  The normal modes, i.e., the eigen solutions to the dispersion relation
equation, are the most fundamental properties of a plasma, which also of key
importance to many nonlinear effects such as parametric and two-plasmon decay,
and Raman scattering. The real part indicates the intrinsic oscillation
frequency while the imaginary part the Landau damping rate. In most of the
literatures, the normal modes of quantum plasmas are obtained by means of small
damping approximation (SDA), which is invalid for high-$k$ modes. In this
paper, we solve the exact dispersion relations via the analytical continuation
(AC) scheme, and, due to the multi-value nature of the Fermi-Dirac
distribution, reformation of the complex Riemann surface is required. It is
found that the change of the topological shape of the root locus in quantum
plasmas is quite different from classical plasmas, in which both real and
imaginary frequencies of high-$k$ modes increase with $k$ in a steeper way than
the typical linear behaviour as appears in classical plasmas. As a result, the
temporal evolution of a high-$k$ perturbation in quantum plasmas is dominated
by the ballistic modes.
