arXiv:nucl-th/0112075 · Physical Review C

Simple approach to the angular momentum distribution in the ground states of many-body systems

Y. M. Zhao, A. Arima, Naotaka Yoshinaga

Published Sep 3, 2002• 30 citations

Abstract

We propose a simple approach to predict the angular momentum I ground state $(I$ g.s.) probabilities of many-body systems that does not require the diagonalization of Hamiltonians with random interactions. This method is found to be applicable to all cases that have been discussed: even and odd fermion systems (both in single-$j$ and many-$j$ shells), and boson (both $\mathrm{sd}$ and $sdg)$ systems. A simple relation for the highest angular momentum g.s. probability is found. Furthermore, it is suggested for the first time that the 0 g.s. dominance in boson systems and in even-fermion systems is given by two-body interactions with specific features.

DOI: 10.1103/physrevc.66.034302

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arXiv:nucl-th/0112075 · Physical Review C

Simple approach to the angular momentum distribution in the ground states of many-body systems

Y. M. Zhao, A. Arima, Naotaka Yoshinaga

Published Sep 3, 2002• 30 citations

arXiv HTML PDF

Abstract

We propose a simple approach to predict the angular momentum I ground state

(I

g.s.) probabilities of many-body systems that does not require the diagonalization of Hamiltonians with random interactions. This method is found to be applicable to all cases that have been discussed: even and odd fermion systems (both in single-

j

and many-

j

shells), and boson (both

\mathrm{sd}

and

sdg)

systems. A simple relation for the highest angular momentum g.s. probability is found. Furthermore, it is suggested for the first time that the 0 g.s. dominance in boson systems and in even-fermion systems is given by two-body interactions with specific features.

DOI: 10.1103/physrevc.66.034302

Related papers

Many-body systems interacting via a two-body random ensemble. I. Angular momentum distribution in the ground states

Y. M. Zhao, A. Arima, Naotaka Yoshinaga

In this paper we discuss in detail the $P(I)'\mathrm{s},$ angular momentum I probabilities in the ground states, of many-body systems interacting via a two-body random ensemble (TBRE). In particular, we extensively apply an approach introduced in an earlier paper and compare the predicted $P(I)'\mathrm{s}$ with those obtained by diagonalizing a TBRE Hamiltonian. We begin with a few solvable cases, such as fermions in a small single- $j$ shell and d boson systems, where elegant agreements between the predicted $P(I)'\mathrm{s}$ and those obtained by diagonalizing a TBRE Hamiltonian are achieved. We find that d boson systems systematically present counterexamples of angular momentum 0 ground state dominance when the number of d bosons is $6\ensuremath{\kappa}\ifmmode\pm\else\textpm\fi{}1$ with $\ensuremath{\kappa}$ a natural number, which suggests that certain fundamental symmetry (say, time reversal invariance) of the Hamiltonian cannot ensure the occurrence of angular momentum 0 ground state dominance. Next, we apply the same approach to more complicated cases, such as even or odd number of fermions in a large single- $j$ shell or a many- $j$ shell, $\mathrm{sd}$ -boson or $\mathrm{sdg}$ -boson systems, etc. We find that the simple approach proposed in an earlier paper is also well applicable, and thus it is a universal approach. The numerical experiments provide a guideline to tell which interactions are essential to produce a sizable $P(I)$ in a many-body system. This disproves a popular idea that the angular momentum 0 ground state (0 g.s.) dominance may be independent of two-body interactions. Some matrix elements which are useful to understand the observed regularities are given or addressed in detail. In this paper we also report a synchronous staggering between the 0 g.s. probabilities of even numbers of fermions in a single- $j$ shells and j g.s. probabilities of odd numbers of fermions in a single- $j$ shell when j is small. The low seniority chain of 0 g.s. using the same set of two-body interactions is confirmed, but it is noted that contribution to the total 0 g.s. probability beyond this chain may be more important for even numbers of fermions in a single- $j$ shell. Some interesting results taking a displaced two-body random ensemble are presented for the I ground state probabilities.In this paper we discuss in detail the $P(I)'\mathrm{s},$ angular momentum I probabilities in the ground states, of many-body systems interacting via a two-body random ensemble (TBRE). In particular, we extensively apply an approach introduced in an earlier paper and compare the predicted $P(I)'\mathrm{s}$ with those obtained by diagonalizing a TBRE Hamiltonian. We begin with a few solvable cases, such as fermions in a small single-$j$ shell and d boson systems, where elegant agreements between the predicted $P(I)'\mathrm{s}$ and those obtained by diagonalizing a TBRE Hamiltonian are achieved. We find that d boson systems systematically present counterexamples of angular momentum 0 ground state dominance when the number of d bosons is $6\ensuremath{\kappa}\ifmmode\pm\else\textpm\fi{}1$ with $\ensuremath{\kappa}$ a natural number, which suggests that certain fundamental symmetry (say, time reversal invariance) of the Hamiltonian cannot ensure the occurrence of angular momentum 0 ground state dominance. Next, we apply the same approach to more complicated cases, such as even or odd number of fermions in a large single-$j$ shell or a many-$j$ shell, $\mathrm{sd}$-boson or $\mathrm{sdg}$-boson systems, etc. We find that the simple approach proposed in an earlier paper is also well applicable, and thus it is a universal approach. The numerical experiments provide a guideline to tell which interactions are essential to produce a sizable $P(I)$ in a many-body system. This disproves a popular idea that the angular momentum 0 ground state (0 g.s.) dominance may be independent of two-body interactions. Some matrix elements which are useful to understand the observed regularities are given or addressed in detail. In this paper we also report a synchronous staggering between the 0 g.s. probabilities of even numbers of fermions in a single-$j$ shells and j g.s. probabilities of odd numbers of fermions in a single-$j$ shell when j is small. The low seniority chain of 0 g.s. using the same set of two-body interactions is confirmed, but it is noted that contribution to the total 0 g.s. probability beyond this chain may be more important for even numbers of fermions in a single-$j$ shell. Some interesting results taking a displaced two-body random ensemble are presented for the I ground state probabilities.

