Ultradilute quasi-two-dimensional Bose-Bose liquid mixtures

Ivan Popari\'c; Jordi Boronat; Kre\v{s}imir D\v{z}elalija; Leandra Vranje\v{s} Marki\'c; Petar Stipanovi\'c

REVIEW 3 major objections 6 minor 40 references

Ultradilute quasi-two-dimensional Bose-Bose liquid mixtures

T0 review · 3 major / 6 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read 2D LHY functional is accurate only for the tightest quasi-2D confinements

desk verdict A credible QMC-based case that 2D LHY functionals fail outside very tight confinement, but the droplet/vortex predictions lean on an untested low-density EoS ansatz. read the letter →

arxiv 2607.25558 v1 pith:LEAGN6J3 submitted 2026-07-28 cond-mat.quant-gas

Leandra Vranješ Markić , Ivan Poparić , Krešimir Dželalija , Petar Stipanović , Jordi Boronat This is my paper · ORCID

classification cond-mat.quant-gas PACS 67.85.-d03.75.Hh05.30.Jp

keywords Bose-BosemixturesquantumdropletsLee-Huang-Yangcorrectionquasi-2DconfinementdiffusionMonteCarlodensityfunctionaltheoryeffectiverangeuniversality

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper shows that the standard Lee-Huang-Yang density functional for 2D Bose-Bose droplets, the 2D LHY functional, is accurate only for very strongly squeezed quasi-2D systems, a much narrower range than commonly assumed. Using diffusion Monte Carlo at zero temperature with two different interaction potentials, the authors establish that equations of state are universal in terms of the s-wave scattering length and effective range down to a harmonic oscillator length of about 9–10 times the intraspecies scattering length, but diverge for tighter confinement. From these equations of state they construct a new 2D quantum-Monte-Carlo-based density functional for each confinement strength, and with it compute droplet density profiles, energies, and vortex stability. Their functional agrees with 3D functionals at moderate squeezing and approaches the bulk Monte Carlo results for large droplets, while the 2D LHY functional overestimates central densities and self-binding energies by large factors in that regime. If correct, the work provides a computationally cheap and systematically more reliable route to quasi-2D droplet physics and changes expectations for vortex studies.

What carries the argument

The key machinery is a confinement-specific 2D density functional built from zero-temperature diffusion Monte Carlo (DMC) equations of state. For each squeezing strength, the bulk energy per particle from DMC is fitted to the three-parameter form E/N = αρ + βρ^γ, with γ between about 1.02 and 1.26 for the tabulated cases, and this expression enters the DFT equation through E_int = ρ E/N. The universal range is established by comparing two interaction potential models (POT1 and POT2) matched to the same scattering length and effective range. The functional is then used, together with the local density approximation, to compute droplet profiles, liquid-drop energies, and vortex-core structure.

What would settle it

A direct diffusion Monte Carlo simulation of a self-bound droplet at f = 0.15 with N ≈ 20,000 atoms, avoiding the density-functional bridge entirely, would settle whether the 2D QMC functional's predicted central density, energy, and vortex stability are correct; alternatively, measuring the low-density tail of the droplet's equation of state and comparing it with the fit E/N = αρ + βρ^γ would test the ansatz that carries the argument.

Watch

Extended reading notes

Core claim

The central quantitative finding is a universality boundary: for 39K Bose-Bose mixtures squeezed by a harmonic trap, equations of state computed with two different potential models that share the same scattering length and effective range agree down to a confinement length a_z ≈ 9–10 a_11 (f = 0.05), but below that the two-parameter description breaks down and details of the interaction potential matter. In the universal regime the paper builds a 2D QMC functional by fitting the DMC energy per particle to E/N = αρ + βρ^γ for each confinement. Using this functional inside a density-functional equation with the local density approximation, the paper shows that the widely used 2D LHY functional

Load-bearing premise

All droplet and vortex predictions are obtained by feeding the fitted form E/N = αρ + βρ^γ into a density functional and applying the local density approximation across the density profile; if that simple fit misrepresents the true equation of state at low densities or near equilibrium, the computed profiles, energies, and vortex-stability conclusions inherit the error.

