Boson-fermion pairing and condensation in two-dimensional Bose-Fermi mixtures

The paper by L. Pisani presents a detailed study of imbalanced Bose-Fermi (BF) mixtures in cold atomic setups. It extends the T-matrix approach, which has been successful for the polaronic theories, to the arbitrary imbalance. The diverse range of observables is discussed in detail.

I have found that the calculations are solid, and the results are very interesting. The analysis of the observables presented is detailed and thoughtful. The paper will impact the field and be noticed by the cold atomic community. I will recommend its publication in the SciPost after the Authors clearly respond to the following points.

1) The Authors state that “However, since the latter choice gives rise to an improper self-energy contribution, to avoid double counting the -matrix in the normal phase Γ(P,Ω) is used instead, thus yielding…”. It is not clear what the improper means here. It looks like the leading contribution (according to the expansion of the Bogoliubov theory) has just been omitted. This approximation needs to be explained in detail.

2) The choice of diagrams is motivated by the well-established polaronic regime with the significant BF imbalance, i.e x<<1. It seems natural that it would produce reasonable results when x is modestly small. How the choice of diagram is justified at x~1? For instance, only particle-particle scattering diagrams are taken into account. What about the particle-hole ones? They do not produce bound states but can renormalize chemical potentials, broaden spectral weights, or modify masses.

2) The described physics has recently been found relevant and fruitful for electron-exciton systems in semiconductor nanostructures. The paper would benefit from discussing whether the obtained results can be extended to the mentioned solid-state setups.

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The manuscript titled "Boson-fermion pairing and condensation in two-dimensional Bose-Fermi mixtures" investigates the phase diagram of a two-dimensional mixture of a Bose gas and a Fermi gas which are coupled via a strong interaction that can feature bound states even in the two-body limit. The work builds on previous analyses conducted in three dimensions which utilized the same type of diagrams. Due to the presence of condensed bosons, the fermionic and the molecular sector are hybridized into a joint excitation and thus the relevant quasiparticles are superpositions of these two sectors. To illuminate this mechanism, the authors derive analytic expressions --valid in the strong-coupling regime-- which showcase the different quasiparticles at play. In the relevant regimes, their fully numerical treatment shows good overlap with their analytic expressions, and they find subtle physical differences (and also similarities) compared to the corresponding physical system in three dimensions. While, as in 3D, they find that increasing the Bose-Fermi interaction strength progressively depletes the condensate, they do not find the condensate fraction to vanish beyond a critical interaction strength (unlike in 3D). As a result, the emerging quasiparticle show some unusual features. Like in 3D, a universal behavior with respect to the ratio between bosonic and fermionic density is found for quantities like the condensate fraction.

The present manuscript is timely, and the physics covered is certainly interesting. I am not aware of previous works that have addressed this phase diagram along with the interplay of pairing and condensation in two dimensions. As mentioned in the main text, an experimental implementation of an analogous system in three dimensions was demonstrated recently and thus two-dimensional implementations are certainly within reach (and have already been studied in the polaron limits). Insights into the qualitative physics in two dimensions along with the differences to analogous systems in three dimensions are thus welcome and provide great progress to the field.

While this manuscript is certainly worthy of publication, there are a few issues/comments/questions I would like to see addressed before recommending publication. I provide the comments below.

Before getting into content feedback, I would like to encourage the authors to proofread the manuscript thoroughly. While reading through the manuscript, I found several instances of unnecessarily long/complicated sentences that required several readings before I was able to understand them. I also found several sentences missing verbs or with singular/plural errors. I invite the authors to conduct another round of proofreading to facilitate readability for all readers.

While the technical analysis conducted in this manuscript is sound (though in some instances I think the presentation may be improved), my main concerns are the diagrammatics used and the physical conclusions one may draw from it. I am fully aware, that the present analysis is challenging enough as is and going beyond it, taking into account higher-order effects or higher degrees of self-consistency is no easy feat. I feel however, that this manuscript lacks transparency in its main drawback, and I would like to see this discussed more. As I am sure the authors are fully aware, in the normal phase this type of T-matrix approach has a central drawback: it treats renormalization of the molecule sector and renormalization of the fermion sector on a different footing. As a result, in the Fermi-polaron limit of n_B=0, n_F>0 in three dimensions the molecule's energy is higher than it should be, and the polaron-to-molecule transition takes place later than it does in more self-consistent approaches. Apart from that, the Ansatz gives qualitatively correct results, especially in observables that do not include the energy (such as the quasiparticle weight). In two dimensions, however, this Ansatz no longer holds a polaron-to-molecule transition, even though several state-of-the-art methods do find one. This of course has implications also for the phase diagram and phase transitions of 2D B-F mixtures, as the present approach cannot reproduce known physics for x->0 at large g. This does not mean that at larger values of x it must produce incorrect results, but it is at least a possibility.

