{"id":"8c8ce010-0713-4286-8380-e547a07496dd","arxiv_id":"2502.07433","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"A new iterative method for adiabatic HFB gives rare-earth moments of inertia and shows that time-odd mean-field corrections vary with nucleus and functional, correlating with effective mass.","lead":"The authors compute moments of inertia for heavy deformed nuclei with a new iterative method that includes time-odd mean fields, finding results close to experiment. They show that the standard correction factor to the Inglis-Belyaev formula is not constant, and that its size is tied to the effective nucleon mass for Skyrme functionals.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Fixed-point iteration (8) is asserted exact, but no convergence proof or residual diagnostics are provided, and the only validation is in an unpublished reference; the exactness of the reported inertias is therefore not yet established.","rationale":"The paper's derivation of Eq. (8) from Eq. (5) is formally correct: at a fixed point the iteration reproduces the linear ATDHFB equation, and the SVD construction gives a valid idempotency-preserving R1 with the required time-odd structure. I therefore do not claim the method is wrong. Credit is due for a parameter-free comparison with data, for showing that the first iteration reproduces the IB formula, and for the effective-mass correlation in Fig. 4, which would be a genuine physical result if the converged inertias are reliable. The load-bearing weakness is that 'exact' is an algorithmic claim: the paper solves Eq. (5) by iteration but gives no convergence proof, no residual diagnostics, and no reproducible code or data, and the only external validation is deferred to an unpublished reference [18]. This is precisely the reader's identified weakest assumption, and it can be settled by an independent numerical test. If the proposed residual/initial-guess test passes, the conditional reservation is lifted; if it fails, the quantitative ratios and the central comparison with IB and experiment would need revision. The concern does not by itself warrant rejection or a stronger verdict, so I keep the reader's verdict unchanged.","tokens_in":10919,"tokens_out":15336,"duration_ms":151610,"concrete_test":"Reproduce the iteration (8) for a representative set, e.g., 166Er with SkM*, UNEDF1, D1S and 186Os with D1S, using (i) 500 iterations rather than ~24, (ii) three initial guesses (Z=0, the IB solution, and a random antisymmetric Z scaled to the IB amplitude), and (iii) a residual norm ||i hbar F_{mu nu} - (E_mu + E_nu) Z_{mu nu} - E_{1,mu nu}(Z)|| / ||i hbar F|| monitored every iteration. If the residual decreases to <1e-6 and the inertia M changes by <0.01% for all cases and initial guesses, the exactness claim is empirically established; if the residual stagnates, oscillates, or depends on the initial guess, the method as presented is not reliable.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that Eq. (8), Z^{(n+1)}_{mu nu} = (i hbar F_{mu nu} - E^{(n)}_{1,mu nu})/(E_mu + E_nu), is an exact iterative solver for the ATDHFB equation. This is true only if the fixed-point iteration converges to the unique solution of the linear response equation (5). The paper neither proves convergence nor provides residual norms or error bounds; it states that 'about two dozen iterations suffice' and cites a to-be-published work for 'perfect agreement' with cranking. The gap is material: Eq. (8) is a stationary splitting of the response operator, and convergence requires the spectral radius of the map Z -> (E_mu + E_nu)^{-1} E_1(Z) to be less than 1. Stability of the HFB minimum does not by itself guarantee contraction of this split, and the paper's own D1S ratios (up to 1.5-1.6, Figs. 3c) show the time-odd correction is large, not perturbatively small. If the iteration cycles, converges slowly, or is stopped prematurely, the quoted ATDHFB inertias and the ratios in Figs. 3 and 4 are not the exact ATDHFB values, so the comparison with IB and data loses its stated basis.