Editorial extensions

If this is right

If the paper is right, vortex and excitation studies of quasi-2D droplets relying on the 2D LHY functional need to be revisited in the moderate-confinement regime, where that functional can flip the sign of the vortex-energy comparison.
The 2D QMC functional provides a computationally efficient alternative that remains accurate across the crossover from 3D to 2D, matching 3D functional results at moderate confinement and the bulk DMC limit for large droplets.
For the tightest confinements (a_z below about 10 a_11), two scattering parameters are no longer sufficient; any theoretical or experimental analysis in that regime must account for details of the interatomic potentials.
The construction used here, computing the bulk equation of state with QMC, fitting a simple empirical form, and using it as a density functional, can be applied to other mixtures and to dipolar molecular droplets, where beyond-LHY effects are stronger.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The paper's fit form E/N = αρ + βρ^γ contains no logarithmic term that the exact 2D equation of state possesses; a natural extension would be to test whether including a log-density term changes droplet-edge properties or the vortex conclusions.
Because the functional was validated on large droplets by showing that saturation density approaches the bulk DMC value, the same quantity it was fitted to, an independent check such as a direct DMC simulation of a finite self-bound droplet would strengthen the case that the LDA-plus-fit bridge is faithful in low-density regions.
If the universality breakdown below a_z ≈ 10 a_11 is a general feature, analogous functional-construction efforts for other mixtures would need to map out their own universality windows before applying the 2D LHY functional.
The vortex-stability reversal between 2D LHY and QMC functionals suggests that experimental searches for vortices in droplets should target intermediate confinements, where the competing functionals give distinct density profiles and where the QMC functionals predict vortex states to be energetically favored.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The thing you should know: this paper makes a solid case that the 2D LHY functional, used in a lot of quasi-2D droplet work, only works at very tight confinement, and it offers a cheap QMC-based alternative. The bulk equation-of-state evidence is the strong part. The weak part is the bridge from that bulk EoS to the droplet and vortex predictions.

What is new: a systematic DMC study of the 3D-to-2D crossover for 39K mixtures, using two potential models with the same a and r_eff, showing universality down to a_z ≈ 9–10 a_11 and building per-confinement 2D functionals. The two-potential check is a sound methodological choice, and the qualitative departure from 2D LHY is corroborated by independent work (Refs. 31, 32, 40). The paper is honest about its limits: it explicitly says it cannot compare directly to Ref. [40], notes that below f = 0.05 two scattering parameters are insufficient, and shows 3D functionals fail under the strongest squeezing.

The soft spot is the DFT step. All droplet and vortex results come from inserting a three-parameter fit E/N = αρ + βργ into the local-density DFT. The fit is anchored around bulk equilibrium densities, but droplet edges and vortex cores sample arbitrarily low densities, where the true 2D EoS has a logarithmic term that this ansatz lacks. The paper gives no sensitivity test against alternative fit forms and no error bars on any DMC quantity, so the factor-4 and factor-10 discrepancy claims carry unknown uncertainty. The large-N validation—saturation density reaching the bulk value—is circular, since the functional was fit to that same bulk EoS. These are real limitations, but they do not overturn the central effect: the bulk comparison in Fig. 3 is less exposed, and the narrow window of 2D LHY validity looks robust across several cross-checks.

This paper is for people working on quasi-2D droplets, vortices, and LHY functionals, and for anyone using 2D LHY in excitation or turbulence studies. It deserves a serious referee. The referees should push for error bars, an alternative low-density fit form, and a check on the vortex ordering, but the work is worth engaging with now.

