Many-body systems interacting via a two-body random ensemble. I. Angular momentum distribution in the ground states

In this paper we discuss in detail the 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)'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)'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 6kappa+/-1 with 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, sd-boson or 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.