Iterative solutions of the ATDHFB equations to determine the nuclear collective inertia

Adrian S\'anchez-Fern\'andez; David Muir; Herlik Wibowo; Jacek Dobaczewski; Jhilam Sadhukhan; Markus Kortelainen; Xuwei Sun

REVIEW 3 major objections 5 minor 1 cited by

Iterative solutions of the ATDHFB equations to determine the nuclear collective inertia

T0 review · 3 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The paper develops an iterative solver for the ATDHFB equations that computes nuclear collective inertia with time-odd mean fields included self-consistently, without ever constructing the two-body stability matrix.

desk verdict A useful proceedings paper on avoiding the stability matrix for collective inertia, with strong ATDHF rotational tests but the ATDHFB claim deferred and convergence unproven. read the letter →

arxiv 2411.18404 v1 pith:FKD6EPH5 submitted 2024-11-27 nucl-th

Xuwei Sun , Jacek Dobaczewski , Markus Kortelainen , David Muir , Jhilam Sadhukhan , Adrian Sánchez-Fernández , Herlik Wibowo This is my paper · ORCID

classification nucl-th MSC 81V3565H10 PACS 21.60.Jz

keywords collectiveinertiaATDHFBtime-oddmeanfieldsmomentofSkyrmedensityfunctionaltheorydynamicalcrankingvibrationalmasstensorfixed-pointiteration

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

The paper develops an iterative way to solve the adiabatic time-dependent Hartree-Fock-Bogoliubov (ATDHFB) equations, which are the standard microscopic route to the collective inertia of a nucleus. The iteration avoids constructing the two-body stability matrix, the step that normally makes ATDHFB calculations prohibitive for deformed or superfluid nuclei. The method includes the time-odd mean fields self-consistently, and the paper verifies it by showing that the resulting rotational inertia matches dynamical cranking values exactly when the full single-particle space is kept. It also computes the vibrational inertia mass tensor for 74Ge by numerical differentiation of the density, reaching about 1% accuracy on the diagonal components. If the method holds, collective inertia for rotation, vibration, and fission becomes practical for a much wider range of nuclei.

What carries the argument

The fixed-point iteration (Eq. 6) expresses the time-odd density correction $\rho_1$ in the Hartree-Fock basis as $(\epsilon_p-\epsilon_h)^{-1} (i\dot{q}\,\partial\rho_0/\partial q - \Gamma_1)$; because the time-odd mean field $\Gamma_1$ is a functional of $\rho_1$, this becomes a self-consistency loop. The loop uses the singular value decomposition of $\rho_1$ to construct the adiabatic basis of paired 'occupied' eigenstates, which lets the code evaluate time-odd densities and currents with the same machinery as the static time-even densities. The inertia is extracted from the collective kinetic energy $K = \tfrac12 \mathrm{Tr}(\dot{\rho}_0 \chi) = \tfrac12 M\dot{q}^2$.

What would settle it

Compute the collective inertia for a fixed deformation and pairing strength by the iterative method and by a direct solution of the ATDHFB equations (or by dynamical cranking at sufficiently small frequency) using the same interaction and basis; a disagreement beyond the numerical tolerance, or a failure of the iteration to converge, would show that the method is not delivering the exact ATDHFB inertia.

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Extended reading notes

Core claim

The central claim is that the ATDHF/ATDHFB equations can be solved by a fixed-point iteration on the particle-hole matrix elements of the time-odd density, given explicitly by Eq. (6), without ever building the stability matrix. Starting from zero time-odd mean field (the Inglis-Belyaev approximation), each step updates the density correction, reconstructs the adiabatic basis via the singular value decomposition of the time-odd density, recomputes the time-odd mean fields, and evaluates the inertia; the iteration stops when the inertia converges. The paper shows that in the full single-particle space the ATDHF moment of inertia of 20Ne coincides exactly with dynamical cranking, that the triaxial 126Ba moments of inertia along all three axes agree with dynamical cranking to the displayed precision and are independent of the nucleus orientation, and that the quadrupole vibrational inertia of 74Ge stabilizes to within 1% as the finite-difference step for the density derivative is reduced.