Referee Report

3 major / 6 minor

Summary. The paper reports diffusion Monte Carlo (DMC) equations of state for ultradilute 39K Bose-Bose mixtures in a one-dimensional harmonic confinement, spanning the crossover from three to two dimensions. Using two different interaction potentials with the same s-wave scattering length and effective range, the authors establish that the EoS is universal in these two parameters down to a_z ≈ 9–10 a_11 (f = 0.05), and that below this confinement two-parameter universality breaks down. The DMC EoS is fitted to E/N = αρ + βργ and used to construct a 2D QMC density functional within the local density approximation; this functional is then applied to compute droplet density profiles, energies, and vortex states. The central conclusions are that the 2D LHY functional is applicable only in a very narrow range of very tight confinements, and that the 2D QMC functional provides more reliable droplet and vortex predictions than 2D LHY in the quasi-2D regime.

Significance. If the results hold, the paper provides a practical QMC-based density functional for quasi-2D Bose-Bose droplets and directly challenges the common practice of using the 2D LHY functional in crossover and quasi-2D settings. The claim that the range of validity of 2D LHY is very narrow has important consequences for predictions of droplet profiles, energies, and vortex stability, and is supported by the paper's two-potential universality check and by independent PIMC results. The manuscript's strengths include fully tabulated model potentials, a systematic test of universality with two potential families, and a direct comparison with prior PIMC and quasi-2D LHY studies. The main vulnerability is the untested low-density extrapolation of the fitted EoS used to build the density functional, which directly affects all droplet and vortex predictions.

major comments (3)

[Results / Eq. (2) / Table VI] All droplet and vortex results (Figs. 2, 4, 5) are obtained from the fitted EoS E/N = αρ + βργ, with parameters in Table VI. The fits are anchored to DMC data near the bulk equilibrium density, but a self-bound droplet edge and a vortex core sample arbitrarily small ρ. There the ansatz yields E/N ≈ αρ (linear and negative), whereas the 2D LHY EoS (Eq. 11) and the expected 2D equation of state contain a nonanalytic ρ² ln ρ term. Since γ > 1 and α < 0 in all cases, the low-density behavior of the functional is an uncontrolled extrapolation. No sensitivity test against alternative fit forms (e.g., including a logarithmic term) or against direct trapped-droplet DMC is provided. The Fig. 5 vortex-state ordering and the quantitative droplet energies in Fig. 4 therefore rest on an ansatz whose low-density shape is not validated. This is the main load-bearing gap in the paper.
[Results / Fig. 3] The statement that saturation densities obtained from the 2D QMC functional 'reach the quasi-2D bulk DMC value, demonstrating functional’s reliability for large droplets' is circular. The functional was constructed by fitting to that same bulk DMC EoS; in the large-N limit the DFT necessarily recovers the minimum of the input EoS. This check confirms internal consistency, but it does not independently validate the functional for inhomogeneous configurations, where the low-density region matters. An independent validation, such as a DMC calculation of a trapped droplet or a direct comparison with PIMC for the same mixture, would be needed to support the droplet-level conclusions.
[Fig. 1 / Table VI / DMC implementation] No statistical error bars are reported for the DMC energies per particle or for the fitted parameters α, β, γ in Table VI. The claim that below f = 0.05 two scattering parameters are 'not sufficient' is based on the difference between POT1 and POT2 results in Fig. 1. Without error bars or fit residuals, the reader cannot judge whether the apparent POT1/POT2 discrepancy at f = 0.01 is statistically significant. Given that this is a load-bearing part of the universality claim, the authors should provide at least representative error bars on the EoS data points and on the fitted parameters.

minor comments (6)

