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Two species $k$-body embedded Gaussian unitary ensembles: $q$-normal form of the eigenvalue density
Manan Vyas, V. K. B. Kota · 2023-06-22 · via math.ST updates on arXiv.org

Eigenvalue density generated by embedded Gaussian unitary ensemble with $k$-body interactions for two species (say $\mathbfπ$ and $\mathbfν$) fermion systems is investigated by deriving formulas for the lowest six moments. Assumed in constructing this ensemble, called EGUE($k:\mathbfπ \mathbfν$), is that the $\mathbfπ$ fermions ($m_1$ in number) occupy $N_1$ number of degenerate single particle (sp) states and similarly $\mathbfν$ 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($k:\mathbfπ \mathbfν$) 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.

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