Density versus chemical potential in holographic probe theories

Abstract

We study the relationship between charge density ($\ensuremath{\rho}$) and chemical potential ($\ensuremath{\mu}$) for an array of Lorentz-invariant $3+1$-dimensional holographic field theories with the minimal structure of a conserved charge. In all cases, at large density, the relationship is well-modeled by a power-law behavior of the form $\ensuremath{\rho}\ensuremath{\propto}{\ensuremath{\mu}}^{\ensuremath{\alpha}}$. For the minimal ingredients of a gravitational field and a probe $U(1)$ gauge field in the bulk, we find general constraints $\ensuremath{\alpha}=1$ for the Maxwell action and $\ensuremath{\alpha}>1$ for the Born-Infeld case, for general background metrics. We show that the constraint $\ensuremath{\alpha}\ensuremath{\ge}1$ can also be understood directly in the field theory from thermodynamic stability and causality. We then determine which values of $\ensuremath{\alpha}$ are realized in a large range of example systems, including $\mathrm{D}p\mathrm{\text{\ensuremath{-}}}\mathrm{D}q$ probe brane constructions and ``bottom-up'' models with gauge and scalar fields.