Nonlocal order parameter of pair superfluids

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%)