{"id":"0a4f18ad-75fb-49c7-b493-19972223065e","arxiv_id":"2501.01771","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Energy spectra of 2D quantum droplets stirred by a rotating barrier show Kolmogorov k^-5/3, Vinen k^-1, and vortex-core k^-3 scaling, with direct energy cascades.","lead":"This paper simulates a two-dimensional quantum droplet stirred by a rotating laser barrier and measures how its energy is distributed across length scales. It reports that droplet turbulence shows the same Kolmogorov and Vinen scaling laws as ordinary Bose gases, with the regime controlled by the stirrer's height and speed.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Central simulations use N=10^4 although the paper's own footnote states the droplet energy is negative only for N≤9700; if N=10^4 is a box-supported gas, the claimed droplet turbulence is not demonstrated.","rationale":"The reader's weakest assumption identifies exactly the concern I find most load-bearing: the paper's own criterion for a droplet state conflicts with the N=10^4 simulations used for the central results. The central claim would be true if the N=10^4 homogeneous state were a self-bound droplet, but footnote 1 gives a direct internal check that at N=10^4 the total energy is positive. That makes the object under study a box-supported gas, not a droplet, and the headline result would reduce to turbulence of a stirred gas unless the scalings are shown to persist at negative-energy droplet parameters. I considered the internal inconsistency in the compressible ultraviolet scaling as the primary concern; it is a clear error in description (k^-7 versus k), but it does not undermine the object of study as directly as the droplet-regime question. I also note that the paper has some independent support: the single-vortex Appendix A calculation provides a clean demonstration of k^-3 vortex-core scaling at N=900, which is evidence for part of the spectral claim. However, it does not address the droplet identity of the turbulent N=10^4 state. The proposed test is minimal and decisive: repeat the representative cases at an atom number explicitly inside the negative-energy regime and compare the spectral scalings and fluxes. If the same turbulence signatures appear there, the central claim survives; if not, the paper should be revised to describe gas turbulence or the regime must be reassigned. Because this is exactly the condition the reader already imposed, I do not move the verdict; it should remain conditional on resolving the droplet-regime issue.","tokens_in":24337,"tokens_out":3778,"duration_ms":41219,"concrete_test":"Rerun the three representative cases (I: v=1.0, V0=2.5; II: v=2.0, V0=8.5; III: v=1.0, V0=8.5) at N=9500 (and optionally N=9000) in the same L=60 box, after verifying numerically that the converged ground-state total energy is negative, and recompute ε_i(k), ε_c(k), and Π(k) at the same evolution times. If the k^-5/3, k^-1, and k^-3 spectral signatures and positive direct incompressible flux survive for these negative-energy droplet parameters, the central claim is robust; if they change or disappear, the N=10^4 results must not be presented as droplet turbulence.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing weakness is internal to the manuscript. Section IV, footnote 1, states that 'the total energy of the system remains negative for N ≤ 9700.' Yet the homogeneous-droplet turbulence analysis in Sec. V—Figs. 2, 5, 6, and 7(a), and all three cases I–III—uses N = 10^4. For a self-bound quantum droplet the total energy should be negative; a positive-energy state in a periodic box is a box-confined gas rather than a droplet. If the N=10^4 state is not a droplet, the central claim that the observed Kolmogorov k^-5/3, Vinen k^-1, and vortex-core k^-3 spectra are properties of turbulent quantum droplets is not established: the simulations would describe a stirred gas in a box. Appendix A supports the k^-3 vortex-core scaling at N=900, but it does not validate the droplet identity of the N=10^4 turbulent state. A secondary internal inconsistency—the compressible ultraviolet scaling reported as k^-7 in Sec. V.C and Fig. 6 for case III, but described as a trend toward k in the abstract and conclusions—is real but less threatening than the regime question.