{"id":"a0d4aae1-7a27-44ba-b87a-4ee4e8d8d7ea","arxiv_id":"2412.13764","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"A GPU-accelerated Ginzburg-Landau simulation resolves the full structure of a single-quantum vortex in superfluid 3He-A, giving quantitative values for the core shifts, tilt angle, energy, and asymmetric superflow.","lead":"This paper reports the first computer calculation that resolves both the tiny hard core and the much larger soft core of a single quantum vortex in superfluid helium-3 A phase, giving numbers for the vortex energy, core positions, and flow pattern. The result matters because it puts a long-studied qualitative picture on a quantitative footing and sharpens proposals to use this vortex as a tabletop model of cosmological phenomena.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No mesh-convergence or domain-size study is reported; the quoted quantitative numbers (η≈55.4°, core locations, 18.2% polar-core energy) could be numerical artifacts, so the central 'quantitative determination' claim is not yet secured.","rationale":"The paper is a serious computational study with a plausible central claim and an analytic current model that reproduces the numerics without fitting. My read is that the biggest gap is not the GL framework itself (which is standard and referenced) but the absence of numerical verification: no mesh-refinement, no domain-size study, no stated tolerances for the L-BFGS minimization, and no error estimates. The quantitative outputs are the paper's main novelty, so this gap is directly load-bearing. The reader's weakest_assumption also flags the GL coefficients at T=0.9Tc; I partially share that concern, but I would not make it the primary issue because even if the coefficients were perfect, the numerical convergence would still need to be established, and because the model-accuracy question is largely a matter of the standard applicability of GL rather than an internal inconsistency. A convergence study is a crisp, feasible check before publication. If it passes, the central claim is much stronger. If it fails, the quantitative values should be downgraded to qualitative. Hence I keep the verdict CONDITIONAL, matching the reader.","tokens_in":12010,"tokens_out":12368,"duration_ms":118603,"concrete_test":"Run the same minimization at p=30 bar, T=0.9Tc with the near-core spacing halved to ξ/6 and the outer radius increased to Rcalc=2000ξ and 4000ξ (keeping boundary conditions and all other settings); then compare η, the hard/soft core positions, and the polar-core energy fraction (using the same 5% amplitude-threshold decomposition). If any of these change by more than ~0.5° in η, ~1ξ in the core positions, or ~1 percentage point in the 18.2% energy fraction, the quantitative claims of §4 and the abstract require revision or explicit error bars.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that the simulation provides quantitative values for the SQV structure. This requires that the discrete GL minimization has converged with respect to both mesh spacing and finite simulation radius. The paper reports one calculation at a near-core mesh of ξ/3, an outer radius Rcalc=1000ξ, and no convergence study anywhere (§2, §4). The hard-core size itself is of order ξ, so the core is sampled by only a few ξ/3 elements, and the soft-core/dipole scale is comparable to Rcalc (the paper gives b≈67ξ≈0.25ξd, implying ξd≈268ξ, so Rcalc≈3.7ξd). The quoted tilt angle η≈55.4°, the hard- and soft-core positions (x=-22.67ξ and x=44.50ξ), and the 18.2% polar-core energy contribution are all presented to 2-3 significant figures without error bars. If a finer mesh shifts the location of the polar core by even a few ξ, or if the energy fraction changes with the 5% threshold used in its definition, the precision implied by 'quantitative determination' is not supported. The analytic current model in §4 is a valuable internal consistency check, but it uses η and the core position from the same numerical solution, so it cannot validate those numbers independently.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript reports a finite-element Ginzburg-Landau minimization of a single-quantum vortex (SQV) in superfluid 3He-A at p=30 bar and T=0.9Tc, with boundary conditions l=d=x and a magnetic field along z. The mesh resolves the hard core with spacing xi/3 out to r=100xi and extends to Rcalc=1000xi with periodic boundary conditions in z. The resulting texture has a tilted radial disgyration with eta~55.4 deg, a hard core shifted to x=-22.67xi, a hyperbolic soft core at x=44.50xi, and a strongly asymmetric superfluid mass current that is largest in the polar hard core. An analytic current model based on an ansatz with constant tilt eta and order-parameter suppression f(r) reproduces the numerical