Physical Review C

Spin I ground state probabilities of integrable systems under random interactions

Yi Lu, Y. M. Zhao, A. Arima

Published Feb 9, 2015• 3 citations

Abstract

In this paper we revisit spin $I$ ground state probabilities (denoted by $P(I))$ of a few integrable systems under random interactions. We obtain explicit and simple formulas of $P(I)$ for $d$-boson systems and fermions in a $j=5/2$ shell.

DOI: 10.1103/physrevc.91.027301

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Physical Review C

Spin I ground state probabilities of integrable systems under random interactions

Yi Lu, Y. M. Zhao, A. Arima

Published Feb 9, 2015• 3 citations

Publisher

Abstract

In this paper we revisit spin

I

ground state probabilities (denoted by

P(I))

of a few integrable systems under random interactions. We obtain explicit and simple formulas of

P(I)

for

d

-boson systems and fermions in a

j=5/2

shell.

DOI: 10.1103/physrevc.91.027301

Related papers

Angular momentum distribution of the ground states in the presence of random interactions: Boson systems

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

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

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Akito Arima

In this talk I shall discuss some regularities of many‐body systems in the presence of random interactions and regularities of a single‐j shell for the J pairing interaction which works only when two particles are coupled to spin J. I shall first explain an empirical rule to predict the spin I ground state probability. Then I shall present some interesting results of a single‐j shell under the J pairing interaction. Last I shall discuss some preliminary results of binding energies in the presence of random two‐body interactions.In this talk I shall discuss some regularities of many‐body systems in the presence of random interactions and regularities of a single‐j shell for the J pairing interaction which works only when two particles are coupled to spin J. I shall first explain an empirical rule to predict the spin I ground state probability. Then I shall present some interesting results of a single‐j shell under the J pairing interaction. Last I shall discuss some preliminary results of binding energies in the presence of random two‐body interactions.

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In this paper we revisit spin

I

ground state probabilities (denoted by

P(I))

of a few integrable systems under random interactions. We obtain explicit and simple formulas of

P(I)

for

d

-boson systems and fermions in a

j=5/2

shell.

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 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.

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.