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

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.