Nonlocal order parameter of pair superfluids

This paper proposes a non-local order parameter "odd parity", which is a modified version of "parity" operator introduced in earlier literature. The authors argue that it characterizes the pair superfluid (PSF) phase, and demonstrate numerically in 1 and 2 dimensions. Indeed the numerical results seem to confirm the picture, and the odd parity operator looks quite interesting and useful.

However, I find the discussion in the paper rather unsatisfactory and cannot recommend publication in the present form.

1. For simplicity let us discuss 1 dimension. By definition, for a fixed string length $j$, the parity operator $O_p(j)$ and the odd parity operator $O_p^{(o)}(j)$ are related as

   $$ O_p(j) = (-1)^{j-1} O_p^{(o)}(j) $$
   . Then how can they detect different phases? I suppose that, the hidden assumption is that the nonlocal order parameter has a definite sign, i.e. it does not change the sign for sufficiently large $j$. So if $O_p(j)$ is non-vanishing but is oscillating and changes sign like $(-1)^j$, we do not regard the parity to be long-ranged but the odd parity is long-ranged (and vice versa). Do you agree? In any case, please clarify. It is also very important to describe how the authors extracted the non-local order parameter $O_p^{(o)}$ from the numerical data $O_p^{(o)}(j)$ for various different $j$'s. This should be one of the essential information the authors are obliged to present.

2. As the authors note, under the three-body constraint, the Bose-Hubbard is mapped to the S=1 system. In the 1D context (S=1 chain), in my understanding, the PSF is nothing but the "XY2" phase identified by Schulz in the seminal paper Ref. [64]. Although the authors do cite Ref. [64], I believe that it would be fair to mention explicitly that the 1D PSF phase was essentially discovered in Ref. [64].

3. Related to the previous point, in Ref. [64] the Antiferromagnetic (Neel) phase was also identified. I believe that the odd parity operator is also non-vanishing in the antiferromagnetic phase, which corresponds to a charge-density wave state in the Bose-Hubbard context. So the non-vanishing odd parity alone does not imply that the system is in PSF phase. For 1D Bose-Hubbard model, according to Fig. 2(b), the authors numerically confirmed the power-law decay of $D(r)$. Moreover, Fig. 3 (b) indicates that $\Delta_2$ always vanishes for the range of $U/t$ studied. So I agree that the system is indeed in the PSF phase when the odd parity is non-vanishing within the model studied by the authors. (However, one should also be able to find the antiferromagnetic phase by modifying the Hamiltonian.) For 2D, Fig. 4(b) might suggest that the system is PSF and not antiferromagnetic/CDW, but I do not feel the evidence is sufficient. How does $D(r)$ look like, for example at $U=-20$??

4. Please give the detailed derivation of Eq. (9) (bosonized expression of the odd parity operator), which is rather important for this paper.

5. I have mostly considered the simplest case especially in 1D. Please also carefully re-examine the cases of the arbitrary filling and the two-species system, following the above comments.

After I formed my own opinion, I also checked the reports by other referees; I think the concern of Referee 2 overlaps my points, especially (1) above.

Ask for major revision

Dear Editors,

This is my reviewer's report on the manuscript https://scipost.org/submissions/2404.15972v2/, by N. Cuzzuol, A. Montorsi, and L. Barbiero, which is under consideration for publication in SciPost Physics. I do not find the paper suitable for publication in its present form, because the text seems to lack accuracy. I suggest a major revision along the lines presented below.

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In this theoretical and numerical study, the Authors analyse the Pair SuperFluid (PSF) Phase for two different Hamiltonians: (i) the single-species Bose-Hubbard model in 1D or 2D (Eq. 1) with on-site attraction and a three-body constraint; (ii) the two-species Bose-Hubbard model (Eq. 11) with on-site interspecies attraction and a hard-core constraint. They introduce a quantity (Eq. 8) which they call the 'odd parity', and show numerically that it is an order parameter for the PSF phase.

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My two more important questions A, B are related to the odd parity defined in Eq. 8 on page 5:

(A) Could the Authors explain in greater detail how their quantity $O_P^{(0)}(j)$, defined in Eq. 8, provides different information from the closely related parity $O_P(j)$, previously defined in Ref. [10: Berg PRB 2008], and explicitly mentioned on p. 4, 3 lines from the bottom of the page ?

