Random matrix ensembles defined by a mean-field one body plus a chaos generating random two-body interaction [called embedded ensembles of (1+2)-body interactions] predict for wave functions, in the chaotic domain, an essentially one-parameter Gaussian forms for the energy dependence of the number of principal components (NPC) and the localization length l(H) (defined by information entropy), which are two important measures of chaos in finite interacting many-particle systems. Numerical embedded ensemble calculations and nuclear shell-model results, for NPC and l(H), are compared with the theory. These analyses clearly point out that for realistic finite interacting many-particle systems, in the chaotic domain, wave-function structure is given by (1+2)-body embedded random matrix ensembles.
Shell-model results for the electric quadrupole
(E2), magnetic dipole
(M1), and Gamow-Teller transition strength sums and occupation numbers are calculated using different valence spaces and compared to the embedded Gaussian orthogonal ensemble (EGOE) predictions. It is shown that transition strength sums can be considered as a new statistic able to distinguish between regular and chaotic motion. Moreover it is established that the EGOE (but not the GOE) provides the good description of the shell-model strength sums in the chaotic domain. To get this result we have studied the behavior of strength sums in order to chaos transitions generated by means of a family of Hamiltonians $H(\ensuremath{\lambda})=h(1)+\ensuremath{\lambda}V(2),$ built from realistic one- and two-body interactions.
Random matrix ensembles defined by a mean-field one-body plus a chaos generating random two-body interaction (called embedded Gaussian orthogonal ensembles of (1+2)-body interactions[EGOE(1+2)]) predict for the entropy defined by the occupation numbers of single-particle states, in the chaotic domain, an essentially one parameter Gaussian form for their energy dependence. Numerical embedded ensemble calculations are compared with the theory. In addition, it is shown that the single-particle entropy, thermodynamic entropy defined by the state density and information entropy defined by wave functions in the mean-field basis for EGOE(1+2) describe the results known for interacting Fermi systems such as those obtained from nuclear shell model.
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.
Employing hamiltonians defined by two-body embedded Gaussian orthogonal ensemble of random matrices(EGOE(2)) plus a mean-field producing one-body part, strength functions (for states defined by the one-body part) are constructed for various values of the strength of the chaos generating two-body part. Numerical calculations for six and seven fermion systems clearly demonstrate Breit-Wigner to Gaussian transition, in the chaotic domain, in strength functions as found earlier in nuclear shell model and Lipkin-Meshkov-Glick model calculations.
Interacting many-particle systems with a mean-field one-body part plus a chaos generating random two-body interaction having strength lambda exhibit Poisson to Gaussian orthogonal ensemble and Breit-Wigner (BW) to Gaussian transitions in level fluctuations and strength functions with transition points marked by lambda = lambda c and lambda = lambda F, respectively; lambda F >> lambda c. For these systems a theory for the matrix elements of one-body transition operators is available, as valid in the Gaussian domain, with lambda > lambda F, in terms of orbital occupation numbers, level densities, and an integral involving a bivariate Gaussian in the initial and final energies. Here we show that, using a bivariate-t distribution, the theory extends below from the Gaussian regime to the BW regime up to lambda = lambda c. This is well tested in numerical calculations for 6 spinless fermions in 12 single-particle states.