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Universal relations in flat band superconducting bipartite lattices
by G. Bouzerar and M. Thumin
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Submission summary
Authors (as registered SciPost users):  Georges Bouzerar 
Submission information  

Preprint Link:  scipost_202308_00024v1 (pdf) 
Date submitted:  20230817 16:33 
Submitted by:  Bouzerar, Georges 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
Unconventional flat band (FB) superconductivity, as observed in van der Waals heterostructures, could open promising avenues towards highT$_c$ materials. Indeed, in FBs, pairings and superfluid weight scale linearly with the interaction parameter, such an unusual behaviour justifies strategies to promote FB engineering. Bipartite lattices (BLs) which naturally host FBs could be particularly interesting candidates. By revealing a hidden symmetry of the quasiparticle eigenstates, we demonstrate that pairings and superfluid weight obey universal relations in BLs. Remarkably, these general properties are insensitive to disorder as long as the bipartite character is protected.
Current status:
Submission & Refereeing History
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Reports on this Submission
Strengths
1  Extensions of results for flatband superconductivity beyond the uniform pairing condition
2  Careful meanfield study of BdG Hamiltonians on biparitite lattices
Weaknesses
1  No discussion of beyond meanfield effects
2  Lack of clear discussion of the novelty of the findings
Report
The authors study the solution of the Bogoliubov de Gennes Hamiltonian on bipartite lattices. They focus on the zerotemperature limit and impose timereversal symmetry. They find bounds on the pairing strength in terms of the ratio between the number of lattice sites in each sublattice and how the superfluid weight depends on the filling of the flat band.
Their findings agree with previous results reported in the literature and are not entirely novel. The main advancement is the identification of a symmetry in the BdG equations on bipartite lattices. This symmetry allows them to extend previous results beyond the uniform pairing condition, i.e., the assumption of equal pairing on all sublattices on which the flat band eigenstates have nonzero weight.
What is lacking and necessary is a better framing in the existing literature. In particular, the authors should better address the regime of validity of their findings. PRB 94, 245149, and Nat. Comm. 6, 8944 show how the BCS wavefunction is an exact zerotemperature ground state of isolated flat bands with onsite attractive interactions. While the latter is restricted to bipartite lattices, the first extends this result beyond bipartite lattices with uniform pairing. Further evidence for the validity of the BCS approximation at zero temperature is provided by the quantum Monte Carlo (QMC) results of PRB 102, 201112; PRL 130, 226001; PRL 128, 087002; PRL 126, 027002.
The authors could use the above references to better argue for the validity of their meanfield approach. For example, the SU(2) symmetry they find at the meanfield level, would at face value forbid any finitetemperature transition in 2D. The QMC results show how this problem is circumvented beyond meanfield.
The authors should address at least the following questions:
 What happens without timereversal symmetry? Which result will break down?
 What will happen in the presence of intrasublattice pairing or longerrange interactions?
 They should distinct between gapless and gapped flat bands. In the latter, the meanfield treatment seems better justified.
 Which results are strictly valid for the flat bands and which for any band of the bipartite lattice?
Lastly, to meet the criteria upheld by sciPost, the authors should specify the relevance of their findings. I think they could stress more the importance of extending previous results beyond the uniform pairing condition.
Some minor remarks on the formatting of the paper:
 The authors should present the noninteracting band structure of the tightbinding model studied in the supplemental material.
 There seems to be an issue with the citation of equations and references. For example, below Eq. (35) the authors refer to Eq. (7) of the supplemental material rather than Eq. (34) of the main text. Similarly, below Eq. (11) they cite Ref. (1) rather than Ref. 13.
 Ref. 9 is repeated in Ref. 10.
Strengths
Presents a detailed and explicit solution of a meanfield model.
Weaknesses
1. Validity beyond meanfield is not clear or even properly addressed.
2. Lots of relevant work is not cited
3. Broad applicability is unclear.
Report
The manuscript introduces and solves a model for BCS pairing in the attractive Hubbard model on bipartite lattices. The bipartite structure introduces a (wellknown) pseudospin SU(2) symmetry that allows some additional analytical tractability. The authors prove various results about this model within the framework of the Bogoliubovde Gennes (BdG) mean field equations, and use this to derive bounds on the superfluid stiffness etc. The key step is the identification of a "hidden symmetry" in the BdG eigenstates, i.e. a property of the meanfield solutions.
I am not persuaded of the validity of the results beyond the meanfield limit assumed without any clear justification. Indeed, there are convincing results (see e.g. PRB 94, 245149) suggesting that the exact BCS wavefunction and hence the BdG solutions are only valid in some special limits. Furthermore given that bipartite lattices often admit signfree Quantum Monte Carlo there exist unbiased numerical studies of the attractiveU Hubbard problem on such lattices even with flat bands (e.g. PRB 102, 201112), but no effort is made by the present authors to compare to these results or address their implications for their work. This lack of any embedding of their work into the very intense and broader range of activity in the field to my mind actually raises the issue to one of poor writing, and as such it is very difficult to assess the significance or originality of the work (which is not to immediately say that it is unoriginal or insignificant, only that a reader is forced to suspend their judgement in the absence of evidence.)
At present I believe the manuscript falls far below the criteria for acceptance in SciPost Physics. At minimum, the authors must explain the extent to which their results are valid outside the meanfield model, or else reframe the paper as a series of exact results on the mean field model, in order to be able to make a fair assessment on whether the paper may be suitable. I believe there may exist some argument to be made that new exact results on a mean field model could be of interest but the bar is high, and if there is not a plausible case to be made of either the importance of these new results or for their validity beyond mean field my recommendation would be to reject; however I am willing to consider that authors may be able to rebut these criticisms in a major revision.
Requested changes
See report.
Report
This manuscript studies some aspects of superconductivity, specifically various "universal" relations between the pairing amplitudes and superfluid stiffness, in “flatband” systems in the presence of *only* an onsite attractiveU Hubbard interaction, and assuming (without any justification; see below) that meanfield theory is applicable. Unfortunately, the paper does make some misleading statements (perhaps unintentionally), especially when it comes to presenting results using the said meanfield theory as if they hold well beyond this uncontrolled and unjustified limit. At the same time, the paper fails to cite many important papers, including an early paper by Tovmasyan et al, Phys. Rev. B 94, 245149 (2016), which beautifully highlighted many important aspects of the flatband problem with onsite attractiveU Hubbard interaction, including where BCS theory is valid. As shown by these authors, the only limit where the BCS wavefunction is the actual groundstate wavefunction, thereby providing some a posteriori justification, is tied to an emergent SU(2) symmetry in the problem. Furthermore, they also derived relations between the superfluid stiffness and pairing amplitudes (they are both related to the interaction strength, as well as some other invariants). However, in this limit even though the stiffness is finite, the transition temperature vanishes.
I do not find this manuscript well written/presented and contributing any fundamental new insights into the problem (though it tries to present results in a “rigorous” fashion, even when the starting point of the approach lacks much rigor). Ultimately, all of the calculations involve various manipulations of the same BdG equations. At this point, based on the large existing literature of similar analysis of the FB problem using meanfield theory, I find that this work is just a straightforward extension of these ideas and does not meet the criteria for acceptance in SciPost.
Interestingly, there is repeated mention of how well BCS meanfield theory apparently agrees with DMRG. However, as has been pointed out in various unbiased, largescale twodimensional QMC simulations, BCS meanfield theory also fails in dramatic ways once the onsite attraction is perturbed by infinitesimal nearestneighbor interactions (Hofmann et al., ‘20, ‘23; Peri et al., ‘21). The authors do not cite any of these papers as well.