Angular-Momentum Variation After Projection

• The paper introduces a framework where angular-momentum variation after projection both diagnoses and variationally optimizes symmetry-restored nuclear states.
• It details methods like VAP and PAV to extract good-J components, improve energy calculations, and manage numerical quadrature efficiently.
• The approach accurately reproduces shell-model results while uncovering insights into intrinsic state decomposition and cross-shell angular-momentum coupling.

Angular-momentum variation after projection denotes the study of how a rotational-symmetry-breaking nuclear many-body state is reorganized once good total angular momentum is restored. In recent nuclear-structure literature, the expression is used in two closely related senses. One is diagnostic: for a product state $|\Phi\rangle$ that is an eigenstate of $J_z$ but not of $J^2$, projection defines the probability distribution of good-$J$ components, $w_J(K)=\langle\Phi_K|P^J_{KK}|\Phi_K\rangle$, and thereby quantifies which irreducible rotational components are present. The other is variational: in variation after projection (VAP), one minimizes a projected energy functional in the symmetry-restored subspace rather than projecting only after solving a symmetry-breaking mean-field or generator-coordinate problem. Both senses are central to contemporary shell-model, projected GCM, HFB/QRPA, and non-orthogonal Slater-determinant methods (Guo et al., 1 Jul 2025, Porro et al., 2024).

1. Definition, scope, and terminology

In the spherical shell model, many-body basis states are products of creation operators $a^\dagger_{jm}$ acting on the vacuum, or equivalently Slater determinants built from single-particle orbitals labeled by $j$ and $m$. Such a product state $|\Phi\rangle$ is typically an eigenstate of $J_z$ with projection $J_z$0, but, because it is a product, it is not an eigenstate of total angular momentum $J_z$1; rotational symmetry is therefore broken in the basis. Angular-momentum projection restores the broken $J_z$2 symmetry by extracting good-$J_z$3 components from $J_z$4 (Guo et al., 1 Jul 2025).

The standard projector is

$J_z$5

with $J_z$6 and $J_z$7 the Wigner $J_z$8 functions. In projected GCM notation, a symmetry-restored state is written

$J_z$9

with $J^2$0, while in non-orthogonal shell-model VAP one uses

$J^2$1

These forms make explicit that projection is not merely a post-processing step but can define the variational space itself (Porro et al., 2024, Dao et al., 11 Jul 2025).

A fundamental distinction is therefore between projection after variation (PAV) and variation after projection (VAP). In PAV one first minimizes an unprojected functional and projects afterward. In VAP one minimizes a projected functional, such as

$J^2$2

so the optimization is carried out directly in a good-$J^2$3 sector. This difference is formal, computational, and physical: it changes the intrinsic state that is selected by the variational principle (Dao et al., 11 Jul 2025).

2. Projection kernels, norm matrices, and the meaning of variation

Given a broken-symmetry product state with fixed $J^2$4 (in practice often $J^2$5), the projected state is

$J^2$6

and the norm kernel is

$J^2$7

Its diagonal element,

$J^2$8

is the weight of total angular momentum $J^2$9 in the intrinsic state. In this diagnostic sense, angular-momentum variation after projection is the full distribution $J$0 rather than a single projected expectation value (Guo et al., 1 Jul 2025).

The projected weights immediately generate standard observables. The intrinsic expectation of $J$1 becomes

$J$2

while the projected energy in a given $J$3 channel is

$J$4

For a set of product states $J$5 with fixed $J$6, diagonalizing the norm matrix

$J$7

yields orthonormal good-$J$8 states and their weights. In projected GCM language this becomes the Hill–Wheeler–Griffin equation,

$J$9

with projected kernels $w_J(K)=\langle\Phi_K|P^J_{KK}|\Phi_K\rangle$0 and $w_J(K)=\langle\Phi_K|P^J_{KK}|\Phi_K\rangle$1 (Guo et al., 1 Jul 2025, Porro et al., 2024).

The identity resolution in projected space,

$w_J(K)=\langle\Phi_K|P^J_{KK}|\Phi_K\rangle$2

justifies the usual interpretation of $w_J(K)=\langle\Phi_K|P^J_{KK}|\Phi_K\rangle$3 as an extractor of good-$w_J(K)=\langle\Phi_K|P^J_{KK}|\Phi_K\rangle$4 components. This is the formal basis for both post hoc spin decomposition and projected variational schemes (Guo et al., 1 Jul 2025).

