Resonant Valance Bond Ground States on Corner-sharing Lattices

The authors study the infinite-$U$ Hubbard model on the sawtooth lattice and the 1D pyrochlore chain, showing that the ground state in the single-hole doped case is a resonating-valence-bond (RVB) state. They also analytically derive the exact ground states and their energies analytically in the thermodynamic limit, which are found to be consistent with the exact diagonalization results for small system sizes. While the results are interesting, there are several concerns regarding the relation to previous literature and numerical accuracy that need to be addressed before I can make a final recommendation

1. Historical remark In the Introduction, the authors state, "It was recently proposed that perhaps a more physically realistic route towards an exact RVB state is via frustration of the kinetic energy of itinerant electrons [10], as opposed to localized spin interactions." However, similar ideas were already discussed in the early 1990s. For instance, please see U. Brandt and A. Giesekus, Phys. Rev. Lett. 68, 2648 (1992); A. Mielke, J. Phys. A 25, 6507 (1992); H. Tasaki, Phys. Rev. B 49, 7763 (1994); A. Giesekus, Phys. Rev. B 52 2476 (1995) and references therein. It seems quite likely that the system studied in Ref. [10] falls into the category of the models studied earlier. The authors should clarify this point to place their results in a proper context.

2. Numerical estimate of the ground state energy At the end of Section 2, the authors extrapolate the ground state energies from $L=12$ and $13$, obtaining $E^{\rm min}_{\rm GS}=-3.2713t$. However, since $H_{\rm eff}$ in Eq. (4) is essentially a noninteracting model, it should not be hard to study much larger system sizes numerically. I have performed my own numerical check and found that the ground state energy for $L=2000$ agrees with $E_{\rm GS}=-(\sqrt{5}+1) t$ to five decimal places.

Furthermore, it may not be appropriate to call $E^{\rm min}_{\rm GS}$ a "lower bound," as it does not constitute a rigorous bound in a mathematical sense. I suggest that the authors clarify their terminology.

1. Section 3.2 I find the application of a non-Abelian generalization of the flux inequality to be interesting. However, it is not clear to me why the authors need to prove the energy level ordering in Eq. (14) in the current way; it seems to naturally follow from the local cluster analysis discussed in Sec. 3.1, as long as we are concerned with the ground state. The authors should clarify the added value or the necessity of the flux inequality analysis in this section.

2. Bethe ansatz approach Regarding the future perspective of applying a nested Bethe ansatz to the hole-doped case, I would like to point out a previous attempt for a related system: S. Gayen and I. Bose, J. Phys.: Condens. Matter 7, 5871 (1995). It should be noted, however, that they employed the standard Bethe ansatz rather than a nested one.

3. Minor comments:

4. Title and abstract: The abbreviation "RVB" typically stands for "resonating valence bond state," as is the case in Anderson's original paper cited as Ref. [1].

5. page 2, Eq. (2): the authors might want to explicitly mention that |vac> denotes the vacuum state.

6. Page 3: the authors state, "The 3-fold degenerate ground states are presented in Fig. 3 (c), ...", but they are in fact 6-fold degenerate when the spin degree of freedom ($\sigma=\uparrow$, $\downarrow$) is taken into account.

7. page 3, footnote 2: 'even thoug' at the end should read 'even though'. Also, the last sentence appears to be incomplete (truncated).

Ask for minor revision

1) insightful combination of analytical consideration and numerical results

1) overly complicated presentation

1) The RVB state stands for the resonatING valence bond state not a resonANT one. I do not want to go into explaining semantic differences between the TWO, though they do exist. Instead I'd like to point that Anderson in his 1973 article as well as in all subsequent publications was using 'Resonating'. The recent authoritative reviews such as, e.g., L. Balents, Nature 464, 199 (2010), also use
"resonating". Why not to stick to the classics? 2) Typo in the title in "valance" 3) The projection operators that impose the infinite U constraint must be explicitly defined together with the Hamiltonian (1) 4) Reading of the paper gives a feeling of an "over-complicated" language for simple things. For example, on p.3, the authors use the 'Bravais lattice' referring to a simple one-dimensional chain. I suggest to use in that instance something like "an index labels unit cells" rather than "sites of the Bravais lattice." There are further examples through the text but I leave their correction to the author.

Ask for minor revision

1- On page 3, in footnote 2, the last sentence appears to be incomplete. 2- I do not see how the normalization factors $\dfrac{1}{\sqrt{6}}$ in Fig. 5 are obtained. 3-I would not object to renaming the “flux inequality” as the “diamagnetic inequality,” since this is the terminology most commonly used in the literature, including in the cited references. 4-The manuscript states that inequality (20) is “obviously” true because inequality (21) holds. However, it is not clear to me (i) why inequality (21) holds and (ii) how (20), in its squared form, follows from it. A few sentences clarifying these steps would be useful.

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