---
title: Angular-Momentum Variation After Projection
url: https://www.emergentmind.com/topics/angular-momentum-variation-after-projection
type: topic
---

# Angular-Momentum Variation After Projection

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 [2507.00940] [2407.01325].

## 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 $M$, but, because it is a product, it is not an eigenstate of total angular momentum $J$; rotational symmetry is therefore broken in the basis. Angular-momentum projection restores the broken $SO(3)$ symmetry by extracting good-$J$ components from $|\Phi\rangle$ [2507.00940].

The standard projector is
$$
P^J_{MK}=\frac{2J+1}{8\pi^2}\int d\Omega\, D^J_{MK}(\Omega)\,R(\Omega),
$$
with $R(\Omega)=e^{-i\alpha J_z}e^{-i\beta J_y}e^{-i\gamma J_z}$ and $D^J_{MK}(\Omega)$ the Wigner $D$ functions. In projected GCM notation, a symmetry-restored state is written
$$
|\Psi_{\nu\sigma}\rangle=\sum_q f_{\nu\sigma}(q)\,\hat P_\sigma |\Phi(q)\rangle,
$$
with $\sigma\equiv(J\,M\,\Pi\,N\,Z)$, while in non-orthogonal shell-model VAP one uses
$$
|\psi^{\pi JM}_n\rangle=\sum_{q,K} C^{\pi J}_{n,qK}\,\mathcal P^J_{MK}P^\pi |\phi_q\rangle.
$$
These forms make explicit that projection is not merely a post-processing step but can define the variational space itself [2407.01325] [2507.09073].

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
$$
E^J[\Phi]=\frac{\langle\Phi|P^J P^\pi \hat H P^J P^\pi|\Phi\rangle}{\langle\Phi|P^J P^\pi P^J P^\pi|\Phi\rangle},
$$
so the optimization is carried out directly in a good-$J$ sector. This difference is formal, computational, and physical: it changes the intrinsic state that is selected by the variational principle [2507.09073].

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

Given a broken-symmetry product state with fixed $K$ (in practice often $K=M$), the projected state is
$$
|JMK\rangle=P^J_{MK}|\Phi\rangle,
$$
and the norm kernel is
$$
N^J_{KK'}=\langle\Phi|P^J_{KK'}|\Phi\rangle.
$$
Its diagonal element,
$$
w_J(K)=\langle\Phi_K|P^J_{KK}|\Phi_K\rangle=\sum_\nu |\langle \nu J K|\Phi_K\rangle|^2,
$$
is the weight of total angular momentum $J$ in the intrinsic state. In this diagnostic sense, angular-momentum variation after projection is the full distribution $\{w_J\}$ rather than a single projected expectation value [2507.00940].

The projected weights immediately generate standard observables. The intrinsic expectation of $J^2$ becomes
$$
\langle\Phi_K|J^2|\Phi_K\rangle=\sum_J J(J+1)\,w_J(K),
$$
while the projected energy in a given $J$ channel is
$$
E_J=\frac{\langle\Phi|H P^J_{KK}|\Phi\rangle}{\langle\Phi|P^J_{KK}|\Phi\rangle}.
$$
For a set of product states $\{|\phi_i\rangle\}$ with fixed $M$, diagonalizing the norm matrix
$$
N^J_{ij}=\langle\phi_i|P^J_{MM}|\phi_j\rangle
$$
yields orthonormal good-$J$ states and their weights. In projected GCM language this becomes the Hill–Wheeler–Griffin equation,
$$
\sum_q [H_\sigma(p,q)-E_{\nu\sigma}N_\sigma(p,q)]\,f_{\nu\sigma}(q)=0,
$$
with projected kernels $H_\sigma(p,q)=\langle\Phi(p)|\hat H \hat P_\sigma|\Phi(q)\rangle$ and $N_\sigma(p,q)=\langle\Phi(p)|\hat P_\sigma|\Phi(q)\rangle$ [2507.00940] [2407.01325].

The identity resolution in projected space,
$$
\sum_{\nu J M} |\nu J M\rangle\langle \nu J M|=\sum_{J M} P^J_{MM},
$$
justifies the usual interpretation of $P^J$ as an extractor of good-$J$ components. This is the formal basis for both post hoc spin decomposition and projected variational schemes [2507.00940].