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript reports a numerical study of turbulence in two-dimensional quantum droplets described by the logarithmic eGPE. A rotating Gaussian barrier stirs a box-confined droplet for three periods and is then removed; the post-stirring dynamics are classified by the vortex sign correlation function C2 and the ratio of incompressible to compressible kinetic energy. Three regimes are identified: vortex dipoles (case I), random vortex distributions with enhanced sound waves (case II), and vortex-antivortex clusters (case III). The central claims are that the incompressible energy spectra show Kolmogorov k^{-5/3} scaling (cases I and III), Vinen-like k^{-1} scaling (case II), and a universal k^{-3} ultraviolet scaling from vortex cores, while the compressible spectra show k^{-3/2} infrared scaling and regime-dependent ultraviolet behavior; positive fluxes are interpreted as a direct energy cascade. A harmonically trapped flat-top droplet is also studied. Appendix A imprints a single vortex and confirms k^{-3} ultraviolet scaling in both scalar BEC and droplet backgrounds.","tokens_in":24598,"tokens_out":3380,"duration_ms":38689,"significance":"If the central claims hold, this is a useful extension of quantum-turbulence phenomenology to the self-bound droplet phase, where the LHY logarithmic nonlinearity changes the background and the equation of motion relative to conventional BECs. The paper has visible strengths: the k^{-3} ultraviolet scaling is explicitly checked in Appendix A by imprinting a single vortex, so that scaling is an output rather than an input; the phase diagrams in Figs. 3 and 4 cover a broad parameter range and give a qualitative map of vortex-dipole, random, and cluster regimes; and the distinction between incompressible and compressible channels is handled with the standard Helmholtz decomposition. The main reservations are whether the N=10^4 homogeneous state used for the central turbulence analysis is actually a self-bound droplet according to the paper's own energy criterion, and whether the quoted scaling exponents are sufficiently quantified, since they are identified from individual simulation trajectories without a stated fitting procedure or uncertainty estimates.","major_comments":[{"comment":"The paper's own footnote in Sec. IV states that the total energy of the droplet remains negative only for N <= 9700, yet the homogeneous-droplet turbulence analysis in Sec. V (Figs. 2, 3, 5, 6, 7(a), and cases I-III) uses N = 10^4. For a self-bound quantum droplet the total energy should be negative; a positive-energy state in a periodic box is a box-confined gas rather than a droplet. If the N=10^4 state is not a droplet, then the central claim that the observed Kolmogorov, Vinen, and vortex-core scalings are properties of turbulent quantum droplets is not established. The authors should either repeat the analysis for N <= 9700 and verify that the spectra and fluxes are unchanged, or provide a quantitative demonstration (e.g., the binding energy, chemical potential, and density profile) that the N=10^4 state remains droplet-like for the purposes of the turbulent dynamics. Appendix A, which uses N=900, does not by itself validate the droplet identity of the N=10^4 turbulent state.","section":"Sec. IV, footnote 1, and Sec. V"},{"comment":"The scaling exponents k^{-5/3}, k^{-1}, k^{-3}, k^{-3/2}, k^{-7/2}, and k^{-7} are identified from single simulation runs, without a stated fitting procedure, fitting ranges, or uncertainty quantification. The phase diagram in Fig. 7 classifies regimes as k^{-1}, k^{-5/3}, 'no clean power law', or 'no vortex generation', but no error bars or statistical measures are given. Because the central claim of the paper is precisely that these power laws appear, the analysis should specify the exact wavenumber intervals used for each fit, the fitting method, and an estimate of the uncertainty (for example, from multiple realizations, time-window averages, or bootstrap resampling of the spectra). Without this, the distinction between genuine power-law regimes and transient or crossover behavior remains insufficiently supported.","section":"Secs. V.B-V.D and Figs. 5-7"},{"comment":"There is an internal inconsistency in the reported compressible ultraviolet scaling. Section V.C and Fig. 6(c) report a k^{-7} decay for case III, while the abstract and the Conclusions state that both cases II and III have a trend toward k scaling signaling thermalization. The text in Sec. V.C explicitly says that the k^{-7} scaling 'is modified as time evolves but again never becomes linear', which contradicts the summary statements. The authors should either correct the summary claims or clearly distinguish between different time windows and wavenumber ranges. The introductory summary in Sec. I, which mentions only k^{-7/2} for dipole-dominated cases and k scaling otherwise, is also not compatible with the k^{-7} case III result.","section":"Abstract, Sec. V.C, Fig. 6(c), and Conclusions"}],"minor_comments":[{"comment":"The intervortex distance l0 = 1/sqrt(Nv) is taken from repulsive Bose gases, and the footnote admits that its validity for droplets is an open issue; since k_l0 is used to identify the