arXiv:nucl-th/0305095 · Physical Review C

General pairing interactions and pair truncation approximations for fermions in a single- j shell

Y. M. Zhao, A. Arima, Joseph N. Ginocchio, Naotaka Yoshinaga

Published Oct 30, 2003• 27 citations

Abstract

We investigate Hamiltonians with attractive interactions between pairs of fermions coupled to angular momentum $J$. We show that pairs with spin $J$ are reasonable building blocks for the low-lying states. For systems with only a $J={J}_{max}$ pairing interaction, eigenvalues are found to be approximately integers for a large array of states, in particular, for those with total angular momenta $I\ensuremath{\leqslant}2j$. For $I=0$ eigenstates of four fermions in a single-$j$ shell we show that there is only one nonzero eigenvalue. We address these observations using the nucleon pair approximation of the shell model and relate our results with a number of currently interesting problems.

DOI: 10.1103/physrevc.68.044320

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In this talk I shall introduce our recent works on general pairing interactions and pair truncation approximations for fermions in a single- j shell, including the spin zero dominance, and features of eigenvalues of fermion systems in a single- j shell interacting by a J -pairing interaction.

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arXiv:nucl-th/0305095 · Physical Review C

General pairing interactions and pair truncation approximations for fermions in a single- j shell

Y. M. Zhao, A. Arima, Joseph N. Ginocchio, Naotaka Yoshinaga

Published Oct 30, 2003• 27 citations

arXiv HTML PDF

Abstract

We investigate Hamiltonians with attractive interactions between pairs of fermions coupled to angular momentum

J

. We show that pairs with spin

J

are reasonable building blocks for the low-lying states. For systems with only a

J={J}_{max}

pairing interaction, eigenvalues are found to be approximately integers for a large array of states, in particular, for those with total angular momenta

I\ensuremath{\leqslant}2j

. For

I=0

eigenstates of four fermions in a single-

j

shell we show that there is only one nonzero eigenvalue. We address these observations using the nucleon pair approximation of the shell model and relate our results with a number of currently interesting problems.

DOI: 10.1103/physrevc.68.044320

Related papers

Classification of states of single- j fermions with J -pairing interaction

Y. M. Zhao, A. Arima

In this paper we show that a system of three fermions is exactly solvable for the case of a single- $j$ in the presence of an angular momentum- $J$ pairing interaction. On the basis of the solutions for this system, we obtain new sum rules for six- $j$ symbols. When the Hamiltonian contains only an interaction between pairs of fermions coupled to spin $J={J}_{\mathrm{max}}=2j\ensuremath{-}1$, the ``non-integer'' eigenvalues of three fermions with angular momentum $I$ around the maximum appear as ``non-integer'' eigenvalues of four fermions if $I$ is around (or larger than) ${J}_{\mathrm{max}}$ . This pattern is also found in five and six fermion systems. A boson system with spin $l$ exhibits a similar pattern.In this paper we show that a system of three fermions is exactly solvable for the case of a single-$j$ in the presence of an angular momentum-$J$ pairing interaction. On the basis of the solutions for this system, we obtain new sum rules for six-$j$ symbols. When the Hamiltonian contains only an interaction between pairs of fermions coupled to spin $J={J}_{\mathrm{max}}=2j\ensuremath{-}1$, the ``non-integer'' eigenvalues of three fermions with angular momentum $I$ around the maximum appear as ``non-integer'' eigenvalues of four fermions if $I$ is around (or larger than) ${J}_{\mathrm{max}}$. This pattern is also found in five and six fermion systems. A boson system with spin $l$ exhibits a similar pattern.

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Nikolaj Thomas Zinner, Klaus Mølmer, C. Özen +2 more

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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.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.

Revista Mexicana de FísicaJan 1, 20010 citations

One-Dimensional Many-Fermion System. I: Exact Solution in a Field Theoretical Framework

T. Kebukawa

The eigenvalue problem in a one-dimensional system of many fermions interacting via a delta-function potential is solved exactly in a field theoretical framework. Simultaneous eigenstates of spin and energy are represented in a systematic and compact form, that is, in such a form that the orbital and spin eigenfunctions are unified in harmony without the aid of Young’s tableau. The eigenstate and energy eigenvalue are specified by two sets of quantum numbers, particle momenta and spin momenta, besides two eigenvalues of a total spin angular momentum and its z-component. It is shown that for given particle momenta and spin momenta the energy eigenvalue is given by the solutions of coupled equations.The eigenvalue problem in a one-dimensional system of many fermions interacting via a delta-function potential is solved exactly in a field theoretical framework. Simultaneous eigenstates of spin and energy are represented in a systematic and compact form, that is, in such a form that the orbital and spin eigenfunctions are unified in harmony without the aid of Young’s tableau. The eigenstate and energy eigenvalue are specified by two sets of quantum numbers, particle momenta and spin momenta, besides two eigenvalues of a total spin angular momentum and its z-component. It is shown that for given particle momenta and spin momenta the energy eigenvalue is given by the solutions of coupled equations.

