Algebraic coefficients of fractional parentage for three particles in the oscillator Jacobi basis

Technical Summary: Algebraic Coefficients of Fractional Parentage for Three Particles in the Oscillator Jacobi Basis

Problem Statement
In translationally invariant few-body nuclear physics, calculations are typically performed using Jacobi coordinates to eliminate spurious center-of-mass excitations. However, enforcing the Pauli principle (antisymmetry) in this basis is computationally challenging. Unlike single-particle bases where particle exchange merely relabels states, exchanging nucleons in the Jacobi basis mixes the coordinates. Consequently, finding the orbital coefficients of fractional parentage (CFP)—the linear combinations of basis states that transform correctly under the permutation group $S(3)$—requires diagonalizing a permutation operator (specifically the class sum $\hat{\Lambda} = \hat{P}_{13} + \hat{P}_{23}$) within every energy ($E$) and angular momentum ($L$) block.

The standard approach involves constructing the matrix of this operator using harmonic oscillator (HO) brackets and applying a numerical eigensolver. This process is computationally expensive ($O(n^3)$ in block dimension $n$), must be repeated for every block, and introduces numerical round-off errors into quantities that should be exact integers. Furthermore, the resulting eigenvectors often require post-hoc analysis to identify their symmetry classes and pairing structures.

Methodology
The authors present a closed-form construction that eliminates the need for numerical diagonalization. The core of the method relies on a specific unitary transformation, $\hat{W} = \hat{H}_1 \hat{\Phi}$, which "dresses" the class operator.

• $\hat{H}_1$: The harmonic oscillator bracket operator with mass parameter $d=1$.
• $\hat{\Phi}$: A fixed complexification operator defined by a diagonal phase matrix $F = \text{diag}(1, i)$ lifted to the oscillator space.

The derivation proceeds by lifting a geometric identity from the Jacobi coordinate plane to the oscillator Fock space. In the plane, conjugating the sum of two transposition reflections ($P_{13} + P_{23}$) by $F$ and the Hadamard mirror $T_1$ (which corresponds to $\hat{H}_1$) transforms the sum into a simple label swap with a real coefficient.

The central identity derived is:
$$\hat{W} \hat{\Lambda} \hat{W}^\dagger = \sigma \hat{J} \hat{Z}$$
where:

• $\sigma = (-1)^{E-L}$ is a constant phase for the block.
• $\hat{J}$ is the label-swap operator (exchanging the two Jacobi quantum number pairs).
• $\hat{Z}$ is a diagonal operator with integer eigenvalues $z_i = 2\cos[\pi(e_1 - e_2)/3] \in \{\pm 1, \pm 2\}$.

This transformation reveals that the class operator, when dressed by $\hat{W}$, becomes a simple permutation of labels multiplied by an integer, rather than a dense matrix requiring diagonalization.

Key Contributions and Results

1. Closed-Form CFPs: The CFPs are explicitly constructed as:
   $$|K_{i,s}\rangle = \sqrt{2} \hat{Q}_s \hat{W}^\dagger |i\rangle$$
   where $\hat{Q}_s$ is a projector selecting a specific $e_2$-parity. The authors demonstrate that every CFP is simply a $\sqrt{2}$-scaled, sign-decorated column of the $d=1$ HO bracket table, restricted to one parity sector.
2. Exact Symmetry Labeling: The eigenvalues of the class operator are exact integers ($\pm 2, \pm 1$) determined directly by the quantum numbers of the state, removing the need for numerical eigenvalue extraction. The symmetry class ($S(3)$ irrep) is assigned immediately:
   • $z_i = \pm 2$ corresponds to totally symmetric $[3]$ or antisymmetric $[111]$ states.
   • $|z_i| = 1$ corresponds to mixed-symmetry $[21]$ pairs.
3. Algorithmic Efficiency: The computational cost of generating CFPs is reduced from $O(n^3)$ (eigensolver) to $O(n^2)$ (matrix assembly and column selection). The eigensolver is effectively removed from the CFP step and relocated entirely to the bracket engine that supplies $\hat{H}_1$.
4. Numerical Validation:
   • The method was implemented in Fortran and tested on 1,301 blocks up to $E=50$ (totaling 912,366 CFPs).
   • The $S(3)$ class assignment was exact in every block.
   • Residual numerical errors were found to originate solely from the SU(3)-based bracket engine used to compute $\hat{H}_1$. In double precision, errors remained below $10^{-8}$ for $E \le 35$. For higher energies, a quadruple-precision build restored errors to $10^{-22}$ or better.
   • The construction was validated against an independent production three-nucleon code (Trys_kunai). Replacing the numerically generated CFPs with the algebraic ones in the full pipeline yielded identical ground-state energies for $^3\text{He}$ (differences $< 10^{-9}$ MeV), confirming the method's physical correctness.

Significance
The paper claims that the eigensolver is unnecessary for determining orbital CFPs in the oscillator Jacobi basis. The coefficients are already "tabulated" within the $d=1$ harmonic oscillator bracket itself. By utilizing the specific unitary $\hat{W}$, the complex bookkeeping of parity, phase, and symmetry classification collapses into a single operator expression.

The significance lies in:

• Uniformity: A single formula applies to all $E$, $L$, and symmetry classes, including cases that previously required separate treatment (e.g., central regions where $e_1=e_2$).
• Precision: The method yields exact integer eigenvalues and avoids the accumulation of round-off errors inherent in iterative diagonalization.
• Performance: While the overall speedup is moderated by the cost of the bracket engine itself, the CFP-specific step sees a 15-fold speedup at $E=50$ compared to the fastest available eigensolvers.

The authors note that the remaining numerical limitations are not due to the CFP construction but to the specific implementation of the bracket engine (SU(3) tower construction). They suggest that a closed-form bracket engine would remove all numerical error from the pipeline. The work also points to potential extensions to the hyperspherical basis and the four-particle case.