Load-bearing premise

The fixed-point iteration is assumed to converge to the unique solution of the ATDHF/ATDHFB equations; the paper gives no proof, convergence rate, or test for multiple fixed points.

Editorial extensions

If this is right

Rotational moments of inertia for arbitrarily deformed nuclei, including triaxial shapes, can now be computed without the numerically prohibitive two-body stability matrix.
The self-consistent time-odd mean fields raise the moment of inertia above the Inglis-Belyaev value by the expected factor of roughly 1.2 to 1.4, so the method removes the main systematic underestimate of the cranking approximation.
The same iterative solver applies to vibrational inertia, with the density-derivative term handled by numerical differentiation at about 1% accuracy for diagonal components.
Because the method is implemented in a Cartesian deformed harmonic-oscillator basis, it is directly applicable to fission paths where deformations are arbitrary and pairing is active.

Reading between the lines

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

If the iteration converges reliably for all deformations along a fission path, the method could replace stability-matrix-based inertia in large-scale fission calculations, where the present bottleneck is exactly the matrix inversion.
The observed sensitivity to the single-particle cutoff in 20Ne suggests that practical truncations in heavier nuclei could bias the moment of inertia, so the full-space limit is the safe operating point.
A natural testable extension is to apply this iteration to odd-mass or well-deformed rare-earth nuclei and compare with measured ground-state band moments of inertia.
The fixed-point iteration has no convergence guarantee in the paper; using a mixing or Anderson acceleration may be needed near level crossings or for strong pairing.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The headline: a sensible iterative method that avoids constructing the stability matrix, with clean ATDHF validation against dynamical cranking for rotations. The ATDHFB part of the story is mostly deferred to a companion paper, and the fixed-point iteration has no convergence analysis. Worth engaging with, but the proceedings format leaves the central paired claim under-supported.

What is actually new: the specific combination of SVD-based adiabatic basis, self-consistent treatment of time-odd fields, and numerical differentiation for the vibrational mass tensor goes beyond earlier iterative work in Refs. [7] and the finite-amplitude implementations in Refs. [9,10]. The rotational tests are the strongest part. In particular, the 20Ne cutoff sensitivity and the 126Ba triaxial orientation invariance are clean numerical checks that the iteration, when it converges, lands on the ATDHF answer. That agreement is not circular in a harmful way; dynamical cranking is a different route through the same functional, so it is an internal consistency test rather than an external benchmark, but it is still meaningful.

The soft spots are real but proportionate. There is no contraction estimate, no spectral-radius bound, and no residual check for Eq. (6); stopping on the scalar M alone can hide large errors in rho1, especially in soft or strongly paired systems. The ATDHFB equations themselves do not appear in this text—they are deferred to Ref. [5]—so one cannot verify that the paired solver is actually solving what the authors claim. The only ATDHFB result, the 74Ge vibrational mass, has no independent benchmark and a 4% spread in the off-diagonal component. That is not fatal, but it tempers the conclusions' phrasing of 'perfect agreement' and 'rapidly converging.' Also, the conclusions say the rotational moments were obtained with the ATDHFB method, but the rotations in the paper are ATDHF without pairing; that looks like a small overstatement. No code or data are included, which is normal for a proceedings, but it limits independent checking.

Who gets value from this: practitioners computing collective inertia for fission or mass tables, especially those who want to avoid the stability matrix. The paper is too short to carry the full weight of the method, but it does point to a concrete implementation in hfodd and gives a credible first validation.

Recommendation: I would referee it. The desk should send it to review. The referee should ask for the ATDHFB equations or a clear pointer to the companion paper, and for a discussion of convergence and the stopping criterion. The rotational validation alone justifies the referee time.