## 3. Antisymmetry, single-$j$ spaces, and allowed total angular momentum

A central recent result is that fermionic antisymmetry is fully absorbed into each single-$j$ sector when angular-momentum projection is evaluated in product bases. For product states of $N$ fermions in a single $j$ shell,
$$
\langle\phi_a| e^{-i\beta J_y} |\phi_b\rangle=\det S^j(\beta),
$$
where $S^j$ is the submatrix of the Wigner small-$d$ matrix $d^j(\beta)$ restricted to the occupied $m$ values in the bra and ket. This determinant formula supplies all rotation kernels needed for the projected norm matrix in a single-$j$ space [2507.00940].

For multiple distinct shells the rotation kernel factorizes blockwise. If $|\phi\rangle=\phi^\dagger_1\phi^\dagger_2|0\rangle$ and $|\phi'\rangle=(\phi'_1)^\dagger (\phi'_2)^\dagger|0\rangle$ belong to two distinct shells $j_1$ and $j_2$, then
$$
\langle\phi'|e^{-i\beta J_y}|\phi\rangle
=
\langle\phi'_1|e^{-i\beta J_y}|\phi_1\rangle\,
\langle\phi'_2|e^{-i\beta J_y}|\phi_2\rangle.
$$
Because the antisymmetrizer commutes with $R(\Omega)$ and hence with $P^J_{MK}$, projection preserves antisymmetry. The theorem proved for the projected kernel shows that after physical good-$J$ states are prepared in each single-$j$ shell, the final many-shell coupling is the ordinary Clebsch–Gordan coupling of angular momenta $J_1,J_2,\ldots$; for different $j$ shells no additional identical-particle restriction survives beyond standard $SU(2)$ selection rules [2507.00940].

This reorganizes the usual shell-model interpretation of allowed total angular momentum. Within a single-$j$ shell, antisymmetry yields the familiar constraints: for two identical nucleons only even $J$ from $0$ to $2j-1$ are allowed; fully paired configurations yield only even $J$; a fully occupied shell gives $J=0$. Across different shells, once each shell’s allowed $J_\alpha$ values are obtained, the total $J$ obeys only
$$
J\in \{|J_1-J_2|,\ldots,J_1+J_2\},
$$
with $M=M_1+M_2$ and parity $\pi=\pi_1\pi_2$. The practical implication is that identical-nucleon constraints do not propagate between distinct $j$ irreducible spaces [2507.00940].

Worked examples make the point explicit. For four identical nucleons in a $j=7/2$ shell, the Slater-space dimension is $\binom{8}{4}=70$; fully paired $M=0$ states have only even-$J$ weights, and the fully occupied shell $N=8$ satisfies $\det d^{7/2}(\beta)=1$, so $w_J=\delta_{J0}$. For two identical nucleons in $j_1=7/2$ and one nucleon in $j_2=3/2$, one has $J_1\in\{0,2,4,6\}$ and $J_2=3/2$, so the allowed total set is the union of the four Clebsch–Gordan ranges: $J=3/2$; $J=1/2,3/2,5/2,7/2$; $J=5/2,7/2,9/2,11/2$; and $J=9/2,11/2,13/2,15/2$ [2507.00940].

## 4. Variation after projection as a variational strategy

In projected GCM for giant monopole resonances, the timing of projection is decisive. The projected ansatz
$$
|\Psi_{\nu\sigma}\rangle=\sum_q f_{\nu\sigma}(q)\,P_\sigma |\Phi(q)\rangle
$$
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,
$$
a_\nu\equiv \langle \tilde\Psi_{0\sigma_0}|\Psi_\nu\rangle,
$$
quantifies this contamination. In $^{28}$Si, $|a_\nu|$ reaches approximately $0.05$, and in $^{46}$Ti and $^{24}$Mg it is reported to be up to three times larger. In the same nucleus, VAP-GCM shifts dominant $J=0$ peaks by approximately $1.4$, $1.0$, $1.5$, and $0.7$ MeV, while dominant $J=2$ peaks shift by approximately $2.4$, $2.1$, $2.5$, and $1.7$ MeV; PAV-GCM, by contrast, leaves $J=0$ energies only modestly changed, typically by approximately $0.5$ 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 [2407.01325].

