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.

翻译：通过推导最低六个矩的公式，研究了两物种（记为$\mathbf{\pi}$和$\mathbf{\nu}$）费米子系统中由$k$体相互作用嵌入高斯幺正系综（EGUE($k:\mathbf{\pi} \mathbf{\nu}$)）生成的本征值密度。在构造该系综时，假设$\mathbf{\pi}$费米子（数量为$m_1$）占据$N_1$个简并单粒子态，类似地$\mathbf{\nu}$费米子（数量为$m_2$）占据$N_2$个简并单粒子态。哈密顿量被假定为保持$(m_1,m_2)$不变的$k$体相互作用。包含有限$(N_1,N_2)$修正的公式及渐近极限公式均表明，本征值密度呈现$q$-正态形式，其中$q$参数由第四矩定义。EGUE($k:\mathbf{\pi} \mathbf{\nu}$)的形式体系与结果被推广至两物种玻色子系统。本文结果表明，近期仅在全同费米子与玻色子系统中建立的$q$-正态本征值密度形式，可推广至两物种费米子与玻色子系统。