We study holographic Wilsonian RG in a general class of asymptotically AdS
backgrounds with a U(1) gauge field. We consider free charged Dirac fermions in
such a background, and integrate them up to an intermediate radial distance,
yielding an equivalent low energy dual field theory. The new ingredient,
compared to scalars, involves a `generalized' basis of coherent states which
labels a particular half of the fermion components as coordinates or momenta,
depending on the choice of quantization (standard or alternative). We apply
this technology to explicitly compute RG flows of charged fermionic operators
and their composites (double trace operators) in field theories dual to (a)
pure AdS and (b) extremal charged black hole geometries. The flow diagrams and
fixed points are determined explicitly. In the case of the extremal black hole,
the RG flows connect two fixed points at the UV AdS boundary to two fixed
points at the IR AdS_2 region. The double trace flow is shown, both numerically
and analytically, to develop a pole singularity in the AdS_2 region at low
frequency and near the Fermi momentum, which can be traced to the appearance of
massless fermion modes on the low energy cut-off surface. The low energy field
theory action we derive exactly agrees with the semi-holographic action
proposed by Faulkner and Polchinski in arXiv:1001.5049 [hep-th]. In terms of
field theory, the holographic version of Wilsonian RG leads to a quantum theory
with random sources. In the extremal black hole background the random sources
become `light' in the AdS_2 region near the Fermi surface and emerge as new
dynamical degrees of freedom.Comment: 37 pages (including 8 pages of appendix), 10 figures and 2 table
