{"id":"a08aa2bf-79f7-4bb1-b2a6-63a58e00ce12","arxiv_id":"2608.06800","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Microscopic BdG calculations for 2D dipolar supersolids identify the three acoustic modes as one shear and two longitudinal branches and show their fluctuations match supersolid hydrodynamics.","lead":"This paper computes, from microscopic equations, how atoms move and flow in the three sound modes of a two-dimensional supersolid crystal. It gives experimentalists a way to identify each sound branch and to separately measure the crystal motion and the superfluid flow.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified; the central claim is supported by independent sum-rule and two-fluid current checks.","rationale":"The paper's strongest claim is that long-wavelength BdG fluctuation amplitudes agree with hydrodynamics specified by elastic coefficients. The reader's weakest assumption is the stationary-point displacement extraction. I examined this and other possible weak points. The displacement extraction is tested by the linearity check in Fig. 3 and, more importantly, by Eqs. (32)-(33), where the same u, combined with phase fluctuations and elastic parameters, reproduces the current calculated directly from BdG; this would fail if u were not the true lattice displacement. The hydrodynamic formulas are quoted from the 1D literature, so the 2D transfer is not shown analytically, but the agreement in the insets of Figs. 4-5 over multiple independent channels, plus the exact sum rules in Fig. 6, gives strong empirical support. The paper also honestly lists limitations: zero temperature, uniform geometry, and no 1D stripe phase. No internal inconsistency or circularity surfaced. I therefore see no load-bearing concern that would change the verdict.","tokens_in":17215,"tokens_out":18430,"duration_ms":175149,"concrete_test":"As a worthwhile verification, independently re-derive Eqs. (44)-(47) for the 2D correlation lengths from the supersolid hydrodynamic Lagrangian rather than importing the 1D result; if the prefactors acquire dimension-dependent factors, the insets' agreement could be accidental. Separately, recompute the strain and defect density decomposition for the honeycomb state by tracking one of the two density maxima in each unit cell instead of the unique minimum; if the cancellation in the lower longitudinal mode survives, the extraction point is not load-bearing.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I read the paper in good faith and find no load-bearing flaw in the central claim that BdG fluctuation amplitudes match hydrodynamic theory. The displacement-extraction procedure flagged by the reader is a genuine methodological premise, but it is not load-bearing here: the paper explicitly verifies linearity for |c| up to about 50 (displacements below 5% of the lattice constant) in Fig. 3, and the two-fluid current relations in Eqs. (32)-(33) provide an independent consistency check that the extracted displacement u is the hydrodynamic one, since δJ, δθ, and u are computed from different matrix elements. The hydrodynamic correlation-length formulas (39)-(49) are quoted from 1D rather than re-derived in 2D, but the numerical agreement across two crystal orders and four observable channels, together with the verified f-sum rule (52), compressibility sum rule (53), and transverse-current sum rule (56), makes the central claim credible. I therefore have no significant objection.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies the long-wavelength acoustic excitations of two-dimensional dipolar supersolids with triangular and honeycomb order. Using Bogoliubov–de Gennes (BdG) calculations, the authors compute the density, phase, current, and lattice-displacement fluctuations of the three acoustic modes. They develop a procedure to extract the lattice displacement from stationary points of the density, which allows a decomposition of density fluctuations into strain and defect-density contributions. The mode analysis identifies one transverse shear branch and two longitudinal branches, with the lower longitudinal branch showing large strain and defect-density fluctuations that substantially cancel in the total density response. The long-wavelength fluctuation amplitudes are compared with hydrodynamic theory specified entirely by the elastic coefficients, and good agreement is found for both crystal types. The calculations also satisfy the f-sum rule, compressibility sum rule, and transverse-current sum rule.","tokens_in":17335,"tokens_out":13519,"duration_ms":121221,"significance":"If the results are correct, this is a significant advance in the microscopic understanding of 2D supersolids. The paper provides the first detailed decomposition of acoustic modes into strain and defect-density contributions, and it identifies experimental signatures in density- and current-sensitive probes. A key strength is that the central claim rests on multiple independent checks: the long-wavelength asymptotes are compared across four observable channels for two different lattice structures; the f-sum rule (52), compressibility sum rule (53), and transverse-current sum rule (56) are all verified numerically; and the two-fluid current relations (32)–(33) are tested using matrix elements computed from different quantities. The displacement-extraction method is a methodological contribution, and its validity is explicitly verified up to about 5% of the lattice