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Moments of inertia of rare-earth nuclei and the nuclear time-odd mean fields within exact solutions of the adiabatic theory

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Pith's one-line read An iterative fixed-point equation replaces stability-matrix inversion for nuclear moments of inertia.

desk verdict A useful systematic survey with a plausible but unproven central claim; the effective-mass correlation is the strongest part. read the letter →

arxiv 2502.07433 v3 pith:DMFE7G5S submitted 2025-02-11 nucl-th

Xuwei Sun , Jacek Dobaczewski , Markus Kortelainen , Jhilam Sadhukhan , Adrián Sánchez-Fernández , Herlik Wibowo This is my paper · ORCID
classification nucl-th PACS 21.60.Jz
keywords rotationalmomentofinertiavibrationalcollectiveadiabatictime-dependentHartree-Fock-Bogoliubovtime-oddmeanfieldsThouless-ValatinInglis-BelyaevformulaSkyrmeandGognydensityfunctionalseffectivemass
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

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.

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 (2)
  1. [Eq. (8) and following paragraph] 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'.
  2. [Vibrational inertia paragraph and Fig. 4] 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.
minor comments (5)
  1. [Summary] The phrase 'bids well' should be 'bodes well'.
  2. [Vibrational inertia paragraph and footnote] 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.
  3. [Ref. [20]] The author name 'W. Satuła' appears corrupted as 'W. Satu/suppress la' in the reference list; this should be corrected.
  4. [Eq. (8) notation] 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.
  5. [Experimental extraction] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Eq. (8) is an algebraic fixed-point form of the linear ATDHFB equation, and the inertia comparisons use externally calibrated functionals; the only self-citations are non-load-bearing.

full rationale

The central derivation is self-contained. The ATDHFB equation (5) is a linear equation for the first-order density R1, and Eq. (8) is obtained by rewriting that equation in the quasiparticle basis and isolating Z on the left; at a fixed point the iteration satisfies (E_mu + E_nu) Z_mu_nu + E1_mu_nu = i hbar F_mu_nu, which is the quasiparticle form of Eq. (5). The paper does not prove convergence or give residual diagnostics, but that is a numerical-verification gap, not a circular reduction: the reported inertias are not defined as the iteration output but as the physical ATDHFB values, and convergence is an algorithmic assumption that can be tested independently. The comparisons to data are also not fitted: SkM*, UNEDF1, and D1S are calibrated to masses, radii, and pairing properties, not to the moments of inertia shown in Figs. 1-3; the SkM* pairing strengths and the time-odd sector adjustments come from Refs. [12] and [28], which are external parameter determinations not based on the present inertia data. The only self-references are Ref. [14] (a companion paper giving details of the method without pairing) and Ref. [18] (an unpublished validation by the same authors); even if those citations are incomplete or deferred, the derivation in this paper does not reduce to them, and no fitted parameter is renamed as a prediction. Because no step of the claimed derivation is equivalent to its inputs by construction, the appropriate finding is no significant circularity.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The central calculation rests on the adiabatic expansion, time-even/odd symmetry, the choice of collective path, the convergence of the fixed-point iteration, and functional parameters taken from earlier fits. No new physical entities are introduced.

free parameters (4)
  • SkM* volume pairing strengths = V_n=-178.83 MeV fm^3, V_p=-211.20 MeV fm^3
    Taken from Ref [12], where they were adjusted to reproduce experimental pairing gaps. They shift absolute IB and ATDHFB inertias.
  • UNEDF1 mixed pairing parameters = V_n^0=-223.278 MeV fm^3, V_p^0=-247.896 MeV fm^3, rho_sat=0.32 fm^-3
    Part of the UNEDF1 functional, fit to masses and radii in the original parameter optimization.
  • Skyrme time-odd coupling constants = not specified in text
    For SkM* and UNEDF1, the time-odd sectors are fixed using the adjustments of Ref [28], fit to other nuclear data. No values are listed in this paper.
  • Isoscalar effective mass m*/m for each functional = seven values in 0.7-1.0 range for SIII, SLy4, SkM*, SkO', UNEDF0, UNEDF1, SkXce
    Input property of each Skyrme functional used in the Fig 4 correlation; values are inherited from the original functional fits.
assumptions (6)
  • domain assumption The collective motion is slow enough for the expansion R = R0 + qdot R1 + ... to truncate at first order in qdot.
    Invoked in Eqs (2)-(4); it is the adiabatic basis of ATDHFB. No quantitative criterion is given for the rotational frequencies considered.
  • domain assumption Zero-order densities and fields are time-even; first-order corrections are time-odd.
    Stated after Eq (5); standard for ATDHFB but restricts the class of solutions.
  • domain assumption For rotation, the collective path is a family of rotated HFB states, |Phi(theta)> = exp(i theta I_x)|Phi(0)>.
    Used to obtain the density derivative dR0/dq for rotation; assumes rigid rotation of the intrinsic state.
  • ad hoc to paper The fixed-point iteration of Eq (8) converges to the exact solution of the linear ATDHFB equation.
    Asserted empirically ('about two dozen iterations suffice') with no proof of convergence or uniqueness.
  • domain assumption Experimental moments of inertia can be extracted from the 2+ excitation energy via I = 3 hbar^2 / E(2+).
    Used in Fig 1; assumes a perfect rotational band and ignores coupling to other modes.
  • standard math Galilean invariance relates the time-odd current term j^2 to the effective-mass term rho tau in Skyrme functionals.
    Used to interpret the Fig 4 correlation; a known property of non-relativistic Skyrme energy density functionals.

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Cite this review

Pith. "Pith review of Moments of inertia of rare-earth nuclei and the nuclear time-odd mean fields within exact solutions of the adiabatic theory." pith.science (2026). https://pith.science/paper/DMFE7G5S

@misc{pith2026250207433,
  author       = {Pith},
  title        = {Pith review of: Moments of inertia of rare-earth nuclei and the nuclear time-odd mean fields within exact solutions of the adiabatic theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DMFE7G5S}},
  note         = {Machine review of arXiv:2502.07433}
}
read the original abstract

We systematically analyse the nuclear moments of inertia determined within the Skyrme and Gogny density functional theories. The time-odd mean fields generated by collective rotation are self-consistently determined by a novel exact iterative solution of the adiabatic time-dependent Hartree-Fock-Bogoliubov (ATDHFB) equations. Although details of the results depend on the functional used, the calculated moments of inertia are in good overall agreement with the experimental data, with no adjustable parameters. To show the essential importance of the time-odd mean fields, we compared the ATDHFB moments of inertia with those obtained from the Inglis-Belyaev formula. For Skyrme density functionals, we find strong correlations between the effective mass and the impact of the time-odd mean fields on the rotational and vibrational collective inertia.

Figures

Figures reproduced from arXiv: 2502.07433 by the authors.

Figure 2
Figure 2. Distributions of the residuals between the IB mome [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 1
Figure 1. Calculated ATDHFB (circles) and IB (triangles) mo [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 3
Figure 3. Ratios of the ATDHFB and IB moments of inertia obtai [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Correlation between the effective mass and the rati [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]

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