Referee Report

3 major / 5 minor

Summary. The manuscript proposes an iterative method for solving the adiabatic time-dependent Hartree-Fock(-Bogoliubov) equations, avoiding explicit construction of the stability matrix, and applies it to compute collective inertia for rotation and vibration. The time-odd density is updated via Eq. (6) in the HF single-particle basis, with the time-odd mean field recalculated at each step until the collective inertia M stops changing. Numerical tests are presented for the rotational moment of inertia of 20Ne and 126Ba compared with dynamical cranking, and for the quadrupole vibrational mass tensor of 74Ge, where the density-derivative term is evaluated by finite differences. The ATDHFB formalism itself is not given in this text: the paper states that the detailed description is deferred to a forthcoming publication [5]. The central claims are that the iterative method reproduces dynamical cranking results exactly in the full particle space and that it can be applied to superfluid nuclei.

Significance. If the method is correct and convergent, it offers a practical route to collective inertia for deformed and superfluid nuclei, which is important for fission and vibrational studies and would bypass the numerically prohibitive stability-matrix calculation. The ATDHF results in Section 3.1 and 3.2 agree with dynamical cranking to a high numerical precision, which is a meaningful implementation check: the comparison is not used to fit any parameter, and the two methods are distinct numerical routes within the same energy functional. The main limitation is that the paired ATDHFB case, which is the headline of the title and abstract, is not actually derived or benchmarked in this manuscript, and the convergence of the fixed-point iteration is not analyzed; the text itself acknowledges the former by deferring the equations to Ref. [5].

major comments (3)

[Section 2, Eq. (6)] The manuscript does not establish that the fixed-point iteration converges to the solution of the ATDHF equation, nor that it converges at all for the cases shown. Written in the form ρ1^{(n+1)} = D^{-1}(b − Lρ1^{(n)}) with D_{ph} = ε_p − ε_h and (Lρ1)_{ph} = Γ1(ρ1)_{ph}, the scheme is a Richardson iteration for the linear system (D + L)ρ1 = b; convergence requires a bound on the iteration matrix. The stopping criterion, 'the desired precision for M' in Section 2, controls only the variation of a scalar integral of ρ1, not the residual of Eq. (6). The conclusion that the algorithm is 'rapidly converging' therefore needs support from either a contraction estimate, a residual check, or an iteration-count/residual analysis for representative deformations and pairing strengths.
[Section 3.3 and Abstract] The only ATDHFB result, the quadrupole vibrational mass tensor of 74Ge, cannot validate the ATDHFB formalism as presented because the ATDHFB equations are not written in this manuscript, being explicitly deferred to Ref. [5]. Moreover, the comparison in Fig. 4 and Table 1 is made against the Inglis-Belyaev formula and against the numerical-differentiation spread, not against an independent solution of the paired ATDHFB equations. Therefore the abstract's claim that 'The ATDHFB equation is solved iteratively' is not substantiated within this text; the central novelty implied by the title remains unverified here.
[Section 3.1, Fig. 1] The 'exact correspondence between the ATDHF and DC moment of inertia in the full single-particle space' is demonstrated only for 20Ne with the SVT functional, and the companion case in Section 3.2 reports agreement only as a few numerical values. Because both calculations are performed in the same code (HFODD) with the same energy functional, the agreement is a self-consistency check of the implementation rather than an independent validation of the physical input; the text should state this caveat explicitly, and at least one more detailed case or a different functional would make the claim more robust.

minor comments (5)

