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Boson-fermion pairing and condensation in two-dimensional Bose-Fermi mixtures
by Leonardo Pisani, Pietro Bovini, Fabrizio Pavan, Pierbiagio Pieri
Submission summary
| Authors (as registered SciPost users): | Pietro Bovini · Fabrizio Pavan · Pierbiagio Pieri · Leonardo Pisani |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2405.05029v2 (pdf) |
| Data repository: | https://zenodo.org/records/14218081 |
| Date submitted: | 2024-11-28 10:58 |
| Submitted by: | Pisani, Leonardo |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We consider a mixture of bosons and spin-polarized fermions in two dimensions at zero temperature with a tunable Bose-Fermi attraction. By adopting a diagrammatic T-matrix approach, we analyze the behavior of several thermodynamic quantities for the two species as a function of the density ratio and coupling strength, including the chemical potentials, the momentum distribution functions, the boson condensate density, and the Tan's contact parameter. By increasing the Bose-Fermi attraction, we find that the condensate is progressively depleted and Bose-Fermi pairs form, with a small fraction of condensed bosons surviving even for strong Bose-Fermi attraction. This small condensate proves sufficient to hybridize molecular and atomic states, producing quasi-particles with unusual Fermi liquid features. A nearly universal behavior of the condensate fraction, the bosonic momentum distribution, and Tan's contact parameter with respect to the density ratio is also found.
Author indications on fulfilling journal expectations
- Provide a novel and synergetic link between different research areas.
- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
Author comments upon resubmission
Thank you for your Editorial Recommendation of July 5th, 2024, asking for a minor revision of our manuscript. We hereby submit a revised version that takes into due account all recommendations by the Referees. Together with this resubmission, we have also sent a point-by-point response to both Referees.
We are really grateful to the Referees for their valuable comments and suggestions, which have helped us to considerably improve our manuscript.
Yours sincerely,
Leonardo Pisani, Pietro Bovini, Fabrizio Pavan, Pierbiagio Pieri
List of changes
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Summary of the changes made
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We have changed the layout of our text to adopt the SciPost latex template.
We refer to the new version when pointing to page, equation, and figure numbers.
Our manuscript has been modified to take into account all the suggestions by the Referees. We have also made some changes to improve the presentation and quoted a few additional references. All of these changes (except minor stylistic/typos corrections) are listed below.
a) Pag. 2, changed “micro-cavities” with “nanostructures” in the 2nd paragraph, added four new sentences to the 2rd paragraph (“The so-called … on more solid ground.”)
b) Pag. 3, re-written the final part of the 5th paragraph (from “For this reason …”)
c) Pag. 5, added a sentence at the end of the first paragraph, re-worded the first two sentences of the last paragraph of Sec. 2.1, re-worded the end of the first paragraph in Sec. 2.2 (“There, the authors … to the boson concentration.”), added a new paragraph at the end of the page (starting with “A warning is however in order.”)
d) Pag. 6, added two new paragraphs (“Let us see now the rationale … controlled approximations are not available”, “The same strategy is then adopted … with (one-component) fermions [76].”), inserted Eq.5 and corresponding discussion
e) Pag. 7, re-worded the paragraph above Eq.7 and the paragraph above Eq.9, added a comment below Eq.9, introduced more specific names for the T-matrices (“many-body” and ”two-body”) below Eq.9
f) Pag. 8, re-written Eq.11 and the paragraph below it
g) Pag. 9, changed the titles of Sec.3 and Sec.3.1, re-worded the last two sentences of the first paragraph of Sec.3 and the first sentence of Sec. 3.1, added the explicit definition of n^0_{\mu_F} below Eq.21, changed \Sigma_{CF} to \Sigma^0_{CF} below Eq.22 and everywhere else
h) Pag. 10, added Eq.24 and its explanation below it, added the sentence “in the many-body T-matrix in the condensed phase” above Eq.25, added a new paragraph (“Here, G_{CF}(P,\Omega), … condensate.”) below Eq.27, re-worded the first sentence below Eq.29, added a whole new paragraph (“One sees from Eq. (29) … value for all intermediate concentrations.”) above Eq.30, changed “pair” with “composite-fermion” before Eq.30