3. Antisymmetry, single-$w_J(K)=\langle\Phi_K|P^J_{KK}|\Phi_K\rangle$5 spaces, and allowed total angular momentum

A central recent result is that fermionic antisymmetry is fully absorbed into each single-$w_J(K)=\langle\Phi_K|P^J_{KK}|\Phi_K\rangle$6 sector when angular-momentum projection is evaluated in product bases. For product states of $w_J(K)=\langle\Phi_K|P^J_{KK}|\Phi_K\rangle$7 fermions in a single $w_J(K)=\langle\Phi_K|P^J_{KK}|\Phi_K\rangle$8 shell,

$w_J(K)=\langle\Phi_K|P^J_{KK}|\Phi_K\rangle$9

where $a^\dagger_{jm}$0 is the submatrix of the Wigner small-$a^\dagger_{jm}$1 matrix $a^\dagger_{jm}$2 restricted to the occupied $a^\dagger_{jm}$3 values in the bra and ket. This determinant formula supplies all rotation kernels needed for the projected norm matrix in a single-$a^\dagger_{jm}$4 space (Guo et al., 1 Jul 2025).

For multiple distinct shells the rotation kernel factorizes blockwise. If $a^\dagger_{jm}$5 and $a^\dagger_{jm}$6 belong to two distinct shells $a^\dagger_{jm}$7 and $a^\dagger_{jm}$8, then

$a^\dagger_{jm}$9

Because the antisymmetrizer commutes with $j$0 and hence with $j$1, projection preserves antisymmetry. The theorem proved for the projected kernel shows that after physical good-$j$2 states are prepared in each single-$j$3 shell, the final many-shell coupling is the ordinary Clebsch–Gordan coupling of angular momenta $j$4; for different $j$5 shells no additional identical-particle restriction survives beyond standard $j$6 selection rules (Guo et al., 1 Jul 2025).

This reorganizes the usual shell-model interpretation of allowed total angular momentum. Within a single-$j$7 shell, antisymmetry yields the familiar constraints: for two identical nucleons only even $j$8 from $j$9 to $m$0 are allowed; fully paired configurations yield only even $m$1; a fully occupied shell gives $m$2. Across different shells, once each shell’s allowed $m$3 values are obtained, the total $m$4 obeys only

$m$5

with $m$6 and parity $m$7. The practical implication is that identical-nucleon constraints do not propagate between distinct $m$8 irreducible spaces (Guo et al., 1 Jul 2025).

Worked examples make the point explicit. For four identical nucleons in a $m$9 shell, the Slater-space dimension is $|\Phi\rangle$0; fully paired $|\Phi\rangle$1 states have only even-$|\Phi\rangle$2 weights, and the fully occupied shell $|\Phi\rangle$3 satisfies $|\Phi\rangle$4, so $|\Phi\rangle$5. For two identical nucleons in $|\Phi\rangle$6 and one nucleon in $|\Phi\rangle$7, one has $|\Phi\rangle$8 and $|\Phi\rangle$9, so the allowed total set is the union of the four Clebsch–Gordan ranges: $J_z$0; $J_z$1; $J_z$2; and $J_z$3 (Guo et al., 1 Jul 2025).

4. Variation after projection as a variational strategy

In projected GCM for giant monopole resonances, the timing of projection is decisive. The projected ansatz

$J_z$4

solves the HWG equation in a symmetry-conserving subspace. When angular momentum is restored only a posteriori, the monopole response is contaminated by unphysical rotational admixtures. A diagnostic overlap,

$J_z$5

quantifies this contamination. In $J_z$6Si, $J_z$7 reaches approximately $J_z$8, and in $J_z$9Ti and $J_z$00Mg it is reported to be up to three times larger. In the same nucleus, VAP-GCM shifts dominant $J_z$01 peaks by approximately $J_z$02, $J_z$03, $J_z$04, and $J_z$05 MeV, while dominant $J_z$06 peaks shift by approximately $J_z$07, $J_z$08, $J_z$09, and $J_z$10 MeV; PAV-GCM, by contrast, leaves $J_z$11 energies only modestly changed, typically by approximately $J_z$12 MeV versus VAP-GCM, but can produce anomalously amplified monopole peaks, including high-energy peaks absent in the unprojected GCM. The analysis led to the conclusion that angular momentum must be restored a priori when rotational and vibrational degrees of freedom coexist (Porro et al., 2024).

