{"id":"6f35a7a8-3134-43f7-b605-0c87e551086b","arxiv_id":"2505.09447","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":3.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"In a numerical model of a two-species quantum droplet, a long-range Pöschl-Teller interaction suppresses phonons, creates a roton, and opens a zero-momentum gap that grows with interaction range and strength.","lead":"This paper numerically models a two-species quantum droplet of Bose atoms with an added Pöschl-Teller interaction whose range and strength can be tuned. It finds that long-range versions of this interaction suppress the phonon branch, create a roton-like minimum, and open a gap at zero momentum that grows with range and strength.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The k=0 gap is likely an artifact of Eq. (7): the PT interaction appears only on the diagonal as C=Ṽ_PT(k)ρ, omitting the off-diagonal terms from the linearized Hartree term; with a self-consistent chemical potential the mode should be gapless.","rationale":"The reader's CONDITIONAL verdict correctly identifies the uniform-density/chemical-potential treatment as the weak point, but the present stress-test finds a more specific and more serious defect: Eq. (7) is not the correct Bogoliubov-de Gennes linearization of the GP equation with the PT potential. The PT term is a genuine two-body potential, not a local diagonal mean field. Its linearized contribution must enter through both the chemical potential and off-diagonal density-fluctuation couplings. As written, the matrix either cancels the PT effect at k=0 (if μ_s is self-consistent) or produces a spurious gap (if μ_s omits the PT Hartree term). In either case, the central claim that a finite-range PT interaction opens a zero-momentum gap is unsupported. This is not merely a matter of missing convergence tests; it is an internal inconsistency in the central calculation. The paper deserves credit for the numerical GP ground-state study and for attempting a two-species Bogoliubov treatment, but the headline result rests on the flawed matrix. The correct spectrum for a finite Ṽ(0) is gapless at k=0 under U(1) symmetry, so the gap cannot survive a consistent derivation. The proposed check would settle the question directly: recompute the k=0 eigenvalue with the correctly linearized PT terms. If the gap disappears, the abstract's main assertion is false and the paper requires major revision or rejection. I therefore recommend REJECT rather than CONDITIONAL, since the identified error attacks the central claim rather than its numerical precision.","tokens_in":7996,"tokens_out":16687,"duration_ms":185703,"concrete_test":"Re-derive Eq. (7) from Eq. (2) for the uniform symmetric droplet with the same parameters as Figs. 5 and 6 (U=50, μ=1, g12=4, gii=1, gLHY=5/2). Replace the diagonal C=Ṽ_PT(k)ρ by the off-diagonal linearization terms Ṽ_PT(k)φ_iφ_j and absorb Ṽ_PT(0)ρ into μ_s as required by the static GP equation. Then compute the lowest eigenvalue at k=0. If the corrected matrix gives ω(0)=0 while the published Eq. (7) gives ω(0)>0, the reported gap is a numerical artifact of the mislinearized PT term.","verdict_should_be":"REJECT","load_bearing_attack":"The load-bearing claim is the zero-momentum gap attributed to the long-range PT interaction. The Bogoliubov matrix in Eq. (7) does not follow from the GP equation (2). Linearizing the PT Hartree term Φψ_i, with Φ=∫V_PT(r−r′)ρ(r′)dr′, gives δ(Φψ_i)=Ṽ_PT(k) φ_i δρ + Ṽ_PT(0)ρ δψ_i. The first term produces off-diagonal couplings proportional to Ṽ_PT(k)φ_iφ_j for every species pair; the second term is part of the equilibrium chemical potential. In Eq. (7), however, the PT contribution is inserted only as C=Ṽ_PT(k)ρ on the diagonal entries H1, H2 and −H1, −H2, with no off-diagonal density-fluctuation terms. If μ_s is chosen self-consistently, including the k=0 Hartree potential, C cancels exactly at k=0 and the PT interaction contributes nothing to the zero-momentum matrix, so the reported U-dependent gap cannot appear. If μ_s omits the PT contribution, the condensate is not stationary and the gap is an artifact of an inconsistent chemical potential. Independently, V_PT(r) is integrable with finite Ṽ_PT(0), so the standard Bogoliubov result ω²=ε_k²+2nṼ(k)ε_k gives ω(0)=0; a genuine k=0 gap requires a singular Ṽ(k) such as the Coulomb 1/k² divergence. The paper's own conclusion notes the PT potential has no singularity, reinforcing that the reported gap is not physically supported. The roton features may also shift when the missing off-diagonal terms are restored, but the zero-momentum gap is the central result in question.