Embedded Gaussian Unitary Ensembles with $U(\Omega) \otimes SU(r)$ Embedding generated by Random Two-body Interactions with $SU(r)$ Symmetry

Following the earlier studies on embedded unitary ensembles generated by random two-body interactions [EGUE(2)] with spin SU(2) and spin-isospin SU(4) symmetries, developed is a general formulation, for deriving lower order moments of the one- and two-point correlation functions in eigenvalues, that is valid for any EGUE(2) and BEGUE(2) ('B' stands for bosons) with $U(\Omega) \otimes SU(r)$ embedding and with two-body interactions preserving $SU(r)$ symmetry. Using this formulation with $r=1$, we recover the results derived by Asaga et al [Ann. Phys. (N.Y.) 297, 344 (2002)] for spinless boson systems. Going further, new results are obtained for $r=2$ (this corresponds to two species boson systems) and $r=3$ (this corresponds to spin 1 boson systems).