Bivariate $t$-distribution for transition matrix elements in Breit-Wigner to Gaussian domains of interacting particle systems

Abstract

Interacting many-particle systems with a mean-field one body part plus a chaos generating random two-body interaction having strength $λ$, exhibit Poisson to GOE and Breit-Wigner (BW) to Gaussian transitions in level fluctuations and strength functions with transition points marked by $λ=λ_c$ and $λ=λ_F$, respectively; $λ_F >> λ_c$. For these systems theory for matrix elements of one-body transition operators is available, as valid in the Gaussian domain, with $λ> λ_F$, in terms of orbitals occupation numbers, level densities and an integral involving a bivariate Gaussian in the initial and final energies. Here we show that, using bivariate $t$-distribution, the theory extends below from the Gaussian regime to the BW regime up to $λ=λ_c$. This is well tested in numerical calculations for six spinless fermions in twelve single particle states.