Errors in user-supplied markup (flagged; corrections coming soon)

----------------------------------------------------------------------
Response: Anonymous Report 1 on 2024-2-2 (Invited Report)
----------------------------------------------------------------------

We thank the referee for the report. We have modified the text accordingly in order to address the referees remarks and criticisms.

We acknowledge that the referee recommends the publication of our work in SciPost Phys. Core. However, we disagree and believe that this assessment is mostly due to the presentation of our results, which did not sufficiently clarified their novelty. For this purpose, we have revised introduction and the text in order to emphasize the results we obtained.

As far as it concerns the relation to SciPost Phys. 14, 136 (2023): we now include quantum quantum fluctuations and study the interplay of quantum fluctuations, long-range interactions, and frustration. The novelty of the present work lies thus not only on the methodology, but also on the results we obtain.



We summarize our results:

We identify a number of features that are not captured by the nearest-neighbor truncation and unveil the effect of the interplay between the long-range dipolar interactions with the lattice geometry.

In the limit of vanishing tunnelling, we show that the ground state is a devil's staircase of solid phases, which can be identified up to an arbitrary precision. This precision, in fact, is only limited by the size of the considered unit cells and the optimization scheme we employ. To the best of our knowledge, we are not aware of studies mapping out quantitatively the complete devil's staircase for two-dimensional systems.

For finite tunnelling, we find solid and supersolid phases, some of which have been reported by numerical studies based on advanced quantum Monte Carlo simulations. Differing from these works, we can avoid the limitation imposed by the constraints on the unit cell and thereby unveil a plethora of solid and supersolid phases which have not been reported before. We characterize the corresponding phases, and unveil a very structured phase diagram. Besides the fundamental interest, our results will be a guide for experimentalists working in the field of dipolar gases as well as frustrated magnets.

Finally, our results thus also provide an important benchmark and guidance for numerical programs, identifying the relevant unit cells, simulation geometries, and observables.

Let us finally state that we fully agree with the referee that "The mean field phase diagrams of dipolar bosons has been worked out by several authors in the phase predicting existence of supersolid phase and devil staircase structures". Nevertheless, the majority of these works truncated the dipolar interactions to the nearest-neighbour, as is the case for some of the ones the referee mentions. Indeed, a result of our paper is in line with advanced numerical studies, and show that the nearest-neighbor approximation for dipolar interactions misses to identify multi-lattice solid and supersolid phases, that our resummation approach instead allows to capture.

We have added these references to the conclusions, in the paragraph where we discuss realizations with tilted dipoles.



Let us further comment on the references mentioned by the referee:

- Thieleman et al., PRA 83, 013627 (2011): The authors focus on nearest-neighbor interactions and a complex hopping as well as cold atoms in a staggered flux. In our work, we study a different model because we have no staggered flux. At the same time, we concentrate on the effects of long-range interactions, thus no truncation in the power-law is performed.

- Zhang et al., New J. Phys. 17 123014 (2015: The authors apply QMC for dipolar hardcore bosons on the square lattice, but restrict to half filling. In our work, we investigate the whole phase diagram at any filling for several 2d lattice geometries.

-Isakov et al., Europhys. Lett. 87 36002 (2009): The authors apply variational QMC and Schwinger-boson mean-field theory for half-filling and anisotropic nearest-neighbor interactions giving rise to a staircase of phases. Again, we focus on the effects of the untruncated long-range interactions.





----------------------------------------------------------------------
Response: Anonymous Report 2 on 2024-2-26 (Invited Report)
----------------------------------------------------------------------

We thank the referee for the report. We have modified the text accordingly in order to address the referees remarks and criticisms.

Below we address the points raised by the referee.

i) Regarding the nearest-neighbour case: the literature states that quantum fluctuations have a larger impact on systems with geometrical frustration. The model on the triangular lattice is subject to geometrical frustration, therefore a larger impact of quantum fluctuation on the phase diagram is expected. We have added a clarification of this point in Sec. 5.

ii) The classical approximation captures the transition at the Heisenberg point correctly since the transition is driven by the change in symmetry of the Hamiltonian from an easy-axis to a rotationally invariant interaction. This change in symmetry is present in the classical spin picture, as well as, the full quantum mechanical problem for the same parameter value $t/V=1/2$. We have added a clarification of this point in Sec. 5.

iii) We have strengthened the emphasis on the quantum phases in the results section. We realised that we used the misleading name "order" for "supersolid" phases in Figs. 5, 6, 10. We correspondingly modified it in the figure captions and in the text. Further, we added text passages discussing the different supersolid phases for the square and the triangular lattice.

In Fig. 7, we have added real and momentum space depictions of the complex supersolid phases on the square lattice. We have added a picture how to understand the supersolid phases on the square lattice in terms of defective checkerboard patterns.

