Many-body Systems Interacting via a Two-body Random Ensemble: average energy of each angular momentum

In this paper, we discuss the regularities of energy of each angular momentum $I$ averaged over all the states for a fixed angular momentum (denoted as $\bar{E}_I$'s) in many-body systems interacting via a two-body random ensemble. It is found that $\bar{E}_I$'s with $I \sim I_{min}$ (minimum of $I$) or $I_{max}$ have large probabilities (denoted as ${\cal P}(I)$) to be the lowest, and that ${\cal P}(I)$ is close to zero elsewhere. A simple argument based on the randomness of the two-particle cfp's is given. A compact trajectory of the energy $\bar{E}_I$ vs. $I(I+1)$ is found to be robust. Regular fluctuations of the $P(I)$ (the probability of finding $I$ to be the ground state) and ${\cal P}(I)$ of even fermions in a single-$j$ shell and boson systems are found to be reverse, and argued by the dimension fluctuation of the model space. Other regularities, such as why there are 2 or 3 sizable ${\cal P}(I)$'s with $I\sim I_{min}$ and ${\cal P}(I) \ll {\cal P}(I_{max})$'s with $I\sim I_{max}$, why the coefficients $C$ defined by $<\bar{E}_I >$=$CI(I+1)$ is sensitive to the orbits and not sensitive to particle number, are found and studied for the first time.