Matrix elements of many-body operators and density correlations (11 pages)

Maksym Serbyn, Zlatko Papić, Dmitry A. Abanin

James P. Finley

Christian Brouder

The matrix element of a general many-body operator between two Slater determinants is calculated explicitly. For this, a split-and-pair method is introduced that provides a convenient expression of Wick's theorem and simplifies many-body calculations. The same method is used to determine the generating function of the matrix elements of many-body operators. The split-and-pair method allows also for the diagonalization of the density correlation operators $:n({x}_{1})\ensuremath{\cdots}n({x}_{k}):$, where $n(x)={\ensuremath{\psi}}^{\ifmmode\dagger\else\textdagger\fi{}}(x)\ensuremath{\psi}(x)$ is the density operator. The relation between the split-and-pair method and quantum group theory is clarified.

Takafumi Kita

V. K. B. Kota, R. Sahu

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.

V. K. B. Kota, R. Sahu

Demonstrating the equivalence between the recent theory of Flambaum and collaborators which is based on smoothed strength functions, with the much earlier formulation due to French and collaborators which is based on embedded random matrix ensembles and smoothed transition strength densities, we derive a theory for matrix elements of one-body transition operators in the quantum chaotic domain of isolated finite interacting particle systems with a mean-field and a chaos generating two-body interaction (V). The role of the bivariate correlation coefficient (zeta) arising out of the noncommutability of V and the transition operator (in the theory of Flambaum et al., zeta=0) is tested in numerical embedded ensemble calculations with a one- plus two-body Hamiltonian generating order-chaos transitions.