The nuclear shell model as a testing ground for many-body quantum chaos

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Atomic nuclei analyzed in the framework of the shell model provide a good example of a many-body quantum system with strong interactions between its constituents. As excitation energy and level density increase, the system evolves in the direction of very complicated (“stochastic”) dynamics. Energy levels and stationary wave functions obtained in realistic shell-model calculations are studied from the viewpoint of signatures of quantum chaos and complexity. The standard characteristics of local level statistics, such as nearest level spacing distribution or spectral rigidity, manifest chaoticity which agrees with the GOE predictions. Going beyond that, we analyze the structure of the eigenfunctions and the distribution function of the eigenvector components using basis-dependent quantitative criteria such as information entropy. The degree of complexity is shown to be a smooth function of excitation energy. The representation dependence provides additional physical information on the interrelation between the eigenbasis and the representation basis. The exceptional role of the mean field basis is discussed. The spreading width and the shape of the strength function of the original simple states are also studied. The generic wave functions in the chaotic region have similar observable properties which can be characterized by the average single-particle occupation numbers. Agreement with the Fermi-Dirac distribution manifests the correspondence between chaotic dynamics and thermalization. The information entropy in the mean field basis gives an equivalent temperature scale which confirms this correspondence. Pairing correlations display a phase transition to the normal state with a long tail of fluctuational enhancement above the level expected for a heated Fermi gas.