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，研究了由两物种（记为$\mathbf{\pi}$和$\mathbf{\nu}$）费米子系统的$k$体相互作用生成的嵌入高斯幺正系综的本征值密度。构建该系综（称为EGUE($k:\mathbf{\pi} \mathbf{\nu}$)）的假设是：$\mathbf{\pi}$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。哈密顿量假定为保持$(m_1,m_2)$不变的$k$体算符。包含有限$(N_1,N_2)$修正的公式以及渐近极限公式均表明，本征值密度呈$q$-正态形式，其中$q$参数由第四矩定义。将EGUE($k:\mathbf{\pi} \mathbf{\nu}$)的形式体系与结果推广至两物种玻色子系统。本研究结果表明，近期才在同类费米子和玻色子系统中建立的本征值密度$q$-正态形式，可扩展至两物种费米子和玻色子系统。