Nuclei at low excitation energy exhibit such well-ordered behavior, such as rotational and vibrational bands as well as pairing gaps, that they are commonly described with algebraic models; but does such order originate in the details of the nuclear force, or are they emergent properties of dense fermion systems? To answer this, I consider ensembles of random two-body shell-model interactions and examine the low-lying spectra and wavefunctions. Some striking regularities emerge: in particular, pairing appears to be a ubiquitous phenomenon, even in the absence of explicit pairing interactions! These results are remarkably insensitive to the detailed definition of the ensemble.
Embedded random matrix ensembles generated by random interactions (of low body rank and usually two-body) in the presence of a one-body mean field, introduced in nuclear structure physics, are now established to be indispensable in describing statistical properties of a large number of isolated finite quantum many-particle systems. Lie algebra symmetries of the interactions, as identified from nuclear shell model and the interacting boson model, led to the introduction of a variety of embedded ensembles (EEs). These ensembles with a mean field and chaos generating two-body interaction generate in three different stages, delocalization of wave functions in the Fock space of the mean-field basis states. The last stage corresponds to what one may call thermalization and complex nuclei, as seen from many shell model calculations, lie in this region. Besides briefly describing them, their recent applications to nuclear structure are presented and they are (i) nuclear level densities with interactions; (ii) orbit occupancies; (iii) neutrinoless double beta decay nuclear transition matrix elements as transition strengths. In addition, their applications are also presented briefly that go beyond nuclear structure and they are (i) fidelity, decoherence, entanglement and thermalization in isolated finite quantum systems with interactions; (ii) quantum transport in disordered networks connected by many-body interactions with centrosymmetry; (iii) semicircle to Gaussian transition in eigenvalue densities with [Formula: see text]-body random interactions and its relation to the Sachdev–Ye–Kitaev (SYK) model for majorana fermions.
Embedded random matrix ensembles operating in nuclear shell model spaces, with nucleons occupying a finite set of single particle orbits and interacting via a two-body interaction, form the basis for statistical shell model. With sufficiently strong interaction, the level densities in shell model spaces take close to a Gaussian form and transition strength distributions close to a bivariate Gaussian form. In practice, partitioning via spherical configurations (
m~) and angular momentum
J (also isospin where appropriate) are essential. The resulting statistical spectroscopy or statistical shell model was applied successfully in the past in some studies of nuclear level densities, orbit occupancies,
β-decay matrix elements and so on. Going beyond these, recently it is recognized that embedded ensembles, in a better approximation, generate in-fact
q-normal form (
q=1 gives Gaussian and
q=0 Wigner's semi-circle) for density of eigenvalues, bivariate
q-normal form for transition strengths and conditional
q-normal form for strength functions. These then allow us to develop statistical shell model with
q-normal forms. These new developments in embedded ensembles and statistical shell model are briefly reviewed in this paper. Also described, using some examples, is the role of the
q parameter in generating statistical properties of general quantum many-particle systems.