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript presents a novel iterative method for solving the adiabatic time-dependent Hartree-Fock-Bogoliubov (ATDHFB) equations for collective inertia, avoiding explicit inversion of the stability matrix. The centerpiece is the fixed-point iteration of Eq. (8), in which the first-order quasiparticle matrix Z is updated from the previous iteration's time-odd mean field E1^{(n)}. The method is applied to rotational moments of inertia of rare-earth even-even nuclei with the Skyrme functionals SkM* and UNEDF1 and the Gogny functional D1S, and to vibrational inertia for 166Er with seven Skyrme functionals. The authors find that ATDHFB inertias are systematically larger than Inglis-Belyaev values, that the ratio depends strongly on nucleus and functional, and that this ratio correlates linearly with the isoscalar effective mass. The comparison with experimental 2+ energies is presented with explicit caveats about pairing uncertainties.","tokens_in":11240,"tokens_out":3213,"duration_ms":33203,"significance":"If the claimed exactness of the iterative solution is established, the method would be a practical way to obtain Thouless-Valatin inertias without inverting the stability matrix, a genuine technical advance for nuclear DFT applications such as fission and spectroscopic calculations. The systematic comparison of ATDHFB and IB inertias across a wide isotopic chain, with three different functionals, provides useful empirical information on the breakdown of the simple cranking enhancement factor. The paper is also honest and specific about pairing uncertainties and about the functional dependence of the results. The correlation of the inertia ratio with the isoscalar effective mass (Fig. 4, R^2 ~ 0.90-0.94) is a clean, falsifiable statement that would be valuable if the underlying vibrational and rotational inertias are reliable. However, the central methodological claim of exactness is currently not backed by a convergence proof or residual diagnostics, and the vibrational derivative step is described only through a missing reference; these points are load-bearing for the quantitative conclusions.","major_comments":[{"comment":"The paper asserts that Eq. (8) is an exact iterative solution of the ATDHFB equation, but it provides no convergence proof, no error bound, and no residual diagnostic. The statement that 'about two dozen iterations suffice' is not a substitute for a stopping criterion, and the only validation cited is 'perfect agreement' with cranking in the unpublished Ref. [18]. Because Eq. (8) is a stationary splitting of the response operator, convergence requires a contraction property that is not guaranteed by HFB stability alone; the large D1S ratios in Fig. 3(c) (up to 1.5-1.6) show that the time-odd correction is not perturbatively small. Please provide either a proof of convergence, or quantitative residual norms for all nuclei and functionals studied, or explicitly soften the claim to 'numerically converged within a stated stopping tolerance' rather than 'exact'.","section":"Eq. (8) and following paragraph"},{"comment":"The vibrational inertia results, which support the effective-mass correlation in Fig. 4(b), rely on a numerical derivative of the HFB densities with respect to the axial quadrupole moment, but the manuscript does not specify the constraint step size, the differentiation formula, or the convergence of this derivative, and the supporting reference appears as '[ ? ]'. Without this information the vibrational ratios are not reproducible and their error is unknown. Please specify the numerical differentiation procedure, give a convergence check, and estimate the resulting uncertainty in the vibrational inertia ratios.","section":"Vibrational inertia paragraph and Fig. 4"}],"minor_comments":[{"comment":"The phrase 'bids well' should be 'bodes well'.","section":"Summary"},{"comment":"Two references are left as unresolved '[ ? ]' placeholders: one for the conference presentation of the iterative method without pairing, and one for the numerical differentiation of HFB densities. These should be filled before publication.","section":"Vibrational inertia paragraph and footnote"},{"comment":"The author name 'W. Satuła' appears corrupted as 'W. Satu/suppress la' in the reference list; this should be corrected.","section":"Ref. [20]"},{"comment":"The symbols E1, F, and Z are defined through the quasiparticle wave functions, but it would aid the reader if the equation explicitly noted that all matrix elements are taken between positive-energy quasiparticle states, as stated only in the following sentence.","section":"Eq. (8) notation"},{"comment":"The experimental moments of inertia are extracted with the rigid rotational formula I = 3\\hbar^2/E_{2+}, which is an approximation for transitional nuclei near N=82 and N=126; a brief caveat about this would be appropriate.","section":"Experimental extraction"}],"recommendation":"major_revision","confidential_remarks":"The main technical claim is plausible but relies on an unpublished validation (Ref. [18]) and on the authors' own preprint (Ref. [14]). If the convergence issue can be settled with a proof or with residual diagnostics, the paper could be a useful contribution. As it stands, the 'exact' label is not yet justified in the manuscript itself. The missing '[ ? ]' references should also be resolved before publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: I agree with the reader — real results, one unproven claim at the center. The systematic ATDHFB/IB ratios across the rare-earth region and the effective-mass correlation in 166Er are worth knowing. The exactness of the iterative solver is asserted, not demonstrated, and the verification sits in an unpublished companion paper.