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript studies a two-species quantum droplet described by a Gross-Pitaevskii (GP) equation that includes contact interactions, a Lee-Huang-Yang (LHY) beyond-mean-field term, and a Pöschl-Teller (PT) interaction whose range and strength can be tuned. The authors first solve the GP equation numerically with an imaginary-time split-step Crank-Nicolson method to obtain ground-state density profiles for various PT parameters, and then compute collective-excitation spectra using Bogoliubov theory in a uniform-density approximation. The central claimed findings are that long-range PT interactions produce a sharp roton minimum, suppress the phonon mode, and generate a zero-momentum gap that increases with the range and strength of the PT interaction.","tokens_in":8346,"tokens_out":11050,"duration_ms":108442,"significance":"The numerical work is transparent and reproducible in principle: the spectra are direct outputs of a LAPACK diagonalization rather than fits to target spectra, and the density calculations use a standard method. If the central claims were correct, the paper would demonstrate a qualitatively new excitation spectrum for quantum droplets with finite-range interactions, which would be of genuine interest to the ultracold-atom community. However, the zero-momentum gap assertion is not supported by the theory as presented: the Bogoliubov matrix in Eq. (7) omits the off-diagonal density-fluctuation terms that arise from linearizing the PT Hartree term, and for a smooth, integrable potential with finite VPT(0) the standard Bogoliubov result is gapless. Thus the paper's main physical conclusion is currently not established.","major_comments":[{"comment":"Equation (7) is not the Bogoliubov linearization of Eq. (2). Linearizing the PT Hartree term, Φψ_i with Φ=∫dr′ V_PT(r−r′)ρ(r′), around a uniform condensate yields two contributions: a diagonal term V_PT(0)ρ δψ_i that is absorbed into the chemical potential, and a density-fluctuation term V_PT(k)φ_i δρ_k that couples the U and V amplitudes of both species. The matrix in Eq. (7), however, contains only the diagonal shift C=V_PT(k)ρ on H1, H2, −H1, −H2 and omits the off-diagonal couplings. If μ_s includes the k=0 Hartree shift, the C term cancels at k=0 and the matrix cannot produce the U-dependent gap; if μ_s excludes it, the condensate is not stationary and the gap is an artifact of an inconsistent chemical potential. In either reading, the central claim of a zero-momentum gap induced by the long-range PT interaction is not supported by the equations presented.","section":"Collective excitation, Eq. (7)"},{"comment":"The reported gap also contradicts a standard theorem of Bogoliubov theory: for an integrable, finite-range potential with finite V_PT(0), the excitation spectrum is gapless, ω(k→0) ~ c k, and a k=0 gap requires a singular Fourier transform such as the Coulomb 1/k² divergence. The paper itself states in the Conclusion that \"there is no singularity in the PT potential,\" which is in direct tension with the reported gap. This tension is not resolved in the manuscript and reinforces that the gap is an artifact of the incomplete linearization in Eq. (7).","section":"Conclusion and standard Bogoliubov theory"},{"comment":"The manuscript never specifies the values of μ_s and the uniform density n used to construct the matrix (7), nor does it verify that they satisfy the stationary GP equation (2) and the Hugenholtz-Pines condition. Figures 5 and 6 are the central results, but without this self-consistency check the diagonalization may be performed at a non-equilibrium density, which can introduce a spurious gap. The authors should state the chemical potential used and demonstrate that it equals the derivative of the GP energy functional at the chosen density.","section":"Collective excitation, chemical potential"},{"comment":"The plane-wave Bogoliubov description assumes an infinite uniform system, but the calculation is applied to a self-bound droplet with N=100000. The claim that surface effects scale as N^{-1/2} is not justified for a three-dimensional droplet, where the surface-to-volume ratio scales as N^{-1/3}; the density profiles in Figs. 2-4 show a visible surface region. The authors should provide a convergence check with N or an estimate of the surface-layer fraction and show that the roton minimum and the purported gap are insensitive to finite-size effects.","section":"Model and calculation, thermodynamic limit"}],"minor_comments":[{"comment":"The caption reads \"in units ofl0\" and should read \"in units of l0\"; it should also explain why the potential is plotted for negative r although the separation r is positive.","section":"Fig. 1 caption"},{"comment":"References [39] and [55] are duplicates (both A. Boudjemaa, Phys. Lett. A 465, 128712 (2023)); reference [34] is cited in support of the gap but appears to concern single-particle excitation shifts and does not establish a gapped Bogoliubov spectrum.","section":"References"},{"comment":"The Fourier transform V_PT(k) is never shown. Since the entire gap and roton discussion depends on the shape of V_PT(k), the authors should provide its analytic expression or a plot.","section":"Results, Fourier transform"},{"comment":"The Conclusion states that \"the phonon mode stiffens [40] with a stronger and longer range of interactions,\" whereas Fig. 5 shows that for long-range interactions (small μ) the phonon mode is missing and only appears at μ=50. This internal inconsistency should be resolved.","section":"Conclusion vs. Fig. 5"},{"comment":"Equation (1) is written with \"U X_{i<j}\" without defining the pair sum; it should define the indices, the domain of r, and the dimensions of U.","section":"Eq. (1)"},{"comment":"The matrix elements in Eq. (7) are not derived in the text; a derivation or a clear reference for the linearization of all terms in Eq. (2), including the PT term, is needed.","section":"Eq. (7), derivation"}],"recommendation":"reject","confidential_remarks":"The load-bearing flaw is in the Bogoliubov linearization: Eq. (7) omits the off-diagonal density-fluctuation terms generated by the PT Hartree term, and the standard Hugenholtz-Pines theorem for a finite-range potential prohibits the reported k=0 gap. In my view the main result cannot be rescued within the current framework; a revision that rederives the excitation matrix from Eq. (2) would most likely find gapless phonons. The citation pattern for the gap also appears questionable, with duplicate references and a reference that does not support the claim."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear [Colleague],\n\nYou should know: the central claim of this paper — the zero-momentum gap in the droplet excitation spectrum — is almost certainly an artifact of an incomplete Bogoliubov derivation. The numerical work is honest and standard, but the headline result should not be trusted.\n\nWhat's here: the authors take the two-species GP equation with contact interactions, LHY quantum fluctuations, and a Pöschl-Teller (PT) two-body potential. They solve for the ground-state density with imaginary-time Crank-Nicolson, then diagonalize a Bogoliubov matrix. The density profiles are plausible, and the parameter scan over U and µ is a reasonable way to move from short- to long-range interactions. The roton minimum for long-range PT interactions is qualitatively consistent with earlier work.\n\nThe problem is in Eq. (7). The matrix includes the PT interaction only as a diagonal term C = Ṽ_PT(k)ρ added to H1, H2, -H1, -H2. Linearizing the Hartree term ∫V(r-r')ρ(r')ψ_i(r) gives both the diagonal shift from the Hartree potential and off-diagonal couplings from the density fluctuation δρ. Those are missing. More importantly, if the chemical potential μ_s is taken from the GP equation that includes the PT interaction, the diagonal shift at k=0 cancels exactly, and the gap disappears. If μ_s omits the PT contribution, the condensate is not stationary and the \"gap\" is just a consequence of an inconsistent chemical potential.\n\nThe standard Bogoliubov result for a non-singular, integrable potential like PT gives ω(k=0)=0. A genuine zero-momentum gap requires a singular Ṽ(k) such as the Coulomb 1/k² divergence. The authors themselves note the PT potential has no singularity (in the Conclusion), which is the right thing to say — but it directly contradicts their gapped spectrum.\n\nSo where does this leave the paper? The roton features may still survive a corrected derivation, and the density profiles are fine as numerically converged results. But the abstract and conclusions are built on the k=0 gap. The authors need to redo the linearization, check the Hugenholtz-Pines condition, and state the density and chemical potential used in Eq. (7). If they do, they might find the roton remains; the gap will not.\n\nI would not cite this paper in its current form, and I would not send it to a referee yet. It deserves a chance after a major revision that fixes the central derivation. For now, the headline result is not credible.