Eigenstate structure in many-body bosonic systems: Analysis using random matrices and $q$-Hermite polynomials

We analyze the structure of eigenstates in many-body bosonic systems by modeling the Hamiltonian of these complex systems using Bosonic Embedded Gaussian Orthogonal Ensembles (BEGOE) defined by a mean-field plus $k$-body random interactions. The quantities employed are the number of principal components (NPC), the localization length ($l_H$) and the entropy production $S(t)$. The numerical results are compared with the analytical formulas obtained using random matrices which are based on bivariate $q$-Hermite polynomials for local density of states $F_k(E|q)$ and the bivariate $q$-Hermite polynomial form for bivariate eigenvalue density $\rho_{biv:q}(E,E_k)$ that are valid in the strong interaction domain. We also compare transport efficiency in many-body bosonic systems using BEGOE in absence and presence of centrosymmetry. It is seen that the centrosymmetry enhances quantum efficiency.