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

Abstract

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

Comments: 21 pages, 3 figures, version accepted for publication in J. Stat. Mech. (2023)