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: $^{20}$Ne at $-40.47231$ MeV with $3$ determinants versus exact $-40.47233$ MeV, $^{24}$Mg at $-87.10405$ MeV with $16$ determinants versus exact $-87.10445$ MeV, $^{28}$Si at $-135.86003$ MeV with $45$ determinants versus exact $-135.86073$ MeV, and $^{26}$Al at $-105.74901$ MeV with $24$ determinants versus exact $-105.74934$ MeV. In $^{48}$Cr the yrast sequence from $J=0$ to $16$ was reproduced to within a few keV with $12$–$57$ determinants, and the drop in $\gamma$-ray spacing from $1.845$ to $1.477$ MeV between $J=10$ and $12$ gave the standard backbending signal. In $^{78}$Ni, DNOSM(VAP) converged to $-372.73275$ MeV, below the largest constrained Lanczos result $-372.71668$ MeV and its extrapolation $-372.72850$ MeV [2507.09073].

A different simplification was established for high-spin VAP based on fixed-$K$ projected states. Instead of using all $(2J+1)$ projected components for each reference determinant, one may minimize
$$
E^J(K)=\frac{\langle \Phi|\hat H\,\hat P^J_{KK}|\Phi\rangle}{\langle \Phi|\hat P^J_{KK}|\Phi\rangle}
$$
with a single chosen $K$. In the reported calculations this produced nearly the same VAP states as full $K$ mixing: in $^{24}$Mg, for $J=2$ and $7$, all independently minimized $E^J(K)$ values coincided with the conventional VAP result, with overlaps mostly above $0.99$ and worst cases above $0.98$; in multi-reference calculations across several $sd$-shell nuclei the overlaps between states obtained with $K=0$ and $K=J$ were at least $0.95$, mostly above $0.98$; in $^{48}$Cr the overlaps between $K=0$ and $K=J$ yrast states exceeded $98\%$ for all $J$. This directly supports the statement that a nuclear state cannot be identified with a single intrinsic state [2102.04044].

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 $^{152}$Dy and $^{194}$Hg, 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 $20$–$25\%$ in the magnitude of $\mathcal J^{(2)}$ 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 [1906.11418].

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

Because AMP and AMP-VAP repeatedly evaluate rotated norm and Hamiltonian kernels, numerical integration on $SO(3)$ 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 $I_{\max}$ and one projects onto $J$, the required degree of exactness is
$$
t=J+I_{\max}.
$$
For the conventional trapezoidal-plus-Gauss–Legendre-plus-trapezoidal scheme, the point count scales as
$$
N_{\mathrm{T+GL+T}}\sim \frac{1}{2}t^3,
$$
whereas the Lebedev-plus-trapezoidal scheme scales as
$$
N_{\mathrm{Lebedev+T}}\sim \frac{1}{3}t^3.
$$
Accordingly, the necessary number of sampling points is reduced by a factor $3/2$ relative to the conventional method. Benchmarks gave energy errors of order $10^{-5}$ MeV in pf-shell tests and, for $^{10}$Be, about $10^{-8}$ MeV at sufficiently high degree; the paper concluded that Lebedev+T is the most efficient among the broadly available quadratures examined [2205.04119].

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 $\alpha$ and $\gamma$ integrations can be inverted exactly on a uniform grid with $N_\alpha=N_\gamma=2J_{\max}+1$. This inversion is algebraically equivalent to the trapezoidal treatment of $z$ rotations and to Fomenko projection in particle-number restoration. In representative triaxial tests, the required kernel evaluations for $| \Delta E_J|<1$ keV were substantially reduced relative to full Gauss–Legendre quadrature: for $^{48}$Cr, $J=12^+$, Gauss–Legendre required $85{,}184$ evaluations, the mixed method $32{,}768$, and full linear algebra $18{,}513$; for $^{57}$Fe, $J=21/2^-$, the numbers were $32{,}768$, $10{,}648$, and $13{,}500$; for low-$J$ $^{48}$Cr states, Gauss–Legendre used $13{,}824$ versus $5{,}832$–$10{,}648$ for the mixed method and $8{,}125$ for full inversion. The inexpensive quantity $f_J=\sum_M N^J_{MM}$ was found to track energy convergence closely and provides a practical truncation monitor [1808.05672].

