  For free fermions at finite density, the Pauli exclusion principle is
responsible for the existence of a Fermi surface and the consequent presence of
low energy spectral weight over a finite range of momenta. We investigate the
extent to which this effect occurs in strongly interacting quantum matter with
a holographic dual. We obtain the low energy current-current spectral weight in
two holographic frameworks at finite density: systems exhibiting semi-local
quantum criticality (with a low temperature entropy density vanishing like s ~
T^eta), and a probe D3/D5 system. For the semi-local theory with 0 < eta < 2 we
find a sharp discontinuity in the transverse spectral weight at a nonzero
momentum k_*. The case eta = 1 is found to have additional symmetries and is
soluble even at nonzero temperature. We show that this case exhibits a robust
linear in temperature resistivity in the presence of random charged impurities.
For the probe D3/D5 system we find an analytic expression for the low energy
spectral weight as a function of momentum. The spectral weight is supported
below a specific momentum k_* and is exponentially suppressed at higher
momenta.
