) on 2022-12-22 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2212.00890v1, delivered 2022-12-22, doi: 10.21468/SciPost.Report.6374
Report
The work by Ding et al. studies the dynamics of impurities immersed in a weakly interacting Bose gas on a two-dimensional square lattice. First the spectral function of a single impurity in the Bose gas is studied. To this end the self-energy is computed with the T-matrix approximation. Both attractively and repulsively bound states (=polarons) are found. Particular focus is then put on the study of effective mutual interactions between impurities, arising from the exchange of bosons in the BEC. Due to the effective attractive interaction a bound state of two polarons forms, known as bipolaron.
The results of the paper are relatively clearly presented and contain new aspects. Given the advancements of quantum simulators, the topic is also timely. A couple of questions and comments arose when reading the paper:
1) Can the authors analyze or at least comment on finite temperature effects? Would the bound state of the impurities be modified in this case?
2) Can the authors comment on whether more complex multi-impurity bound states can arise in this setting? Given the simplifications, which have been used to reduce the Bethe-Salpeter equation to a easily-tractable equation, it may be possible to consider also higher order bound states.
3) What are the expectations when the bosons are driven near a Mott transition or even into an insulator?
4) A minor comment: In the beginning of Chapter 6 the question of whether a bound state is formed for the effective attractive interactions has been asked. Is there a reason not to expect the bound state to form, given the system is in 2D and the potential is attractive? If this is the expectation, then I'd suggest to reformulate.
I believe, that the paper could benefit from a discussion of the above-mentioned comments, as some extra focus is put on the central aspects of the lattice. Once taken into account, the papers can be accepted in Sci Post.