[End Matter / Tables I–V] Typographical errors: 'Paramatres' should be 'Parameters', 'unists' should be 'units'. Please proofread the End Matter.
[Fig. 1 caption] The sentence 'The liquid distribution in the direction of squeezing follows has the single-particle gaussian shape' is grammatically awkward. Also, the inset would benefit from a definition of the plotted quantity (probability density versus z/a_z, presumably).
[Table VI] The confinement is given only as a_z in µm. Since the text uses the dimensionless factor f = a_z/0.639 µm, it would help to list f alongside a_z in the table, or at least define the corresponding f values in the caption.
[Fig. 2] The top and bottom panels show different squeezings, but the atom number N is only stated in the text for one case. Please label or state N explicitly for each curve so the comparison is unambiguous.
[References] Ref. [33] is cited as Phys. Rev. A 109, 013313 (2014); PR A volume 109 corresponds to 2024, so the year appears to be a typo. Also, Ref. [35] and [39] would benefit from article titles, which are currently omitted.
[Comparison with Ref. [40]] The comparison with the PIMC results of Ref. [40] is indirect: that work uses a symmetric mixture, different scattering parameters, and no finite-range effects. The agreement with 3D functionals is suggestive but not a quantitative test of the proposed 2D QMC functional. This should be stated more cautiously.

Circularity Check

1 steps flagged · score 2.0 of 10

Central benchmark against external 2D LHY is independent; only the large-N saturation check is a self-consistency loop.

fitted input called prediction [Results, discussion of Fig. 3]
"The saturation densities of droplets obtained with the 2D QMC functional in the limit N→∞ reach the quasi-2D bulk DMC value, demonstrating functional’s reliability for large droplets."

The 2D QMC functional is built from E_int = ρ E/N, where E/N = αρ + βρ^γ is fitted to the same quasi-2D bulk DMC equation of state (Table VI). In the LDA, the saturation density of a large droplet is the density minimizing E/N, so it is determined by the same fitted parameters. Its agreement with the bulk DMC equilibrium density is therefore a consistency check of the fit, not an independent validation. Presenting it as 'demonstrating functional’s reliability' elevates a construction-derived quantity to an apparent independent check. However, this loop is confined to the validation statement: the droplet profiles, energies, and vortex comparisons are computed from the same functional and are not used as inputs to the fit.

full rationale

The paper's central claim — that the 2D LHY functional has limited applicability in quasi-2D systems — is benchmarked against external results: the 2D LHY functional of Ref. [11], the 3D LHY functional, and the independent PIMC results of Ref. [40]. The DMC bulk equations of state are obtained from first-principles many-body calculations, and the universality check uses two different potential models with the same scattering parameters; this is not circular. The functional ansatz E/N = αρ + βργ is explicitly presented as a fit, not smuggled in via citation, and the droplet/vortex results are genuine consequences of that fitted functional rather than inputs. The only reduction-by-construction I find is the large-N saturation 'validation': because the functional is fitted to the bulk EoS, the droplet saturation density in the LDA is the minimum of that same fitted E/N, so the agreement is tautological. This is a minor self-consistency loop and does not affect the external comparisons that support the paper's main conclusion. The low-density extrapolation of the polynomial ansatz is a correctness risk, not a circularity, because it concerns the quality of the model rather than a logical equivalence between input and output.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The paper postulates no new physical entities; POT1 and POT2 are computational interaction models tuned to known scattering parameters, not new physics. Load-bearing inputs: hand-tuned model potentials (2 families × 2 fields), 3-parameter EoS fits per state point (Table VI), the LDA reduction, and prior results from Refs [11], [12], and the group's own 3D QMC functional [33,35]. The main unstated risks are the untested fit ansatz and the model dependence of the universality window.

free parameters (2)

