Diffusion and instabilities in large-N holographic Fermi liquids: the vector fluctuations of the electron star

We study the hydrodynamic response of the AdS electron star in the vector sector, and compute the correlation functions and the transverse conductivity of the dual field theory. The system exhibits hydrodynamic behavior at low temperatures and near the critical temperature where the electron star undergoes the phase transition to the RN black hole. However, at intermediate temperatures the hydrodynamics does not exist. Remarkably, the system has an instability, i.e. a pole on the positive imaginary frequency axis at finite temperature. This instability is found both from analytical arguments and from numerics. Its physical meaning is so far unclear but it might mean that the ideal fluid limit for the star is a false vacuum.