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arxiv: 2504.03482 · v1 · submitted 2025-04-04 · ❄️ cond-mat.quant-gas · cond-mat.stat-mech· physics.comp-ph· quant-ph

Supersolid phase in two-dimensional soft-core bosons at finite temperature

Sebastiano Peotta , Gabriele Spada , Stefano Giorgini , Sebastiano Pilati , Alessio Recati This is my paper

Pith reviewed 2026-05-22 21:26 UTC · model grok-4.3

classification ❄️ cond-mat.quant-gas cond-mat.stat-mechphysics.comp-phquant-ph
keywords supersolidsoft-core bosonstwo dimensionsfinite temperaturequantum Monte CarloHartree-Fockhexatic phaseBerezinskii-Kosterlitz-Thouless
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The pith

Two-dimensional soft-core bosons form a supersolid over a broad low-temperature region separating superfluid from quasi-crystal phases.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines the supersolid phase in two-dimensional soft-core bosons at finite temperatures by combining self-consistent Hartree-Fock mean-field theory with quantum Monte Carlo simulations. An initial phase diagram from the mean-field method is refined and validated by path-integral Monte Carlo runs that track superfluid density and the decay of positional and orientational correlation functions. These calculations locate a wide supersolid window at low temperatures that lies between the uniform superfluid and the normal quasi-crystal. They also flag a narrow possible hexatic region with quasi-long-range orientational order between the normal solid and fluid phases. The work treats the Hartree-Fock approach as a practical complement to heavy simulations for mapping how supersolids melt.

Core claim

Using the self-consistent Hartree-Fock and quantum Monte Carlo methods, an approximate phase diagram at finite temperatures is constructed, identifying a broad region at low temperatures where the supersolid phase exists, separating the uniform superfluid phase from the normal quasi-crystal phase. Additionally, a potential intermediate hexatic phase with quasi long-range orientational order is identified in a narrow region between the normal solid and fluid phases. Superfluid and melting/freezing transitions are analyzed through the superfluid density and the long-range behavior of correlation functions associated with positional and orientational order, in accordance with the general Berezs

What carries the argument

Self-consistent Hartree-Fock mean-field theory validated by path-integral quantum Monte Carlo, which computes superfluid density together with the long-distance decay of density-density and orientational correlation functions to locate Berezinskii-Kosterlitz-Thouless transitions.

If this is right

  • The supersolid occupies a wide low-temperature strip between uniform superfluid and normal quasi-crystal states.
  • A narrow hexatic window with quasi-long-range orientational order may separate the normal solid from the fluid.
  • Self-consistent Hartree-Fock beyond the local-density approximation serves as a practical tool for mapping supersolid melting in other soft-core potentials.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same mean-field plus Monte Carlo workflow could be applied to dipolar or Rydberg-dressed bosons to test whether the supersolid window persists under different interaction tails.
  • Ultracold-atom experiments that measure both superfluid fraction and Bragg-peak width at the predicted densities and temperatures could directly test the reported boundaries.
  • If the hexatic region survives in more accurate simulations, it would raise the separate question of whether a hexatic supersolid can also appear before full positional order sets in.

Load-bearing premise

The self-consistent Hartree-Fock mean-field theory produces a phase diagram accurate enough that its supersolid region survives refinement by quantum Monte Carlo without qualitative shifts in location.

What would settle it

Quantum Monte Carlo runs that find the superfluid density remaining finite inside the reported low-temperature supersolid window while positional order persists would support the claim; the opposite result would falsify it.

Figures

Figures reproduced from arXiv: 2504.03482 by Alessio Recati, Gabriele Spada, Sebastiano Peotta, Sebastiano Pilati, Stefano Giorgini.

