Bivariate moments of the two-point correlation function for embedded Gaussian unitary ensemble with $k$-body interactions

Embedded random matrix ensembles with $k$-body interactions are well established to be appropriate for many quantum systems. For these ensemble the two point correlation function is not yet derived though these ensembles are introduced 50 years back. Two-point correlation function in eigenvalues of a random matrix ensemble is the ensemble average of the product of the density of eigenvalues at two eigenvalues say $E$ and $E^\prime$. Fluctuation measures such as the number variance and Dyson-Mehta $\Delta_3$ statistic are defined by the two-point function and so also the variance of the level motion in the ensemble. Recently, it is recognized that for the embedded ensembles with $k$-body interactions the one-point function (ensemble averaged density of eigenvalues) follows the so called $q$-normal distribution. With this, the eigenvalue density can be expanded by starting with the $q$-normal form and using the associated $q$-Hermite polynomials $He_\zeta(x|q)$. Covariances $\overline{S_\zeta S_{\zeta^\prime}}$ (overline representing ensemble average) of the expansion coefficients $S_\zeta$ with $\zeta \ge 1$ here determine the two-point function as they are a linear combination of the bivariate moments $\Sigma_{PQ}$ of the two-point function. Besides describing all these, in this paper derived are formulas for the bivariate moments $\Sigma_{PQ}$ with $P+Q \le 8$, of the two-point correlation function, for the embedded Gaussian unitary ensembles with $k$-body interactions [EGUE($k$)] as appropriate for systems with $m$ fermions in $N$ single particle states. Used for obtaining the formulas is the $SU(N)$ Wigner-Racah algebra. These formulas with finite $N$ corrections are used to derive formulas for the covariances $\overline{S_\zeta S_{\zeta^\prime}}$ in the asymptotic limit.