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arXiv:nucl-th/0206040 · Physical Review C

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

Published Dec 31, 2002• 20 citations
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Abstract

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.

DOI
10.1103/physrevc.66.064322

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In this paper we discuss in detail the P(I)′s,P(I)'\mathrm{s},P(I)′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)′sP(I)'\mathrm{s}P(I)′s with those obtained by diagonalizing a TBRE Hamiltonian. We begin with a few solvable cases, such as fermions in a small single-jjj shell and d boson systems, where elegant agreements between the predicted P(I)′sP(I)'\mathrm{s}P(I)′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\ensuremathκ\ifmmode±\else\textpm\fi16\ensuremath{\kappa}\ifmmode\pm\else\textpm\fi{}16\ensuremathκ\ifmmode±\else\textpm\fi1 with \ensuremathκ\ensuremath{\kappa}\ensuremathκ 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-jjj shell or a many-jjj shell, sd\mathrm{sd}sd-boson or sdg\mathrm{sdg}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)P(I)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-jjj shells and j g.s. probabilities of odd numbers of fermions in a single-jjj 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-jjj 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(I(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-jjj and many-jjj shells), and boson (both sd\mathrm{sd}sd and sdg)sdg)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.
In this paper, we report our systematic calculations of angular momentum I ground state probabilities P(I)P(I)P(I) of boson systems with spin l in the presence of random two-body interactions. It is found that the P(0)P(0)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)P(0)P(0) dominance is partly connected to the even number of identical particles. It is also noticed that the P(Imax){P(I}_{\mathrm{max}})P(Imax​)'s of bosons with spin l do not follow the 1/N1/N1/N (N=l+1,(N=l+1,(N=l+1, referring to the number of independent two-body matrix elements) relation. The properties of the P(I)P(I)P(I)'s obtained in boson systems with spin l are discussed.
In this paper, we discuss the regularities of average energies with a fixed angular momentum I (denoted as EI′s){E}_{I}'\mathrm{s})EI′​s) in many-body systems interacting via a two-body random ensemble. It is found that EI′s{E}_{I}'\mathrm{s}EI′​s with $I\ensuremath{\sim}{I}_{\mathrm{min}}$ (minimum of I)I)I) or $I\ensuremath{\sim}{I}_{\mathrm{max}}$ (maximum of I)I)I) have large probabilities [denoted as P(I)]\mathcal{P}(I)]P(I)] to be the smallest in energy, and P(I)\mathcal{P}(I)P(I) is close to zero elsewhere. A simple argument assuming the randomness of the two-particle coefficients of fractional parentage is given to explain these observations. A compact trajectory of the energy EI{E}_{I}EI​ vs I(I+1)I(I+1)I(I+1) is found to be robust. Other regularities, such that there are two or three sizable P(I)′s\mathcal{P}(I)'\mathrm{s}P(I)′s with $I\ensuremath{\sim}{I}_{\mathrm{min}}$ but $\mathcal{P}(I)\ensuremath{\ll}\mathcal{P}{(I}_{\mathrm{max}})'\mathrm{s}$ with $I\ensuremath{\sim}{I}_{\mathrm{max}},$ and that the coefficients C defined by 〈EI〉min=CI(I+1)〈{E}_{I}{〉}_{\mathrm{min}}=CI(I+1)〈EI​〉min​=CI(I+1) are sensitive to the orbits and not sensitive to particle number, etc., are discovered and studied for the first time.
We apply a two-body random ensemble to a hypothetical system of spinless fermions occupying a one-dimensional rotationally invariant orbit in order to study the statistical tendencies of spins of ground states for a random two-body interaction. We also investigate the origin of the dominance of spin-zero ground states of a single-j shell in the nuclear shell model, which can be interpreted geometrically as a particular case of the simple hypothetical model we consider. Assuming a specific symmetry, or the doublet structure due to time-reversal symmetry, it is found that the ensemble has a preference for a ground state whose spin is zero. We demonstrate how the time-reversal symmetry and its doublet structure affect this ground state tendency.
For single-jjj shells with j=7/2,j=7/2,j=7/2, 9/2, and 11/2,11/2,11/2, we relate the large probability of I+{I}^{+}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, vvv 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+{I}^{+}I+ ground states are noticed and discussed for four-particle systems. Several counter examples of the 0+{0}^{+}0+ ground state (0g.s.) predominance are noticed for the first time.