Charged Deformation of a Hairy Black Brane: Perturbative Solution and Thermodynamics

Abstract

We study electrically charged black branes in four-dimensional Einstein-Maxwell-scalar theory with Maxwell coupling $Z(ϕ)=e^{3ϕ/\sqrt7}$, whose potential has an AdS$_4$ fixed point only in the infrared. For nonzero hair parameter $α$ the scalar grows logarithmically at large radius and the geometry is not asymptotically AdS, so the holographic dictionary is a working assumption. With the scale-invariant multi-trace boundary condition of the neutral branes, giving conformal thermodynamics, we work to second order in the charge, obtaining the Maxwell field and susceptibility in closed form, plus the thermodynamics, condensate, incoherent conductivity and cumulant ratio $T^2χ_4/χ_2=6c(α)<0$ for all $α$ considered. Independent perturbative computations and first-law closed forms agree within $4\times10^{-6}$; nonlinear solutions reproduce second-order results to $1.2\times10^{-5}$ and $c$ to $6\times10^{-4}$. They are locally thermodynamically stable where constructed, and the pressure is analytic in $(μ_Q/T)^2$ near zero density. No regular extremal horizon exists at finite scalar value, and $ρ/s<\sqrt{21}/(2π)$ exactly. The extremal limit is an AdS$_2\times\mathbb R^2$ throat where the scalar grows as $\ln\ln(1/u)$ in the AdS$_2$ radial coordinate $u$; the bound is saturated there, and extremal values are approached as powers of $1/\ln(μ_Q/T)$. With a Dirichlet condition on the scalar source, the same solutions describe a non-conformal theory with scale $Λ$, whose charged branes exist only below $T_{\max}(μ_Q)$, which falls from $0.194Λ$ at zero density to zero at $μ_Q=0.506Λ$. This boundary is a diffusive instability where the charge susceptibility diverges at nonzero density, and the bulk viscosity violates Buchel&#39;s bound at all densities. Quasinormal modes with the normalizable large-radius condition reproduce its hydrodynamics.