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In this paper, we report our systematic calculations of angular momentum I ground state probabilities $P(I)$ of boson systems with spin l in the presence of random two-body interactions. It is found that the $P(0)$ dominance is usually not true for a system with an odd number of bosons, while it is valid for an even number of bosons, which indicates that the $P(0)$ dominance is partly connected to the even number of identical particles. It is also noticed that the ${P(I}_{\mathrm{max}})$ 's of bosons with spin l do not follow the $1/N$ $(N=l+1,$ referring to the number of independent two-body matrix elements) relation. The properties of the $P(I)$ 's obtained in boson systems with spin l are discussed.In this paper, we report our systematic calculations of angular momentum I ground state probabilities $P(I)$ of boson systems with spin l in the presence of random two-body interactions. It is found that the $P(0)$ dominance is usually not true for a system with an odd number of bosons, while it is valid for an even number of bosons, which indicates that the $P(0)$ dominance is partly connected to the even number of identical particles. It is also noticed that the ${P(I}_{\mathrm{max}})$'s of bosons with spin l do not follow the $1/N$ $(N=l+1,$ referring to the number of independent two-body matrix elements) relation. The properties of the $P(I)$'s obtained in boson systems with spin l are discussed.

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Eigenvalue correlations and the distribution of ground state angular momenta for random many-body quantum systems

José Barea, R. Bijker, Alejandro Frank

The observed preponderance of ground states with angular momentum $L=0$ in many-body quantum systems with random two-body interactions is analyzed in terms of correlation coefficients (covariances) among different eigenstates. It is shown that the geometric analysis of Chau et al. can be interpreted in terms of correlations (covariances) between energy eigenvalues, thus providing an entirely statistical explanation of the distribution of ground state angular momenta of randomly interacting quantum systems that, in principle, is valid for both fermionic and bosonic systems. The method is illustrated for the interacting boson model.The observed preponderance of ground states with angular momentum $L=0$ in many-body quantum systems with random two-body interactions is analyzed in terms of correlation coefficients (covariances) among different eigenstates. It is shown that the geometric analysis of Chau et al. can be interpreted in terms of correlations (covariances) between energy eigenvalues, thus providing an entirely statistical explanation of the distribution of ground state angular momenta of randomly interacting quantum systems that, in principle, is valid for both fermionic and bosonic systems. The method is illustrated for the interacting boson model.

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Naotaka Yoshinaga, A. Arima, Y. M. Zhao

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Towards understanding the probability of 0+ ground states in even-even many-body systems

Y. M. Zhao, A. Arima

For single- $j$ shells with $j=7/2,$ 9/2, and $11/2,$ we relate the large probability of ${I}^{+}$ ground states to the largest (smallest) coefficients ${\ensuremath{\alpha}}_{I(v\ensuremath{\beta})}^{J}=〈\mathrm{nv}\ensuremath{\beta}I|{A}^{J\ifmmode\dagger\else\textdagger\fi{}}\ensuremath{\cdot}{A}^{J}|\mathrm{nv}\ensuremath{\beta}I〉,$ where n is the particle number, $v$ is the seniority, \ensuremath{\beta} is an additional quantum number, and I is the angular momentum of the state. Interesting regularities of the probabilities of ${I}^{+}$ ground states are noticed and discussed for four-particle systems. Several counter examples of the ${0}^{+}$ ground state (0g.s.) predominance are noticed for the first time.For single-$j$ shells with $j=7/2,$ 9/2, and $11/2,$ we relate the large probability of ${I}^{+}$ ground states to the largest (smallest) coefficients ${\ensuremath{\alpha}}_{I(v\ensuremath{\beta})}^{J}=〈\mathrm{nv}\ensuremath{\beta}I|{A}^{J\ifmmode\dagger\else\textdagger\fi{}}\ensuremath{\cdot}{A}^{J}|\mathrm{nv}\ensuremath{\beta}I〉,$ where n is the particle number, $v$ is the seniority, \ensuremath{\beta} is an additional quantum number, and I is the angular momentum of the state. Interesting regularities of the probabilities of ${I}^{+}$ ground states are noticed and discussed for four-particle systems. Several counter examples of the ${0}^{+}$ ground state (0g.s.) predominance are noticed for the first time.

