Random Interactions and Coherent Nuclear Structure

The interplay of regular and chaotic elements in quantum many-body dynamics was extensively studied in the framework of random matrix theory and in realistic models of atoms, nuclei, condensed matter and quantum fields. The existence in finite systems of exact conservation laws, such as angular momentum, parity or isospin, raises new questions, for instance, how quantum chaos is influenced by these indestructible symmetries, and what are (if any) the correlations between the blocks of states with different exact quantum numbers governed by the same Hamiltonian. The nuclear shell model with the effective two-body forces in a restricted Hilbert space is the best available theoretical tool for calculating the properties of the low-lying states. Recently [1,2] the low-lying spectra were studied with the shell-model techniques but using, instead of effective interactions, randomly generated (but rotationally-invariant) two-body matrix elements. Some of the results resemble the pattern of actual nuclear spectra. One particularly interesting observation was that of predominance of spin J = 0 in the ground state in spite of the low statistical weight of states with J = 0 in Hilbert space. This result is robust and insensitive to precise statistical properties of the random interaction. A simple mechanism of random coupling of individual particle spins was suggested in Ref. [3] to explain the preponderance of J = 0 (and, in some cases, J = Jmax) in the ground state. In average the yrast-line in a randomly interacting fermionic system acquires a random sign of the effective moment of inertia which leads to the large probabilities of the edge values of the total spin. Our studies of the wave functions with realistic and random interactions show that the overlap of the 0+ ground state wave functions generated by random interactions with those obtained for realistic interactions is very small. Also the associated transition probabilities B(E2) to the 21 state are low. The implication is that the order which is present in actual nuclear states is almost entirely due to the coherent (non-random) aspects of the nuclear Hamiltonian. We need to stress that here we look for the signatures of the coherent phenomena not in the spin ordering which might be a consequence of geometrical constraints, but in the collectivity of the wave functions. As a generic system we take that of eight particles in the sd-shell, the case corresponding to the 24Mg nucleus. The geometry of the system is much richer than that of the schematic single-j model studied in Ref. [3]; it includes also isospin variables. In the sd model space there are 63 independent matrix elements under constraints of rotational and isospin invariance. We use two interactions, one from Ref. [4] and SDPOTA from Ref. [5], denoted below as (W ) and (P ), respectively. Being based on different approaches [a fit of individual matrix elements (W ) and a fit of a potential (P )], these sets agree in predicting the ground state with quantum numbers JπT = 0+0. Our conclusions are essentially the same for both realistic interactions; their ground state wave functions overlap by 98%. Earlier it was suggested [2] that the wave functions generated by the random interactions carry significant pairing correlations. Having this conjecture in mind, four models of random interactions were considered: