Localization-Delocalization Transitions in Bosonic Random Matrix Ensembles

Abstract

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 $\pm 2\sigma$ to $\pm 3\sigma$ 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.

11 pages, 6 figures, Contribution to the Special Issue "Many-Body Localization" in Annalen der Physik

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• Statistical Nuclear Spectroscopy with $q$-normal and bivariate $q$-normal distributions and $q$-Hermite polynomials
• Thermalization in many-fermion quantum systems with one- plus random $k$-body interactions
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