In axially symmetric implementations the numerical structure simplifies because only $K=0$ contributes. In DRHBc+AMP, the projected norm and Hamiltonian kernels reduce to one-dimensional $\theta$ integrals involving $d^J_{00}(\theta)$. Using a Dirac Woods–Saxon basis with continuum coupling, the implementation reported that $n_\theta=6$ Gauss–Legendre points were sufficient for relative accuracies better than $10^{-4}$ in energies and $10^{-5}$ in $B(E2)$ values, and DRHBc+AMP excitation energies and $E2$ strengths for $^{24}$Mg agreed within approximately $1\%$ with MDC-RHB+AMP [2107.05925].

## 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 $\beta$ 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 $60\%$, and sometimes increased them when projection quenched the first low-lying GT peak. In $^{64}$Fe at $\beta_2=0.12$, the half-life dropped from $0.46$ s in the needle approximation to $0.25$ s with exact AMP, a reduction of about $46\%$. The same work showed that projection shifts the $J=0$ minima of $^{64-68}$Fe from spherical HFB shapes to prolate shapes near $\beta_2\approx0.15$, thereby modifying $Q_{\beta^-}$ and hence the phase-space integrals [2510.16313].

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,
$$
P^J_{MK}
=
\sum_{J_\pi J_\nu K_\pi K_\nu M_\pi M_\nu}
\langle J_\pi K_\pi\,J_\nu K_\nu|JK\rangle
\langle J_\pi M_\pi\,J_\nu M_\nu|JM\rangle
P^{J_\pi}_{M_\pi K_\pi}P^{J_\nu}_{M_\nu K_\nu},
$$
the projected VAP shell-model wave function can be decomposed into orthogonal $(J_\pi,J_\nu)$ sectors. In the reported $sd$-shell examples, even-even ground states were not fully paired in the sense that $C(J_\pi=0,J_\nu=0)<1$ throughout, and in $^{20}$Ne and $^{24}$Mg the $(2,2)$ component exceeded the $(0,0)$ component. For the yrast $2^+$ state of $^{24}$Mg the dominant components were $(0,2)$, $(2,0)$, and $(2,2)$; for the $5^+$ ground state of $^{26}$Al the dominant component was $(5/2,5/2)$. 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 [2601.15002].

Several recurrent misconceptions are corrected by these developments. One is that identical-particle antisymmetry imposes additional cross-shell restrictions on total $J$; the factorization theorem shows that, after single-$j$ projection, coupling across different shells proceeds as for non-identical particles, subject only to ordinary triangle and parity rules [2507.00940]. 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 [2407.01325]. A third is that a projected nuclear eigenstate can be assigned a unique intrinsic state; fixed-$K$ high-spin VAP shows that nearly equivalent projected states can arise from different projected bases and different intrinsic representatives [2102.04044].

The present body of work also defines clear limitations. Some formulations focus on symmetry restoration and angular-momentum structure without specifying the interaction $H$ or providing large-scale benchmarks beyond toy cases; in the many-nucleon reorganization study, numerical examples of $w_J$ beyond simple model spaces were not given, and performance claims for very large spaces remain qualitative [2507.00940]. 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 [2107.05925]. In projected small-amplitude dynamics, the projected QRPA or PQRPA remains a stated future goal rather than an established large-scale tool [2407.01325]. Finally, efficient projection still depends on careful quadrature design, overlap conditioning, and, in inversion-based schemes, the absence of near-singular $\beta$ meshes for selectively pruned $J$ sets [1808.05672].

Angular-momentum variation after projection has thus become both a diagnostic language for the $J$ 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-$j$ 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 $K$ mixing, and explicitly projected variational spaces [2507.00940] [2507.09073].

Source: https://www.emergentmind.com/topics/angular-momentum-variation-after-projection