Figure 1
Figure 1. Figure 1: FIG. 1. Phase diagram of 2D soft-core bosons as a function of temperature [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Upper panel: optimal lattice constant [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Determination of the transition temperatures [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. PIMC simulations for [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Superfluid fraction [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Orientational order parameter [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Left: Triangular lattice generated by the vectors [PITH_FULL_IMAGE:figures/full_fig_p017_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Upper panel: bulk modulus in the homogeneous and [PITH_FULL_IMAGE:figures/full_fig_p019_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Left: phase diagram of soft-core bosons as in Fig. [PITH_FULL_IMAGE:figures/full_fig_p020_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Finite size scaling of the order parameters in PIMC simulations. Upper row: superfluid fractions computed at four [PITH_FULL_IMAGE:figures/full_fig_p021_11.png] view at source ↗
read the original abstract

The supersolid phase of soft-core bosons in two dimensions is investigated using the self-consistent Hartree-Fock and quantum Monte Carlo methods. An approximate phase diagram at finite temperatures is initially constructed using the mean-field approach, which is subsequently validated through precise path-integral simulations, enabling a microscopic characterization of the various phases. Superfluid and melting/freezing transitions are analyzed through the superfluid density and the long-range behavior of correlation functions associated with positional and orientational order, in accordance with the general picture of Berezinskii-Kosterlitz-Thouless transitions. A broad region at low temperatures is identified where the supersolid phase exists, separating the uniform superfluid phase from the normal quasi-crystal phase. Additionally, a potential intermediate hexatic phase with quasi long-range orientational order is identified in a narrow region between the normal solid and fluid phases. These findings establish self-consistent Hartree-Fock theory beyond the local density approximation as an effective tool, complementary to computationally intensive quantum Monte Carlo simulations, for investigating the melting of the supersolid phase and the possible emergence of the hexatic superfluid phase in bosonic systems with various interaction potentials.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript investigates the supersolid phase of two-dimensional soft-core bosons at finite temperature via self-consistent Hartree-Fock mean-field theory to construct an approximate phase diagram, followed by validation and refinement with quantum Monte Carlo (QMC) path-integral simulations. Transitions are characterized using superfluid density and the long-range behavior of positional and orientational correlation functions in accordance with Berezinskii-Kosterlitz-Thouless physics. The central claims are the existence of a broad supersolid region at low temperatures separating the uniform superfluid from the normal quasi-crystal phase, plus a narrow potential hexatic phase with quasi-long-range orientational order between the normal solid and fluid phases. The work positions self-consistent Hartree-Fock theory (beyond local-density approximation) as a computationally efficient complement to QMC.

Significance. If the mean-field diagram is shown to be qualitatively reliable and the QMC validation confirms the reported phases without major boundary shifts, the results would establish the supersolid and possible hexatic phases in this 2D bosonic system while demonstrating the practical utility of the combined mean-field plus QMC approach for soft-core interactions. The explicit use of superfluid density and correlation functions for BKT-consistent analysis is a methodological strength that supports microscopic characterization.

major comments (2)
  1. [QMC validation section] QMC validation section: the abstract states that the mean-field phase diagram is 'subsequently validated through precise path-integral simulations,' yet no error bars, full data tables, or explicit exclusion criteria for parameter points are described; this directly affects assessment of whether the reported narrow hexatic region and broad supersolid region are robust or sensitive to post-hoc choices.
  2. [Phase diagram and transitions analysis] Phase diagram and transitions analysis: the central claim that mean-field provides a qualitatively reliable diagram (with QMC only refining boundaries) requires evidence that QMC checks include systematic finite-size scaling of superfluid density and orientational correlations across multiple points in the proposed supersolid region; without this, 2D BKT fluctuation effects that can narrow or eliminate ordered phases in mean-field predictions are not demonstrably ruled out.
minor comments (2)
  1. The terminology 'normal quasi-crystal phase' and 'normal solid phase' should be explicitly related or distinguished to avoid ambiguity in the phase diagram description.
  2. Figure captions would benefit from explicit listing of the temperature, density, and interaction-strength values corresponding to each panel or data set.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below and agree that additional details will improve clarity and strengthen the evidence for the reported phases. Revisions will be made accordingly.

read point-by-point responses

  1. Referee: [QMC validation section] QMC validation section: the abstract states that the mean-field phase diagram is 'subsequently validated through precise path-integral simulations,' yet no error bars, full data tables, or explicit exclusion criteria for parameter points are described; this directly affects assessment of whether the reported narrow hexatic region and broad supersolid region are robust or sensitive to post-hoc choices.

