Density versus chemical potential in holographic field theories

We study the relationship between charge density ({\rho}) and chemical potential ({\mu}) for an array of Lorentz invariant 3 + 1 dimensional holographic field theories with the minimal structure of a conserved charge. The systems considered include Dp-Dq probe brane constructions and probe and backreacted 'bottom-up' models with gauge and scalar fields. In all cases, at large density, the relationship is well modelled by a power law behaviour of the form {\rho} $\propto$ {\mu}^{\alpha}. A variety of powers {\alpha} are found in the brane systems while in most of the bottom-up models {\alpha} is determined by the underlying conformal symmetry. Further, it is demonstrated that basic thermodynamical and causality constraints demand {\alpha} \geq 1, a condition that was realized in each system considered.