The authors mention this point in the conclusion, but I feel that in the interest of transparency it must be stated also in other places in the main text, especially when the diagrammatic is motivated due to its success in 3D or when results for large g and smaller values of x are discussed. It is also not clear to me how, when having a vanishing condensate fraction for x=0, this is regenerated upon going to x>0. Of course, I cannot discount the possibility that in 2D no phase transition exists, and one may rather see somewhat of a crossover, but this cannot be inferred from the analysis conducted here and I feel that the reader should be made aware of this potential shortfall BEFORE they get to the conclusion. I strongly encourage the authors to make mention of this, especially when pointing out differences between two and three dimensions, which may possibly only be due to this simple shortfall. I don't think mentioning it lessens the great value of their work.

In Figure 14, no correspondence between the condensate fraction and the Z factor is found. This is highly surprising to me as from a simple Chevy Ansatz one can see a close correspondence between the two. Is this difference because the chosen values of x were not small enough? If true, this is a major difference between 2D and 3D. Certainly one which has nothing to do with the absence of a polaron-molecule transition. Could the authors illuminate/investigate this further? For x->0 or at least for x=0, these two observables should be the same, so I am very surprised they are different.

Other feedback (in no particular order):

1. While I am familiar with the diagrammatics used, I am afraid many readers might not find them easily accessible. I feel that confusion might originate from the differences between the Gamma and T vertex in Figure 1 and how these diagrams are obtained. I suggest to either a) expand on an explanation of how the diagrammatics can be obtained (possibly in an appendix), b) provide a reference where this is done, or c) conduct parts of the explanation in a two-channel language (done recently in Ref. 53), where their different roles are more clear. I am not aware of Ref 73 or Ref 88 providing a more accessible explanation. In section 2B the Gamma vertex is referred to as a T-matrix; while I am aware that for n0=0 they are the same thing, this can easily add to confusion. I suggest to use distinct wording when referring to these vertices. Furthermore, I think a few references to T matrices in the normal phase might be helpful where the T matrix is first introduced.

2. I appreciate the analysis conducted in section 3 to illuminate the nature of the quasiparticles resulting from hybridization, however in part I find it very hard to follow due to its very technical nature and the non-trivial effects of hybridization. The analysis in 3A is easy to follow, I would possibly ask that the E^+ and E^- excitations are related also to the undressed states they correspond to for vanishing condensate density. The analysis in section 3B I found very hard to follow and I wonder whether it is necessary in the place it occurs now. It is my understanding that section 3 serves to illuminate the underlying quasiparticle qualitatively, mainly to shine light on the effects of hybridization:

In section 3A the T-matrix is analyzed, and one finds that it mixes unpaired atoms with molecules. The quantum fluctuations taken into account are a self-energy renormalization of the molecule (contained within Gamma). As a result, the two excitations found within T are a result of the mixing of a non-renormalized atom with a renormalized molecule. These two excitations would also show up in the same way if one considered a corresponding fermionic Green's function (of course the distribution of quasiparticle weight between them would be different). However, the fermionic Green's function in this work is considered on different footing and mixes a renormalized molecule with a renormalized fermion, resulting in two (slightly) different excitations. I am however not sure that this different, second set of excitations, introduced in 3B and analyzed after Eq. 32 for \Delta_0/E_F small is actually needed in this detail. Maybe the part with \Delta_0/E_F small is better suited for an appendix? When references to the analytical expressions were made later in the text, I could only really find references to 3A and parts up to Eq. 32.

The reason I bring this up, is that I found this part (after Eq. 32 until before Eq. 47) extremely challenging to follow, due to its technical nature and I am not sure which physical insights are conveyed in it. Furthermore, it had the effect that in section 3D and onwards I found it challenging to follow which analytical results were being referred to, those mainly from 3A or those from 3B.

4. Are there cheap ways to enforce a polaron-to-molecule transition, along with the corresponding physics? For example, in the analytical results obtained in Section 3. If yes, how do the results obtained from that look like?

5. Using the same diagrammatics in 3D, in Figure 7 of Ref. 77 a peculiar bosonic distribution function was observed, which vanished identically below a certain momentum. This was due to the bosons participating in fermionic molecule formation and as a result, the bosonic distribution function showed remnants of a fermionic effect. Can something analogous be observed here in 2D?

6. I find the used terminology of dressed/undressed dangerous. As is, "dressed" refers to the effects of hybridization (which one may see as resulting from a Green's matrix inversion in a two-channel language) and "undressed" refers to effects without hybridization. However, the undressed propagator still contains quantum fluctuations. In section 3A the adjective "bare" is used below equation 25. However, that molecule still contains quantum fluctuations, albeit in a mean-field fashion through Sigma_CF. Maybe the authors mean undressed? In any case, I feel these ambiguities can be a source of confusion for many readers and I would ask that the authors explain exactly what they mean when they use adjectives like dressed/undressed/bare.

7. I believe an analytical expression for Eq. 5 was provided in Ref. 106

8. Where is the condensate factor introduced? It briefly appears in the caption to Figure 1 and in the main text it starts appearing in Eq. 6, but there is no proper mention of what it actually is there.