current. The polar core is estimated to contribute ~18.2% of the vortex energy for an outer cutoff at the intervortex distance corresponding to Omega=1 rad/s. The four (nu,s) vortex variants are found to be degenerate within 0.04%. The authors conclude that this is the first quantitative determination of the SQV structure and note connections to eccentric fractional skyrmions in spinor BECs.","tokens_in":12188,"tokens_out":9292,"duration_ms":82699,"significance":"If the quantitative claims survive the numerical verification requested below, this is a genuinely useful contribution: it is, to my knowledge, the first GL calculation that resolves both the coherence-length hard core and the dipole-length soft core of the SQV in the same simulation. The paper is self-contained methodologically: it gives the tetrahedral discretization, analytic integrals for bulk and gradient terms, the functional gradients, and the GPU L-BFGS minimization, which would allow reproduction. The analytic current model of Sec. 4 is a valuable physical explanation of the asymmetric flow and channeling through the polar core. The connection to eccentric fractional skyrmions in spinor BECs gives the work broader relevance. The predicted texture and current profile are, in principle, falsifiable via NMR signatures and transverse-current experiments, which adds value. The main limitations are an absence of numerical convergence studies and of error estimates for the headline numbers; these are fixable but currently prevent the paper from supporting 'quantitative determination' as strongly as claimed.","major_comments":[{"comment":"The central quantitative claims are reported for a single mesh with near-core spacing xi/3 and Rcalc=1000xi, and no mesh-convergence or domain-size study is presented. With b~67xi~0.25xi_d, the dipole length is xi_d~268xi, so Rcalc~3.7xi_d and the soft-core texture cannot be assumed free of finite-size effects. The values eta~55.4 deg, x=-22.67xi, x=44.50xi, and the 18.2% energy fraction are quoted to 2-3 significant figures without error bars. Please add systematic studies varying the near-core spacing (e.g. xi/6 and xi/9) and the outer radius (e.g. 1500xi and 2000xi), and report the resulting changes in these observables; a minimization stopping criterion should also be stated.","section":"Section 2 and Section 4"},{"comment":"The 18.2% polar-core energy contribution depends on two choices that are not given sensitivity tests: the definition of the hard core as the region where the order-parameter amplitude deviates by more than 5% from bulk, and the outer cutoff rv in Eq. (19) evaluated at Omega=1 rad/s. Since previous estimates are 'a few percent,' the discrepancy makes the new value physically interesting only if its dependence on these choices is shown. Please report the energy fraction for thresholds (e.g. 3%, 7%, 10%) and for at least one other realistic rotation rate or cutoff.","section":"Section 4 (polar-core energy)"},{"comment":"The statement that the model current matches 'without fitting parameters' should be qualified. The model uses eta=55.39 deg and the hard-core position x=-22.67xi taken directly from the minimized texture, so the agreement is an internal consistency check of the ansatz rather than an independent validation of those two numbers. The paper should state this explicitly and, if possible, show the model current for eta varied within the numerical uncertainty to indicate the precision of the match.","section":"Section 4, Eqs. (26)-(27)"},{"comment":"The quantitative outputs inherit the accuracy of the GL functional and its coefficients at p=30 bar and T=0.9Tc, a temperature not very close to Tc. No discussion of the expected systematic error of the GL approximation, nor of sensitivity to the beta coefficients or the magnetic-field term, is given. This is not a reason to doubt the qualitative structure, but it limits the precision that can be claimed for eta and the core positions; please add an explicit statement of expected GL-level uncertainty and, if feasible, a calculation at one additional temperature.","section":"Section 4, GL input (Ref. [31])"}],"minor_comments":[{"comment":"The typeset expression 'A. µm' in the last term appears to be a subscript artifact; the index structure of this K3 term should be corrected or clarified.","section":"Eq. (17)"},{"comment":"The symbol nu is used both for the circulation quantum number and for the +/-1 vortex/antivortex sign in Eq. (21); using a different symbol, such as sigma, would avoid confusion.","section":"Eqs. (3), (18), (19), (21)"},{"comment":"The caption of Fig. 5 gives eta=55.39 deg, while the text and Fig. 1 give eta~55.4 deg; the precision of the value should be made consistent, and an error estimate should be attached if the value is meant