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Density versus chemical potential in holographic field theories

We study the relationship between charge density (ρ) and chemical potential (μ) 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 ρ $\propto$ μ^α. A variety of powers α are found in the brane systems while in most of the bottom-up models α is determined by the underlying conformal symmetry. Further, it is demonstrated that basic thermodynamical and causality constraints demand α \geq 1, a condition that was realized in each system considered.