\n\nBest,","headline":"The zero-momentum gap is likely an artifact of incomplete linearization; the paper's headline claim is not credible, though the roton trend may survive a corrected derivation.","tokens_in":8980,"tokens_out":7757,"would_cite":false,"duration_ms":74489,"reading_group":"no","serious_thinker":"yes","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["03.75.Kk","03.75.Mn"],"model":"deepseek-v4-flash","headline":"A two-species quantum droplet with a tunable Pöschl-Teller interaction develops a finite excitation gap at zero momentum and a sharp roton minimum when the interaction is long-ranged, and both features grow with the interaction's range…","keywords":["quantum droplet","Pöschl-Teller interaction","Bogoliubov excitation spectrum","roton minimum","zero-momentum gap","two-species Bose-Einstein condensate","Lee-Huang-Yang correction","Gross-Pitaevskii equation"],"falsifier":"One decisive check is to compute the excitation spectrum for the same $\\mu$ and $U$ at several particle numbers with fixed density: if the zero-momentum gap shrinks toward zero or the roton minimum flattens as $N$ grows, the claimed long-range gap is a finite-size artifact. An experimental version would measure the Bragg-scattering spectrum of a two-species droplet with a tunable long-range interaction; the prediction would fail if low-momentum excitations remain gapless whenever the PT interaction is present.","tokens_in":1758,"feed_emoji":"💧","tokens_out":2458,"duration_ms":111215,"temperature":0.7,"pith_summary":"This paper studies a self-bound two-species Bose droplet whose atoms interact through the Pöschl-Teller potential, a repulsive interaction whose width is set by a single parameter. By solving the Gross-Pitaevskii equation with the Lee-Huang-Yang correction and linearizing around the uniform droplet, the authors map how the collective excitation spectrum changes as the interaction is tuned from short-range to long-range. The central result is that a long-range Pöschl-Teller interaction removes the usual phonon branch, opens a finite gap at zero momentum, and produces a sharp roton minimum; both the gap and the sharpness of the roton grow with the interaction's range and strength. If correct, this gives a single model potential in which a droplet's superfluid response can be tuned from phononic to roton-dominated by adjusting one parameter.","feed_headline":"Long-range interactions open a gap in droplet spectra","feed_subtitle":"Tuning the Pöschl-Teller range shifts droplet excitations from phonons to a gapped roton minimum.","key_machinery":"The load-bearing object is the Pöschl-Teller pair potential $V_{\\mathrm{PT}}(r)=U\\,2\\mu/\\cosh^2(\\mu r)$, in which $\\mu$ controls the inverse width (so the range is $1/\\mu$) and $U$ controls the strength. Its continuous deformation from a delta-like contact interaction to a long-range potential lets a single model interpolate between regimes. In the excitation calculation, the potential enters through its Fourier transform $\\tilde{V}_{\\mathrm{PT}}(k)$ in the density-dependent matrix element $C=\\tilde{V}_{\\mathrm{PT}}(k)\\rho$; because this term is momentum dependent, it reshapes the spectrum, producing the roton minimum and, at $k=0$, the finite gap. The ground state is obtained from the Gross-Pitaevskii equation with the Lee-Huang-Yang correction using imaginary-time propagation, and the uniform density justifies the plane-wave Bogoliubov ansatz.","core_discovery":"The paper's central claim is that when atoms in a two-species quantum droplet interact through a repulsive Pöschl-Teller potential of long range (small $\\mu$), the Bogoliubov excitation spectrum is not the usual phononic one: it has a finite gap at zero momentum, no phonon branch, and a sharp roton minimum at finite momentum. As the potential is made more short-ranged (larger $\\mu$), the roton weakens and moves to higher momentum and eventually disappears, while a phonon mode reappears. Increasing the interaction strength $U$ makes the roton sharper, shifts it to higher momentum, and widens the zero-momentum gap. The same calculations show that the ground-state density of the droplet decreases with $U$ and saturates as $\\mu$ becomes large, consistent with the PT potential becoming delta-function-like. The paper presents this as a distinctive feature of the Pöschl-Teller interaction, different from dipolar and Coulomb long-range interactions.","pith_inferences":["A natural next test is to measure the dynamic structure factor of a two-species condensate dressed with a soft-core potential that approximates the PT shape; a finite gap at zero momentum would appear as an energy threshold and the roton as a finite-momentum peak.","If the zero-momentum gap survives the thermodynamic limit, it would be a rare case of a gapped excitation in a self-bound superfluid, which would require reconciling the droplet's broken gauge symmetry with the usual expectation of a gapless Goldstone mode.","The finite-$N$ droplet used in the numerics ($N=10^5$) could in principle introduce surface effects; repeating the calculation at several $N$ values and checking convergence