Localization-Delocalization Transitions in Bosonic Random Matrix Ensembles

Localization to delocalization transitions in eigenfunctions are studied for finite interacting boson systems by employing one- plus two-body embedded Gaussian orthogonal ensemble of random matrices [EGOE(1+2)]. In the first analysis, considered are bosonic EGOE(1+2) for two-species boson systems with a fictitious ($F$) spin degree of freedom [called BEGOE(1+2)-$F$]. Numerical calculations are carried out as a function of the two-body interaction strength ($\lambda$). It is shown that, in the region (defined by $\lambda>\lambda_c$) after the onset of Poisson to GOE transition in energy levels, the strength functions exhibit Breit-Wigner to Gaussian transition for $\lambda>\lambda_{F_k}>\lambda_c$. Further, analyzing information entropy and participation ratio, it is established that there is a region defined by $\lambda\sim\lambda_t$ where the system exhibits thermalization. The $F$-spin dependence of the transition markers $\lambda_{F_k}$ and $\lambda_t$ follow from the propagator for the spectral variances. These results, well tested near the center of the spectrum and extend to the region within $\pm2\sigma$ to $\pm3\sigma$ from the center ($\sigma^2$ is the spectral variance), establish universality of the transitions generated by embedded ensembles. In the second analysis, entanglement entropy is studied for spin-less BEGOE(1+2) ensemble and shown that the results generated are close to the recently reported results for a Bose-Hubbard model.