Hidden Pseudogap and Excitation Spectra in a Strongly Coupled Two-Band Superfluid/Superconductor
Round 1
Reviewer 1 Report
Bose-Einstein-condensation (BEC) crossover, within the many-body T-matrix approximation. They analyze the
single-particle density of states and the spectral functions showing how the pseudogap state in the shallow band is
hidden by the deep band contribution. The paper presents an interesting analysis of the pseudogap in a two-band superconductor, but requires an extended revision before the acceptance of the manuscript. My comments are the following:
-The authors introduce the two-band superconductor model with a deep and a shallow band, but they never discuss the origin of the character of these two bands in the context of the iron based superconductors. Naively, I would have expected that the masses relative to the two bands would be different in FeSe, so the authors could add a paragraph in the manuscript;
-In the hamiltonian (1) the sum on j goes from 1 to 2; the indices j_1,j_2 are not defined. Afterwards they should signal that both \hbar and k_B are taken to be one.
-The T-matrix has only the diagonal contributions? I would have expected T to be a tensor.
-In Eq. (7) the authors introduce the dressed propagator, but how is taken the self-energy? The effect of \Sigma can dramatically change the spectrum of the bare bands. Could the author show how the spectrum is modified? Of course, this is relevant for the calculation of the spectral function.
-In Eq.(9) the A_r is not defined. Is it the tip spectral function?
-The inset in Fig.2 right doesn't contain the labels.
-Below Eq.(18) the wavevector region is k \le k_F, not p \le k_F
-Finally I see a considerable overlap with the results presented in Phys. Rev. B 102, 220504(R) (2020). Could the author better enphasize the novel aspects compared to the PRB manuscript? The authors could better contextualize their results and analysis for the iron-based superconductors.
Author Response
We thank Reviewer 1 for carefully reading our manuscript and giving us useful comments. In what follows, we would like to reply point by point to each comment raised by Reviewer 1.
Reviewer 1: The authors introduce the two-band superconductor model with a deep and a shallow band, but they never discuss the origin of the character of these two bands in the context of the iron based superconductors. Naively, I would have expected that the masses relative to the two bands would be different in FeSe, so the authors could add a paragraph in the manuscript;
Our reply: Thank you very much for drawing our attention to the origin of shallow and deep bands in iron-based superconductors as well as the mass difference between two bands. In the revised manuscript, we have added a paragraph discussing the band configuration in the end of Section 2.
Reviewer 1: In the hamiltonian (1) the sum on j goes from 1 to 2; the indices j_1,j_2 are not defined. Afterwards they should signal that both \hbar and k_B are taken to be one.
Our reply: We have added explicit definitions of the indices j_1 and j_2. We have also mentioned that \hbar and k_B are taken to be one in the end of Section 1.
Reviewer 1: The T-matrix has only the diagonal contributions? I would have expected T to be a tensor.
Our reply: Indeed, the T-matrix has also off-diagonal components. However, they are not needed for the calculation of the self-energies because the band indices are conserved in the external lines since the bare propagator <c^{\dagger}_j c_{j’}> = 0 vanishes when j \neq j’.
We mentioned this point in the formalism.
Reviewer 1: In Eq. (7) the authors introduce the dressed propagator, but how is taken the self-energy? The effect of \Sigma can dramatically change the spectrum of the bare bands. Could the author show how the spectrum is modified? Of course, this is relevant for the calculation of the spectral function.
Our reply: The self-energy is taken as shown in Eq. (7). The choice of this self-energy to describe pairing fluctuations in the normal state was motivated in our previous works [36,38]. In the absence of \Sigma, the spectrum exhibits two bare quadratic bands. On the other hand, as shown in Fig. 3, the self-energy corrections associated with pairing fluctuations lead to the pseudogap opening around k=k_F.
Reviewer 1: In Eq.(9) the A_r is not defined. Is it the tip spectral function?
Our reply: A_r is defined below Eq. (10). It is the spectral function in the reference normal metal.
Reviewer 1: The inset in Fig.2 right doesn't contain the labels.
Our reply: In the revised manuscript, we have added the labels in the inset of Fig.2. We thank Reviewer 1 for pointing it out.