constant in Fig. 3. A caveat is that the hydrodynamic coefficients and the BdG modes are computed from the same eGPE ground state, so the agreement is a strong consistency check rather than an independent ab initio prediction; the paper should state this more clearly.","major_comments":[],"minor_comments":[{"comment":"The correlation-length formulas are quoted from the 1D theory of Ref. [27] without a derivation for the 2D case; please provide a brief derivation or explicitly state that the numerical agreement presented here is the verification of the 2D forms.","section":"Sec. IV.C, Eqs. (39)–(49)"},{"comment":"The sentence about the density maximum being unique applies to the triangular lattice; for honeycomb the tracked points are density minima. Please clarify this and justify that the minimum follows the acoustic displacement rather than an optical internal distortion.","section":"Sec. II.D.1"},{"comment":"The phase fluctuation is defined with a denominator \\Psi_0(\\rho,0); please comment on the numerical stability of this quantity in regions of low density, even though the density does not vanish.","section":"Eq. (14) and Sec. II.C"},{"comment":"The derivation of the transverse current fluctuation amplitude from the sum rule (56) is only sketched; a few lines of explanation would make the logic transparent.","section":"Sec. IV.C after Eq. (48)"},{"comment":"There is a typo: 'by vitue' should be 'by virtue'.","section":"Footnote 1"},{"comment":"The hydrodynamic asymptotes are marked with crosses, but the figure labels do not make clear which cross corresponds to which band; please label the asymptotes directly.","section":"Insets of Figs. 4 and 5"}],"recommendation":"minor_revision","confidential_remarks":"The manuscript is within the scope of the journal. The central claim is supported by the numerics; the only substantive caveat is that the hydrodynamic coefficients and BdG modes derive from the same eGPE, so the agreement is a consistency check. This should be framed clearly in the text. I recommend minor revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Blakie gives us mode-resolved fluctuation amplitudes for the three acoustic branches of 2D dipolar supersolids, and the central result holds up: the BdG density, phase, current, and displacement fluctuations match hydrodynamic theory at long wavelengths. The genuinely new pieces are the lattice-displacement extraction from the microscopic density perturbation, the strain/defect decomposition that reveals the large cancellation in the lower longitudinal mode, and the demonstration that the transverse shear mode is visible only through current probes. The paper also earns credit for checking itself: the f-sum rule, compressibility sum rule, and transverse-current sum rule are all satisfied, and the two-fluid current relations (32)-(33) hold across the full q range, not just asymptotically.\n\nThe soft spots are real but minor. The asymptotic formulas (39)-(42) are imported from the 1D theory rather than re-derived in 2D; the numerical agreement across two crystal orders and four channels makes this acceptable, but a derivation would tighten the paper. More substantively, the hydrodynamic comparison is a consistency check rather than an independent prediction: the elastic coefficients in Table I are computed from the same eGPE ground state that generates the BdG modes. That does not undercut the result, but it does mean the paper is not a test of supersolid hydrodynamics, it is a demonstration that the BdG amplitudes are consistent with it. The displacement extraction assumes small displacements, and the paper verifies linearity up to about 5% of the lattice constant in Fig. 3, so that assumption is handled honestly. The zero-temperature uniform-geometry scope is a limitation, but the conclusions section is explicit about it and suggests box-trap or toroidal realization.\n\nOverall, this is a focused, careful extension of the 1D program to 2D. It will be useful to anyone working on dipolar supersolids, and it deserves a serious referee. I'd send it out.","headline":"A credible, carefully checked extension of the 1D supersolid fluctuation program to 2D; the hydrodynamic comparison is a consistency check rather than an independent test, but the new mode-resolved signatures are worth having.","tokens_in":17872,"tokens_out":2132,"would_cite":true,"duration_ms":20189,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"In two-dimensional dipolar supersolids, the long-wavelength density, phase, current, and lattice-displacement fluctuations of all three acoustic modes are quantitatively described by hydrodynamic theory whose only inputs are the…","keywords":["supersolids","dipolar Bose-Einstein condensates","Bogoliubov-de Gennes theory","hydrodynamic theory","acoustic excitations","lattice displacement","elastic coefficients","two-fluid current"],"falsifier":"A direct falsifier would be a long-wavelength measurement of the dynamical structure factor of a 2D dipolar supersolid resolving the two longitudinal branches: if the lower branch's density response were comparable to the upper branch's instead of strongly suppressed, the predicted strain/defect cancellation would be wrong. Alternatively, a numerical BdG calculation in which the quasimomentum is pushed to where the density stationary points of neighboring cells merge (large displacement) should show the $q\\to 0$ hydrodynamic asymptotes (Eqs. (39)--(49)) failing as the displacement