Bivariate q -normal distribution for transition matrix elements in quantum many-body systems

Abstract

Recently, it was established, via lower order moments, that the univariate q-normal distribution, which is the weighting function for q-Hermite polynomials, describes the ensemble-averaged eigenvalue density from many-particle random matrix ensembles generated by k-body interactions (Vyas and Kota 2019 J. Stat. Mech. 103103). These ensembles are generically called embedded ensembles of k-body interactions [EE(k)] and their Gaussian orthogonal ensemble (GOE) and Gaussian unitary ensemble (GUE) versions are called embedded Gaussian orthogonal ensemble (EGOE), EGOE(k), and embedded Gaussian unitary ensemble (EGUE), EGUE(k), respectively. Going beyond the earlier work, this paper examines the lower-order bivariate reduced moments of the distribution of the square of the transition matrix elements (usually called transition strengths) generated by the action of a transition operator O (such as a dipole operator in atoms and molecules, a quadrupole operator in atomic nucleic, etc) on the eigenstates generated by a k-body Hamiltonian H(k). To this end, H is represented by EGOE(k) [or EGUE(t)] and O, that is to say, a t-body, by an independent EGOE(t) [or EGUE(t)]. Given this, it is shown that the ensembleaveraged distribution of transition strengths follows a bivariate q-normal, giving a first application of the bivariate q-normal distribution in the statistical physics of quantum systems. Also presented are the formulas for the bivariate correlation coefficient ρ and the q value, that define a bivariate q-normal as a function of

Statistical Nuclear Spectroscopy with $q$-normal and bivariate $q$-normal distributions and $q$-Hermite polynomials

“…Remarkably, it is seen that the lower order moments (up to 8th order) of the eigenvalue density generated by EE(k) are essentially identical to the lower order moments given by q-normal distribution [51] with the fourth moment determining the value of the q parameter. Similarly, it is shown that the lower order bivariate moments (verified up to order 6) of the transition strength densities (see Section V for definition) generated by EE are indeed essentially same as those of the bivariate qnormal distribution [52]. Going further, it is also seen that the lower order moments of strength functions or local density of states (also called partial densities) generated by EE are close to those from the conditional q-normal distribution [53].…”

“…Going further, it is also seen that the lower order moments of strength functions or local density of states (also called partial densities) generated by EE are close to those from the conditional q-normal distribution [53]. All these results are also supported by several numerical calculations using EE for both fermion and boson systems; see [51][52][53][54]. An important property of the q distributions is that they are bounded unlike a Gaussian.…”

“…with the centroid linear in y and variance independent of y. In addition, we have the following formulas for the skewness and excess [49,52],…”

“…Representing H by a EGOE(k) and O by another independent EGOE(t), it is shown recently that I O (E i , E f ) is close to a q-bivariate normal distribution (in all past applications bivariate Gaussian form is used [10,16,38]). Although this is established in [52] using transition operators that are t-body in nature (then initial and final spaces are same), from the bivariate cumulants derived in [72] for β decay and double β decay type operators and particle transfer operators, it can be argued that the bivariate q-normal form applies in general. Results in the Tables 2, 3 and 4 in [72] show that the cumulants k rs ≈ k sr (cumulants are shape parameters and related in a simple manner to the moments µ rs [56]) as needed for a symmetrical distribution like bivariate q-normal; see Eq.…”

“…(11). With H represented by EGOE(k) and O a t-body operator represented by an independent EGOE(t), the bivariate correlation coefficient ξ is given by [52,72] (appropriate for identical nucleons),…”