  We construct superfluid black hole solutions with two chemical potentials. By
analogy with QCD, the two chemical potentials correspond to the baryon and
isospin symmetries, respectively. We consider two systems: the back-reacted
U(2) Einstein-Yang-Mills theory in 4+1 dimensions and the 9+1-dimensional D3/D7
brane setup with two coincident D7-brane probes. In the D7-brane model, the
identification of baryon and isospin chemical potential is explicit since the
dual field theory is explicitly known. Studying the phase diagram, we find in
both systems a quantum phase transition at a critical ratio of the two chemical
potentials. However the quantum phase transition is different in the two
systems: In the D3/D7 brane setup we always find a second order phase
transition, while in the Einstein-Yang-Mills theory, depending on the strength
of the back-reaction, we obtain a continuous or first order transition. We
expect the continuous quantum phase transition to be BKT-like. We comment on
the origin of this differing behavior in these apparently very similar models
and compare to phenomenological systems.