extraction becomes ill-defined.","tokens_in":16966,"feed_emoji":"🌊","tokens_out":6895,"duration_ms":60368,"temperature":0.7,"pith_summary":"The paper establishes that the microscopic acoustic fluctuations of two-dimensional dipolar supersolids—triangular and honeycomb—are completely captured at long wavelengths by hydrodynamic theory, with the elastic coefficients of the crystal as the only inputs. To see this, the author develops a way to extract the lattice displacement field directly from the computed density perturbation, by tracking the stationary points of the density in each unit cell. With that displacement in hand, each of the three gapless modes can be identified as one transverse shear wave and two longitudinal waves, and the density fluctuation of each mode splits into a piece from crystal strain and a piece from particle transport relative to the lattice. The key surprise is that the lower longitudinal mode has large strain and defect-density fluctuations that mostly cancel in the total density response, so it is nearly invisible to density probes but visible to current probes. A sympathetic reader would care because this gives experimentalists precise signatures—which probe sees which mode—and because it extends the earlier sound-speed agreement between Bogoliubov theory and hydrodynamics to the full fluctuation amplitudes.","feed_headline":"Hidden counterflow mode of 2D supersolids exposed","feed_subtitle":"Tracking lattice motion shows three sound branches, one of which hides large strain behind a near-zero density signal.","key_machinery":"The central object is the lattice displacement field $\\mathbf{u}(\\boldsymbol{\\rho},t)$, extracted from the microscopic density perturbation by solving $\\nabla\\varrho = 0$ at each shifted lattice site (Eqs. (20)--(24)): with the Hessian $H$ of the ground-state density at the site, the displacement amplitude is $\\tilde{\\mathbf{u}} = -A H^{-1} \\nabla\\delta\\varrho$. This field carries the argument because it lets the total density fluctuation be decomposed as $\\delta\\tilde{\\varrho} = \\delta\\tilde{\\varrho}_u + \\delta\\tilde{\\varrho}_\\Delta$, where $\\delta\\tilde{\\varrho}_u = -i\\rho\\,\\mathbf{q}\\cdot\\tilde{\\mathbf{u}}$ is the strain-induced piece and $\\delta\\tilde{\\varrho}_\\Delta$ is the defect-density piece from particle transport relative to the lattice. The same displacement, together with the phase fluctuation, resolves the current into superfluid ($\\propto \\nabla\\theta$) and normal ($\\propto \\partial_t \\mathbf{u}$) components a la Andreev--Lifshitz. The hydrodynamic predictions (Eqs. (39)--(49)) express all long-wavelength fluctuation amplitudes in terms of elastic coefficients and sound speeds, and the paper verifies them against the BdG matrix elements.","core_discovery":"The central claim is that the long-wavelength limits of the Bogoliubov–de Gennes fluctuation amplitudes—for the areal density, phase, planar current, and lattice displacement—match the predictions of supersolid hydrodynamics, whose input parameters are the superfluid density, the density stiffness, the density-strain coupling, and the Lamé elastic coefficients. The match is verified in the insets of Figs. 4 and 5 for both a triangular and a honeycomb supersolid, and it covers the asymptotic $q\\to 0$ behavior of the density, strain, defect-density, phase, longitudinal-current, and transverse-current fluctuations. Along the way the paper introduces a displacement-extraction procedure: at each lattice site the displacement is found by requiring that the perturbed density's gradient vanish at the shifted site, giving $\\mathbf{u} = -H^{-1}\\,\\nabla\\delta\\varrho$. This turns the density fluctuation into a sum of a strain part and a defect-density part, and it reveals that the three acoustic bands organize identically in both geometries: a transverse shear mode ($\\nu=0$), a lower longitudinal counterflow mode ($\\nu=1$) whose strain and defect parts cancel, and an upper longitudinal coflow mode ($\\nu=2$) that dominates density and phase response.","pith_inferences":["If the cancellation in the lower longitudinal mode survives at finite temperature, the mode would appear as a \"dark\" density excitation that carries momentum and energy but almost no density contrast—an analogue of second sound in superfluid helium, potentially observable through velocity or displacement imaging rather than density imaging.","The displacement-extraction procedure suggests a practical data-analysis recipe for experiments: from a time series of high-resolution in-situ density images of a 2D supersolid, one could fit the per-cell density peaks to obtain $\\mathbf{u}(\\boldsymbol{\\rho},t)$ and then separate strain from defect motion, turning images into a direct readout of the two-fluid currents.","Since the hydrodynamic parameters are determined by elastic coefficients, the fluctuation amplitudes provide a new route to measure those elastic coefficients: rather than measuring sound speeds alone, one can fit the $q\\to 0$ amplitudes of density and current fluctuations to extract $\\rho_s$, $\\alpha_{\\rho u}$, and the Lamé parameters.","The avoided crossing between the $\\nu=2$ and $\\nu=4$ bands at larger $q$ suggests that the simple three-mode hydrodynamic description will fail before the Brillouin zone edge; the paper's sum-rule analysis indicates where higher bands must be included, which may set the momentum scale for when supersolid hydrodynamics breaks