\n\nWhat is new: a survey of Gd-Os with SkM*, UNEDF1, and D1S showing the ATDHFB/IB ratio varies with nucleus, neutron number, and functional — no fixed 1.2–1.3 enhancement factor survives. The cleanest result is Fig. 4: for seven Skyrme functionals, the rotational and vibrational inertia ratios in 166Er correlate linearly with isoscalar effective mass (R² ≈ 0.9), which points at the j² time-odd channel. The iterative method is also neat; Eq. (8) is a simple stationary splitting and the SVD construction is a clever way to rebuild the first-order density matrix. If it converges, it avoids stability-matrix inversion and opens a practical route to fission inertia.\n\nThe soft spot is exactly the one the stress test identifies: convergence of the fixed point is not proved. Eq. (8) is a map Z -> (E_mu+E_nu)^{-1}(...), and stability of the HFB minimum does not by itself guarantee contraction in this split. The D1S ratios reaching 1.5–1.6 mean the correction is not perturbatively small, so this is not a formality. The paper says 'about two dozen iterations suffice' and defers 'perfect agreement' with cranking to a to-be-published reference. No residual diagnostics or error bounds are given. That is fixable — add residuals and a convergence tolerance, or include the companion paper — but as written the label 'exact' outruns the evidence. Code and data are only 'available on request,' which also slows independent checks. The overlap with arXiv:2411.18404 should be clarified, though the survey results here appear new.\n\nNone of that changes the main physics message. The comparison to data is honest, pairing uncertainties are stated, and the functionals were not fitted to these inertias, so the agreement is predictive. I would send this to a serious referee, asking for a convergence appendix and a decision on the companion paper. It is a useful paper for DFT and fission practitioners, and the effective-mass correlation is likely to be cited.","headline":"A useful systematic survey with a plausible but unproven central claim; the effective-mass correlation is the strongest part.","tokens_in":11754,"tokens_out":3556,"would_cite":true,"duration_ms":33694,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["21.60.Jz"],"model":"deepseek-v4-flash","headline":"An iterative fixed-point equation replaces stability-matrix inversion for nuclear moments of inertia.","keywords":["rotational moment of inertia","vibrational collective inertia","adiabatic time-dependent Hartree-Fock-Bogoliubov","time-odd mean fields","Thouless-Valatin inertia","Inglis-Belyaev formula","Skyrme and Gogny density functionals","effective mass"],"falsifier":"Take a small deformed superfluid nucleus and solve the ATDHFB equation in a truncated quasiparticle space both by direct inversion of the full two-body stability matrix and by iterating Eq. (8); if the two inertias disagree beyond numerical precision, or if the iteration oscillates or fails to settle for some functional, the quoted values are not exact ATDHFB inertias.","tokens_in":10759,"feed_emoji":"⚛️","tokens_out":13744,"duration_ms":117057,"temperature":0.7,"pith_summary":"Rare-earth nuclei resist rotation through an inertia that depends on how the nuclear mean field responds when time-reversal symmetry is broken. The standard Inglis-Belyaev formula drops the time-odd part of that response and underestimates the inertia, which is why calculations historically multiply it by empirical factors around 1.3. This paper claims that the full adiabatic time-dependent Hartree-Fock-Bogoliubov equation for the first-order density response can be solved exactly by a fixed-point iteration, with no stability-matrix inversion, and applies the method to gadolinium-to-osmium isotopes with two Skyrme and one Gogny functional. The resulting self-consistent moments of inertia agree broadly with the measured first 2+ energies with no adjustable parameters, and