[Section 2, Eq. (6)] The notation ρ1,ph^{(n+1)} and Γ1,ph^{(n)} should specify the basis and the finite Hilbert-space truncation used in HFODD; in particular, degenerate or near-degenerate particle-hole energy denominators ε_p − ε_h require a comment on numerical stability.
[Conclusions] The statement that a 'rapidly converging iterative algorithm' was developed is not supported by any quantitative iteration counts, convergence rates, or residual norms in the manuscript.
[Section 3.3, Fig. 4] The label 'ADB' in Fig. 4 appears to be a typo for 'ATDHFB' or 'ADB' should be defined; also, the figure caption does not explain the horizontal-axis units or the range shown.
[References] Reference [10] lists 'Phys. Rev. C 109, L051301 (2014)'; volume 109 corresponds to 2024, so the year appears to be a typo.
[Section 3.2, Fig. 3] The six 'different orientations' mentioned in the text are not defined in the figure caption or in the text; please specify the orientation angle or the rotation of the principal axes used.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the iterative ATDHF equations are derived explicitly and benchmarked against independent dynamical-cranking calculations.

full rationale

The derivation chain in this paper is not circular. The iterative update in Eq. (6) is obtained by algebraic rearrangement of the ATDHF equation (3) in the Hartree-Fock basis, with an explicit energy denominator epsilon_p - epsilon_h, and no quantity is defined in terms of the target inertia. The fixed-point procedure starts from Gamma_1 = 0, iterates Eq. (6), and terminates when the scalar inertia M is stable; this is a convergence-criterion issue, not a circularity. In Sections 3.1 and 3.2 the ATDHF rotational inertias are compared with dynamical-cranking results obtained from a separate angular-momentum constraint at a small cranking frequency, and no parameter is fitted to force agreement. The statement that 'the ATDHF calculation must include all particle states' to match the DC method is a completeness requirement, not a fitted constant. The finite-difference evaluation of the density derivative in Eq. (11) is tested for convergence as the step Delta decreases, rather than tuned to reproduce a target. The self-citations [5] and [12] disclose forthcoming method details and code documentation, but the present paper supplies the working equations for the ATDHF case and the numerical benchmark is an independent calculation within the same code and energy functional. That shared framework limits external validation, but it does not make the derivation equivalent to its input. No circular step can be exhibited from the text.

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

The ledger tracks inputs the central claim rests on. The pairing strengths and finite-difference step are numerical inputs from prior fits or hand choices; the adiabatic approximation and the Skyrme functional are domain assumptions; the convergence of the fixed-point iteration is an unproven numerical assumption that is load-bearing. No new physical entities are introduced.

free parameters (4)

V_n0 (neutron pairing strength) = -178.83 MeV fm^3
Volume pairing strength for SkM* used in the 74Ge ATDHFB calculation; adopted from prior fits, not fitted in this paper, but the numerical inertia depends on it.
V_p0 (proton pairing strength) = -211.20 MeV fm^3
Same as above for protons.
Finite-difference step Δ = 0.02 b
Chosen by hand for the density derivatives in Eq. (11); diagonal components stabilize within 1% but the off-diagonal tensor element varies by ~4% across sign choices.
Cranking frequency ω_y = 0.001 MeV
Used in the dynamical cranking reference calculations; must be small enough to stay in the linear regime, and the comparison depends on this choice.

assumptions (4)

domain assumption Adiabatic approximation: collective motion is slow compared to single-particle motion
Stated in the introduction as the basis for using ATDHF/ATDHFB; if violated, the ATDHF equation (3) does not apply.
standard math Time-dependent density decomposition ρ(t)=e^{iχ}ρ0 e^{-iχ} and the resulting ATDHF equation (3)
Standard result from Baranger and Veneroni (Ref. [6]), used as the starting point; no derivation in this paper.
domain assumption The Skyrme energy density functional (SVT for 20Ne/126Ba and SkM* for 74Ge) with specified parameters accurately describes the nuclei
The calculated inertia values inherit any error in the chosen phenomenological functional; the paper does not test functional dependence.
ad hoc to paper Convergence of the fixed-point iteration to the unique ATDHF/ATDHFB solution
No proof or numerical evidence of convergence beyond the stated stopping criterion on M is given; this assumption is load-bearing for the method's validity.