In the shell model, VAP with non-orthogonal Slater determinants has been used to solve the secular problem variationally in a compact projected basis. For the USDB interaction, DNOSM(VAP) reproduced exact ground-state energies with very small determinant counts: $J_z$13Ne at $J_z$14 MeV with $J_z$15 determinants versus exact $J_z$16 MeV, $J_z$17Mg at $J_z$18 MeV with $J_z$19 determinants versus exact $J_z$20 MeV, $J_z$21Si at $J_z$22 MeV with $J_z$23 determinants versus exact $J_z$24 MeV, and $J_z$25Al at $J_z$26 MeV with $J_z$27 determinants versus exact $J_z$28 MeV. In $J_z$29Cr the yrast sequence from $J_z$30 to $J_z$31 was reproduced to within a few keV with $J_z$32–$J_z$33 determinants, and the drop in $J_z$34-ray spacing from $J_z$35 to $J_z$36 MeV between $J_z$37 and $J_z$38 gave the standard backbending signal. In $J_z$39Ni, DNOSM(VAP) converged to $J_z$40 MeV, below the largest constrained Lanczos result $J_z$41 MeV and its extrapolation $J_z$42 MeV (Dao et al., 11 Jul 2025).

A different simplification was established for high-spin VAP based on fixed-$J_z$43 projected states. Instead of using all $J_z$44 projected components for each reference determinant, one may minimize

$J_z$45

with a single chosen $J_z$46. In the reported calculations this produced nearly the same VAP states as full $J_z$47 mixing: in $J_z$48Mg, for $J_z$49 and $J_z$50, all independently minimized $J_z$51 values coincided with the conventional VAP result, with overlaps mostly above $J_z$52 and worst cases above $J_z$53; in multi-reference calculations across several $J_z$54-shell nuclei the overlaps between states obtained with $J_z$55 and $J_z$56 were at least $J_z$57, mostly above $J_z$58; in $J_z$59Cr the overlaps between $J_z$60 and $J_z$61 yrast states exceeded $J_z$62 for all $J_z$63. This directly supports the statement that a nuclear state cannot be identified with a single intrinsic state (Gao, 2021).

For high-spin superdeformed bands, full AMP-VAP remains numerically prohibitive in realistic spaces, and multicranked configuration mixing after projection has been used as a practical surrogate. In $J_z$64Dy and $J_z$65Hg, mixing projected states built from several cranked HFB or PN-VAP configurations recovered the Thouless–Valatin moment of inertia and corrected the single-configuration Yoccoz behavior. The method improved over single-configuration AMP by approximately $J_z$66–$J_z$67 in the magnitude of $J_z$68 and qualitatively corrected its spin dependence, while the first simultaneous use of angular-momentum and particle-number projection in this multicranked setting showed that if VANP intrinsic states are employed, particle-number projection must also be retained in the mixing kernels to avoid severe overestimation of couplings (Ushitani et al., 2019).

5. Numerical quadrature, inversion, and large-scale implementation

Because AMP and AMP-VAP repeatedly evaluate rotated norm and Hamiltonian kernels, numerical integration on $J_z$69 is a major algorithmic issue. For triaxial states, exactness of the quadrature is controlled by the maximal angular content of the intrinsic state. If the intrinsic wave function is effectively band-limited to $J_z$70 and one projects onto $J_z$71, the required degree of exactness is

$J_z$72

For the conventional trapezoidal-plus-Gauss–Legendre-plus-trapezoidal scheme, the point count scales as

$J_z$73

whereas the Lebedev-plus-trapezoidal scheme scales as

$J_z$74

Accordingly, the necessary number of sampling points is reduced by a factor $J_z$75 relative to the conventional method. Benchmarks gave energy errors of order $J_z$76 MeV in pf-shell tests and, for $J_z$77Be, about $J_z$78 MeV at sufficiently high degree; the paper concluded that Lebedev+T is the most efficient among the broadly available quadratures examined (Shimizu et al., 2022).

A complementary approach is projection through linear algebra. Sampling the rotated kernels on an Euler-angle mesh produces linear systems for the projected kernels, and the $J_z$79 and $J_z$80 integrations can be inverted exactly on a uniform grid with $J_z$81. This inversion is algebraically equivalent to the trapezoidal treatment of $J_z$82 rotations and to Fomenko projection in particle-number restoration. In representative triaxial tests, the required kernel evaluations for $J_z$83 keV were substantially reduced relative to full Gauss–Legendre quadrature: for $J_z$84Cr, $J_z$85, Gauss–Legendre required $J_z$86 evaluations, the mixed method $J_z$87, and full linear algebra $J_z$88; for $J_z$89Fe, $J_z$90, the numbers were $J_z$91, $J_z$92, and $J_z$93; for low-$J_z$94 $J_z$95Cr states, Gauss–Legendre used $J_z$96 versus $J_z$97–$J_z$98 for the mixed method and $J_z$99 for full inversion. The inexpensive quantity $J^2$00 was found to track energy convergence closely and provides a practical truncation monitor (Johnson et al., 2018).