the ratio of exact to Inglis-Belyaev inertia varies with neutron number and functional, so no single enhancement factor is justified. For Skyrme functionals the ratio correlates strongly with the isoscalar effective mass, tying the time-odd response to the $\\rho\\tau-j^2$ current terms.","feed_headline":"New iteration computes exact nuclear moments of inertia","feed_subtitle":"An exact iterative solver captures the time-odd fields standard formulas ignore, so correction factors are no longer needed.","key_machinery":"The machinery is the fixed-point equation (8) together with the density-reconstruction step that closes the self-consistent loop. Writing the first-order density matrix in the quasiparticle basis as a Hermitian matrix $Y$, the singular value decomposition $Z=U\\Omega V^+$ of the antisymmetric matrix $Z$ gives eigenvectors of $Y$ in opposite-sign pairs, so $\\tilde R_1$ is assembled from the two sets of quasiparticle amplitudes $\\chi_1=(A_1,B_1)$ and $\\phi_1=(B_1^*,A_1^*)$ with occupation factors $\\pm\\omega_\\mu$. Standard HFB routines then compute the time-odd mean fields, and Eq. (8) provides the next $Z$; this is structurally the same loop as a static HFB iteration, which is what makes the exact solution affordable. The quasiparticle energies $E_\\mu,E_\\nu>0$ in the denominator keep each step a simple division rather than a matrix inversion.","core_discovery":"The central claim is that the linearized equation of adiabatic collective motion, $$i\\hbar \\frac{\\partial R_0}{\\partial q}=[H_0,\\tilde R_1]+[\\tilde H_1,R_0],$$ admits an exact iterative solution in the quasiparticle basis through the update $$$Z^{{(n+1)}}$_{\\mu\\nu}=\\frac{(i\\hbar F-$E_1^{{(n)}}$)_{\\mu\\nu}}{E_\\mu+E_\\nu}.$$ At each iteration the antisymmetric matrix $Z$ is decomposed by a singular value decomposition, the first-order density matrix $\\tilde R_1$ is reconstructed from paired quasiparticle wave functions, the time-odd mean fields $\\tilde H_1$ are recomputed by the standard HFB algorithm, and the loop repeats until the collective inertia $M^{(n)}=\\frac{i\\hbar}{2}\\mathrm{Tr}\\big(\\frac{\\partial R_0}{\\partial q}[R_0,\\tilde R_1^{(n)}]\\big)$ stabilizes. The first iteration with $\\tilde H_1=0$ reproduces the Inglis-Belyaev inertia, while the converged value includes the full time-odd response and is therefore the Thouless-Valatin inertia within ATDHFB. Because the update involves only one-body matrices and a diagonal energy denominator, the two-body stability matrix is never formed or inverted.","pith_inferences":["If the iteration is a contraction, the same loop can be applied to all five quadrupole collective coordinates at once, potentially delivering full five-dimensional adiabatic inertia at a fraction of the cost of linear-response methods.","The linear relation with effective mass suggests a practical cross-check: a fit of Skyrme time-odd coupling constants to measured rotational bands should also constrain the isoscalar effective mass, because both enter through the same Galilean-invariant combination $\\rho\\tau-j^2$.","A formal error bound for Eq. (8) and a published direct-inversion benchmark in a small configuration space would turn the empirical convergence claim into a rigorous exactness statement.","Because the method needs no adjustable parameters and no stability matrix, it could equally be applied to shape coexistence and fission paths, where the inertia matrix is presently the main uncertainty."],"forward_implications":["The full Thouless-Valatin moment of inertia of a deformed superfluid nucleus can now be computed in roughly two dozen iterations of one-body matrices, making systematic isotope-by-isotope calculations practical.","The common practice of scaling Inglis-Belyaev inertias by a fixed 1.2-1.3 factor is not supported; the ATDHFB/IB ratio varies with element, neutron number, and functional, reaching about 1.5-1.6 for D1S osmium isotopes and falling well below 1.2 for mid-shell UNEDF1.","For the seven Skyrme functionals tested on $^{166}$Er, the ATDHFB/IB ratio of both rotational and vibrational inertia decreases linearly with the isoscalar effective mass (slopes $-0.582$ and $-0.323$, $R^2\\simeq0.90$), showing that the time-odd current term $\\rho\\tau-j^2$ controls the size of the effect.","Pairing uncertainties shift the IB and ATDHFB values together, so the present data cannot decisively rank the functionals; nonetheless, the low cost of the method opens the door to including rotational inertia in future functional parameter fits."],"supporting_citations":[{"why":"Establishes