arXiv:nucl-th/0108052Physical Review CSep 4, 200130 citations

We propose a simple approach to predict the angular momentum I ground state

(I

j

and many-

j

shells), and boson (both

\mathrm{sd}

and

sdg)

In this paper we discuss in detail the

P(I)'\mathrm{s},

angular momentum I probabilities in the ground states, of many-body systems interacting via a two-body random ensemble (TBRE). In particular, we extensively apply an approach introduced in an earlier paper and compare the predicted

P(I)'\mathrm{s}

with those obtained by diagonalizing a TBRE Hamiltonian. We begin with a few solvable cases, such as fermions in a small single-

j

shell and d boson systems, where elegant agreements between the predicted

P(I)'\mathrm{s}

and those obtained by diagonalizing a TBRE Hamiltonian are achieved. We find that d boson systems systematically present counterexamples of angular momentum 0 ground state dominance when the number of d bosons is $6\ensuremath{\kappa}\ifmmode\pm\else\textpm\fi{}1$ with $\ensuremath{\kappa}$ a natural number, which suggests that certain fundamental symmetry (say, time reversal invariance) of the Hamiltonian cannot ensure the occurrence of angular momentum 0 ground state dominance. Next, we apply the same approach to more complicated cases, such as even or odd number of fermions in a large single-

j

shell or a many-

j

shell,

\mathrm{sd}

-boson or

\mathrm{sdg}

-boson systems, etc. We find that the simple approach proposed in an earlier paper is also well applicable, and thus it is a universal approach. The numerical experiments provide a guideline to tell which interactions are essential to produce a sizable

P(I)

in a many-body system. This disproves a popular idea that the angular momentum 0 ground state (0 g.s.) dominance may be independent of two-body interactions. Some matrix elements which are useful to understand the observed regularities are given or addressed in detail. In this paper we also report a synchronous staggering between the 0 g.s. probabilities of even numbers of fermions in a single-

j

shells and j g.s. probabilities of odd numbers of fermions in a single-

j

shell when j is small. The low seniority chain of 0 g.s. using the same set of two-body interactions is confirmed, but it is noted that contribution to the total 0 g.s. probability beyond this chain may be more important for even numbers of fermions in a single-

j

shell. Some interesting results taking a displaced two-body random ensemble are presented for the I ground state probabilities.

In this paper, we report our systematic calculations of angular momentum I ground state probabilities

P(I)

of boson systems with spin l in the presence of random two-body interactions. It is found that the

P(0)

dominance is usually not true for a system with an odd number of bosons, while it is valid for an even number of bosons, which indicates that the

P(0)

dominance is partly connected to the even number of identical particles. It is also noticed that the

{P(I}_{\mathrm{max}})

's of bosons with spin l do not follow the

1/N

(N=l+1,

referring to the number of independent two-body matrix elements) relation. The properties of the

P(I)

's obtained in boson systems with spin l are discussed.

The observed preponderance of ground states with angular momentum

L=0

in many-body quantum systems with random two-body interactions is analyzed in terms of correlation coefficients (covariances) among different eigenstates. It is shown that the geometric analysis of Chau et al. can be interpreted in terms of correlations (covariances) between energy eigenvalues, thus providing an entirely statistical explanation of the distribution of ground state angular momenta of randomly interacting quantum systems that, in principle, is valid for both fermionic and bosonic systems. The method is illustrated for the interacting boson model.

For single-

j

shells with

j=7/2,

9/2, and

11/2,

we relate the large probability of

{I}^{+}

ground states to the largest (smallest) coefficients ${\ensuremath{\alpha}}_{I(v\ensuremath{\beta})}^{J}=〈\mathrm{nv}\ensuremath{\beta}I|{A}^{J\ifmmode\dagger\else\textdagger\fi{}}\ensuremath{\cdot}{A}^{J}|\mathrm{nv}\ensuremath{\beta}I〉,$ where n is the particle number,

v

is the seniority, \ensuremath{\beta} is an additional quantum number, and I is the angular momentum of the state. Interesting regularities of the probabilities of

{I}^{+}

ground states are noticed and discussed for four-particle systems. Several counter examples of the

{0}^{+}

ground state (0g.s.) predominance are noticed for the first time.