EoS fit parameters α, β, γ (per confinement, per field) = Table VI: 11 rows; e.g., B=56.337 G, a_z=0.032 µm: α=−0.446218 ℏ²/2m, β=0.499704 ℏ² a_11^{2γ−2}/2m, γ=1.029605
Three-parameter fit E/N = αρ + βργ to the QMC EoS for each state point. The 2D QMC density functional used for all droplet, energy-scaling, and vortex results is constructed from these fits, so the fitted values are load-bearing.
POT1/POT2 model-potential parameters = Tables I–V: V0, V1, R0, R1, r0, r1, r2 per magnetic field
Two ad hoc potential families (square-well, Lennard-Jones-like, Gaussian-repulsion variants) chosen by hand and tuned to reproduce the experimental scattering length and effective range. The QMC EoS and the claimed universality window depend on these model choices; the two-model agreement is the evidence for universality.

assumptions (5)

domain assumption A two-body potential with the given a and r_eff reproduces the low-energy physics of the 39K mixture (universality hypothesis used in constructing interactions)
In Method/End Matter, potentials are built to reproduce experimental scattering parameters from Ref [4]; the central universality claim is the test of this hypothesis, but the construction presupposes it.
domain assumption DMC ground-state energies are exact within statistical noise for the Hamiltonian (1)
Standard QMC assumption; the paper cites the established second-order DMC method [38] but shows no convergence analysis (time step, population bias, box-size extrapolation).
domain assumption Local density approximation: E_int = ρ E/N(ρ) applied pointwise in the DFT equation (2) remains valid across the droplet profile
All droplet/vortex results (Figs. 2–5) rely on LDA with the bulk-derived functional; validity is supported by agreement with 3D functionals down to a_z ≈ 0.1 µm but not independently certified.
ad hoc to paper The ansatz E/N = αρ + βργ captures the QMC EoS over all densities sampled by droplets, including the low-density edge and vortex core
The fit form is introduced in Results without derivation or alternative-form sensitivity test; γ ∈ [1.02, 1.26] and the absence of a log term leave the low-density behavior uncertain.
domain assumption The confinement-to-scattering mapping ε_ij = (A/π a_z²) exp(√(2π) a_z / a_ij), A ≈ 0.905, from Ref [12] correctly defines the 2D LHY benchmark [11]
Used in End Matter Eqs. (10)–(11) to evaluate the 2D LHY functional that the paper benchmarks against; imported from prior literature without re-derivation.

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Pith. "Pith review of Ultradilute quasi-two-dimensional Bose-Bose liquid mixtures." pith.science (2026). https://pith.science/paper/LEAGN6J3

@misc{pith2026260725558,
  author       = {Pith},
  title        = {Pith review of: Ultradilute quasi-two-dimensional Bose-Bose liquid mixtures},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LEAGN6J3}},
  note         = {Machine review of arXiv:2607.25558}
}

abstract

We study ultradilute $^{39}$K Bose-Bose bulk mixtures and droplets in an external harmonic potential that confines them in one spatial direction towards the two-dimensional (2D) limit. Equations of state for several confinements are obtained with quantum Monte Carlo (QMC) at $T=0$, using interaction potentials that include information on the $s$-wave scattering length $a$ and the effective range $r_{\rm eff}$. Performing the calculations using two different interaction potential models we have determined the range of confinements for which equations of state are universal in terms of $a$ and $r_{\rm eff}$. Based on the QMC equation of state, we develop a 2D QMC density functional for each confinement strength and use it together with the local density approximation to determine properties of the self-bound drops. For moderate squeezing, energies and droplet profiles obtained using the 2D QMC functional agree well with those obtained using 3D functionals, while offering a substantial reduction in computational cost, and a consistent approach in crossover to 2D. Noticeably, our results approach 2D mean-field (MF) + Lee-Huang-Yang (LHY) predictions only for the most strongly confined systems for which universality in terms of $a$ and $r_{\rm eff}$ is observed. This implies a very narrow range of confinements for which 2D LHY functionals are applicable, which has important consequences for the study of vortices.

Figures

Figures reproduced from arXiv: 2607.25558 by the authors.

