Two species $k$-body embedded Gaussian unitary ensembles: $q$-normal form of the eigenvalue density

Eigenvalue density generated by embedded Gaussian unitary ensemble with $k$-body interactions for two species (say $\mathbf{\pi}$ and $\mathbf{\nu}$) fermion systems is investigated by deriving formulas for the lowest six moments. Assumed in constructing this ensemble, called EGUE($k:\mathbf{\pi} \mathbf{\nu}$), is that the $\mathbf{\pi}$ fermions ($m_1$ in number) occupy $N_1$ number of degenerate single particle (sp) states and similarly $\mathbf{\nu}$ fermions ($m_2$ in number) in $N_2$ number of degenerate sp states. The Hamiltonian is assumed to be $k$-body preserving $(m_1,m_2)$. Formulas with finite $(N_1,N_2)$ corrections and asymptotic limit formulas both show that the eigenvalue density takes $q$-normal form with the $q$ parameter defined by the fourth moment. The EGUE($k:\mathbf{\pi} \mathbf{\nu}$) formalism and results are extended to two species boson systems. Results in this work show that the $q$-normal form of the eigenvalue density established only recently for identical fermion and boson systems extends to two species fermion and boson systems.