From Noether's theorem we know symmetries lead to conservation laws. What is left to nature is the ordering of conserved quantities; for example, the quantum numbers of the ground state. In physical systems the ground state is generally associated with 'low' quantum numbers and symmetric, low-dimensional irreps, but there is no a priori reason to expect this. By constructing random matrices with nontrivial point-group symmetries, I find the ground state is always dominated by extremal low-dimensional irreps. Going further, I suggest this explains the dominance of J = 0 g.s. even for random two-body interactions.
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.
Regular structures generated by random interactions in energy centroids defined over irreducible representations (irreps) of some of the group symmetries of the interacting boson models sdIBM, sdgIBM, sdIBM-T, and sdIBM-ST are studied by deriving trace propagation equations for the centroids. It is found that, with random interactions, the lowest and highest group irreps in general carry most of the probability for the corresponding centroids to be lowest in energy. This generalizes the result known earlier, via numerical diagonalization, for the more complicated fixed spin J centroids where simple trace propagation is not possible.
Regular structures generated by random interactions in energy centroids defined over irreducible representations (irreps) of some of the group symmetries of the interacting boson models sdIBM, sdgIBM, sdIBM-T and sdIBM-ST are studied by deriving trace propagations equations for the centroids.It is found that, with random interactions, the lowest and highest group irreps in general carry most of the probability for the corresponding centroids to be lowest in energy.This generalizes the result known earlier, via numerical diagonalization, for the more complicated fixed spin (J) centroids where simple trace propagation is not possible.