Analytic fermionic Green’s functions from holography

Abstract

We find exact, analytic solutions of the Dirac equation for a charged, massless fermion in the background of a charged, dilatonic black hole in ${\mathrm{AdS}}_{5}$. The black hole descends from type IIB supergravity, where it describes D3-branes with equal angular momenta in two of the three independent planes of rotation orthogonal to the world-volume. The Green's function near the Fermi surface for a strongly coupled fermionic system can be extracted holographically from an exact solution of the Dirac equation at zero frequency but nonzero momentum. There can be several Fermi momenta, and they take the form ${k}_{F}=q\ensuremath{-}n\ensuremath{-}1/2$ (in units of the chemical potential), where $q$ is the charge of the spinor, and $n$ is a non-negative integer that labels the Fermi surfaces. Much as for holographic Fermi surfaces based on the Reissner-Nordstr\"om-${\mathrm{AdS}}_{5}$ solution, the dispersion relation of the excitations near the Fermi surface is determined by the geometry close to the horizon, and one can obtain Fermi liquid, marginal Fermi liquid, and non-Fermi liquid behaviors depending on the value of ${k}_{F}$.