Random matrix theory for transition strength densities in finite quantum systems: Results from embedded unitary ensembles

Abstract

Embedded random matrix ensembles are generic models for describing statistical properties of finite isolated interacting quantum many-particle systems. For the simplest spinless fermion (or boson) systems, with say m fermions (or bosons) in N single particle states and interacting via k-body interactions, we have EGUE(k) [embedded GUE of k-body interactions) with GUE embedding 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 (then the initial and final systems are same), nuclear beta and double beta decay (then the initial and final systems are different), particle addition to or removal from a given system and so on. Towards developing a complete statistical theory for transition strength densities (transition strengths multiplied by the density of states at the initial and final energies), we have derived formulas for the lower order bivariate moments of the strength densities generated by a variety of transition operators. Firstly, for a spinless fermion system, using EGUE(k) representation for a Hamiltonian that is k-body and an independent EGUE(t) representation for a transition operator that is t-body and employing the embedding U(N) algebra, finite-N formulas for moments up to order four are derived, for the first time, for the transition strength densities. Secondly, formulas for * Corresponding author. Phone: +91-79-26314464, Fax: +91-79-26314460Email addresses: vkbkota@prl.res.in (V.K.B. Kota), manan@fis.unam.mx (Manan Vyas) April 6, 2015 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. In addition, moments formulas are also derived for a transition operator that removes k 0 number of particles from a system of m spinless fermions. In the dilute limit, these formulas are shown to reduce to those for the EGOE version derived using the asymptotic limit theory of Mon and French [Ann. Phys. (N.Y.) 95 (1975) 90]. Numerical results obtained using the exact formulas for two-body (k = 2) Hamiltonians (in some examples for k = 3 and 4) and the asymptotic formulas clearly establish that in general the smoothed (with respect to energy) 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. Preprint submitted to Annals of Physics

Random matrix theory for transition strengths: Applications and open questions

“…ζ = 0 implies GOE representation and from many numerical examples we have ζ ∼ 0.6 − 0.9 implying EGOE representation [29,26]. Also, large ζ value implies that the strength distribution will be in a narrow (E i , E F ) region.…”

“…For nuclei of interest, using the numerical results in [26], it is seen that ζ ∼ 0.6 − 0.8. In the applications presented in the next Section, ζ is used as a parameter.…”

“…by deriving formulas for the bivariate moments M PQ = O † H Q OH P , and thereby for the bivariate cumulants k rs with r + s = 2, 4, with independent EGUE representations for H and O operators [26]. This independence implies that we are removing the H − O correlated part from the transition operator O.…”

“…Secondly, for the correlation coefficient ζ, there is not yet any valid form involving configurations. Therefore, the only plausible way forward currently is to estimate ζ as a function of (m p , m n ) using random matrix theory given in [26]. Then, the definition of ζ is…”

“…(6) are approximated, following the random matrix theory results in [26], to the corresponding state density centroids and variances giving,…”