Regarding the triangular lattice, we are grateful for the reference to the quantum Monte Carlo study PRL 104, 125302 (2010). We have added a discussion of the reference in comparison to our results in Sec. 7. In the conclusion we highlighted the application of the discussed method to determine relevant observables and unit cells for further numerical studies.

----------------------------------------------------------------------
List of changes
----------------------------------------------------------------------

line 19 "square lattice" -> "square and triangular lattice"

line 55 "important" -> "prominent"

line 56 "Due" -> "Thanks"

line 57 "allow for applications as quantum simulators of" -> "permit to shed light on"

line 58 "of" -> "on the predictions of"

line 79 added "three lattice geometries often discussed in the literature:"

line 82 to 99 paragraph rephrased

line 109 "In the following, we introduce the model in Sec. 2 in the particle as well as the spin language." -> "This paper is organised as follows. In Sec. 2 we introduce the model in the particle as well as the spin language."

line 112 added "In Sec. 2.3 we define the three lattices geometries that are considered in this work (square, honeycomb and triangular lattice)." due to the shift of the paragraph

line 115 "We further discuss how we characterise ground states from the results of the mean-field calculation, as well as the inherent limitations of the approach." -> "We then detail how we characterise ground states from the results of the mean-field calculation, and critically discuss the inherent limitations of the approach."

line 122 "discuss" -> "analyse"

line 128 "investigate" -> "consider"

line 129 "and in a grand-canonical ensemble" -> "with a variable particle number"

line 133 to 136 paragraph rephrased

line 140 "From former studies in the atomic limit without hopping on a one-dimensional chain [61,62] and two-dimensional lattices [34] a devil’s staircase of crystalline phases is expected considering the full long-range interaction." -> A former study analysing the full long-range interations in the atomic limit t = 0 predicts a devil staircase of crystalline phases [34].

line 146 added "By"

line 148 "where" -> "Here,"

line 149 to 152 paragraph rephrased

line 155 "for any value of the exponent $\alpha$"

line 163 "and the absence of any longitudinal field at the symmetry line where" -> "about the symmetry line where the longitudinal field vanishes,"

Sec. 2.3 is former Sec 3

line 211 "superfluid" -> "superfluid phase"

line 213 "the long-range" -> "the corresponding order by long-range"

Eq. (9) indices changed

line 225 and 227 $f=n/m$ corrected

line 230 "unit cells with more than one site" -> "composed by multiple sites"

line 231 "against finite" -> "against quantum fluctuations at finite"

line 242 added "In order to identify supersolids with a complex sublattice structure, a careful choice of an order parameter is required, see Sec. 5.2.1."

Caption Table 1 "momentum" -> "wave-vector"

Caption Table 1 "dependent" -> "determined by"

Caption Table 1 "complex sublattice structure" -> "with more than two sublattices"

line 266 "pattern" -> "pattern minimizing energy"

line 297 "In addition" -> "In addition,"

Caption Fig. 2 "second (first)" -> "first (second)" - statement corrected

Caption Fig.2 "illustrate" -> "display"

line 383 "The phase transition between phases is first-order, if the two phases have a different diagonal long-range order." -> "First-order phase transitions separate two phases with different diagonal long-range order."

line 406 added a paragraph on the Heisenberg point

line 415 added a paragraph on the impact of frustration

line 430 "present" -> "analyse"

line 432 "The classical spin approach ground states ... resulting from the numerical optimization procedure, which were used to create Figs. 5-7, can be found in Ref. [79]." -> "We refer to Ref. [64] for the data on the classical spin approach ground states ... resulting from the numerical optimization procedure. This data was used to create Figs. 5-8."

line 457 "in some" -> "even though in some"

line 466 "discuss" -> "analyse"

Fig. 5 has changed. The misleading term "order" has been replaced by the more accurate term "supersolid".

Fig. 6 has changed. The misleading term "order" has been replaced by the more accurate term "supersolid". The translational vectors depicted for the "5/14 solid" and "4/7 - 1/14 - 5/14 supersolid" have been wrong and were replaced by the correct ones.

Fig. 7 has been added.

line 503 to 516 paragraphs rephrased

line 542 "In comparison" -> "In contrast to"

line 544 to 557 paragraph rephrased

line 577 to 582 paragraph rephrased

Fig. 10 has changed. The misleading term "order" has been replaced by the more accurate term "supersolid".

line 593 to 595 rephrased and shortened paragraph

line 596 to 626 added a paragraph discussing the reference L. Pollet et al. [36] (2010)

Sec. 7 Conclusion has been reformulated extensively

line 683 "KPS acknowledges" -> "KPS and JAK acknowledge"

The reference Su2023 has been updated from the arXiv to the published version.

The reference Adelhardt2024Review was added.

The reference Koziol2024 was added.

The reference Lan2018 added.

The reference Pollet2007 was added.

The reference Pollet2010 was added.

The reference Prokofev1998 was added.

The reference Suthar2020 was added.

The reference Zhang2022 was added.