Thermoelectric DC conductivities and Stokes flows on black hole horizons

TL;DR

This work shows that the DC thermo-electric response of a broad class of holographic CFTs with broken translation invariance can be obtained by solving a decoupled, generalized Stokes problem for a charged fluid on the black hole horizon. The authors derive horizon equations from horizon constraints, define horizon current fluxes that are radially conserved, and express boundary DC conductivities entirely in terms of horizon data. They provide closed-form horizon solutions for Q-lattices and one-dimensional lattices, and analyze perturbative lattices about AdS-Reissner–Nordström to extract leading-order DC conductivities, uncovering a Wiedemann–Franz-like relation. The formalism also offers a perspective in the language of differential forms and cohomology, with clear topological interpretations of the sources and currents, and lays out a pathway to general static spacetimes and potential extensions to magnetic fields and multiple horizons.

Abstract

We consider a general class of electrically charged black holes of Einstein-Maxwell-scalar theory that are holographically dual to conformal field theories at finite charge density which break translation invariance explicitly. We examine the linearised perturbations about the solutions that are associated with the thermoelectric DC conductivity. We show that there is a decoupled sector at the black hole horizon which must solve generalised Stokes equations for a charged fluid. By solving these equations we can obtain the DC conductivity of the dual field theory. For Q-lattices and one-dimensional lattices we solve the fluid equations to obtain closed form expressions for the DC conductivity in terms of the solution at the black hole horizon. We also determine the leading order DC conductivity for lattices that can be expanded as a perturbative series about translationally invariant solutions.