Bounding the Temperatures of Black Holes Dual to Strongly Coupled Field Theories on Flat Spacetime

We show that AdS black holes dual to field theories on flat spacetime, as used in applications of the AdS/CFT correspondence to strong interaction and condensed matter physics, have temperatures with positive lower bounds. There are two distinct effects involved. For low chemical potentials in the dual field theory, the cooling black hole makes a transition to a state corresponding to confinement in the field theory. For high chemical potentials, it becomes unstable to a non-perturbative string effect. This allows a holographic sketch of the field theory phase diagram, one which is in qualitative agreement with the phenomenological understanding of the theory at [relatively] low temperatures. It also puts an interesting upper bound on the temperature-normalized chemical potential $\bar{\mu}$ of the field theory, if it describes a plasma: in the normalization of Myers et al., $\bar{\mu}$ must be less than approximately 0.49. Thus, the extent to which a chemical potential can worsen violations of the KSS bound is severely restricted.