    Authors: We agree that the QMC validation section would benefit from greater transparency. In the revised manuscript we will add error bars to all QMC-derived points in the phase diagram and include a supplementary table that lists the simulation parameters, measured quantities, and the criteria used to select or exclude parameter points for validation. This will allow readers to evaluate the robustness of the supersolid and hexatic regions directly. revision: yes

  2. Referee: [Phase diagram and transitions analysis] Phase diagram and transitions analysis: the central claim that mean-field provides a qualitatively reliable diagram (with QMC only refining boundaries) requires evidence that QMC checks include systematic finite-size scaling of superfluid density and orientational correlations across multiple points in the proposed supersolid region; without this, 2D BKT fluctuation effects that can narrow or eliminate ordered phases in mean-field predictions are not demonstrably ruled out.

    Authors: We acknowledge the importance of explicit finite-size scaling to confirm that BKT fluctuations do not eliminate the ordered phases. Our existing QMC data include finite-size scaling of the superfluid density at several representative points inside the supersolid region. To address the referee's concern we will add systematic scaling plots for both superfluid density and orientational correlations at additional points across the supersolid region, thereby demonstrating that the mean-field diagram remains qualitatively reliable after accounting for finite-size effects. revision: yes

Circularity Check

0 steps flagged

Numerical phase diagram from independent mean-field and QMC methods shows no circular reduction

full rationale

The paper constructs an approximate phase diagram via self-consistent Hartree-Fock mean-field theory and validates it with path-integral quantum Monte Carlo simulations, analyzing transitions through superfluid density and correlation functions. These are standard numerical techniques applied to the Hamiltonian without any derivation chain that reduces outputs to inputs by construction, fitted parameters renamed as predictions, or load-bearing self-citations. The BKT framework referenced is external standard knowledge. The approach is self-contained against external benchmarks and does not invoke uniqueness theorems or ansatze from prior author work.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard assumptions of quantum many-body theory and BKT transition phenomenology; no new free parameters or invented entities are introduced beyond the model Hamiltonian parameters.

axioms (1)
  • domain assumption Berezinskii-Kosterlitz-Thouless transition framework applies to the superfluid and melting transitions in 2D
    Invoked when analyzing superfluid density and long-range behavior of correlation functions.

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Works this paper leans on

76 extracted references · 76 canonical work pages

  1. [1]

    (D1) The expectation values ⟨ˆb† k ˆbk′⟩ appearing in the functional are initially set to zero

    Initialization: the energy functional (B19) is mini- mized at fixed density ρ = 1 A X k |ψk|2 . (D1) The expectation values ⟨ˆb† k ˆbk′⟩ appearing in the functional are initially set to zero. The minimiza- tion produces an initial guess for the condensate wave function ψk and the chemical potential µ, which is the Lagrange multiplier associated to the den...

    work page

  2. [2]

    Initialization: given the condensate wave function, the self-consistency equations (B14)-(B16) with ⟨ˆb† k ˆbk′⟩ = 0 are used to provide an initial guess for Γk,k′

    work page

  3. [3]

    The quasiparticle Hamiltonian H0 is constructed using the chemical potential and the HF potential obtained in the previous steps, see (B13)

    work page

  4. [4]

    (D2) The matrix elements of the density matrix R are the expectation values ⟨ˆb† k ˆbk′⟩

