  The free energy of effective spin or "Polyakov line" models with a chemical
potential, based on the U(N) group, does not depend on the chemical potential.
In a mean field-inspired expansion, we show how the condition of unit
determinant, taking U(N) to SU(N), reintroduces the chemical potential, and
allows us to express the free energy, as a function of mean field variational
parameters, in terms of an expansion in the baryon (rather than the quark)
fugacity at each lattice site. We solve the SU(3) mean field equations
numerically to determine the phase diagram and compute observables. We also
calculate the first corrections to the leading order mean field results, and
find that these can significantly shift the endpoint of a line of first order
transitions. The problem of deriving an effective spin model from full QCD is
discussed.