the adiabatic time-dependent Hartree-Fock theory whose linearized equation is the object the paper solves.","marker":"[1]"},{"why":"Supplies the HFB quasiparticle formalism, the stability matrix whose inversion is avoided, and the standard HFB iteration adapted here.","marker":"[2]"},{"why":"Formulates the cranking approximation for collective inertia that serves as the baseline and as the Inglis-Belyaev route.","marker":"[3]"},{"why":"Gives the original Inglis formula for the rotational moment of inertia.","marker":"[4]"},{"why":"Extends the Inglis formula to paired systems, defining the Inglis-Belyaev inertia used throughout the comparisons.","marker":"[5]"},{"why":"Provides a finite-amplitude linear-response computation of the Thouless-Valatin rotational inertia and the volume-pairing strengths adopted for SkM*.","marker":"[12]"},{"why":"Shows an existing route to five-dimensional adiabatic inertia with improved inertial functions, the practical alternative the iterative method is compared with.","marker":"[13]"},{"why":"Companion paper presenting the iterative ATDHFB solutions on which the present implementation and convergence observations build.","marker":"[14]"},{"why":"Gives the quasiparticle-basis expression of the ATDHFB equation from which the iterative update (8) is written.","marker":"[19]"}],"fun_headline_variants":["Exact moments of inertia from iterative ATDHFB","Time-odd fields captured: exact nuclear inertia without parameters","Iterative solver computes rare-earth moments beyond Inglis-Belyaev","Self-consistent rotation fields yield exact collective inertia","No adjustable parameters: exact ATDHFB inertia from iterative SVD"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the fixed-point iteration (8) converges to the exact solution of the linear ATDHFB equation for every functional and nucleus studied; the paper supports this empirically by reporting that about two dozen iterations suffice and by an unpublished comparison with cranking, without giving a convergence proof or error bound.","fun_headline_variants_meta":{"raw":{"variants":["Exact moments of inertia from iterative ATDHFB","Time-odd fields captured: exact nuclear inertia without parameters","Iterative solver computes rare-earth moments beyond Inglis-Belyaev","Self-consistent rotation fields yield exact collective inertia","No adjustable parameters: exact ATDHFB inertia from iterative SVD"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000348,"raw_usage":{"total_tokens":1916,"prompt_tokens":972,"completion_tokens":944,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":588,"completion_tokens_details":{"reasoning_tokens":862}},"tokens_in":588,"tokens_out":944,"duration_ms":8144,"temperature":1.0,"reasoning_tokens":862,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-08T12:46:49.735186+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a small deformed superfluid nucleus and solve the ATDHFB equation in a truncated quasiparticle space both by direct inversion of the full two-body stability matrix and by iterating Eq. (8); if the two inertias disagree beyond numerical precision, or if the iteration oscillates or fails to settle for some functional, the quoted values are not exact ATDHFB inertias.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the original Inglis formula for the rotational moment of inertia."},{"cited_title":"Beliaev, Concerning the calculation of the nuclear mo- ment of inertia , Nuclear Physics 24 (2) (1961) 322–325","cited_arxiv_id":null,"evidence_quote":"Extends the Inglis formula to paired systems, defining the Inglis-Belyaev inertia used throughout the comparisons."},{"cited_title":"Petr´ ık, M","cited_arxiv_id":null,"evidence_quote":"Provides a finite-amplitude linear-response computation of the Thouless-Valatin rotational inertia and the volume-pairing strengths adopted for SkM*."},{"cited_title":"Washiyama, N","cited_arxiv_id":null,"evidence_quote":"Shows an existing route to five-dimensional adiabatic inertia with improved inertial functions, the practical alternative the iterative method is compared with."},{"cited_title":"Iterative solutions of the ATDHFB equations to determine the nuclear collective inertia","cited_arxiv_id":"2411.18404","evidence_quote":"Companion paper presenting the iterative ATDHFB solutions on which the present implementation and convergence observations build."},{"cited_title":"Dobaczewski, J","cited_arxiv_id":null,"evidence_quote":"Gives the quasiparticle-basis expression of the ATDHFB equation from which the iterative update (8) is written."}],"review_version":1}