In axially symmetric implementations the numerical structure simplifies because only $J^2$01 contributes. In DRHBc+AMP, the projected norm and Hamiltonian kernels reduce to one-dimensional $J^2$02 integrals involving $J^2$03. Using a Dirac Woods–Saxon basis with continuum coupling, the implementation reported that $J^2$04 Gauss–Legendre points were sufficient for relative accuracies better than $J^2$05 in energies and $J^2$06 in $J^2$07 values, and DRHBc+AMP excitation energies and $J^2$08 strengths for $J^2$09Mg agreed within approximately $J^2$10 with MDC-RHB+AMP (Sun et al., 2021).

6. Physical consequences, decomposition of projected states, and limitations

The physical content of angular-momentum projection is not restricted to rotational spectra. In deformed pnQRPA for $J^2$11 decay, exact angular-momentum projection applied after variation and after the QRPA altered both transition strengths and phase space. For neutron-rich Fe isotopes, exact projection reduced calculated half-lives relative to the needle approximation by up to about $J^2$12, and sometimes increased them when projection quenched the first low-lying GT peak. In $J^2$13Fe at $J^2$14, the half-life dropped from $J^2$15 s in the needle approximation to $J^2$16 s with exact AMP, a reduction of about $J^2$17. The same work showed that projection shifts the $J^2$18 minima of $J^2$19Fe from spherical HFB shapes to prolate shapes near $J^2$20, thereby modifying $J^2$21 and hence the phase-space integrals (Chen et al., 18 Oct 2025).

Projection also exposes internal neutron–proton angular-momentum coupling. Using a new identity that decomposes the conventional full-system projector into proton and neutron projectors coupled by Clebsch–Gordan coefficients,

$J^2$22

the projected VAP shell-model wave function can be decomposed into orthogonal $J^2$23 sectors. In the reported $J^2$24-shell examples, even-even ground states were not fully paired in the sense that $J^2$25 throughout, and in $J^2$26Ne and $J^2$27Mg the $J^2$28 component exceeded the $J^2$29 component. For the yrast $J^2$30 state of $J^2$31Mg the dominant components were $J^2$32, $J^2$33, and $J^2$34; for the $J^2$35 ground state of $J^2$36Al the dominant component was $J^2$37. This decomposition was then promoted from an analysis tool to an enlarged variational basis, improving ground-state energies modestly for even-even nuclei and considerably for odd-mass and odd-odd nuclei (Chen et al., 21 Jan 2026).

Several recurrent misconceptions are corrected by these developments. One is that identical-particle antisymmetry imposes additional cross-shell restrictions on total $J^2$38; the factorization theorem shows that, after single-$J^2$39 projection, coupling across different shells proceeds as for non-identical particles, subject only to ordinary triangle and parity rules (Guo et al., 1 Jul 2025). A second is that projection can safely be postponed until after a variational secular equation is solved; PGCM calculations of monopole response show that this can introduce spurious rotation–vibration coupling (Porro et al., 2024). A third is that a projected nuclear eigenstate can be assigned a unique intrinsic state; fixed-$J^2$40 high-spin VAP shows that nearly equivalent projected states can arise from different projected bases and different intrinsic representatives (Gao, 2021).

The present body of work also defines clear limitations. Some formulations focus on symmetry restoration and angular-momentum structure without specifying the interaction $J^2$41 or providing large-scale benchmarks beyond toy cases; in the many-nucleon reorganization study, numerical examples of $J^2$42 beyond simple model spaces were not given, and performance claims for very large spaces remain qualitative (Guo et al., 1 Jul 2025). Full AMP-VAP in relativistic density-functional frameworks is described as technically very complicated because it requires variational derivatives of mixed local densities and currents at every Euler angle (Sun et al., 2021). In projected small-amplitude dynamics, the projected QRPA or PQRPA remains a stated future goal rather than an established large-scale tool (Porro et al., 2024). Finally, efficient projection still depends on careful quadrature design, overlap conditioning, and, in inversion-based schemes, the absence of near-singular $J^2$43 meshes for selectively pruned $J^2$44 sets (Johnson et al., 2018).

Angular-momentum variation after projection has thus become both a diagnostic language for the $J^2$45 content of broken-symmetry states and a unifying variational principle for symmetry-restored nuclear structure calculations. Its recent reformulations replace coefficients of fractional parentage by determinant-based projectors, reinterpret multi-shell coupling through single-$J^2$46 sectors, expose spurious rotational admixtures in post-variational projection, and support compact non-orthogonal projected bases that can reproduce exact shell-model results. This suggests a continued convergence of shell-model, projected-GCM, and projected-response methodologies around symmetry-restored kernels, controlled $J^2$47 mixing, and explicitly projected variational spaces (Guo et al., 1 Jul 2025, Dao et al., 11 Jul 2025).