  We study N=4 super Yang Mills theory at finite U(1)_R charge density (and
temperature) using the AdS/CFT Correspondence. The ten dimensional backgrounds
around spinning D3 brane configurations split into two classes of solution. One
class describe spinning black holes and have previously been extensively
studied, and interpreted, in a thermodynamic context, as the deconfined high
density phase of the dual field theory. The other class have naked
singularities and in the supersymmetric limit are known to correspond to
multi-centre solutions describing the field theory in the Coulomb phase. We
provide evidence that the non-supersymmetric members of this class represent
naked, spinning D-brane distributions describing the Coulomb branch at finite
density. At a critical density a phase transition occurs to a spinning black
hole representing the deconfined phase where the higgs vevs have evaporated. We
perform a free energy calculation to determine the phase diagram of the Coulomb
branch at finite temperature and density.