literally.","section":"Fig. 5 and Sec. 4"},{"comment":"The statement that external rotation is neglected 'as it is not expected to have an effect on the length scales of a single vortex' needs a brief justification, since the energy fraction computed in the same section explicitly uses an outer cutoff from rotation.","section":"Section 4"},{"comment":"The four degenerate vortex variants are reported to be equivalent within 0.04%, but no absolute energy per unit length is given; providing the vortex energy in units such as Delta_A^2 xi^2 would allow quantitative comparison with future calculations.","section":"Section 4"},{"comment":"A data or code availability statement for the minimized order-parameter textures would aid reproducibility and is recommended.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"The paper is within the scope of the journal and the qualitative picture is convincing. The missing convergence study is, in my view, the single most important obstacle to acceptance. I am not asking for a full quasiclassical treatment, but the numerical precision claims need to be backed by mesh/domain checks and threshold/cutoff sensitivity. If the authors provide those, the paper should be suitable for publication. I also note that the manuscript does not mention data or code availability; providing the minimized order-parameter fields would strengthen reproducibility."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Massimo—this one is worth reading. Rantanen, Thuneberg and Eltsov have done the first numerical calculation that resolves both the coherence-length hard core and the dipole-length soft core of the single-quantum vortex in 3He-A. The qualitative structure—tilted radial disgyration, eccentric fractional skyrmion—was predicted before, but the numbers are new: tilt angle about 55.4 degrees, hard core shifted to x=-22.67xi, soft core at x=44.50xi, and an asymmetric mass current that channels through the polar core. If these numbers hold up, they give the texture and flow inputs needed for NMR calculations and analog-gravity work.\n\nThe paper does several things well. The finite-element GL minimization is described in enough detail to reproduce the method. The analytic current model in Sec. 4 is a legitimate internal consistency check, and the match to the numerics is indeed parameter-free in the sense that the only inputs are the tilt and core position taken from the same simulation. The citation pattern is fair; the paper credits the earlier structural proposals and the spinor-BEC naming.\n\nThe soft spots are real but not fatal. There is no convergence study—no variation of mesh spacing or outer radius. The mesh is xi/3 near the core, which samples a coherence-length-sized core with only a few points, so the 2-3 significant figures quoted for the core shifts and tilt are not supported unless convergence is shown. The energy fraction of 18.2% depends on the arbitrary 5% amplitude threshold and the choice of outer cutoff; that number should be read as order-of-magnitude. The GL coefficients at 0.9Tc are standard for this community, but a sensitivity study would tell you how much the numbers move if the coefficients vary.\n\nThe stress-test note worries that the quantitative claim is 'not yet secured.' I disagree with the strength of that. The structure is not a numerical artifact; the qualitative picture is robust and the current-channeling effect follows directly from the tilt and the polar core. What is not yet secured is the precision of the quoted values. That is a request for a convergence study and error bars, not a rejection.\n\nWho is this for? Anyone working on vortices in 3He or on analog-gravity models using polar-core textures. It deserves a serious referee; I would send it out and request a convergence study and a clearer handling of the threshold and cutoff dependence. With that, the quantitative claims will be grounded.","headline":"First simultaneous resolution of hard and soft cores of the 3He-A single-quantum vortex, with a clean analytic cross-check; needs a convergence study before the headline numbers are taken as precise.","tokens_in":12821,"tokens_out":2166,"would_cite":true,"duration_ms":19884,"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":"The internal structure of a single-quantum vortex in superfluid 3He-A has been calculated for the first time at both core scales, giving a 55.4° disgyration tilt, offset hard and soft cores, and a polar core that carries 18.2% of the…","keywords":["Superfluid 3He-A","Quantized vortices","Ginzburg-Landau","eccentric fractional skyrmion","radial disgyration","polar core","superfluid mass current","Mermin-Ho relation"],"falsifier":"A nuclear magnetic resonance measurement of a single-quantum vortex in 3He-A: if the measured satellite spectrum does not