Reviewer 1: Below Eq.(18) the wavevector region is k \le k_F, not p \le k_F
Our reply: We have corrected it as k \leq k_F. We thank Reviewer 1 for pointing it out.
Reviewer 1: Finally I see a considerable overlap with the results presented in Phys. Rev. B 102, 220504(R) (2020). Could the author better emphasize the novel aspects compared to the PRB manuscript? The authors could better contextualize their results and analysis for the iron-based superconductors.
Our reply: In the end of Section 3 (Results), we have emphasized the difference between the present manuscript and our previous work published in PRB 102, 220504(R) (2020). In this new work submitted to Condensed Matter, we have addressed the total density of states and single-particle spectra which are relevant for the recent observations in FeSe. The key importance of the interplay between overlapped spectral weights and pseudogap opening in the two-band BCS-BEC crossover has been discussed firstly in this paper, in contrast with the previous PRB paper where each spectral weight was investigated independently.
We would like to thank Reviewer 1 for the very useful comments and hope that the revised manuscript and the reply document meet the approval of Reviewer 1.
Reviewer 2 Report
In the work 'Hidden Pseudogap and Excitation Spectra in a Strongly Coupled Two-Band Superfluid/Superconductor, the authors investigate a three-dimensional two-band model across the crossover from BEC to BCS.
The model generalizes the well-known single band mode including another band which is coupled through a contact interactionterm to the other one. This generalization has applications to superfluid Fermi gases in the ultracold regime, as well as to iron-based superconductors. The authors compute single-excitation properties in the normal state, namely the tota density of states and most importantly the total spectral function A(k, \omega). The latter is related to the pseudogap energy \Delta_pg which is induced by pairing fluctuations. The analysis is done varying the scattering length a_22 in the second band for a fixed interband couplings.
I believe that the results are interesting and timely, given recent amount of theoretical and experimental work in the two related fields of ultracold superfluid gases and multi-band superconductors.
I have some minor comments regarding the writing of the manuscript and the presentation of the formalism. This work employs a T-matrix formalism, similar to the a recent interesting work by some of the authors Phys. Rev. B 2020, 102, 203 220504(R). I would suggest the authors to include few further details about the derivation of eq.(4) which is a central quantity to derive the subsequent results. This would make the paper self-contained.
I have two questions related to the results in sec.3:
-
How sensitive are the results of the total density of states to temperature? The small dips in the density of states are expected to disappear at larger temperature, T_pg? I guess this is related to figs.3,5 of the PRB above. Is this feature observable directly in experiments?
-
How are the results expected to change when a_11 is set closer to unitarity? Is the formalism still applicable?
In conclusion I find this analysis interesting and it deserves publication after the comments regarding the inclusion of further technical details and questions are answered.
Author Response
We thank Reviewer 2 for carefully reading our manuscript and giving us positive comments for publication in Condensed Matter.
In what follows, we would like to reply point by point to the comments raised by Reviewer 2.
Reviewer 2: I would suggest the authors to include few further details about the derivation of eq.(4) which is a central quantity to derive the subsequent results. This would make the paper self-contained.
Our reply: In the revised manuscript, we have given additional details of the two-band T-matrix in Section 2. We thank Reviewer 2 for suggesting it.
Reviewer 2: How sensitive are the results of the total density of states to temperature? The small dips in the density of states are expected to disappear at larger temperature, T_pg? I guess this is related to figs.3,5 of the PRB above. Is this feature observable directly in experiments?
Our reply: Yes, since the pseudogap in Fig.2 left (where \tilde{V}_12=0) is associated with the pseudogap in the shallow band, it will disappear at the pseudogap temperature of the second band T=T_2^* which is shown in Figs. 3 and 5 of the PRB paper (noting that the details of the parameter sets are different). On the other hand, in Fig. 2 right with \tilde{V}_12=1, the small pseudogap is deeply related not only to the shallow band but also to the deep band. In this case, the pseudogap structure is suppressed around the pseudogap temperature of the deep band T=T_1^* which is close to T_c, while the overall dip structure disappears around T=T_2^*. We have mentioned these interesting points in the Section 3.
Reviewer 2: How are the results expected to change when a_11 is set closer to unitarity? Is the formalism still applicable?