i) Pag. 11, added Eq.34 and comments above and below it, changed “two-body correlator” with “many-body T-matrix” at the end of Sec.3.1, re-worded the paragraph containing Eqs. 35 and 36
j) Pag. 12, re-organized the end of Sec.3.2, in particular moving text and equations from former Sec.IIIB (from former Eq.33 to 46b) to the new Appendix D2 while re-organizing and re-wording the rest; re-worded the sentence after Eq.44
k) Pag. 13, added a comment at the end of the paragraph after Eq.47, re-written the last two paragraphs of the page
l) Pag. 14, re-written the 1st paragraph, added a second inset in Fig.3(b) updating the caption, re-worded the last sentence of the page
m) Pag. 15, added a sentence “So, in this regime, …”, modifies the first sentence of the 2nd paragraph, re-written the last two sentences of the 2nd paragraph, inserted new paragraph “The Luttinger theorem states …” at the end of the page
n) Pag. 16, inserted new paragraph (2nd paragraph: “In a BF mixture with a condensate …”), updated the caption of Fig.5, completely re-written the last paragraph of the page with several new sentences
o) Pag. 17, completely re-written all paragraphs before Section 4 with several new sentences, added a new sentence (last sentence of the page)
p) Pag. 19, added Eq.49
q) Pag. 20, added insets in Fig.9 and their explanations in the caption
r) Pag. 21, re-worded a sentence in the caption of Fig.11, added two new sentences after Eq. 51
s) Pag. 22, substituted “asymptotic limit” with “large-momentum behavior” in the caption of Fig.12, added two new sentences in the 2nd paragraph: “to check to what extent …” and “The use of the same thermodynamic…”,, added “internal part of the” before “molecular wave-function” in the first sentence of the 3rd paragraph
t) Pag. 23, replaced “asymptotics” with “approximation” in the caption of Fig.13, added a new sentence immediately after Eq.55
u) Pag. 24, added a new curve and a new set of points in Fig.14(a) updating the corresponding caption, re-written a sentence in the 1st paragraph, inserted a new paragraph (2nd paragraph), re-worded the last sentence of Sec.4.3
v) Pag. 25, added two new sentences (“As an independent check…to possible numerical errors”) near the end of the 3rd paragraph, added a new 4th paragraph (“As a matter of fact, … followed by the limit x -> 0.”), added four new sentences at the end of the last paragraph
w) Pag. 26, changed the notation of Eqs.56-57
x) Pag. 27, reworded the last sentence of the last paragraph at the end of the page
y) Pag. 28, re-written the first two sentences of the 4th paragraph in Sec.5 “A further difference with …”
z) Pag. 29, added words “in particular at small values of x …” at the end of the second sentence of the 2nd paragraph, added a new paragraph at the end of the page (”It is worth also discussing the relevance … sufficient to guarantee stability.“)
aa) Pag. 30, added two new paragraphs (“As a final remark …are available online [131].”) before the Acknowledgements
bb) Pag. 30-31, renamed, re-organized, and re-written Appendix A
cc) Pag. 32, added Fig.17 and its caption
dd) Pag. 34-35, added Appendix D1 containing the whole content of former Appendix D
ee) Pag. 35-37, added Appendix D2 containing material from former Sec.IIIB (from former Eq.33 to 46b)
ff) Bibliography, added the following references 42, 44, 48, 49, 95, 96, 100, 101, 102, 111, 112, 113, 114, 115, 128, 129, 130, 131
gg) Bibliography, updated the following references (due to publication or change of version): former 34 now 39, former 45 now 46, former 53 now 56
hh) Bibliography, removed former reference 40
Current status:
Reports on this Submission
Report
The authors have responded to my points in a satisfactory manner and they have made corresponding changes, that -in my eyes- greatly improve the quality of the manuscript as well as its accessibility.
The physics is interesting and with experimental realizations of such systems within reach this work is certainly timely. The present manuscript sets a foundation for the exploration of two-dimensional strongly-coupled Bose-Fermi mixtures, that future works will be able to draw from. As a result, I can recommend publication of the manuscript in its present from.
Below, I provide additional feedback that I encourage the authors to consider/incorporate if it is feasible and applicable. These points, I think, could enhance the insights obtained from this manuscript, but, if the authors deem these points not applicable or feasible, then my points should not delay publication. The manuscript in present form is already suitable for publication.
