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Quantum phases of hardcore bosons with repulsive dipolar densitydensity interactions on twodimensional lattices
by J. A. Koziol, G. Morigi, K. P. Schmidt
This is not the latest submitted version.
Submission summary
Authors (as registered SciPost users):  Jan Alexander Koziol · Giovanna Morigi · Kai Phillip Schmidt 
Submission information  

Preprint Link:  https://arxiv.org/abs/2311.10632v1 (pdf) 
Data repository:  https://zenodo.org/records/10126774 
Date submitted:  20231120 14:09 
Submitted by:  Koziol, Jan Alexander 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approaches:  Theoretical, Computational 
Abstract
We analyse the groundstate quantum phase diagram of hardcore Bosons interacting with repulsive dipolar potentials. The bosons dynamics is described by the extendedBoseHubbard Hamiltonian on a twodimensional lattice. The ground state results from the interplay between the lattice geometry and the longrange interactions, which we account for by means of a classical spin meanfield approach limited by the size of the considered unit cells. This extended classical spin meanfield theory accounts for the longrange densitydensity interaction without truncation. We consider three different lattice geometries: square, honeycomb, and triangular. In the limit of zero hopping the ground state is always a devil's staircase of solid (gapped) phases. Such crystalline phases with broken translational symmetry are robust with respect to finite hopping amplitudes. At intermediate hopping amplitudes, these gapped phases melt, giving rise to various lattice supersolid phases, which can have exotic features with multiple sublattice densities. At sufficiently large hoppings the ground state is a superfluid. The stability of phases predicted by our approach is gauged by comparison to the known quantum phase diagrams of the BoseHubbard model with nearestneighbour interactions as well as quantum Monte Carlo simulations for the dipolar case on the square lattice. Our results are of immediate relevance for experimental realisations of selforganised crystalline ordering patterns in analogue quantum simulators, e.g., with ultracold dipolar atoms in an optical lattice.
Current status:
Reports on this Submission
Strengths
1) The manuscript is clearly written. A brief review of the method used is provided. All relevant physics is neatly summarized in the first sections of the manuscript.
2) Citations are carefully chosen and very relevant to the topic.
3) Results are for the most part clearly described.
4) The results of this work are interesting and relevant to current experiments with ultracold gases.
5) The manuscript provides novel in depth analysis of the strongly interacting regime (where the method presented in most accurate). A plethora of solid phases are presented which had not been observed before with approximationfree methods.
Weaknesses
1) On the one hand, significant emphasis is given to QUANTUM phases, such as supersolid phases, in the introductory part of the manuscript. On the other hand, most of the description of results is dedicated to solid phases. Only supersolid phases at 1/2 (for bipartite) and 1/3,1/4 (for nonbipartite) are marked in the phase diagrams. While the 1/4 supersolid in triangular lattice is new, the others were already known. It would be better to further describe novel supersolids such as the ones mentioned for square lattice (for honeycomb lattice limitations of the method are discussed). For example, what are supersolids "with more complex sub lattice structure"? Maybe a picture of density patterns of this supersolid would help. If it is already there, then it is not clear that it does refer to a supersolid phase. Moreover, the authors should comment about how their results compare with QMC study of hardcore purely repulsive dipolar bosons in triangular lattice (PRL 104, 125302 (2010)).
Report
Overall, this is an interesting, wellwritten manuscript which presents results on manybody hamiltonians relevant to current experimental efforts in ultracold gases. It meets the criteria for publication of SciPost and I therefore recommend it for publication once the requested changes are addressed.
Requested changes
1) Do authors have an explanation on why their method does not compare as well with approximationfree numerical methods in the case of nonbipartite triangular lattice?
2) Why is the method so much more accurate at the Heisenberg point?
3) Can the authors address the point mentioned in the Weaknesses Section of this report?
Strengths
1) A detailed meanfield analysis of dipolar bosons particularly in a triangular lattice where QMC data does not exist.
Weaknesses
1) The paper seems to be a possible and somewhat expected application of the formalism developed in Ref 34 of the work (by the same authors).
2) 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. See for example Phys. Rev. A 83, 013627 (2011), New J. Phys. 17 123014 (2015) or Europhys. Lett. 87 36002 (2009). In fact there are many other similar results in the literature.
Report
I believe, based on the weaknesses mentioned, the paper in its present form is suitable for publication in Scipost Phys. Core. Whereas the analysis is detaiked and it deserves publication in some form, I do not see sufficiently new results which allows consideration in Scipost.
Author: Jan Alexander Koziol on 20240712 [id 4617]
(in reply to Report 1 on 20240202)
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, longrange 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 nearestneighbor truncation and unveil the effect of the interplay between the longrange 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 twodimensional 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 nearestneighbour, 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 nearestneighbor approximation for dipolar interactions misses to identify multilattice 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 nearestneighbor 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 longrange interactions, thus no truncation in the powerlaw 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 Schwingerboson meanfield theory for halffilling and anisotropic nearestneighbor interactions giving rise to a staircase of phases. Again, we focus on the effects of the untruncated longrange interactions.
Author: Jan Alexander Koziol on 20240712 [id 4618]
(in reply to Report 2 on 20240226)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 nearestneighbour 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 easyaxis 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.