9. In Eq. 8 the convergence factor appears but there is no mention of it.

10. In Eq. 2, it is v_0 while other times it is \nu_0.

11. I believe the renormalization/regularization in Eq.2 would benefit from a reference.

12. Is Omega a real or an imaginary frequency?

13. Are there things one can say about T>0?

14. The authors set m_B=m_F, what is the role of mass ratio in this phase diagram?

15. At the end of section 2A where the quantum depletion is mentioned, I think a reference to condensation in a repulsive Bose gas would be helpful, or alternatively an explanation of what is meant by "quantum depletion determined by eta_B"

16. Is the self-energy integral in Eq. 10 the same as the one introduced in Eq. 8? If yes, why not write Sigma_B=Sigma_11+ Sigma_BF?

17. "The direct boson-boson repulsion is neglected in the present regime, since it is expected to produce negligible effect", I am guessing this is for realistic experimental values?

18. I could not find any detail as to how Eq. 19 and 20 were obtained exactly

19. Is there a simple expression for n^0_{\mu F} below Eq. 20?

20. When G^0_CF is introduced, should the reader know already what it is or is it simply a Green's function of the form in Eq. 21 which is rescaled to fulfill the frequency sum rule of Green's functions?

21. Above Eq. 29 it is stated that the pole from G^0_B does not contribute. I am guessing this is due to the sign of the bosonic chemical potential. Was the sign of the bosonic potential already mentioned at this point? Does it ever change?

22. What does the sentence below Eq. 30 about neglecting E^\pm altogether with respect to mu_B mean?

23. Section 3B, I feel it could be made clearer where small Delta_0/E_F is presumed and where one goes back to considering Eq. 32.

24. "Before passing to the exact evaluation of Eq. (32) in closed form, it is instructive for its physical interpretation to analyze the limit of small Delta_0/F, which is expected to occur either when x -> 1 (depletion of the condensate density due to increase of molecule number) or x-> 0 (reduction of the condensate density due to decrease of boson number)." How can this be understood intuitively? I understand that for x->0 we have n_B->0 and since n_0<n_B we also have n_0->0. But how can this be intuitively understood for x->1? If I understand correctly, then for all n_B<n_F, all bosons could potentially bind into molecules, leaving n_0=0. Why is this only expected for x->1?

25. I believe Eqs. 41,43 and 44 were never formally related to n_F(k).

26. Why does the part in Eq. 45 that regards the population of composite fermions not have a corresponding theta function? Is there a way to understand Eqs. 45 and 46 in terms of two Fermi seas filling up? Or possible in terms of particle branches that are present in the two-body limit/ the non-interacting limit/ the Fermi polaron limit, which are then populated (albeit with modified quasiparticle properties)?

27. What is on the x-axis in the inset of Figure 3B? I presume x? I think Figure 3 would benefit from showing the condensate fraction.

28. In the paragraph before the one starting with "In the crossover region, all chemical potentials are comparable in size", how can one understand the characterizations of the particle in terms of their chemical potentials? Is there an intuitive way to understand why for example \tilde{\mu}_{CF}>0 means degenerate molecules.

29. I could not find it explicitly: Which method was used to obtain the results shown in Figure 3,4,5,6? Is it Eqs. 50-52 to obtain chemical potentials and condensate density or are additional Eqs. involved?

30. "In Fig. 5 the quasi-particle weight \nu^2_p of the occupied states \Theta(-E^-) is displayed for a number of concentrations corresponding to...". What does this mean? If I understand correctly then \nu^2_p*\Theta(-E^-) is shown. In Figure 5, the qp-weight is also shown for unoccupied states, however for unoccupied states it is set to 0. That is not precisely the same thing. Similarly, in the caption of Figure 5 only \nu^2_p is mentioned, but from Eq. 27 it is not clear to me that the function has a step-like drop (that only occurs after multiplying with Theta). Can this be formulated more precisely?

31. I think the paragraph below Figure 5 would benefit from explicitly stating what the Luttinger theorem is, how it breaks down, and how it is compensated in n_CF. I am guessing Theta has larger support and in return \nu^2_p is smaller, such that n_CF remains constant?

32. Eq. 61 is first referenced below Figure 8, a long time before it is first introduced.

33. I think more detail surrounding Eq.57 is needed. How can one see that a=1? What are the values of a, b as function of g? As is there is no real way to follow the argument made here.

34. "using the same thermodynamic parameters for both sets of curves". What does this mean? Same chemical potentials/condensate densities? If yes, how is this justified, shouldn't the different methods come up with different chemical potentials?

35. "The boundary between them can be roughly estimated by looking at the smallest coupling at which \mu_F vanishes as a function of x...". Is this a coincidence or is there physical meaning to \mu_F=0? I guess this goes back to my question of how to characterize particles in terms of their chemical potentials.

36. The notation inf Eqs 60,61 can be confusing, I would suggest switching to n_{F,k\gg k_F}(k) or something of that sort.

37. I found it hard to follow what happens in Appendix B. I think a Feynman diagram would help.

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