down."],"forward_implications":["In a density-sensitive probe (e.g., Bragg spectroscopy), the upper longitudinal band will dominate the low-momentum response; the lower longitudinal band will be almost invisible despite carrying large internal strain and counterflow.","A transverse, shear-sensitive probe (measuring the transverse current) is the only way to see the $\\nu=0$ mode at long wavelengths; density and phase probes are blind to it.","The strain/defect decomposition gives a direct microscopic definition of \"defect density\" in a supersolid, making the Andreev--Lifshitz two-fluid picture—superflow plus lattice motion—computable from first-principles BdG wavefunctions.","The same machinery transfers to any 2D supersolid with a periodic ground state, including soft-core models and tilted-dipole systems, as long as the density stationary points remain trackable.","Sum-rule verification (the $f$-sum rule, the compressibility sum rule, and the transverse-current response) means the computed fluctuations can serve as a quantitative benchmark for interpreting current experiments in box traps and toroidal geometries."],"supporting_citations":[{"why":"Supplies the effective Lagrangian for supersolids that the hydrodynamic fluctuation predictions are based on.","marker":"[44]"},{"why":"Provides the hydrodynamic theory with elastic coefficients and density-density correlation functions used for the asymptotic predictions.","marker":"[45]"},{"why":"Derived the fluctuation asymptotics (Eqs. (39)-(42)) for one-dimensional supersolids that this paper extends to two dimensions.","marker":"[27]"},{"why":"Supplies the elastic-parameter computation and sound-speed comparisons used to fix the hydrodynamic inputs.","marker":"[16]"},{"why":"Provides the Bogoliubov-de Gennes framework and sound-speed comparisons for two-dimensional supersolids used here.","marker":"[23]"},{"why":"Gives the Andreev-Lifshitz two-fluid current form that the paper tests against its microscopic current fluctuations.","marker":"[17]"},{"why":"Justifies the presence of three gapless Nambu-Goldstone bands in a two-dimensional supersolid.","marker":"[21]"}],"fun_headline_variants":["Hidden counterflow mode of 2D supersolids exposed","Three sound branches in 2D supersolids, one hides strain","Counterflow branch hides large strain in 2D supersolids","2D supersolid acoustics: how to spot the hidden mode"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The decomposition into strain and defect-density fluctuations assumes each lattice cell keeps a single identifiable density stationary point that moves smoothly under the perturbation, i.e., the lattice displacement must stay small compared to the lattice constant and the Hessian at that point must remain invertible; if the perturbation is strong enough to merge or wipe out these stationary points, the displacement field and the mode identification break down.","fun_headline_variants_meta":{"raw":{"variants":["Hidden counterflow mode of 2D supersolids exposed","Three sound branches in 2D supersolids, one hides strain","Counterflow branch hides large strain in 2D supersolids","2D supersolid acoustics: how to spot the hidden mode"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000534,"raw_usage":{"total_tokens":2585,"prompt_tokens":979,"completion_tokens":1606,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":595,"completion_tokens_details":{"reasoning_tokens":1531}},"tokens_in":595,"tokens_out":1606,"duration_ms":10969,"temperature":1.0,"reasoning_tokens":1531,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T20:35:06.738342+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A direct falsifier would be a long-wavelength measurement of the dynamical structure factor of a 2D dipolar supersolid resolving the two longitudinal branches: if the lower branch's density response were comparable to the upper branch's instead of strongly suppressed, the predicted strain/defect cancellation would be wrong. Alternatively, a numerical BdG calculation in which the quasimomentum is pushed to where the density stationary points of neighboring cells merge (large displacement) should show the $q\\to 0$ hydrodynamic asymptotes (Eqs. (39)--(49)) failing as the displacement extraction becomes ill-defined.","supporting_citations":[{"cited_title":"Baillie and P","cited_arxiv_id":null,"evidence_quote":"Supplies the effective Lagrangian for supersolids that the hydrodynamic fluctuation predictions are based on."},{"cited_title":"Zhang, T","cited_arxiv_id":null,"evidence_quote":"Provides the hydrodynamic theory with elastic coefficients and density-density correlation functions used for the asymptotic predictions."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Derived the fluctuation asymptotics (Eqs. (39)-(42)) for one-dimensional supersolids that this paper extends to two dimensions."},{"cited_title":"Zhang and F","cited_arxiv_id":null,"evidence_quote":"Supplies the elastic-parameter computation and sound-speed comparisons used to fix the hydrodynamic inputs."},{"cited_title":"Leggett, On the superfluid fraction of an arbitrary many- body system atT= 0, J","cited_arxiv_id":null,"evidence_quote":"Provides the Bogoliubov-de Gennes framework and sound-speed comparisons for two-dimensional supersolids used here."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the Andreev-Lifshitz two-fluid current form that the paper tests against its microscopic current fluctuations."}],"review_version":1}