  We analyze the phase diagram of N=4 supersymmetric Yang-Mills theory with
fundamental matter in the presence of a background magnetic field and nonzero
baryon number. We identify an isolated quantum critical point separating two
differently ordered finite density phases. The ingredients that give rise to
this transition are generic in a holographic setup, leading us to conjecture
that such critical points should be rather common. In this case, the quantum
phase transition is second order with mean-field exponents. We characterize the
neighborhood of the critical point at small temperatures and identify some
signatures of a new phase dominated by the critical point. We also identify the
line of transitions between the finite density and zero density phases. The
line is completely determined by the mass of the lightest charged quasiparticle
at zero density. Finally, we measure the magnetic susceptibility and find hints
of fermion condensation at large magnetic field.