**Figure 3.** FIG. 3. The equilibrium densities of the bulk liquid, obtained [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

**Figure 2.** FIG. 2. Density profiles of droplets obtained with the 2D [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: FIG. 4. Energy per particle of the droplets as a function of [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]

**Figure 5.** Figure 5: FIG. 5. Density profiles of droplets with 100000 atoms in [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]

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Works this paper leans on

40 extracted references

[40]

Spada, S

G. Spada, S. Pilati, and S. Giorgini, Phys. Rev. Lett.133 083401 (2024). END MA TTER Interaction potential models In our calculations we considered two sets of interac- tion potential models, POT1 and POT2, with the same scattering lengthsaand effective rangesr eff . The first model POT1 is given by V (1,1)(r) =    −V0, r≤R 0, V1, R 0 < r≤R 1, 0, R 1...

2024
[1]

D. S. Petrov, Quantum mechanical stabilization of a col- lapsing Bose-Bose mixture, Phys. Rev. Lett115, 155302 (2015)

2015
[2]

T. D. Lee, K. Huang, and C. N. Yang, Eigenvalues and eigenfunctions of a Bose system of hard spheres and its low-temperature properties, Phys. Rev.106, 1135 (1957)

1957
[3]

D. M. Larsen, Binary mixtures of dilute Bose gases with repulsive interactions at low temperature, Annals of Physics24, 89 (1963)

1963
[4]

Cabrera, L

C.R. Cabrera, L. Tanzi, J. Sanz, B. Naylor, P. Thomas, P. Cheiney, and L. Tarruell, Science359, 301 (2018)

2018
[5]

Semeghini, G

G. Semeghini, G. Ferioli, L. Masi, C. Mazzinghi, L. Wol- swijk, F. Minardi, M. Modugno, G. Modugno, M. In- guscio, and M. Fattori, Self-bound quantum droplets of atomic mixtures in free space, Phys. Rev. Lett120, 235301 (2018)

2018
[6]

D’Errico, A

C. D’Errico, A. Burchianti, M. Prevedelli, L. Salasnich, F. Ancilotto, M. Modugno, F. Minardi, and C. Fort, Ob- servation of quantum droplets in a heteronuclear bosonic mixture, Phys. Rev. Research1, 033155 (2019)

2019
[7]

Z. Guo, F. Jia, L. Li, Y. Ma, J. M. Hutson, X. Cui, and D. Wang, Lee-Huang-Yang effects in the ultracold mixture of 23Na and 87Rb with attractive interspecies interactions, Phys. Rev. Res.3, 033247 (2021)

2021

Show all 40 references

[8]

Cavicchioli, C

L. Cavicchioli, C. Fort, F. Ancilotto, M. Modugno, F. Mi- nardi, A. Burchianti, Dynamical Formation of Multiple Quantum Droplets in a Bose-Bose Mixture, Phys. Rev. Lett.134, 093401 (2025)

2025
[9]

Langen, J

T. Langen, J. Boronat, J. S´ anchez-Baena, R. Bomb ´ ın, T. Karman, and F. Mazzanti, Phys. Rev. Lett.134, 053001 (2025)

2025
[10]

Zhang, W

S. Zhang, W. Yuan, N. Bigagli, H. Kwak, T. Karman, I. Steven, and S. Will, Nature651, 601 (2026)

2026
[11]

D. S. Petrov and G. E. Astrakharchik, Phys. Rev. Lett. 117, 100401 (2016)

2016
[12]

D. S. Petrov and G. V. Shlyapnikov, Interatomic colli- sions in a tightly confined Bose gas, Phys. Rev. A64, 012706 (2001)

2001
[13]

Y. Li, Z. Chen, Z. Luo, C. Huang, H. Tan, W. Pang, and B. A. Malomed, Two-dimensional vortex quantum droplets, Phys. Rev. A98, 063602 (2018)

2018
[14]

M. N. Tengstrand, P. St¨ urmer, author E. Karabulut, and S. Reimann, Rotating Binary Bose-Einstein Condensates and Vortex Clusters in Quantum Droplets, Physical Re- view Letters123, 160405 (2019)

