We propose a simple approach to predict the angular momentum I ground state
(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-
j and many-
j shells), and boson (both
sd and
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 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 $6\ensuremath{\kappa}\ifmmode\pm\else\textpm\fi{}1$ 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,
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.
We present our results on properties of ground states for nucleonic systems in the presence of random two-body interactions. In particular, we calculate probability distributions for parity, seniority, spectroscopic (i.e., in the laboratory frame) quadrupole moments, and discuss $\ensuremath{\alpha}$ clustering in the ground states. We find that the probability distribution for the parity of the ground states obtained by a two-body random ensemble simulates that of realistic nuclei with $A\ensuremath{\geqslant}70$: positive parity is dominant in the ground states of even-even nuclei, while for odd-odd nuclei and odd-mass nuclei we obtain with almost equal probability ground states with positive and negative parity. In addition, assuming pure random interactions, we find that, for the ground states, low seniority is not favored, no dominance of positive values of spectroscopic quadrupole deformation is observed, and there is no sign of $\ensuremath{\alpha}$-clustering correlation, all in sharp contrast to realistic nuclei. Considering a mixture of a random and a realistic interaction, we observe a second-order phase transition for the $\ensuremath{\alpha}$-clustering correlation probability.
The observed preponderance of ground states with angular momentum
L=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.
The problem of random interactions leading to regular spectra in shell model type simulations is described. The key results are reviewed alnog with a selection of the explanations. A model system of N particles on 2 spin levels having random 2-body collisions that conserve angular momentum is examined. Preliminary results are described, including the ground state spin distributions peaked at extreme values of angular momentum, signatures of rotational bands, and smooth parabolic yrast lines. A simple random matrix theory analysis shows signatures of quantum chaos in the level spacing distribution and the Δ3 statistic.