    From the quasiparticle Hamiltonian, the one-body density matrix at temperature T (β = 1/kBT is the inverse temperature) is computed as R = 1 eβH0 − 1 . (D2) The matrix elements of the density matrix R are the expectation values ⟨ˆb† k ˆbk′⟩. Note that at zero temperature the HF approximation reduces to the solution of the standard GP equation (step 1 abov...

    work page

  5. [5]

    (D3) In contrast to (D1), the density contribution of the quasiparticle excitations is now included in the to- tal density

    The energy functional (B19), in which the expec- tation values ⟨ˆb† k ˆbk′⟩ are the ones computed in the previous step, is minimized at fixed density ρ = 1 A X k |ψk|2 + ⟨ˆb† k ˆbk⟩ . (D3) In contrast to (D1), the density contribution of the quasiparticle excitations is now included in the to- tal density. The minimization produces updated values for the ...

    work page

  6. [6]

    The condensate wave function ψk and the expecta- tion values ⟨ˆb† k ˆbk′⟩ obtained respectively at steps 5 and 4 are used in the self-consistency equa- tions (B14)-(B16) to obtain an updated HF poten- tial Γk,k′

    work page

  7. [7]

    Otherwise, the iteration continues from step 3

    If the value of the grand potential Ω m.f., the con- densate wave function and the HF potential have converged within a given threshold, the procedure terminates. Otherwise, the iteration continues from step 3. If one seeks a self-consistent solution at fixed chemical po- tential rather than fixed total density ρ, the only mod- ification is that the chemi...

    work page

  8. [8]

    A. F. Andreev and I. M. Lifshitz, Quantum theory of defects in crystals, Sov. Phys. JETP 29, 1107 (1969)

    work page 1969

  9. [9]

    Watanabe and T

    H. Watanabe and T. Brauner, Spontaneous breaking of continuous translational invariance, Phys. Rev. D 85, 21 6 7 8 W/ϵ0 0.0 0.2 0.4 0.6 0.8 1.0ρs/ρ N-K kBT=3.0ϵ0 6 7 8 W/ϵ0 N-K kBT=4.0ϵ0 N=240 N=448 N=1560 N=4032 6 7 8 W/ϵ0 N-K kBT=5.0ϵ0 6 7 8 W/ϵ0 0.00 0.02 0.04 0.06 0.08 0.10|ψ6|2 kBT=3.0ϵ0 6 7 8 W/ϵ0 kBT=4.0ϵ0 N=240 N=448 N=1560 N=4032 6 7 8 W/ϵ0 kBT=5...

    work page 2012

  10. [10]

    Boninsegni and N

    M. Boninsegni and N. V. Prokof’ev, Colloquium: Super- solids: What and where are they?, Rev. Mod. Phys. 84, 759 (2012)

    work page 2012

  11. [11]

    Kim and M

    E. Kim and M. H. W. Chan, Probable observation of a supersolid helium phase, Nature 427, 225 (2004)

    work page 2004

  12. [12]

    Kim and M

    E. Kim and M. H. W. Chan, Observation of superflow in solid helium, Science 305, 1941 (2004)

    work page 1941

  13. [13]

    Balibar, The enigma of supersolidity, Nature 464, 176 (2010)

    S. Balibar, The enigma of supersolidity, Nature 464, 176 (2010)

    work page 2010

  14. [14]

    D. Y. Kim and M. H. W. Chan, Absence of supersolidity in solid helium in porous vycor glass, Phys. Rev. Lett. 109, 155301 (2012)

    work page 2012

  15. [15]

    Prokof’ev and B

    N. Prokof’ev and B. Svistunov, Supersolid state of mat- ter, Phys. Rev. Lett. 94, 155302 (2005)

    work page 2005

  16. [16]

    E. P. Gross, Quantum theory of interacting bosons, An- nals of Physics 9, 292 (1960)

    work page 1960

  17. [17]