2019
[15]

Y. V. Kartashov, B. A. Malomed, L. Torner, Metastabil- ity of Quantum Droplet Clusters, Phys. Rev. Lett.122, 193902 (2019)

2019
[16]

Y. V. Kartashov, B. A. Malomed, and L. Torner, Struc- tured heterosymmetric quantum droplets, Phys. Rev. Res.2, 033522 (2020)

2020
[17]

Examilioti and G

P. Examilioti and G. M. Kavoulakis, J. Phys. B: At. Mol. Opt. Phys.53, 175301 (2020)

2020
[18]

Dong and Y

L. Dong and Y. V. Kartashov, Rotating multidimensional quantum droplets, Phys. Rev. Lett.126, 244101 (2021)

2021
[19]

St¨ urmer, M

P. St¨ urmer, M. N. Tengstrand, R. Sachdeva and S. M. Reimann, Breathing mode in two-dimensional binary self-bound Bose-gas droplets, Phys. Rev. A103, 053302 (2021). 6

2021
[20]

Nikolaou, G

S. Nikolaou, G. M. Kavoulakis, and M. Ogren, Rotating quantum droplets confined in a harmonic potential, Phys. Rev. A108, 053309 (2023)

2023
[21]

Gu and X

Q. Gu and X. Cui, Self-bound vortex lattice in a rapidly rotating quantum droplet, Physical Review A 108, 063302 (2023)

2023
[22]

Nikolaou, G

S. Nikolaou, G. M. Kavoulakis, M. and ¨Ogren, Rotating quantum droplets confined in an anharmonic potential, Phys. Rev. A,109, 043304 (2024)

2024
[23]

Y. Fei, X. Du, X.-L. Chen and Y. Zhang, Collective ex- citations in two-dimensional harmonically trapped quan- tum droplets, Phys. Rev. A109, 053309 (2024)

2024
[24]

S. K. Jha, M. K. Verma, S. I. Mistakidis, P. K. Mishra, Energy spectra and fluxes of two-dimensional turbulent quantum droplets, Phys. Rev. Fluids10, 064618 (2025)

2025
[25]

Paredes, J

A. Paredes, J. Guerra-Carmenate, J. R. Salgueiro, D. Tommasini, H. Michinel, Traveling bubbles and vor- tex pairs within symmetric two-dimensional quantum droplets, Phys. Rev. E111, 054217 (2025)

2025
[26]

Paredes, J

A. Paredes, J. Guerra-Carmenate, and H. Michinel, Fate of traveling waves at the boundary of quantum droplets, Phys. Rev. E112, 054205 (2025)

2025
[27]

S. Wang, L. Duan, L. Dong, Z.-Y. Yang, Dynamics of vortices in uniform Bose quantum liquids, Phys. Rev. A 112, 063304 (2025)

2025
[28]

T. Ilg, J. Kumlin, L. Santos, D. S. Petrov, and H. P. B¨ uchler, Dimensional crossover for the beyond- mean-field correction in Bose gases, Phys. Rev. A98, 051604(R) (2018)

2018
[29]

P. Zin, M. Pylak, T. Wasak, M. Gajda, and Z. Idziaszek, Quantum Bose-Bose droplets at a dimensional crossover, Phys. Rev. A98, 051603(R) (2018)

2018
[30]

P. Zin, M. Pylak, Z. Idziaszek, and M. Gajda, Self- consistent description of Bose–Bose droplets: modified gapless Hartree–Fock–Bogoliubov method, New Journal of Physics24, 113038 (2022)

2022
[31]

P. Zin, M. Pylak, and M. Gajda, Self-consistent descrip- tion of Bose-Bose droplets: Harmonically trapped quasi- two-dimensional droplets, Phys. Rev. A106, 013320 (2022)

2022
[32]

