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.
Predictions of the embedded Gaussian orthogonal random matrix ensemble of one plus two-body interactions, in the Gaussian domain, for the number of principal components and localization length, measuring complexity and chaos in transition strengths originating from an eigenstate with energy
E, are tested successfully for the first time for
E2 and
M1 transition strengths. Shell-model results for the
2p1f-shell nucleus
46V are used as the example.
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.
Finite interacting Fermi systems with a mean-field and a chaos generating two-body interaction are modeled by one plus two-body embedded Gaussian orthogonal ensemble of random matrices with spin degree of freedom [called EGOE(1+2)-s]. Numerical calculations are used to demonstrate that, as lambda , the strength of the interaction (measured in the units of the average spacing of the single-particle levels defining the mean-field), increases, generically there is Poisson to GOE transition in level fluctuations, Breit-Wigner to Gaussian transition in strength functions (also called local density of states) and also a duality region where information entropy will be the same in both the mean-field and interaction defined basis. Spin dependence of the transition points lambda_{c} , lambdaF, and lambdad , respectively, is described using the propagator for the spectral variances and the formula for the propagator is derived. We further establish that the duality region corresponds to a region of thermalization. For this purpose we compared the single-particle entropy defined by the occupancies of the single-particle orbitals with thermodynamic entropy and information entropy for various lambda values and they are very close to each other at lambda=lambdad.
There is a newly emerging understanding that in the chaotic domain of isolated finite interacting many particle systems smoothed densities define the statistical description of these systems and these densities follow from embedded (two-body) random matrix ensembles and their various deformations. These ensembles predict that the smoothed form of matrix elements of a transition operator between the chaotic eigenstates weighted by the densities at the two ends (i.e. the bivariate strength density) will be a bivariate Gaussian with the bivariate correlation coefficient arising out of the non-commutability of the hamiltonian and the transition operator involved. The ensemble theory extends to systems with a mean-field and a chaos generating two-body interaction (as in nuclei, atoms and diffusive quantum dots). These developments in many-body quantum chaos are described with special reference to one-body transition operators.