    Tanzi, E

    L. Tanzi, E. Lucioni, F. Fam` a, J. Catani, A. Fioretti, C. Gabbanini, R. N. Bisset, L. Santos, and G. Modugno, Observation of a Dipolar Quantum Gas with Metastable Supersolid Properties, Phys. Rev. Lett. 122, 130405 (2019)

    work page 2019

  18. [18]

    B¨ ottcher, J.-N

    F. B¨ ottcher, J.-N. Schmidt, M. Wenzel, J. Hertkorn, M. Guo, T. Langen, and T. Pfau, Transient Supersolid Properties in an Array of Dipolar Quantum Droplets, Phys. Rev. X 9, 011051 (2019)

    work page 2019

  19. [19]

    Chomaz, D

    L. Chomaz, D. Petter, P. Ilzh¨ ofer, G. Natale, A. Traut- mann, C. Politi, G. Durastante, R. M. W. van Bijnen, A. Patscheider, M. Sohmen, M. J. Mark, and F. Fer- laino, Long-Lived and Transient Supersolid Behaviors in 22 Dipolar Quantum Gases, Phys. Rev. X 9, 021012 (2019)

    work page 2019

  20. [20]

    Tanzi, S

    L. Tanzi, S. M. Roccuzzo, E. Lucioni, F. Fam` a, A. Fioretti, C. Gabbanini, G. Modugno, A. Recati, and S. Stringari, Supersolid symmetry breaking from com- pressional oscillations in a dipolar quantum gas, Nature 574, 382 (2019)

    work page 2019

  21. [21]

    M. Guo, F. B¨ ottcher, J. Hertkorn, J.-N. Schmidt, M. Wenzel, H. P. B¨ uchler, T. Langen, and T. Pfau, The low-energy Goldstone mode in a trapped dipolar super- solid, Nature 574, 386 (2019)

    work page 2019

  22. [22]

    Natale, R

    G. Natale, R. M. W. van Bijnen, A. Patscheider, D. Pet- ter, M. J. Mark, L. Chomaz, and F. Ferlaino, Excitation spectrum of a trapped dipolar supersolid and its experi- mental evidence, Phys. Rev. Lett. 123, 050402 (2019)

    work page 2019

  23. [23]

    M. A. Norcia, C. Politi, L. Klaus, E. Poli, M. Sohmen, M. J. Mark, R. N. Bisset, L. Santos, and F. Ferlaino, Two-dimensional supersolidity in a dipolar quantum gas, Nature 596, 357 (2021)

    work page 2021

  24. [24]

    Bland, E

    T. Bland, E. Poli, C. Politi, L. Klaus, M. A. Norcia, F. Ferlaino, L. Santos, and R. N. Bisset, Two-dimensional supersolid formation in dipolar condensates, Phys. Rev. Lett. 128, 195302 (2022)

    work page 2022

  25. [25]

    Casotti, E

    E. Casotti, E. Poli, L. Klaus, A. Litvinov, C. Ulm, C. Politi, M. J. Mark, T. Bland, and F. Ferlaino, Ob- servation of vortices in a dipolar supersolid, Nature 635, 327 (2024)

    work page 2024

  26. [26]

    Sohmen, C

    M. Sohmen, C. Politi, L. Klaus, L. Chomaz, M. J. Mark, M. A. Norcia, and F. Ferlaino, Birth, life, and death of a dipolar supersolid, Phys. Rev. Lett. 126, 233401 (2021)

    work page 2021

  27. [27]

    S´ anchez-Baena, C

    J. S´ anchez-Baena, C. Politi, F. Maucher, F. Ferlaino, and T. Pohl, Heating a dipolar quantum fluid into a solid, Nat. Commun. 14, 1868 (2023)

    work page 2023

  28. [28]

    Todoshchenko, M

    I. Todoshchenko, M. Kamada, J.-P. Kaikkonen, Y. Liao, A. Savin, M. Will, E. Sergeicheva, T. S. Abhilash, E. Kauppinen, and P. J. Hakonen, Topologically-imposed vacancies and mobile solid 3He on carbon nanotube, Nat. Commun. 13, 5873 (2022)

    work page 2022

  29. [29]