Progress of Theoretical PhysicsMay 1, 19853 citations

Eigenstates of the L=0 , Charge- and Spin-Independent Pairing Hamiltonian. I. Seniority-Zero States

R. W. Richardson

Equations for all the seniority-zero eigenstates of $2N$ nucleons in an arbitrary charge- and spin-independent potential well and interacting through charge- and spin-independent pairing forces are derived. These equations are solved exactly for a large number of states of this system. The interaction in this Hamiltonian is effective in the $L=0$ states of the two-nucleon system, and its strength is independent of the remaining quantum numbers of the two nucleons. We solve our equations exactly for those states whose wave functions are totally symmetric functions of the spin-isospin coordinates of the $N$ $L=0$ pairs of nucleons in the state. The wave functions of these states factor into a spin-isospin-dependent part and a spatially dependent part. The spin-isospin-dependent part of one of these wave functions is an eigenvector of three tridiagonal matrices which insure that the state is a spin, isospin, and supermultiplet eigenstate, respectively. Explicit expressions are given for the eigenvalues of these three matrices in terms of the quantum numbers of the state. The spatially-dependent part of one of these wave functions is given explicitly in terms of $N$ parameters which we call pair energies. These pair energies are shown to satisfy $N$ coupled algebraic equations which depend parametrically upon the supermultiplet quantum numbers of the state. An expression for the occupation probabilities of the levels of the single-particle well is given. Throughout this work, an arbitrary splitting of the single-particle levels is treated exactly.Equations for all the seniority-zero eigenstates of $2N$ nucleons in an arbitrary charge- and spin-independent potential well and interacting through charge- and spin-independent pairing forces are derived. These equations are solved exactly for a large number of states of this system. The interaction in this Hamiltonian is effective in the $L=0$ states of the two-nucleon system, and its strength is independent of the remaining quantum numbers of the two nucleons. We solve our equations exactly for those states whose wave functions are totally symmetric functions of the spin-isospin coordinates of the $N$ $L=0$ pairs of nucleons in the state. The wave functions of these states factor into a spin-isospin-dependent part and a spatially dependent part. The spin-isospin-dependent part of one of these wave functions is an eigenvector of three tridiagonal matrices which insure that the state is a spin, isospin, and supermultiplet eigenstate, respectively. Explicit expressions are given for the eigenvalues of these three matrices in terms of the quantum numbers of the state. The spatially-dependent part of one of these wave functions is given explicitly in terms of $N$ parameters which we call pair energies. These pair energies are shown to satisfy $N$ coupled algebraic equations which depend parametrically upon the supermultiplet quantum numbers of the state. An expression for the occupation probabilities of the levels of the single-particle well is given. Throughout this work, an arbitrary splitting of the single-particle levels is treated exactly.

Physical ReviewJul 20, 196739 citations

We investigate Hamiltonians with attractive interactions between pairs of fermions coupled to angular momentum

J

. We show that pairs with spin

J

are reasonable building blocks for the low-lying states. For systems with only a

J={J}_{max}

pairing interaction, eigenvalues are found to be approximately integers for a large array of states, in particular, for those with total angular momenta

I\ensuremath{\leqslant}2j

. For

I=0

eigenstates of four fermions in a single-

j

In this paper we show that a system of three fermions is exactly solvable for the case of a single-

j

in the presence of an angular momentum-

J

pairing interaction. On the basis of the solutions for this system, we obtain new sum rules for six-

j

symbols. When the Hamiltonian contains only an interaction between pairs of fermions coupled to spin $J={J}_{\mathrm{max}}=2j\ensuremath{-}1$, the ``non-integer'' eigenvalues of three fermions with angular momentum

I

around the maximum appear as ``non-integer'' eigenvalues of four fermions if

I

is around (or larger than)

{J}_{\mathrm{max}}

. This pattern is also found in five and six fermion systems. A boson system with spin

l

exhibits a similar pattern.

We study a trapped system of fermions with an attractive zero-range two-body interaction using the Shell-Model Monte Carlo method. The method provides {\em ab initio} results in the low

N

limit where mean-field theory is not applicable. The energy and pairing properties are presented as functions of interaction strength, particle number, and temperature. In the interesting region where typical matrix elements of the two-body interaction are comparable to the level spacing of the trap we find large odd-even effects and signatures of shell structure. As a function of temperature, we observe the disappearance of these effects as in a phase transition.

Equations for all the seniority-zero eigenstates of

2N

nucleons in an arbitrary charge- and spin-independent potential well and interacting through charge- and spin-independent pairing forces are derived. These equations are solved exactly for a large number of states of this system. The interaction in this Hamiltonian is effective in the

L=0

states of the two-nucleon system, and its strength is independent of the remaining quantum numbers of the two nucleons. We solve our equations exactly for those states whose wave functions are totally symmetric functions of the spin-isospin coordinates of the

N

L=0

pairs of nucleons in the state. The wave functions of these states factor into a spin-isospin-dependent part and a spatially dependent part. The spin-isospin-dependent part of one of these wave functions is an eigenvector of three tridiagonal matrices which insure that the state is a spin, isospin, and supermultiplet eigenstate, respectively. Explicit expressions are given for the eigenvalues of these three matrices in terms of the quantum numbers of the state. The spatially-dependent part of one of these wave functions is given explicitly in terms of

N

parameters which we call pair energies. These pair energies are shown to satisfy

N

coupled algebraic equations which depend parametrically upon the supermultiplet quantum numbers of the state. An expression for the occupation probabilities of the levels of the single-particle well is given. Throughout this work, an arbitrary splitting of the single-particle levels is treated exactly.