Momentum analyticity of the holographic electric polarizability in 2+1 dimensions

The static electric polarization of a holographic field theory dual to the Einstein-Maxwell theory in the background of $AdS_4$ with a Reissner-Nordst\"{o}m (AdS-RN) black hole is investigated. We prove that the holographic polarization is a meromorphic functions in complex momentum plane and locate analytically the asymptotic distribution of the poles along two straight lines parallel to the imaginary axis for a large momentum magnitude. The results are compared with the numerical result on Friedel-like poles of the same holographic model reported in the literature and with the momentum singularities of the one-loop polarization in weak-coupling spinor QED$_3$ and scalar QED$_3$ with the similarities and differences discussed.