Motivated by a scarcity of simple and analytically tractable models of
superconductivity from strong repulsive interactions, we introduce a simple
tight-binding lattice model of fermions with repulsive interactions that
exhibits unconventional superconductivity (beyond BCS theory). The model
resembles an idealized conductor-dielectric-conductor trilayer. The Cooper pair
consists of electrons on opposite sides of the dielectric, which mediates the
attraction. In the strong coupling limit, we use degenerate perturbation theory
to show that the model reduces to a superconducting hard-core Bose-Hubbard
model. Above the superconducting critical temperature, an analog of pseudo-gap
physics results where the fermions remain Cooper paired with a large
single-particle energy gap.
Dear Editor and Referees,
Thank you for carefully reviewing our work and for the detailed comments and suggestions. We have taken the revision requests very seriously and we are submitting a new draft with major revisions. We believe that the new draft establishes a clear theoretical and physical motivation and that we have fully addressed the concerns of the referees in the responses below.
Kevin Slagle and Yong Baek Kim
Response to Referee 1:
Physically, there is no fine tuning. The physics is robust to all local perturbations. The exact degeneracy in the 4-site model was just a trick so that we could do degenerate perturbation theory. We now emphasize and explain this just before Section 4. If \mu is not exactly equal to s/2, then this just shifts the chemical potential mu' in the hard-core boson model, which is more explicit in the new draft.
Weakness 2 and requested changes 2 and 3:
We originally stated that the Cooper pair can be thought of as being nodelss p_z-wave since there is antisymmetry in the layer index (in analogy to the nodeless d-wave pairing in Ref. 33 (Ref. 37 in the new draft)). (The sites 1 and 2 in the 4-site model are displaced in the z-direction.) However, after talking to more of the community, it seems that this may be misleading. We now refer to the Cooper pair as s-wave, where the antisymmetry is in the alpha=1,2 index. The Cooper pair is a boson, and the condensate is a bosonic condensate as usual.
We mistakenly previously derived the effective boson model while also assuming s/U << t/s. In the new draft, we drop this rather excessive assumption. Dropping this assumption just adds the \epsilon to the definition of \mu' in Equation 8 of the new draft. Now, all ignored terms are much smaller than O(t^2/s) and are safe to ignore.
In all three of these subsections, we explain how to obtain the exact degeneracy. Theoretically, obtaining a degeneracy is easy by just tuning the chemical potential. The hard part is making sure that the degenerate states have a difference in fermion number equal to two and a large pair-binding energy (see Figure 2b). The solutions are exact in the limits considered in each section. See also our response to weakness 2 of the other Referee.
The spinless nature of the pairing state in our model is actually an advantage since the unimportance of spin could allow a superconducting realization of our model to be more robust to magnetic fields. As mentioned at the beginning of Section 4.3, spinless fermions could be effectively obtained by applying an in-plane magnetic field. However, Section 4.3 shows that a hare-core boson model (where the bosons have 4 spin states) can also be obtained in a similar way using spinful fermions. It is very likely that this hard-core boson model also hosts superconductivity. The new draft makes this claim more explicit, and also includes a more detailed example in Section 4.3.1 that reproduces the hard-core boson found in Section 3, which is known to exhibit superconductivity.
Yes, we are making reference to the fullerene crystals. The similarity to the 4-site model is not related to spinless fermions, but the fact that both models (with the right chemical potential) consist of clusters with low-energy states that have a difference in fermion number equal to two.
We hope the referee now finds that our results are accurate and that a conductor-dielectric-conductor trilayer may be a feasible realization of our toy model.
Requested change 1:
We added two paragraphs to the introduction that discuss the t-J and Hubbard models and help motivate our model.
Requested change 2 and 3:
See weakness 2.
Requested change 4:
All of the figures have captions. Many of the equations have small graphics that accompany them. We have added a little extra clarification below Equation 6 in the new draft.
Requested change 5:
We reworded that paragraph to be more explicit. We just meant that the leading order process resulted in a term of order t^2/s.
Requested change 6:
Thank you bringing this typo to our attention. It is fixed now.
Requested change 7:
Tc~t/2 is more precise. We corrected the footnote.
Response to Referee 2:
Indeed, the original motivation of this work was primarily theoretical: to uncover the simplest analytically-tractable theoretical model of unconventional superconductivity. In the new draft, we added additional physical motivation for a conductor-dielectric-conductor trilayer. See Figure 1 and the second paragraph of Section 5.2.
Perhaps our motivation for this section was not clear. In Section 4, our claim is that those extensions to our model also result in hard-core boson models in the strong interaction limit, and that the hard-core boson models are known to or are very likely to superconduct. We have provided sufficient analysis to support this claim. We have made our motivation for this section more clear in the new draft.
In the new draft, we have added additional discussion to the introduction to discuss this relation.
The model is significantly simpler and more analytically tractable than previous strongly interacting models of superconductivity. However, it is just a toy model, and future work is needed to explore the possibility of a conductor-dielectric-conductor trilayer realization. We added comparisons to other models in the introduction.
1) The introduction has been expanded to discuss previous models of unconventional superconductivity in order to motivate our model.
2) We now physically motivate our model as a toy model for a conductor-dielectric-conductor trilayer in the abstract, introduction, Figure 1, after Equations 2 and 6, and the Discussion (Section 5.2).
3) In response to Referee 1 and private communications, we have clarified the nature of the superconducting order parameter below Equation 3.
4) Figure 2 and Equation 5 have been added to help emphasize the generality of the 4-site model to smaller repulsion.
5) A comment regarding fine tuning was added at the end of Section 3.
6) We further emphasized our motivations in the abstract and Sections 4 and 5.
7) Section 4.3.1 was added.
8) In response to weakness 3 that Referee 1 pointed out, the perturbation theory in he appendix has been slightly reformulated.
A detailed markup of changes to the text can be found at goo.gl/5Wg2bE