Kolmogorov scaling range, a sensitivity analysis with respect to this definition would strengthen the quantitative conclusions.","section":"Sec. V.B, footnote 4"},{"comment":"The phase diagrams show C2 and zeta as smooth color maps, but each parameter point appears to come from a single simulation; adding a brief statement about run-to-run variability or averaging protocols would improve reproducibility.","section":"Fig. 3 and Fig. 4"},{"comment":"The red markers in Fig. 7 are labeled with a theta symbol in the legend, but the caption does not define what value of theta corresponds to the 'not a clean power law' classification.","section":"Sec. V.D, Fig. 7"},{"comment":"There are several typographical errors, including 'striring' (Sec. V.A), 'interacomponent' (Sec. II), and inconsistent notation for the k^1 scaling, which is written both as 'k' and as 'k^1' in different places; a careful proofread is recommended.","section":"Throughout"},{"comment":"The flat-top droplet section uses a harmonic trap and N=2e4, but the criterion from footnote 1 is not discussed for this trapped geometry; a sentence clarifying the droplet binding condition in the presence of the trap would avoid confusion.","section":"Sec. VI"}],"recommendation":"major_revision","confidential_remarks":"The central physical scenario is interesting and the numerical infrastructure appears appropriate, but the N=10^4 vs N<=9700 issue is a load-bearing inconsistency that must be resolved before the droplet-specific interpretation can be accepted. The lack of uncertainty quantification on the scaling exponents is also likely to be raised by any turbulence-oriented referee. I would not recommend rejection, because the issues are fixable within the manuscript's scope."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The genuinely new thing here is the setting: nobody has looked at vortex and wave turbulence in 2D quantum droplet backgrounds. The authors use the eGPE, stir with a rotating Gaussian barrier, and classify the response with the sign correlation function and the incompressible-to-compressible energy ratio. That parameter sweep is systematic, and the three regimes (dipoles, random pairs, clusters) are clearly illustrated. The paper also does one thing I particularly like: Appendix A imprints a single vortex in both a BEC and a droplet and confirms the k^-3 ultraviolet scaling, which means the small-scale vortex-core result is not just read off the noisy turbulent runs. That is real support for the central claim.\n\nThe soft spots are real but not equally soft. The most load-bearing is the internal inconsistency: footnote 1 says the total energy is negative only for N≤9700, yet the main homogeneous-droplet simulations all use N=10^4. If that state has positive energy, it is a box-confined gas rather than a self-bound droplet, and the headline claim of droplet turbulence is not established. This needs to be fixed before I would trust the regime interpretation. The secondary inconsistency in the compressible ultraviolet scaling (k^-7 in Sec. V.C and Fig. 6 vs. a 'trend toward k' in the abstract/conclusions) is sloppy but less threatening. The bigger methodological weakness is that every scaling exponent is identified from a single run, with no error bars, no ensemble averaging, and no stated fitting procedure. Given how noisy decaying turbulence spectra are, I would not bet heavily on the precise values of the k^-5/3, k^-1, and k^-3/2 exponents yet. The borrowed intervortex distance l0 = 1/sqrt(Nv) is at least flagged as an open issue for droplets, which is honest.\n\nThe citation pattern is fine and the paper engages properly with the BEC turbulence literature. No code or data is shipped, which is a miss but not disqualifying.\n\nBottom line: this is a solid first study of a new physical setting, with one internal inconsistency that could undercut the central claim and a general lack of statistics on the scaling exponents. It deserves a serious referee, but the referee should be asked to push on the droplet identity of the N=10^4 state and on the statistical support for the spectra. If those are cleared up, this would be a useful contribution to the ultracold-atoms subfield.","headline":"First numerical study of turbulent energy spectra in 2D quantum droplets; plausible and worth refereeing, but the N=10^4 state may not be self-bound per the paper's own footnote, and the scaling claims lack error bars.","tokens_in":25104,"tokens_out":1063,"would_cite":false,"duration_ms":13150,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Stirred two-dimensional quantum droplets show Kolmogorov and Vinen energy spectra with a direct cascade and a universal vortex-core tail.","keywords":["quantum droplets","two-dimensional turbulence","energy