  We investigate quantum phase transitions in a 2+1 dimensional gauge theory at
finite chemical potential $\chi$ and magnetic field $B$. The gravity dual is
based on 4D $\mathcal{N}=2$ Fayet-Iliopoulos gauged supergravity and the
solutions we consider---that are constructed analytically---are extremal,
dyonic, asymptotically $AdS_4$ black-branes with a nontrivial radial profile
for the scalar field. We discover a line of second order fixed points at
$B=B_c(\chi)$ between the dyonic black brane and an extremal "thermal gas"
solution with a singularity of good-type, according to the acceptability
criteria of Gubser [1]. The dual field theory is the ABJM theory [2] deformed
by a triple trace operator $\Phi^3$ and placed at finite charge and magnetic
field. This line of fixed points might be useful in studying the various
strongly interacting quantum critical phenomena such as the ones proposed to
underlie the cuprate superconductors. We also find curious similarities between
the behaviour of the VeV $\langle \Phi \rangle$ under B and that of the quark
condensate in 2+1 dimensional NJL models.