J. C. Pelayo, G. Bougas, T. Fogarty, T. Busch, S.I.Mistakidis, SciPost Phys.18, 129 (2025)

2025
[33]

S. Ares, R. Baraˇ c, P. Stipanovi´ c, L. Vranjeˇ s Marki´ c, J. Boronat, Squeezing Bose-Bose liquid drops, Phys. Rev. A109013313 (2014)

2014
[34]

Staudinger, F

C. Staudinger, F. Mazzanti, and R. E. Zillich, Self-bound Bose mixtures, Phys. Rev. A98, 023633 (2018)

2018
[35]

Cikojevi´ c, L

V. Cikojevi´ c, L. Vranjeˇ s Marki´ c, G. E. Astrakharchik and J. Boronat, Phys. Rev. A,99, 023618 (2019)

2019
[36]

Cikojevi´ c, L

V. Cikojevi´ c, L. Vranjeˇ s Marki´ c, and J. Boronat, New J. Phys.22053045 (2020)

2020
[37]

Cikojevi´ c, L

V. Cikojevi´ c, L. Vranjeˇ s Marki´ c, M. Pi, M. Barranco, and J. Boronat, Towards a quantum Monte Carlo–based density functional including finite-range effects: Excita- tion modes of a 39K quantum droplet, Phys. Rev. A102, 033335 (2020)

2020
[38]

Boronat and J

J. Boronat and J. Casulleras, Phys. Rev. B49, 8920 (1994)

1994
[39]

Popari´ c, L

I. Popari´ c, L. Vranjeˇ s Marki´ c, and J. Boronat, Phys. Rev. A,112, 053317 (2025)

2025

Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.

[1] [40]

Spada, S

G. Spada, S. Pilati, and S. Giorgini, Phys. Rev. Lett.133 083401 (2024). END MA TTER Interaction potential models In our calculations we considered two sets of interac- tion potential models, POT1 and POT2, with the same scattering lengthsaand effective rangesr eff . The first model POT1 is given by V (1,1)(r) =    −V0, r≤R 0, V1, R 0 < r≤R 1, 0, R 1...

2024

[2] [1]

D. S. Petrov, Quantum mechanical stabilization of a col- lapsing Bose-Bose mixture, Phys. Rev. Lett115, 155302 (2015)

2015

[3] [2]

T. D. Lee, K. Huang, and C. N. Yang, Eigenvalues and eigenfunctions of a Bose system of hard spheres and its low-temperature properties, Phys. Rev.106, 1135 (1957)

1957

[4] [3]

D. M. Larsen, Binary mixtures of dilute Bose gases with repulsive interactions at low temperature, Annals of Physics24, 89 (1963)

1963

[5] [4]

Cabrera, L

C.R. Cabrera, L. Tanzi, J. Sanz, B. Naylor, P. Thomas, P. Cheiney, and L. Tarruell, Science359, 301 (2018)

2018

[6] [5]

Semeghini, G

G. Semeghini, G. Ferioli, L. Masi, C. Mazzinghi, L. Wol- swijk, F. Minardi, M. Modugno, G. Modugno, M. In- guscio, and M. Fattori, Self-bound quantum droplets of atomic mixtures in free space, Phys. Rev. Lett120, 235301 (2018)

2018

[7] [6]

D’Errico, A

C. D’Errico, A. Burchianti, M. Prevedelli, L. Salasnich, F. Ancilotto, M. Modugno, F. Minardi, and C. Fort, Ob- servation of quantum droplets in a heteronuclear bosonic mixture, Phys. Rev. Research1, 033155 (2019)

2019

[8] [7]

Z. Guo, F. Jia, L. Li, Y. Ma, J. M. Hutson, X. Cui, and D. Wang, Lee-Huang-Yang effects in the ultracold mixture of 23Na and 87Rb with attractive interspecies interactions, Phys. Rev. Res.3, 033247 (2021)

2021