A discussion of the seniority quantum number in many-body systems is presented. The analysis is carried out for bosons and fermions simultaneously but is restricted to identical particles occupying a single shell. The emphasis of the paper is on the possibility of partial conservation of seniority which turns out to be a peculiar property of spin-9/2 fermions but prevalent in systems of interacting bosons of any spin. Partial conservation of seniority is at the basis of the existence of seniority isomers, frequently observed in semi-magic nuclei, and also gives rise to peculiar selection rules in one-nucleon transfer reactions. Key words: quantum mechanics, many-body systems, seniority, nuclear shell model, interacting bosons
Background: In quantum mechanics, entanglement is the most striking phenomenon which has no counterpart in classical systems. Although different approaches have already been developed to study correlations in the case of indistinguishable particles, the exploration of the so-called mode entanglement is still in its initial stage in nuclear physics.Purpose: We aim to study mode entanglement in the seniority model, derive analytic formulas for the one-body reduced density matrix of states with seniority $\ensuremath{\nu}=0,\phantom{\rule{0.16em}{0ex}}1,\phantom{\rule{0.16em}{0ex}}2$, and 3, and also determine the particle number dependence of the one-body reduced density matrix for arbitrary seniority. In addition, we compare the predictions of the seniority model with the results of the shell model to gain insight into the structure and correlations of the ground and lowest yrast states.Methods: In the seniority model the analytic results are calculated using the quasispin formalism. The numerical shell model calculations are carried out in the standard shell model framework using an inert core. The applied realistic effective interactions are derived earlier using the
G-matrix formalism. The density matrix renormalization group method is also applied in order to directly calculate the mode entropies.Results: In the seniority model simple analytical expressions are given for the mode entropies. The peculiar behavior of the half-filled shells and the seniority-zero states is revealed. Numerical results are presented for the lightest stable calcium isotopes and for
94Ru nucleus.Conclusions: For
94Ru, the seniority model accounts for the
0g9/2 mode entropies, but seniority mixing is important for certain yrast states. Interaction induced quantum fluctuations decrease the occupation of the
0f5/2,
1p3/2, and
1p1/2 shells, and result in non-negligible mode entropies on these shells, too, clearly outside the scope of the simple
(0g9/2)4 seniority model. The
0f7/2 shell based seniority model is more accurate for the
Ca isotopes, but seniority mixing is substantial for some
44Ca yrast states, too. Mode and one-body entanglement entropies are useful tools to investigate the structure of quantum correlations in nuclei.
The use of the seniority quantum number in many‐body systems is reviewed. A brief summary is given of its introduction by Racah in the context of atomic spectroscopy. Several extensions of Racah’s original idea are discussed: seniority for identical nucleons in a single‐j shell, its extension to the case of many, non‐degenerate j shells and to systems with neutrons and protons. To illustrate its usefulness to this day, a recent application of seniority is presented in Bose‐Einstein condensates of atoms with spin.
We derive an approximate boson-quasifermion mapping of a single-j shell-model algebra, applicable to the seniority classification scheme, as an expansion in the ratio of the seniority quantum number to the level degeneracy, with selective summation of higher order terms. The procedure used is an extension of the algebraic technique first applied to the purely bosonic part of the mapping by Bonatsos, Klein, and Li. Only the maps of the generators are needed to derive a Hamiltonian for the interacting boson fermion model (IBFM), and these can be obtained in Hermitian form without having to go through the intermediary of a mapping of the fermion operators. Application of the results is illustrated by a fresh discussion of the so-called exchange interaction of the IBFM. A consistent mapping of the single fermion in terms of bosons and quasifermions is also derived that includes a consideration of the unusual properties of the quasifermions.
The particle-hole symmetry (equivalence) of the full shell-model Hilbert space is straightforward and routinely used in practical calculations. In this work I show that this symmetry is preserved in the subspace truncated up to a certain generalized seniority and give the explicit transformation between the states in the two types (particle and hole) of representations. Based on the results, I study particle-hole symmetry in popular theories that could be regarded as further truncations on top of the generalized seniority, including the microscopic interacting boson (fermion) model, the nucleon-pair approximation, and other models.
Two advances are reviewed in the application of pairing-force theory in the nuclear shell model. The first exploits a discovery that a wide range of two-nucleon interactions conserve seniority as a good quantum number. Asa consequence, the eigenstates of a hamiltonian with such an interaction belong to irreducible representations of a compact unitary-symplectic group. This makes it possible to extend the simply-solvable models with J = 0 pairing forces to a much richer set of models and still obtain states uniquely classified by their seniority and angular momentum quantum numbers. Moreover, many of the low-lying energy levels of such models can be obtained algebraically; in technical terms, the models are in some cases completely solvable and in other cases partially solvable by algebraic methods. The second advance exploits the discovery that, in a coherent state representation, states of good nucleon number can be projected analytically from BCS vacuum and excited quasiparticle states. This makes it possible to perform calculations in a number-projected BCS basis without losing much of the advantage of working in a quasiparticle scheme.
We study mode-entanglement in the seniority model, derive analytic formulas for the one-body reduced density matrix of states with seniority
ν=0,1,2, and
ν=3, and also determine the particle number dependence of the one-body reduced density matrix for arbitrary seniority. We carry out numerical calculations for the lightest calcium isotopes and for
94Ru nucleus, and investigate the structure of their ground and low energy yrast states. We investigate the fulfillment of several predictions of the seniority model regarding the behavior of one-mode entropies, which we compare with the results of configuration interaction (CI) and density matrix renormalization group (DMRG) computations. For
94Ru, the seniority model accounts for the
0g9/2 mode entropies, but seniority mixing is important for certain yrast states. Interaction induced quantum fluctuations decrease the occupation of the
0f5/2,
1p3/2 and
1p1/2 shells, and amount in finite mode entropies on these shells, too, clearly outside the scope of the simple
(0g9/2)4 seniority model. The
0f7/2 shell based seniority model is more accurate for the light
Ca isotopes, but seniority mixing is substantial for some
44Ca yrast states, too.