of the gap and roton depth would separate the bulk prediction from a finite-size artifact."],"forward_implications":["For long-range PT interactions ($\\mu$ small), the droplet's low-momentum spectrum is gapped rather than phononic; if confirmed, Bragg spectroscopy should show a threshold at zero momentum instead of a linear sound mode.","Tuning $\\mu$ from small to large moves the spectrum from gapped-roton to phonon-dominated, so a single potential parameter controls the superfluid response of the droplet.","Increasing $U$ lowers the droplet density, sharpens the roton, moves it to higher momentum, and increases the zero-momentum gap.","Because the PT potential has no singularity, it offers a cleaner theoretical model for long-range interaction effects than dipolar or Coulomb potentials, where infrared behavior complicates the analysis."],"supporting_citations":[{"why":"Supplies the Pöschl-Teller interaction potential between Bose atoms that the paper uses as its tunable long-range interaction.","marker":"[35]"},{"why":"Earlier application of the Pöschl-Teller potential to quantum gases supports its use as a finite-range model interaction.","marker":"[36]"},{"why":"Provides the two-species droplet parameter regime that yields a uniform ground-state density.","marker":"[3]"},{"why":"Basis of the Lee-Huang-Yang beyond-mean-field correction that stabilizes the droplet against mean-field collapse.","marker":"[15]"},{"why":"Prior observation of a zero-momentum gap and phonon stiffening in long-range interacting gases that the droplet spectrum is compared against.","marker":"[40]"},{"why":"Reference for the zero-momentum gap used to identify the PT-induced gap in the spectra.","marker":"[39]"},{"why":"Supplies the Bogoliubov fluctuation ansatz and the two-species excitation matrix diagonalized to obtain the spectra.","marker":"[54]"},{"why":"Provides the imaginary-time split-step method used to find the droplet ground-state density.","marker":"[52]"}],"fun_headline_variants":["Roton gap emerges in quantum droplet spectra","Long-range PT potential gaps droplet excitations","Phonons vanish, roton gap grows in droplets","Gapped roton from long-range droplet interactions","Pöschl-Teller range tunes droplet roton gap"],"cache_read_input_tokens":10880,"weakest_assumption_plain":"The main load-bearing premise is that the droplet is effectively infinite and uniform, so the plane-wave Bogoliubov calculation from a homogeneous density captures the true bulk excitation spectrum and surface effects are negligible.","fun_headline_variants_meta":{"raw":{"variants":["Roton gap emerges in quantum droplet spectra","Long-range PT potential gaps droplet excitations","Phonons vanish, roton gap grows in droplets","Gapped roton from long-range droplet interactions","Pöschl-Teller range tunes droplet roton gap"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000138,"raw_usage":{"total_tokens":1119,"prompt_tokens":873,"completion_tokens":246,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":489,"completion_tokens_details":{"reasoning_tokens":173}},"tokens_in":489,"tokens_out":246,"duration_ms":2618,"temperature":1.0,"reasoning_tokens":173,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T21:32:07.874971+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"One decisive check is to compute the excitation spectrum for the same $\\mu$ and $U$ at several particle numbers with fixed density: if the zero-momentum gap shrinks toward zero or the roton minimum flattens as $N$ grows, the claimed long-range gap is a finite-size artifact. An experimental version would measure the Bragg-scattering spectrum of a two-species droplet with a tunable long-range interaction; the prediction would fail if low-momentum excitations remain gapless whenever the PT interaction is present.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the Pöschl-Teller interaction potential between Bose atoms that the paper uses as its tunable long-range interaction."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Earlier application of the Pöschl-Teller potential to quantum gases supports its use as a finite-range model interaction."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Basis of the Lee-Huang-Yang beyond-mean-field correction that stabilizes the droplet against mean-field collapse."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Prior observation of a zero-momentum gap and phonon stiffening in long-range interacting gases that the droplet spectrum is compared against."},{"cited_title":"Banerjee, D","cited_arxiv_id":null,"evidence_quote":"Supplies the Bogoliubov fluctuation ansatz and the two-species excitation matrix diagonalized to obtain the spectra."}],"review_version":1}