Exact dispersion relation of the quantum Langmuir wave

The normal modes, i.e., the eigensolutions to the dispersion relation equation, are the most fundamental properties of a plasma. The real part indicates the intrinsic oscillation frequency while the imaginary part the Landau damping rate. In most of the literature, the normal modes of quantum plasmas are obtained by means of small damping approximation, which is invalid for high-k modes. In this paper, we solve the exact dispersion relations via the analytical continuation 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 topological shape of the root locus in quantum plasmas is quite different from classical ones, in which both real and imaginary frequencies of high-k modes increase with k steeper than the typical linear behavior in classical plasmas. As a result, the time-evolving behavior of a high-k initial perturbation becomes ballistic-like in quantum plasmas.