match the spectrum computed from the predicted l-vector texture (η ≈ 55.4°, hard and soft cores separated by about 67ξ), the quantitative structure presented here is refuted.","tokens_in":11698,"feed_emoji":"🌀","tokens_out":9469,"duration_ms":74901,"temperature":0.7,"pith_summary":"This paper reports the first numerical calculation of the internal structure of a single-quantum vortex in the A phase of superfluid helium-3 that resolves both the tiny hard core (coherence length, about ten nanometres) and the much larger soft core (dipole length, about ten micrometres) in a single simulation. The calculation pins down the l-vector texture around the vortex, giving a tilt angle of roughly 55.4 degrees for the radial disgyration, and shows the hard and soft cores are offset from the vortex centre by -22.67ξ and +44.50ξ respectively. It also reveals that the superfluid mass current is highly asymmetric and flows mostly through the polar-phase hard core, and an analytic model reproduces this current without any fitting parameters. The polar core is found to carry about 18.2 percent of the total vortex energy, much larger than the few percent estimated earlier.","feed_headline":"First full core map of a 3He-A vortex, from nm to µm","feed_subtitle":"A single simulation resolves both core scales and fixes the 55° disgyration tilt, core offsets, and an 18% polar-core energy share.","key_machinery":"The load-bearing object is the tilted radial disgyration, Eq. (21): $\\hat{l} = s[\\hat{y}\\sin\\phi + \\cos\\phi(\\hat{x}\\cos\\eta + \\hat{z}\\nu\\sin\\eta)]$, with a constant tilt angle $\\eta$, surrounding a polar-phase hard core where the $\\hat{n}$ component vanishes. The Mermin-Ho relation $\\nabla\\times\\mathbf{v}_s = (\\hbar/4m_3)\\sum_{ijk}\\epsilon_{ijk}\\hat{l}_i(\\nabla\\hat{l}_j\\times\\nabla\\hat{l}_k)$ ties circulation to the skyrmion number $N = \\frac{1}{4\\pi}\\int \\hat{l}\\cdot(\\partial_x\\hat{l}\\times\\partial_y\\hat{l})\\,dx\\,dy$ through $\\nu = 2N$; for the single-quantum vortex $N = 1/2$, which requires the hard core and disgyration. The analytic current model, Eqs. (26)-(27), derived from this texture plus the core suppression function $f(r)$, reproduces the numerical superflow without fitting parameters and shows the polar core acts as a channel for the mass current.","core_discovery":"The single-quantum vortex in $^3$He-A is an eccentric fractional skyrmion: the orbital angular momentum unit vector $\\hat{l}$ rotates over half the unit sphere, forcing a hard core in the polar phase and a tilted radial disgyration around it, together with a hyperbolic soft core on the opposite side. Minimizing the Ginzburg-Landau free energy on a tetrahedral mesh that resolves both the coherence-length hard core and the dipolar-length soft core yields quantitative values: the disgyration is tilted out of the $xy$-plane by $\\eta \\approx 55.4^\\circ$, the hard core sits at $x = -22.67\\xi$, the soft core at $x = 44.50\\xi$, and the polar core contributes $18.2\\%$ of the total vortex energy when the circulating flow outside the simulation box is included. The paper's new result is the mass current: $j_y(x,y) = -(4m_3 K \\Delta_A^2/\\hbar)\\sin\\eta\\,\\sin^2\\phi/r$ away from the core and $j_y(x,0) = -(4m_3 K \\Delta_A^2/\\hbar)\\sin\\eta\\, f'(|x|)$ on the axis through the core, where $f(r)$ is the suppression of the $\\hat{n}$ amplitude inside the hard core. This analytic form reproduces the numerical flow without fitting parameters, establishing that the asymmetric channeling of superflow through the polar core is a direct consequence of the tilted $\\hat{l}$-texture, analogous to a magnetization current.","pith_inferences":["If the analytic current model is exact in the Ginzburg-Landau limit, the total current through the polar core depends only on the tilt angle $\\eta$ and not on the shape of the core suppression function; this could be tested in spinor Bose-Einstein condensates, where analogous eccentric fractional skyrmion structures can be imaged directly.","The 18.2% polar-core energy share, if correct at 30 bar and 0.9Tc, suggests that at other pressures or temperatures the single-quantum vortex may change stability relative to the double-quantum vortex; a phase-diagram scan across the p-T plane would reveal such a transition.","The asymmetric current channeling implies a transverse force on the vortex line (a Hall-like response) that should enter the equations of vortex motion; if measured, it would provide an independent check of the model beyond NMR."],"forward_implications":["The quantitative texture allows computing experimental signatures such as the NMR response of the single-quantum vortex, enabling direct comparison with existing and future observations.","The phase diagram of vortices in $^3$He-A can now be