    Todoshchenko, M

    I. Todoshchenko, M. Kamada, J.-P. Kaikkonen, Y. Liao, A. Savin, E. Kauppinen, E. Sergeicheva, and P. J. Hako- nen, Quantum degeneracy in mesoscopic matter: Casimir effect and Bose-Einstein condensation, Phys. Rev. B109, 224519 (2024)

    work page 2024

  30. [30]

    A. R. P. Lima and A. Pelster, Beyond mean-field low- lying excitations of dipolar Bose gases, Phys. Rev. A 86, 063609 (2012)

    work page 2012

  31. [31]

    W¨ achtler and L

    F. W¨ achtler and L. Santos, Quantum filaments in dipo- lar Bose-Einstein condensates, Phys. Rev. A 93, 061603 (2016)

    work page 2016

  32. [32]

    Saito, Path-Integral Monte Carlo Study on a Droplet of a Dipolar Bose–Einstein Condensate Stabilized by Quantum Fluctuation, J

    H. Saito, Path-Integral Monte Carlo Study on a Droplet of a Dipolar Bose–Einstein Condensate Stabilized by Quantum Fluctuation, J. Phys. Soc. Jpn. 85, 053001 (2016)

    work page 2016

  33. [33]

    Macia, J

    A. Macia, J. S´ anchez-Baena, J. Boronat, and F. Maz- zanti, Droplets of trapped quantum dipolar bosons, Phys. Rev. Lett. 117, 205301 (2016)

    work page 2016

  34. [34]

    B¨ ottcher, M

    F. B¨ ottcher, M. Wenzel, J.-N. Schmidt, M. Guo, T. Lan- gen, I. Ferrier-Barbut, T. Pfau, R. Bomb´ ın, J. S´ anchez- Baena, J. Boronat, and F. Mazzanti, Dilute dipolar quan- tum droplets beyond the extended Gross-Pitaevskii equa- tion, Phys. Rev. Res. 1, 033088 (2019)

    work page 2019

  35. [35]

    Pomeau and S

    Y. Pomeau and S. Rica, Dynamics of a model of super- solid, Phys. Rev. Lett. 72, 2426 (1994)

    work page 1994

  36. [36]

    Boninsegni, Supersolid phases of cold atom assem- blies, J

    M. Boninsegni, Supersolid phases of cold atom assem- blies, J. Low Temp. Phys. 168, 137 (2012)

    work page 2012

  37. [37]

    Saccani, S

    S. Saccani, S. Moroni, E. Vitali, and M. Boninsegni, Bose soft discs: A minimal model for supersolidity, Mol. Phys. 109, 2807 (2011)

    work page 2011

  38. [38]

    Saccani, S

    S. Saccani, S. Moroni, and M. Boninsegni, Phase diagram of soft-core bosons in two dimensions, Phys. Rev. B 83, 092506 (2011)

    work page 2011

  39. [39]

    Macr` ı, F

    T. Macr` ı, F. Maucher, F. Cinti, and T. Pohl, Elemen- tary excitations of ultracold soft-core bosons across the superfluid-supersolid phase transition, Phys. Rev. A 87, 061602 (2013)

    work page 2013

  40. [40]

    Saccani, S

    S. Saccani, S. Moroni, and M. Boninsegni, Excitation Spectrum of a Supersolid, Phys. Rev. Lett. 108, 175301 (2012)

    work page 2012

  41. [41]

    E. Poli, D. Baillie, F. Ferlaino, and P. B. Blakie, Excita- tions of a two-dimensional supersolid, Phys. Rev. A 110, 053301 (2024)

    work page 2024

  42. [42]

    D. R. Nelson and J. M. Kosterlitz, Universal jump in the superfluid density of two-dimensional superfluids, Phys. Rev. Lett. 39, 1201 (1977)

    work page 1977

  43. [43]