My feedback:
I still believe that one of the key results of this work, the condensate fraction differing from the quasiparticle weight, could be illuminated more.
The correspondence in 3D is fairly strong, which suggest that it isn’t a coincidence and also begs the question what changes in 2D.
I do not find the argument given by the authors on how „there is no reason why the limit for x->0 of the condensate fraction and the polaron residue Z should coincide“ particularly convincing. Both quantities are well defined in the thermodynamic limit. The field theory does not know about a particle number, it only knows about particle densities. The condensate fraction is obtained at nB>0, while the polaron quasiparticle weight is obtained from the weight of the quasiparticle pole in the bosonic Green’s/spectral function at nB=0 (which is also well defined at nB>0). Importantly the quasiparticle weight obtained from the spectral function has the same physical interpretation as the quasiparticle weight in the polaron wave function (Chevy) Ansatz.
In both cases one has taken V->infinity before specifying the chemical potentials (and condensate densities) which eventually yield the corresponding densities nB and nF.
Furthermore, it was my understanding that the universality came from, as the authors note also in this work, the bosons being nearly independent of each other, while interacting with the medium. Thus, it would at least be physically intuitive that the probabilities for a boson to be in a p=0 mode are related in these cases. My understanding was that the correspondence between the quasiparticle weight and the condensate fraction was a reflection of that. So if a universality is observed here, then I would still expect some sort of correspondence between probabilities to be in effect. Thus it would be insightful to illuminate where the „remaining probability“ goes.
-The curves shown in Figure 14 are for eta=0, however for eta=0 we have that nB(0) is finite. For eta\neq 0, nB(0) diverges. Could it be that there is some sort of delta function for p=0 that contributes to the fraction of p=0 bosons? Perhaps because in 2D one only has a single factor of p in the measure instead of p^2 in 3D? Do plots like Figure 14a also exist for eta>0? Could it be that there is stronger correspondence for eta\neq 0?
-How does the bosonic quasiparticle weight at x>0 (not just at x=0) compare to the condensate fraction?
-I may be mistaken about this: Have the authors considered reconstructing the quantum effective action from the renormalizations employed here? In particular the effective potential? After a short, (not very careful) analysis I obtain that \frac{\delta \Gamma[\phi[J]]}{\delta \phi[J]} |_{J=0}= \phi[0] G^{-1}_B[\phi[0]]. Where J is the source field, \phi[J] is the source-dependent boson field and \phi[0] is the stationary field at vanishing source, which here is proportional to \phi[0]\propto \sqrt{rho} \delta(…).
It would seem that the Hugenholtz-Pines condition employed here is a necessary condition for the field to be stationary. However, I don’t think it necessarily implies that \phi[0] minimizes the effective potential? This could either mean that the value obtained for rho is not unique and there is a second value of rho that fulfills the Hugenholtz-Pines condition, along with the other fixing conditions AND additionally yields a lower value of the effective potential. Alternatively, it could also mean that the field additionally condenses in a different mode, for example p>0 (though I don’t think this is the case here).
Have the authors considered checking if there is a larger value of \rho that fulfills the Hugenholtz-Pines condition? I would guess that actually computing the effective potential and comparing values is quite cumbersome, but checking whether there is a second solution to the Hugenholtz-Pines condition should be feasible.
-I believe a formal consideration of the effective potential/ effective action might also yield more insights into the possible correspondence between quasiparticle weight and condensate fraction
-I may be mistaken, but is there a possibility that the used Hugenholtz-Pines condition is only a low-order approximation of a „more“ accurate condition? Eq. 4.10 in Ref. 107 and the text below seem to indicate that Eq. 6.2 of Ref. 107 is only an approximation. Though it seems to be increasingly valid at low density, which however begs the question of whether this would refer to boson or fermion density in this case? I am not sure how this point fits with my previous points, but I thought it might be better to include it regardless.
Recommendation
Publish (surpasses expectations and criteria for this Journal; among top 10%)
Report
The Authors have responded to all my points and made the corresponding changes in the paper.
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 recommend its publication in SciPost in the present form.
Recommendation
Publish (easily meets expectations and criteria for this Journal; among top 50%)