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New angular momentum coupling method based on Wigner rotation theory

Junchao Guo, Yang Sun

TL;DR

The paper addresses the challenge of constructing total angular momentum for many-nucleon states in spherical shell-model spaces, where antisymmetry and rotational symmetry must be reconciled. It introduces angular momentum projection (AMP) to restore rotational symmetry and shows that the projection matrix can be computed and diagonalized to yield good-$J$ states, with a determinant expression $\langle \phi_a|e^{-i\beta J_y}|\phi_b\rangle=\det S^j$ in a single-$j$ shell. For multiple $j$-shells, the authors prove that the rotation matrix is block-diagonal and that the total projection reduces to a sum over intermediate angular momenta with CG-coupling coefficients, i.e. $\langle \phi'|\hat{P}^J_{M'M}|\phi\rangle=\sum_{J_1,J_2}\langle \phi'_1|\hat{P}^{J_1}_{M_1'M_1}|\phi_1\rangle \langle \phi'_2|\hat{P}^{J_2}_{M_2'M_2}|\phi_2\rangle \langle J_1M_1'J_2M_2'|JM'\rangle \langle J_1M_1J_2M_2|JM\rangle$. This leads to a recurrence that enables constructing many-body $|JM\rangle$ states across several shells while absorbing antisymmetry constraints into the single-$j$ spaces, potentially replacing the traditional $J$-scheme and reducing computational cost. The framework builds on Wigner rotation theory and relates to beyond-mean-field methods, with broad applicability to nuclear structure, reactions, and fission computations.

Abstract

We present a new method for constructing the total angular momentum of many-nucleon states. We find that the restrictions imposed by the fermion antisymmetry on the total state are fully absorbed into the single-j space when the broken rotational symmetry of the product state is restored by angular momentum projection. For different j-shells, any total angular momentum that obeys the selection rule is allowed, just as for non-identical particles. The method based on this reorganization is conceptually different from the traditional J and m schemes and may help to improve the efficiency of angular momentum coupling in nuclear many-body calculations.

New angular momentum coupling method based on Wigner rotation theory

TL;DR

The paper addresses the challenge of constructing total angular momentum for many-nucleon states in spherical shell-model spaces, where antisymmetry and rotational symmetry must be reconciled. It introduces angular momentum projection (AMP) to restore rotational symmetry and shows that the projection matrix can be computed and diagonalized to yield good- states, with a determinant expression in a single- shell. For multiple -shells, the authors prove that the rotation matrix is block-diagonal and that the total projection reduces to a sum over intermediate angular momenta with CG-coupling coefficients, i.e. . This leads to a recurrence that enables constructing many-body states across several shells while absorbing antisymmetry constraints into the single- spaces, potentially replacing the traditional -scheme and reducing computational cost. The framework builds on Wigner rotation theory and relates to beyond-mean-field methods, with broad applicability to nuclear structure, reactions, and fission computations.

Abstract

We present a new method for constructing the total angular momentum of many-nucleon states. We find that the restrictions imposed by the fermion antisymmetry on the total state are fully absorbed into the single-j space when the broken rotational symmetry of the product state is restored by angular momentum projection. For different j-shells, any total angular momentum that obeys the selection rule is allowed, just as for non-identical particles. The method based on this reorganization is conceptually different from the traditional J and m schemes and may help to improve the efficiency of angular momentum coupling in nuclear many-body calculations.

Paper Structure

This paper contains 11 sections, 4 theorems, 51 equations.

Table of Contents

  1. Introduction
  2. Angular momentum projection
  3. A warm-up example of AMP: a single-$j$ case
  4. AMP for manybody states of multiple $j$-shells
  5. discussions
  6. Conclusions
  7. Basic elements of AMP
  8. Derivation of Equations (\ref{['det']}) and (\ref{['blockM']})
  9. New derivations of some related propositions
  10. Particle-hole symmetry in angular momentum
  11. Pairing and angular momentum

Key Result

Lemma 1

Suppose there are two different $j$-shells, $j_1$ and $j_2$. By defining we write $\ket{\phi_1} = \phi_1^\dagger\ket{0}$ as a general $s$-particle state in $j_1$-shell and $\ket{\phi_2}$ a $t$-particle state in $j_2$-shell. Similarly, we write $\ket{\phi_1^\prime}$ and $\ket{\phi_2^\prime}$ as another $s$-particle state in $j_1$ and $t$-particle state in $j_2$, respectiv and the matrix of the ro

Theorems & Definitions (8)

  • Lemma 1
  • proof
  • Corollary 1
  • Corollary 2
  • Theorem 1
  • proof
  • proof
  • proof