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(B) The Authors write (abstract, l. 7; and again p. 2, 4 lines from end) that the odd parity is the "unique order parameter" of the PSF phase. However, Ref. [44: Bonnes PRL 2011] also introduce the order parameter <b_i^2> in a similar context. What are the differences in the roles of these two quantities ?

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The following remarks (1-4) are more minor, but may help in making the manuscript more accessible to a non-specialised audience.

(1) Concerning the second considered model (Eq. 11): Figure 6a seems to suggest that the species A and B are trapped in two spatially-separated optical lattices, but this does not explicitly appear in the Hamiltonian: does this subtlety in the proposed realisation play a role ? Why do the paired bosons belong to sites of the two lattices which differ by one lattice spacing rather than to corresponding sites ?

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(2) The Authors use two different methods: bosonisation (Sec. 2.1) and DMRG (Sec. 3). Matrix Product States are also mentioned once in Sec. 3.2. Would the Authors consider including a section entitled "Methods", in which they discuss the strengths and limitations of these approaches ?

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(3) Two keywords seem to be used without a definition or reference: (a) "snake-like path", on p. 7, paragraph 1, line 6; (b) "C_4 symmetric 2D limit", on p. 7, paragraph 3, line 2.

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(4) I am surprised at the Authors' usage of "normal superfluid" as a single keyword (abstract, l. 13; and again p. 2, paragraph 3, l. 9). Is this standard terminology? If so, could the Authors provide a reference using it?

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Ask for major revision

The paper by Cuzzuol et al. is a vry interesting paper a kind if exotic superfluids, the so calls air sulerfluids. The authors introduce a novel nonlocal order parameter, named odd parity. They show that pair superfluids can be rigorously defined in terms of it.Moreover , this order parameter is experimentally accessible. As a case of study, they investigate a constrained Bose-Hubbard model at different densities, both in one and two spatial dimensions. Here, pair superfluidity occurs for relatively strong attractive interactions. The odd parity operator acts as the unique order parameter for such a phase. They apply also their approach to a two-component Bose-Hubbard Hamiltonian. Such a system is at the centre of interestinf investigations in ultracold atoms.

The paper is very clear and very well written. The studies are very profound and cover all relevant aspects of the problem. I recommend publication in SciPOst.

I have only minor suggestions:
a)it would be useful to core a review on non-standard Hubbardmodels where pair supefluidity is discussed thoroughly in various context:

1. Omjyoti Dutta, Mariusz Gajda, Philipp Hauke, Maciej Lewenstein, Dirk-Sören Lühmann, Boris A. Malomed, Tomasz Sowiński, Jakub Zakrzewski, Non-standard Hubbard models in optical lattices, Rep. Prog. Phys. 78, 066001 (2015), arXiv:1406.0181.
2. Titas Chanda, Luca Barbiero, Maciej Lewenstein, Manfred J. Mark, and Jakub Zakrzewski, Recent progress on quantum simulations of non-standard Bose-Hubbard modes, arXiv:2405.07775

no weaknesses

The paper by Cuzzuol et al. is a vry interesting paper a kind if exotic superfluids, the so calls air sulerfluids. The authors introduce a novel nonlocal order parameter, named odd parity. They show that pair superfluids can be rigorously defined in terms of it.Moreover , this order parameter is experimentally accessible. As a case of study, they investigate a constrained Bose-Hubbard model at different densities, both in one and two spatial dimensions. Here, pair superfluidity occurs for relatively strong attractive interactions. The odd parity operator acts as the unique order parameter for such a phase. They apply also their approach to a two-component Bose-Hubbard Hamiltonian. Such a system is at the centre of interestinf investigations in ultracold atoms.

The paper is very clear and very well written. The studies are very profound and cover all relevant aspects of the problem. I recommend publication in SciPOst.

I have only minor suggestions:

Minor suggestions

a)it would be useful to core a review on non-standard Hubbardmodels where pair supefluidity is discussed thoroughly in various context:

1. Omjyoti Dutta, Mariusz Gajda, Philipp Hauke, Maciej Lewenstein, Dirk-Sören Lühmann, Boris A. Malomed, Tomasz Sowiński, Jakub Zakrzewski, Non-standard Hubbard models in optical lattices, Rep. Prog. Phys. 78, 066001 (2015), arXiv:1406.0181.
2. Titas Chanda, Luca Barbiero, Maciej Lewenstein, Manfred J. Mark, and Jakub Zakrzewski, Recent progress on quantum simulations of non-standard Bose-Hubbard modes, arXiv:2405.07775

Publish (surpasses expectations and criteria for this Journal; among top 10%)