    S. C. Kapfer and W. Krauth, Two-Dimensional Melting: From Liquid-Hexatic Coexistence to Continuous Transi- tions, Phys. Rev. Lett. 114, 035702 (2015)

    work page 2015

  44. [44]

    B. I. Halperin and D. R. Nelson, Theory of Two- Dimensional Melting, Phys. Rev. Lett. 41, 121 (1978)

    work page 1978

  45. [45]

    D. R. Nelson and B. I. Halperin, Dislocation-mediated melting in two dimensions, Phys. Rev. B19, 2457 (1979)

    work page 1979

  46. [46]

    A. P. Young, Melting and the vector Coulomb gas in two dimensions, Phys. Rev. B 19, 1855 (1979)

    work page 1979

  47. [47]

    K. Zahn, R. Lenke, and G. Maret, Two-stage melting of paramagnetic colloidal crystals in two dimensions, Phys. Rev. Lett. 82, 2721 (1999)

    work page 1999

  48. [48]

    Gasser, C

    U. Gasser, C. Eisenmann, G. Maret, and P. Keim, Melt- ing of Crystals in Two Dimensions, ChemPhysChem 11, 963 (2010)

    work page 2010

  49. [49]

    K. J. Strandburg, Two-dimensional melting, Rev. Mod. Phys. 60, 161 (1988)

    work page 1988

  50. [50]

    Alba-Simionesco, B

    C. Alba-Simionesco, B. Coasne, G. Dosseh, G. Dudziak, K. E. Gubbins, R. Radhakrishnan, and M. Sliwinska- Bartkowiak, Effects of confinement on freezing and melt- ing, J. Phys. Condens. Matter 18, R15 (2006)

    work page 2006

  51. [51]

    G. M. Bruun and D. R. Nelson, Quantum hexatic order in two-dimensional dipolar and charged fluids, Phys. Rev. B 89, 094112 (2014)

    work page 2014

  52. [52]

    Mullen, H

    K. Mullen, H. T. C. Stoof, M. Wallin, and S. M. Girvin, Hexatically ordered superfluids, Phys. Rev. Lett. 72, 4013 (1994)

    work page 1994

  53. [53]

    Pilati, S

    S. Pilati, S. Giorgini, and N. Prokof’ev, Critical temper- ature of interacting Bose gases in two and three dimen- sions, Phys. Rev. Lett. 100, 140405 (2008)

    work page 2008

  54. [54]

    Doniach, Quantum fluctuations in two-dimensional su- perconductors, Phys

    S. Doniach, Quantum fluctuations in two-dimensional su- perconductors, Phys. Rev. B 24, 5063 (1981)

    work page 1981

  55. [55]

    Giorgini, L

    S. Giorgini, L. P. Pitaevskii, and S. Stringari, Thermody- namics of a Trapped Bose-Condensed Gas, J. Low Temp. Phys. 109, 309 (1997)

    work page 1997

  56. [56]

    Holzmann, W

    M. Holzmann, W. Krauth, and M. Naraschewski, Preci- sion Monte Carlo test of the Hartree-Fock approximation for a trapped Bose gas, Phys. Rev. A 59, 2956 (1999)

    work page 1999

  57. [57]

    R. P. Feynman,Statistical Mechanics: A Set of Lectures, Advanced Book Classics (Westview Press, Boulder, 1998)

    work page 1998

  58. [58]

    Blaizot and G

    J.-P. Blaizot and G. Ripka, Quantum Theory of Finite Systems (MIT Press, Cam- bridge, Mass., 1986). 23

    work page 1986

  59. [59]

    Peotta, Superconductivity, generalized random phase approximation and linear scaling methods, New J

    S. Peotta, Superconductivity, generalized random phase approximation and linear scaling methods, New J. Phys. 24, 113019 (2022)

    work page 2022

  60. [60]

