  Using analytical methods, we derive and extend previously obtained numerical
results on the low temperature properties of holographic duals to
four-dimensional gauge theories at finite density in a nonzero magnetic field.
We find a new asymptotically AdS_5 solution representing the system at zero
temperature. This solution has vanishing entropy density, and the charge
density in the bulk is carried entirely by fluxes. The dimensionless magnetic
field to charge density ratio for these solutions is bounded from below, with a
quantum critical point appearing at the lower bound. Using matched asymptotic
expansions, we extract the low temperature thermodynamics of the system. Above
the critical magnetic field, the low temperature entropy density takes a simple
form, linear in the temperature, and with a specific heat coefficient diverging
at the critical point. At the critical magnetic field, we derive the scaling
law s ~ T^{1/3} inferred previously from numerical analysis. We also compute
the full scaling function describing the region near the critical point, and
identify the dynamical critical exponent: z=3.
  These solutions are expected to holographically represent boundary theories
in which strongly interacting fermions are filling up a Fermi sea. They are
fully top-down constructions in which both the bulk and boundary theories have
well known embeddings in string theory.
