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arXiv:nucl-th/0206041 · Physical Review C

Many-body systems interacting via a two-body random ensemble. II. Average energy of each angular momentum

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

Published Dec 31, 2002• 15 citations
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Abstract

In this paper, we discuss the regularities of average energies with a fixed angular momentum I (denoted as ${E}_{I}'\mathrm{s})$ in many-body systems interacting via a two-body random ensemble. It is found that ${E}_{I}'\mathrm{s}$ with $I\ensuremath{\sim}{I}_{\mathrm{min}}$ (minimum of $I)$ or $I\ensuremath{\sim}{I}_{\mathrm{max}}$ (maximum of $I)$ have large probabilities [denoted as $\mathcal{P}(I)]$ to be the smallest in energy, and $\mathcal{P}(I)$ 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 ${E}_{I}$ vs $I(I+1)$ is found to be robust. Other regularities, such that there are two or three sizable $\mathcal{P}(I)'\mathrm{s}$ 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 $〈{E}_{I}{〉}_{\mathrm{min}}=CI(I+1)$ are sensitive to the orbits and not sensitive to particle number, etc., are discovered and studied for the first time.

DOI
10.1103/physrevc.66.064323

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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∼IminI\ensuremath{\sim}{I}_{\mathrm{min}}I\ensuremath∼Imin​ (minimum of I)I)I) or I\ensuremath∼ImaxI\ensuremath{\sim}{I}_{\mathrm{max}}I\ensuremath∼Imax​ (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∼IminI\ensuremath{\sim}{I}_{\mathrm{min}}I\ensuremath∼Imin​ but P(I)\ensuremath≪P(Imax)′s\mathcal{P}(I)\ensuremath{\ll}\mathcal{P}{(I}_{\mathrm{max}})'\mathrm{s}P(I)\ensuremath≪P(Imax​)′s with I\ensuremath∼Imax,I\ensuremath{\sim}{I}_{\mathrm{max}},I\ensuremath∼Imax​, 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.
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 $6\ensuremath{\kappa}\ifmmode\pm\else\textpm\fi{}1$ 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-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 study the behavior of energy centroids (denoted as EI‾\overline{{E}_{I}}EI​​) of spin I states in the presence of random two-body interactions, for systems ranging from very simple systems (e.g., single-j shell for very small j) to very complicated systems (e.g., many-j shells with different parities and with isospin degree of freedom). Regularities of EI‾\overline{{E}_{I}}EI​​'s discussed in terms of the so-called geometric chaoticity (or quasi-randomness of two-body coefficients of fractional parentage) in earlier works are found to hold even for very simple systems in which one cannot assume geometric chaoticity. It is shown that the inclusion of isospin and parity does not ``break'' the regularities of EI‾\overline{{E}_{I}}EI​​'s.
The observed preponderance of ground states with angular momentum L=0L=0L=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.
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.
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.