  We use the AdS/CFT correspondence to study N=4 supersymmetric SU(Nc)
Yang-Mills theory, in the limits of large Nc and large 't Hooft coupling,
coupled to a number Nf of massless hypermultiplet fields in the fundamental
representation of the gauge group. We identify a U(1) subgroup of the
R-symmetry under which the fermions in the hypermultiplet are charged but the
scalars are not. All the hypermultiplet fields are also charged under a U(1)
baryon number symmetry. We introduce an external magnetic field for the baryon
number U(1), which triggers the spontaneous breaking of the U(1) R-symmetry,
and we then introduce a chemical potential for the U(1) R-charge, producing a
state with a nonzero density of the U(1) R-charge. The system should then
exhibit superconductivity of the U(1) R-charge. The dual supergravity
description is a number Nf of D7-branes in AdS5 x S5 with angular momentum on
the S5 and a worldvolume magnetic field. We study the zero-temperature
thermodynamics of the system, and find that for sufficiently large magnetic
field the system prefers to be in the symmetry-broken phase. For smaller
magnetic fields we find a discontinuous free energy, indicating that our
gravitational setup does not capture all equilibrium states of the field
theory.
