Random Matrix Theory for Transition Strength Densities in Finite Quantum Systems: Results from Embedded Unitary Ensembles

Embedded random matrix ensembles are generic models for describing statistical properties of finite isolated interacting quantum many-particle systems. For the simplest spinless systems, with say $m$ particles in $N$ single particle states and interacting via $k$-body interactions, we have EGUE($k$) and the embedding algebra is $U(N)$. A finite quantum system, induced by a transition operator, makes transitions from its states to the states of the same system or to those of another system. Examples are electromagnetic transitions (same initial and final systems), nuclear beta and double beta decay (different initial and final systems), particle addition to/removal from a given system and so on. Towards developing a complete statistical theory for transition strength densities, we have derived formulas for lower order bivariate moments of the strength densities generated by a variety of transition operators. For a spinless fermion system, using EGUE($k$) representation for Hamiltonian and an independent EGUE($t$) representation for transition operator, finite-$N$ formulas for moments up to order four are derived, for the first time, for the transition strength densities. Formulas for the moments up to order four are also derived for systems with two types of spinless fermions and a transition operator similar to beta decay and neutrinoless beta decay operators. Moments formulas are also derived for transition operator that removes $k_0$ number of particles from $m$ fermion system. Numerical results obtained using the exact formulas for two-body ($k=2$) Hamiltonians (in some examples for $k=3,4$) and the asymptotic formulas clearly establish that in general the smoothed form of the bivariate transition strength densities take bivariate Gaussian form for isolated finite quantum systems. Extensions of these results to bosonic systems and EGUE ensembles with further symmetries are discussed.