    Tam and S

    M. Tam and S. Peotta, Geometry-independent superfluid weight in multiorbital lattices from the generalized ran- dom phase approximation, Phys. Rev. Res. 6, 013256 (2024)

    work page 2024

  61. [61]

    Capogrosso-Sansone, S

    B. Capogrosso-Sansone, S. Giorgini, S. Pilati, L. Pollet, N. Prokof’ev, B. Svistunov, and M. Troyer, The Beliaev technique for a weakly interacting Bose gas, New J. Phys. 12, 043010 (2010)

    work page 2010

  62. [62]

    D. R. Nelson, Study of melting in two dimensions, Phys. Rev. B 18, 2318 (1978)

    work page 1978

  63. [63]

    L. D. Landau, L. P. Pitaevskij, A. M. Kosevich, and E. M. Lifˇ sic,Theory of Elasticity: Volume 7, 3rd ed. (Elsevier Butterworth-Heinemann, 1986)

    work page 1986

  64. [64]

    Rakic, A

    M. Rakic, A. F. Ho, and D. K. K. Lee, Elastic proper- ties and thermodynamic anomalies of supersolids (2024), arXiv:2403.13727

    work page arXiv 2024

  65. [65]

    D. M. Ceperley, Path integrals in the theory of condensed helium, Rev. Mod. Phys. 67, 279 (1995)

    work page 1995

  66. [66]

    Boninsegni, N

    M. Boninsegni, N. Prokof’ev, and B. Svistunov, Worm Algorithm for Continuous-Space Path Integral Monte Carlo Simulations, Phys. Rev. Lett. 96, 070601 (2006)

    work page 2006

  67. [67]

    Spada, S

    G. Spada, S. Giorgini, and S. Pilati, Path-Integral Monte Carlo Worm Algorithm for Bose Systems with Periodic Boundary Conditions, Condens. Matter 7, 30 (2022)

    work page 2022

  68. [68]

    M. P. Allen and D. J. Tildesley, Computer Simulation of Liquids, 2nd ed. (Oxford university press, Oxford, 1987)

    work page 1987

  69. [69]

    Bagchi, H

    K. Bagchi, H. C. Andersen, and W. Swope, Computer Simulation Study of the Melting Transition in Two Di- mensions, Phys. Rev. Lett. 76, 255 (1996)

    work page 1996

  70. [70]

    E. P. Bernard and W. Krauth, Two-Step Melting in Two Dimensions: First-Order Liquid-Hexatic Transition, Phys. Rev. Lett. 107, 155704 (2011)

    work page 2011

  71. [71]

    Huang, T

    P. Huang, T. Sch¨ onenberger, M. Cantoni, L. Heinen, A. Magrez, A. Rosch, F. Carbone, and H. M. Rønnow, Melting of a skyrmion lattice to a skyrmion liquid via a hexatic phase, Nat. Nanotechnol. 15, 761 (2020)

    work page 2020

  72. [72]

    E. L. Pollock and D. M. Ceperley, Path-integral com- putation of superfluid densities, Phys. Rev. B 36, 8343 (1987)

    work page 1987

  73. [73]

    D. M. Ceperley and E. L. Pollock, Path-integral sim- ulation of the superfluid transition in two-dimensional $ˆ{4}\mathrm{He}$, Phys. Rev. B 39, 2084 (1989)

    work page 2084

  74. [74]

    P. Keim, G. Maret, and H. H. von Gr¨ unberg, Frank’s constant in the hexatic phase, Phys. Rev. E 75, 031402 (2007)

    work page 2007

  75. [75]

    P. B. Blakie, Superfluid fraction tensor of a two- dimensional supersolid, J. Phys. B: At. Mol. Opt. Phys. 57, 115301 (2024)

    work page 2024

  76. [76]

    R. R. Puri, Mathematical Methods of Quantum Optics, edited by W. T. Rhodes, Springer Series in Optical Sci- ences, Vol. 79 (Springer Berlin Heidelberg, Berlin, Hei- delberg, 2001)

    work page 2001