A Simple Mechanism for Unconventional Superconductivity in a Repulsive Fermion Model

The authors introduce a tight-binding model of fermions with repulsive interactions that exhibits unconventional SC.

(1) A main result is that the model produces pseudo-gap-like physics above the transition temperature, with fermions remaining paired and with a large single-particle energy gap.

(2) The main model—a 4 site model of spinless fermions, is introduced in Eq.(2). The discussion that follows shows that this model has features that mimic superconductivity (at least in terms of energetics), and in a certain limit produces and effective attractive interaction.

(1) The model is not physical motivated, i.e., the sort of miscroscopic crystal and local orbital structures that would lead to this are not clearly motivated. It seems the motivation is simply to produce a model with the desired features. Once this is established, the authors go on to study a grid of coupled 4-site models form a two-dimensional square lattice.

Section 4 of the paper goes on to address the lack of realism of the model by discussion some extensions of it.
(2) Mostly this section uses order of magnitude estimates to argue what the behavior of the system will be in various parameter limits. Very few numerical calculations or numerically generated phase diagrams are exhibited.

(3) I also found missing a detailed discussion of how this work relates to prior work on repulsive SC, such as in Ref [25-27].

In summary, I found the paper did not present a clear message of what this model accomplishes and where it falls short. It would also be helpful to better relate this model to other repulsive fermion models of SC for comparison/contrast. How is this model “better” than prior models of repulsive Hamiltonians supporting SC?

I recommend the authors address the weaknesses before publication.

1) The manuscript produces a simple model to understand superconductivity from repulsive interaction.
2) Overall well-written and well-organized work, which is easy to follow. Building up a case for superconductivity from simple local clusters to lattices and then extending with some perturbations.

1) The manuscript starts with a simple 4-site model with a degenerate ground state acting as a hard-core boson state with 0 or 1 bosons, respectively. However, this is only achieved at the extreme fine-tuning point of mu = s. Then, coupling such 4-site models into a lattice the authors again reach a hard-core model to order t^2/s (Eq. 7) at the fine-tuning point mu = s/2. The authors then try to improve on the model to make it more general. However, to me a large limitation seems to be this extreme fine-tuning. What happens if mu is not exactly s or s/2? Are the results approximately valid or is it crucial to have an exact ground state degeneracy?
2) The manuscript derives effective attractive Hamiltonians for hard-core bosons and then use previous results to proclaim a superfluid state which is argued to be a superconductor as well. It would be good if the manuscript is more explicit on this in order to clarify the superconducting /superfluid state. For example on p. 1 it is mentioned that a bosonic bound state is formed, which then condenses. However, on p. 2 it seems to be stated that the condensate has p_z symmetry, which would then mean overall Fermi-Dirac statistics for the condensate (since it’s spinless)? In short, will the condensate be bosonic or fermionic and what are the implications for the order parameter symmetry?
3) How unimportant are the ignored terms in the effective Hamiltonians? For example in Eq. 4 terms of order s^2/U are ignored and similarly in Eq. 7. In fact, from the derivation of Eq. 7 in the Appendix it is not even obvious that higher order terms are smaller. Can this also complicate the fine-tuning that is need (see point 1)?
4) In sections 4.1-4.3 perturbations are added to the system, but how do these effect the exact degeneracy that is needed for the ground state in the very first step of the argument (see also point 1)? For example in Eq. 8, why not solve the 4-site model exactly and show that the ground state is still exactly degenerate?
5) A large limitation to achieve anything similar to this model in real materials seems to be the spinless nature of the pairing state. Section 4.3 is spent trying to remedy this, but beyond looking at only a single cluster, the results seems to only be speculations?
6) One of the few actual materials mentioned for this work is doped C60. Are these the fulleride crystals? It would be very interesting if the authors can elaborate a bit more on the said similarities between this work and superconductivity in the fullerides. For example, how can the fullerides be considered to be spinless superconductors?

The manuscript provides an interesting idea to generate superconductivity form repulsive interactions by explicitly showing how a clever cluster can generate an attractive model for hard-core bosons. Extending this to lattices requires for some necessary approximations and one weakness is that it is not clear how accurate this result is then (see e.g. points 1,3 under weaknesses). Also, while the authors brings up a set of possible perturbations and show that the results still hold, it is still very unclear how feasible it is to even try to apply this model framwork to any sort of material. For example the spinless property is problematic, but so is the intricate orbital/site structure needed. In fact, many superconductors with multiple low-energy orbitals have an intricate dependence, even for the order parameter, on the orbital structure, for example producing odd-parity interorbital pairing states in Cu-doped Bi2Se3. The competition with such states are seemingly not taken into account here.

1) In the introduction there is no mentioning of mechanisms causing pairing from extremely strong repulsive interactions, such as the t-J model. For completeness such references would be appropriate.
2) Below Eq. 3, it is never explained why this is a p_z Cooper pair, and in what direction is z?
3) Why call the state nodeless and not just gapped?
4) Several figure texts are missing.
5) The logic leading up to Eq. 7 seems to be reversed: saying H’ must act twice and thus perturbation theory to order t^2/s is enough. I believe the final result might be ok, but the argument is strange.
6) Why is n_5 missing in the last line in Eq. 10?
7) In Section 5 it is first stated that Tc~t/2, but in the footnote it says Tc~t, which one is correct?