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The ground states of all even-even nuclei have angular momentum, I, equal to zero, I = 0, and positive parity, π = +. This feature was believed to be a consequence of the attractive short-range interaction between nucleons. However, in the presence of two-body random interactions, the predominance of I π = 0 + ground states (0 g.s.) was found to be robust both for bosons and for an even number of fermions. For simple systems, such as d bosons, sp bosons, sd bosons, and a few fermions in single-j shells for small j, there are a few approaches to predict and/or explain spin I ground state (I g.s.) probabilities. An empirical approach to predict I g.s. probabilities is available for general cases, such as fermions in a single-j (j > 7/2) or many-j shells and various boson systems, but a more fundamental understanding of the robustness of 0 g.s. dominance is still out of reach. Further interesting results are also reviewed concerning other robust phenomena of manybody systems in the presence of random two-body interactions, such as the odd-even staggering of binding energies, generic collectivity, the behavior of average energies, correlations, and regularities of many-body systems interacting by a displaced twobody random ensemble.

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In this paper we investigate the possible I 4 dependence [I = (N − Z)/A] in "experimental" symmetry energy extracted from nuclear masses. Our results show that the inclusion of the I 4 term in mass formulas leads to sizable changes for symmetry energy coefficients of the I 2 terms.

We propose a simple approach to predict the angular momentum I ground state (Ig.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 0g.s. dominance in boson systems and in even-fermion systems is given by two-body interactions with specific features.PACS number: 05.30. Fk, 21.60Cs, 24.60.Lz Typeset using REVT E X 1

We investigate Hamiltonians with attractive interactions between pairs of fermions coupled to angular momentum J. We show that pairs with spin J are reasonable building blocks for the low-lying states. For systems with only a J = J max pairing interaction, eigenvalues are found to be approximately integers for a large array of states, in particular for those with total angular momenta I ≤ 2j. For I = 0 eigenstates of four fermions in a single-j shell we show that there is only one non-zero eigenvalue. We address these observations using the nucleon pair approximation of the shell model and relate our results with a number of currently interesting problems.