spectra","Kolmogorov scaling","Vinen scaling","vortex turbulence","extended Gross-Pitaevskii equation","Lee-Huang-Yang corrections"],"falsifier":"Compute the total energy of the N=$10^{4}$ box-trapped ground state used for cases I-III. If it is positive, the state is not a self-bound droplet and the central claim fails; a clean test would repeat the stirring protocol at lower atom numbers in a larger box and check whether $k^{{-5/3}}$, $k^{{-1}}$ and $k^{{-3}}$ scalings and the positive direct flux survive.","tokens_in":1713,"feed_emoji":"🌀","tokens_out":1778,"duration_ms":68912,"temperature":0.7,"pith_summary":"The paper claims that a two-dimensional quantum droplet, after being stirred for three rotations by a moving repulsive barrier and then released, develops vortex turbulence whose kinetic-energy spectra reproduce classic scaling laws of quantum fluids. The incompressible part follows Kolmogorov $k^{{-5/3}}$ scaling when vortices organize into dipoles or clusters, Vinen-like $k^{{-1}}$ scaling when vortices are randomly distributed, and a $k^{{-3}}$ tail at large wavenumbers coming from the vortex cores. The compressible part shows $k^{{-3/2}}$ infrared scaling, indicating weak wave turbulence, and in one regime a trend toward k at high wavenumbers, suggesting thermalization. Energy fluxes are mostly positive, meaning a direct cascade from large to small length scales. The result matters because it extends quantum-turbulence phenomenology from ordinary Bose-Einstein condensates to droplets stabilized by quantum fluctuations.","feed_headline":"Turbulent quantum droplets show Kolmogorov and Vinen spectra","feed_subtitle":"Stirred droplets show a direct energy cascade and vortex-core scaling, a first for quantum-fluctuation-stabilized matter.","key_machinery":"The argument rests on the two-dimensional extended Gross-Pitaevskii equation with a logarithmic nonlinearity that encodes mean-field interactions plus Lee-Huang-Yang quantum fluctuations. Turbulence is diagnosed by decomposing the density-weighted velocity field into incompressible (vortical) and compressible (acoustic) parts, computing angle-averaged kinetic energy spectra and fluxes, and classifying vortex arrangements with the second-order sign correlation function C_2 and the ratio of incompressible to compressible kinetic energy. The spectra are obtained through the angle-averaged Wiener-Khinchin theorem, and the fluxes through a spectral decomposition of the energy transfer across wavenumbers.","core_discovery":"The authors' central discovery is that a self-bound two-dimensional droplet driven by a rotating Gaussian obstacle enters clearly separated turbulent response regimes determined by the barrier height and speed, as diagnosed by the incompressible and compressible kinetic energy spectra. In the homogeneous droplet environment, vortex dipoles and vortex clusters yield Kolmogorov $k^{{-5/3}}$ infrared scaling, random vortex distributions yield Vinen $k^{{-1}}$ scaling, and all cases show $k^{{-3}}$ ultraviolet scaling from vortex cores. The compressible spectra exhibit $k^{{-3/2}}$ infrared scaling in all cases, with the random-distribution case approaching k in the ultraviolet and indicating thermalization. The flux analysis shows a direct energy cascade that is largest during the second stirring period and suppressed once stirring stops.","pith_inferences":["The k^{-3} ultraviolet scaling is presented as independent of background, suggesting a universal vortex-core signature that could be tested by phase-imprinting single vortices into droplets versus scalar condensates and comparing spectra at fixed healing length.","Because the homogeneous results are obtained in a finite box that nearly holds the droplet, the infrared scalings may inherit finite-size effects; a larger box or a free-space flat-top droplet would clarify whether k^{-5/3} and k^{-1} regimes are intrinsic or box-induced.","The observed boundary deformation and vortex escape in the flat-top droplet offer a testable experimental fingerprint: absorption imaging of a stirred potassium-39 droplet mixture should show irregular edges when vortex-antivortex pairs dominate."],"forward_implications":["If the central claim is correct, vortex turbulence in droplet environments displays a universal k^{-3} vortex-core scaling at large wavenumbers, matching scalar Bose-Einstein condensates.","The infrared scaling of the incompressible kinetic energy depends on vortex organization: k^{-5/3} for dipoles or clusters, and k^{-1} for random vortex distributions.","The mostly positive incompressible energy flux indicates a direct cascade from large to small length scales in the homogeneous droplet.","The compressible k^{-3/2} infrared scaling implies that weak wave turbulence coexists with vortex