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Cite this review

Pith. "Pith review of Iterative solutions of the ATDHFB equations to determine the nuclear collective inertia." pith.science (2026). https://pith.science/paper/FKD6EPH5

@misc{pith2026241118404,
  author       = {Pith},
  title        = {Pith review of: Iterative solutions of the ATDHFB equations to determine the nuclear collective inertia},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FKD6EPH5}},
  note         = {Machine review of arXiv:2411.18404}
}

read the original abstract

An iterative adiabatic time-dependent Hartree-Fock-Bogoliubov (ATDHFB) method is developed within the framework of Skyrme density functional theory. The ATDHFB equation is solved iteratively to avoid explicitly calculating the stability matrix. The contribution of the time-odd mean fields to the ATDHF(B) moment of inertia is incorporated self-consistently, and the results are verified by comparing them with the dynamical cranking predictions. The inertia mass tensor is calculated with the density-derivative term evaluated by numerical differentiation.

Figures

Figures reproduced from arXiv: 2411.18404 by the authors.

**Figure 1.** The ATDHF Moment of inertia of 20Ne (solid line) compared with the value evaluated from the dynamical cranking (DC) calculation (dashed line) for different single-particle cutoffs. The Skyrme interaction SVT is used. the DC method, the ATDHF calculation must include all particle states. The exact correspondence between the ATDHF and DC moment of inertia in the full single-particle space demonstrates the accuracy of… view at source ↗

**Figure 2.** Distributions of the current densities (9), panels (a) and (b), and spin densities (10), panels (c) and (d), of 20Ne rotating along the y-axis, projected on the x − y plane at z = ±2.0 fm. As the nucleus rotates along the y-axis, at z = 2.0 and z = −2.0 fm the current flows in opposite directions. The spin density is mostly aligned along the y-axis and being parity-even, at z = ±2.0 fm it shows symmetric distribut… view at source ↗

**Figure 3.** Moments of inertia of 126Ba rotating along the y-, z-, and x-axes with different orientations. The iterative ATDHF method is capable of evaluating the moment of inertia of arbitrarily-deformed nuclei. In this section, we investigate the moment of inertia of the triaxial deformed nucleus 126Ba. The calculations are performed with 16 harmonic oscillator shells. The SVT interaction is used. The calculated intrinsic qua… view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: The diagonal components of the IB and ATDHFB (ADB) mass tensor B(a0) and B(a2) for 74Ge at a0 = 3.5 b and a2 = 1.5 b. The density-derivative terms are evaluated with positive numerical differences δa0 = δa2 ≡ ∆. The IB and the ATDHFB inertia tend to stabilize as ∆ de…

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Moments of inertia of rare-earth nuclei and the nuclear time-odd mean fields within exact solutions of the adiabatic theory
nucl-th 2025-02 conditional novelty 6.0 of 10

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.

Reference graph

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [7]

Dobaczewski and J

J. Dobaczewski and J. Skalski, Nuclear Physics A369 (1981) 123-140

work page 1981

[2] [5]

Sun et al., to be published

X. Sun et al., to be published

work page

[3] [1]

Ring and P

P. Ring and P. Schuck, The Nuclear Many-Body Problem (Springer- Verlag, Berlin, 1980)

work page 1980

[4] [2]

D. R. Inglis, Phys. Rev. 103 (1956) 1786

work page 1956

[5] [3]

S. T. Beliaev, Nuclear Physics 24 (1961) 322

work page 1961

[6] [4]

Libert, M

J. Libert, M. Girod, and J.-P. Delaroche, Phys. Rev. C 60 (1999) 054301

work page 1999

[7] [6]

Baranger and M

M. Baranger and M. V´ en´ eroni, Annals of Physics 114 (1978) 123-200

work page 1978

[8] [8]

Z. P. Li et al., Phys. Rev. C 86 (2012) 034334. ATDHFB printed on November 28, 2024 9

work page 2012