completed theoretically, because the SQV energy and structure are known quantitatively.","The strong flow through the polar core provides a concrete setting for studying core-bound fermion states and Weyl quasiparticles in a controlled laboratory system.","The eccentric fractional skyrmion structure confirms the analogy with spinor Bose-Einstein condensate vortices, where such structures have been predicted and observed.","The 18.2% polar-core energy share, much larger than earlier few-percent estimates, may change the relative stability of single- and double-quantum vortices under different conditions."],"supporting_citations":[{"why":"Supplies the Mermin-Ho relation linking superfluid vorticity to the l-vector texture, the foundation for the skyrmion-number argument.","marker":"[1]"},{"why":"Provided the earlier suggestion of the tilted radial disgyration and the η parameterization that the paper now determines quantitatively.","marker":"[24]"},{"why":"Provides the Ginzburg-Landau coefficients and theory details used for the numerical minimization at p = 30 bar, T = 0.9Tc.","marker":"[31]"},{"why":"Gives the superfluid mass current formula used to compute the asymmetric flow.","marker":"[34]"},{"why":"Earlier asymptotic model of the vortex energy and l-texture that the paper extends and revises with full numerical results.","marker":"[5]"},{"why":"Earlier rough estimate of the polar core energy contribution (a few percent) that the paper replaces with 18.2%.","marker":"[11]"},{"why":"Identifies the polar phase inside the hard core of the disgyration, which is the central core state in the SQV.","marker":"[22]"}],"fun_headline_variants":["3He-A vortex: both cores resolved in one GPU simulation","Eccentric fractional skyrmion: exact core geometry and flow","Vortex core map reveals 55° tilt and 18% energy share","Single simulation captures hard and soft cores of 3He-A vortex","Vortex mass current predicted without fitting parameters"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"All quantitative numbers rest on the Ginzburg-Landau free energy functional with coefficients from reference [31] being accurate at p = 30 bar and T = 0.9Tc, a temperature not very close to Tc, and the paper reports no sensitivity, convergence, or mesh-resolution tests to support the quoted precision.","fun_headline_variants_meta":{"raw":{"variants":["3He-A vortex: both cores resolved in one GPU simulation","Eccentric fractional skyrmion: exact core geometry and flow","Vortex core map reveals 55° tilt and 18% energy share","Single simulation captures hard and soft cores of 3He-A vortex","Vortex mass current predicted without fitting parameters"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000227,"raw_usage":{"total_tokens":1486,"prompt_tokens":971,"completion_tokens":515,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":587,"completion_tokens_details":{"reasoning_tokens":428}},"tokens_in":587,"tokens_out":515,"duration_ms":4959,"temperature":1.0,"reasoning_tokens":428,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T12:49:50.204911+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A nuclear magnetic resonance measurement of a single-quantum vortex in 3He-A: if the measured satellite spectrum does not match the spectrum computed from the predicted l-vector texture (η ≈ 55.4°, hard and soft cores separated by about 67ξ), the quantitative structure presented here is refuted.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the Mermin-Ho relation linking superfluid vorticity to the l-vector texture, the foundation for the skyrmion-number argument."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provided the earlier suggestion of the tilted radial disgyration and the η parameterization that the paper now determines quantitatively."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the Ginzburg-Landau coefficients and theory details used for the numerical minimization at p = 30 bar, T = 0.9Tc."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the superfluid mass current formula used to compute the asymmetric flow."},{"cited_title":"Jour- nal of Low Temperature Physics 42, 503–514 (1981) https://doi.org/10.1007/ BF00117428","cited_arxiv_id":null,"evidence_quote":"Earlier asymptotic model of the vortex energy and l-texture that the paper extends and revises with full numerical results."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Earlier rough estimate of the polar core energy contribution (a few percent) that the paper replaces with 18.2%."},{"cited_title":"Journal of Low Temperature Physics 25, 225–243 (1976) https://doi.org/ 10.1007/BF00654831","cited_arxiv_id":null,"evidence_quote":"Identifies the polar phase inside the hard core of the disgyration, which is the central core state in the SQV."}],"review_version":1}