[Paper Review] Multiple multi-orbit pairing algebras in shell model and interacting boson models

V. K. B. Kota|arXiv (Cornell University)|Jul 12, 2017

Quantum Mechanics and Non-Hermitian Physics2 references3 citations

TL;DR

This paper introduces multiple multi-orbit pairing algebras in shell model and interacting boson models by generalizing the quasi-spin SU(2) algebra through arbitrary phase choices (αj = ±1) in the pair creation operator. It derives generalized seniority selection rules for electromagnetic transitions and shows that the Arvieu-Moszkowski phase choice maximizes correlation with realistic effective interactions, extending to identical fermion and boson systems with applications to quantum phase transitions.

ABSTRACT

In nuclei with valence nucleons are say identical nucleons and say these nucleons occupy several-$j$ orbits, then it is possible to consider pair creation operator $S_+$ to be a sum of the single-$j$ shell pair creation operators $S_+(j)$ with arbitrary phases, $S_+=\sum_j α_j S_+(j); α_j=\pm 1$. In this situation, it is possible to define multi-orbit or generalized seniority that corresponds to the quasi-spin $SU(2)$ algebra generated by $S_+$, $S_-=(S_+)^\dagger$ and $S_0=(\hat{n} -Ω)/2$ operators; $\hat{n}$ is number operator and $Ω=[\sum_j (2j+1)]/2$. There are now multiple pairing quasi-spin $SU(2)$ algebras. Also, the $α_j$'s and the generators of the corresponding generalized seniority generating sympletic algebras $Sp(2Ω)$ in $U(2Ω) \supset Sp(2Ω)$ have one-to-one correspondence. Using these, derived are the special seniority selection rules for electromagnetic transitions. A particular choice for $α_j$'s as advocated by Arvieu and Moszkowski (AM) in the past gives pairing Hamiltonians having maximum correlation with well known effective interactions. The various results derived for identical fermion systems are shown to extend to identical boson systems with the bosons occupying several-$\ell$ orbits as for example in $sd$, $sp$, $sdg$ and $sdpf$ IBM's. The quasi-spin algebra here is $SU(1,1)$ and the generalized seniority quantum number is generated by $SO(2Ω)$ in $U(2Ω) \supset SO(2Ω)$. The different $SO(2Ω)$ algebras here will be important in the study of quantum phase transitions and order-chaos transitions in nuclei.

Motivation & Objective

• To generalize the quasi-spin SU(2) algebra in shell models by allowing arbitrary phase choices (αj = ±1) in the multi-orbit pair creation operator.
• To establish a one-to-one correspondence between phase choices αj and generators of generalized seniority Sp(2Ω) algebras in U(2Ω) ⊃ Sp(2Ω).
• To derive selection rules for electromagnetic transitions based on the generalized seniority quantum number and phase-dependent quasi-spin operators.
• To demonstrate that the Arvieu-Moszkowski (AM) phase choice yields maximum correlation with realistic effective interactions using spectral distribution methods.
• To extend the formalism to interacting boson models (IBMs) with identical bosons, identifying SU(1,1) and SO(2Ω) algebras for generalized seniority and transition selection rules.

Proposed method

• Define the generalized pair creation operator as $ S_{+} = igoplus_j ar{ heta}_j S_{+}(j) $, where $ ar{ heta}_j = \pm 1 $, enabling multiple SU(2) algebras for r orbits.
• Construct the quasi-spin generators $ S_+ $, $ S_- $, and $ S_0 = (\hat{n} - \Omega)/2 $, forming SU(2) algebras for each phase configuration $ \{\alpha_j\} $.
• Establish correspondence between phase sets $ \{\alpha_j\} $ and generators of generalized seniority Sp(2Ω) algebras in the U(2Ω) ⊃ Sp(2Ω) chain.
• Derive conditions under which one-body operators of rank k are scalar or vector under the generalized SU(2) algebra, leading to selection rules.
• Use spectral distribution method to compute correlation coefficients between pairing Hamiltonians and realistic effective interactions, identifying optimal αj sets.
• Extend formalism to interacting boson models (IBMs) using SU(1,1) quasi-spin algebra and SO(2Ω) generalized seniority algebras, with tensorial E1 and E2 operators.

Experimental results

Research questions

• RQ1 How many distinct quasi-spin SU(2) algebras can be constructed for r j-orbits with arbitrary phase choices αj = ±1?
• RQ2 What is the one-to-one correspondence between phase sets {αj} and generators of generalized seniority Sp(2Ω) algebras in U(2Ω) ⊃ Sp(2Ω)?
• RQ3 What selection rules for electromagnetic transitions arise from the generalized seniority quantum number in multi-orbit pairing algebras?
• RQ4 Which phase choice {αj} maximizes correlation with realistic effective interactions, as measured by spectral distribution correlation coefficients?
• RQ5 How do the generalized pairing algebras extend to interacting boson models (IBMs) with identical bosons in multiple ℓ-orbits?

Key findings

• For r j-orbits, there exist $ 2^{r-1} $ distinct quasi-spin SU(2) algebras, each corresponding to a different choice of phase set {αj}.
• The phase set {αj} is in one-to-one correspondence with the generators of generalized seniority Sp(2Ω) algebras in the U(2Ω) ⊃ Sp(2Ω) chain.
• Electromagnetic transition matrix elements obey selection rules dependent on the phase choice {αj}, with specific constraints for operators of rank k.
• The Arvieu-Moszkowski (AM) phase choice maximizes correlation with realistic effective interactions, as confirmed by spectral distribution method correlation coefficients.
• The formalism extends to interacting boson models (IBMs), where SU(1,1) quasi-spin and SO(2Ω) generalized seniority algebras govern transition rules and Hamiltonian structure.
• The generalized pairing Hamiltonian $ H_p^G $ can be constructed to interpolate between different pairing algebras in IBMs, enabling study of quantum phase and order-chaos transitions.