Our reply: Yes, essentially there is no obstacle for exploring the situation where a_11 is close to the unitarity in our formalism, considering that the physics of the crossover regime centered around unitarity is described in a reasonable, qualitative, manner by a diagrammatic interpolation between two controlled limits (BCS and BEC), as known in the context of the BCS-BEC crossover theories and their comparison with experiments or numerical simulations, see Ref. [8] of our revised manuscript for details. In the revised manuscript, we explicitly mentioned it when introducing the formalism.
We thank Reviewer 2 again for the constructive comments and suggestions.
Reviewer 3 Report
This manuscript performs a theoretical study on the STS and ARPES signals of a model two-band superconductor based on the T-matrix approximation, aiming to clarify the origin of the missing pseudogap behavior in FeSe. The main conclusion can be summarized as follows. Suppose one band is in the weak-coupling regime and the other is in the strong-coupling one. If the interband pairng interaction is weak, the pseudogap behavior in the strong-coupling band can easily be masked by the conventional density of states in the weak-couplind band. This one may anticipate without any elaborate calculations. However, I am not at all convinced after reading through the manuscript whether the mechanism is really relavent to FeSe. Is the isotropic two-band model appropriate for FeSe? How can the authors justify the values of the parameters chosen for the numerical study? Is there any evidence that the interband pairing interaction is negligible? It seems that the authors have adopted a model that can be handled easily for their numerical study without due consideration on its relevance to the material. With these observations, I cannot recommend publication of the manuscript in Condensed Matter.
Author Response
We thank Reviewer 3 for carefully reading our manuscript and giving us useful comments.
Indeed, our model is rather simplified compared to realistic FeSe superconducting materials. However, our aim was to discuss the two-band BCS-BEC crossover phenomena observed recently in these systems in connection with ultracold atoms, where both theoretical and experimental aspects of the BCS-BEC crossover physics are well established. The isotropic two-band model described by Eq. (1) is directly relevant to Yb Fermi gases in the presence of an orbital Feshbach resonance. The differential conductance corresponding to STS will also be addressed in ultracold atom experiments (see also arXiv:2010.00582) and will enable one to verify our predictions directly in the near future. On the other hand, we believe that a more elaborate model, better tailored to FeSe materials, would not drastically change the fundamental BCS-BEC crossover physics we are after.
For this reason, we believe that our simplified model is the first right choice for understanding on a common basis strong-coupling physics occurring throughout the two-band BCS-BEC crossover in both iron-based superconductors and ultracold Fermi gases. In the near future, we and other research groups will move to the analysis of the fluctuations and pseudogap physics adopting a more refined model, closer to the realist electronic structures and effective pairing interactions of iron-based (e.g., FeSe) superconductors as suggested by the Reviewer 3.　This will require, however, much more involved diagrammatic numerical calculations to quantify spectral properties of interest.
Regarding the masking effect which, according to Reviewer 3 could be anticipated without any elaborate calculations, we emphasize that in the case with strong interband interactions, such masking effect can be easily overwhelmed by strong pairing fluctuations. The choice of the interband pair-exchange couplings made in our manuscript is thus motivated by the experimental observation of such masking effect in experiments.
Furthermore, we wish to emphasize that the chemical potential which has to be determined self-consistently by solving numerically the number equation (15) plays a crucial role in the particle transfer between the two bands. As a consequence, the delicate and of key importance interplay between the masking effect and the pseudogap opening is not trivial at all, deserving a quantitative analysis. Such effects have never been discussed in the existing literature in the context of the two-band BCS-BEC crossover.
We have added in the revised manuscript the discussions associated with the above-mentioned issues in Section 2 as well as in Section 3.
We thank Reviewer 3 again. We hope that our revisions to the manuscript and our reply could meet the approval of Reviewer 3 for publication of our work in Condensed Matter.
Round 2
Reviewer 1 Report
The authors have properly taken into account my previous reccomendations and the manuscript can now be accepted for publication.
Reviewer 3 Report
The authors revised the manuscript appropriately so that the relevance of the simple theoretical model to FeSe superconductors can be seen to an enough extent. I now recommend publication of the manuscript in Condensed Matter.