turbulence in stirred droplets.","In the flat-top droplet, faster stirring converts a vortex-dipole regime into one with many vortex-antivortex pairs, accompanied by observable distortion of the droplet boundary."],"supporting_citations":[{"why":"Supplies the two-dimensional extended Gross-Pitaevskii model with logarithmic nonlinearity that defines the droplet background.","marker":"[54]"},{"why":"Provides the angle-averaged Wiener-Khinchin spectral method used to compute kinetic energy spectra.","marker":"[67]"},{"why":"Establishes the k^{-3} ultraviolet scaling for vortex cores in two-dimensional quantum turbulence that the paper reproduces.","marker":"[37]"},{"why":"Introduces the second-order vortex sign correlation function used to classify vortex dipoles, random distributions, and clusters.","marker":"[78]"},{"why":"Provides the spectral-flux definition and rotating Bose-Einstein condensate turbulence methodology that the paper adapts to droplets.","marker":"[25]"},{"why":"Demonstrates direct energy cascade and Kolmogorov scaling in two-dimensional compressible quantum turbulence, the comparison basis for the observed cascade.","marker":"[26]"},{"why":"Reviews droplet properties and the Lee-Huang-Yang stabilization that justify treating the box-filled state as a droplet.","marker":"[55]"},{"why":"Reports Vinen-like k^{-1} scaling in fast-rotating two-dimensional Bose-Einstein condensates, the reference for the random-vortex regime.","marker":"[33]"},{"why":"Supplies the k^{-7/2} wave-turbulence scaling at large momenta that the compressible spectra are compared against.","marker":"[80]"}],"fun_headline_variants":["Turbulent quantum droplets hit Kolmogorov and Vinen scaling","Quantum droplet turbulence shows direct energy cascade","Vortex clusters drive Kolmogorov and Vinen spectra in droplets","Stirred droplets reveal quantum turbulence with Kolmogorov scaling","Direct cascade and vortex scaling in 2D quantum droplets"],"cache_read_input_tokens":27264,"weakest_assumption_plain":"The homogeneous-droplet simulations use N=$10^{4}$ atoms, yet the paper's own footnote states the total energy is negative only for atom numbers up to 9700; if the N=$10^{4}$ box-filling state is a gas with positive energy, the claimed droplet-specific turbulence is not demonstrated.","fun_headline_variants_meta":{"raw":{"variants":["Turbulent quantum droplets hit Kolmogorov and Vinen scaling","Quantum droplet turbulence shows direct energy cascade","Vortex clusters drive Kolmogorov and Vinen spectra in droplets","Stirred droplets reveal quantum turbulence with Kolmogorov scaling","Direct cascade and vortex scaling in 2D quantum droplets"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000529,"raw_usage":{"total_tokens":2543,"prompt_tokens":930,"completion_tokens":1613,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":546,"completion_tokens_details":{"reasoning_tokens":1531}},"tokens_in":546,"tokens_out":1613,"duration_ms":13241,"temperature":1.0,"reasoning_tokens":1531,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T22:20:37.335774+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the total energy of the N=$10^{4}$ box-trapped ground state used for cases I-III. If it is positive, the state is not a self-bound droplet and the central claim fails; a clean test would repeat the stirring protocol at lower atom numbers in a larger box and check whether $k^{{-5/3}}$, $k^{{-1}}$ and $k^{{-3}}$ scalings and the positive direct flux survive.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the angle-averaged Wiener-Khinchin spectral method used to compute kinetic energy spectra."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the second-order vortex sign correlation function used to classify vortex dipoles, random distributions, and clusters."},{"cited_title":"Sivakumar, P","cited_arxiv_id":null,"evidence_quote":"Provides the spectral-flux definition and rotating Bose-Einstein condensate turbulence methodology that the paper adapts to droplets."},{"cited_title":"Numasato, M","cited_arxiv_id":null,"evidence_quote":"Demonstrates direct energy cascade and Kolmogorov scaling in two-dimensional compressible quantum turbulence, the comparison basis for the observed cascade."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Reviews droplet properties and the Lee-Huang-Yang stabilization that justify treating the box-filled state as a droplet."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Reports Vinen-like k^{-1} scaling in fast-rotating two-dimensional Bose-Einstein condensates, the reference for the random-vortex regime."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the k^{-7/2} wave-turbulence scaling at large momenta that the compressible spectra are compared against."}],"review_version":1}