  Using high-precision numerical analysis, we show that 3+1 dimensional gauge
theories holographically dual to 4+1 dimensional Einstein-Maxwell-Chern-Simons
theory undergo a quantum phase transition in the presence of a finite charge
density and magnetic field. The quantum critical theory has dynamical scaling
exponent z=3, and is reached by tuning a relevant operator of scaling dimension
2. For magnetic field B above the critical value B_c, the system behaves as a
Fermi liquid. As the magnetic field approaches B_c from the high field side,
the specific heat coefficient diverges as 1/(B-B_c), and non-Fermi liquid
behavior sets in. For B<B_c the entropy density s becomes non-vanishing at zero
temperature, and scales according to s \sim \sqrt{B_c - B}. At B=B_c, and for
small non-zero temperature T, a new scaling law sets in for which s\sim
T^{1/3}. Throughout a small region surrounding the quantum critical point, the
ratio s/T^{1/3} is given by a universal scaling function which depends only on
the ratio (B-B_c)/T^{2/3}.
  The quantum phase transition involves non-analytic behavior of the specific
heat and magnetization but no change of symmetry. Above the critical field, our
numerical results are consistent with those predicted by the Hertz/Millis
theory applied to metamagnetic quantum phase transitions, which also describe
non-analytic changes in magnetization without change of symmetry. Such
transitions have been the subject of much experimental investigation recently,
especially in the compound Sr_3 Ru_2 O_7, and we comment on the connections.
