In this talk, I will summarize recent progress in the description of thermoelectric transport using gauge/gravity duality. I will first review thermoelectric transport in hydrodynamics, where momentum conservation implies infinite zero-frequency conductivities. By a change of basis of the conserved currents, a universal, finite conductivity can be extracted. It can be computed holographically. I will discuss its low-temperature scaling in terms of critical exponents characterizing time and space anisotropy and anomalous dimensions for the free energy and conserved current. When momentum is almost conserved, the zero-frequency delta functions broaden into Drude-like peaks. A holographic computation precisely identifies the redistribution of the low-frequency spectral weight between two contributions originating from the non-conservation of momentum and intrinsic dissipation respectively. It also sheds some light on how to construct effective theories of thermoelectric transport when momentum is not conserved.

In this talk, we focus on strongly-coupled, translation-invariant holographic phases at finite density. We show that they can be classified according to the scaling behavior of the metric, the electric potential and the electric flux, introducing to new scaling exponents (cohesion and conduction). Solutions fall into two classes, depending on whether they break relativistic symmetry or not. We show that the dimensions of IR operators are governed by the new scaling exponents, as well as the low-frequency scaling of the optical conductivity. We show that thermodynamically stable phases are always gapless. Finally, we examine a refinement of the holographic entanglement entropy sensitive to the IR behaviour of the electric flux, and show that the minimal surface thus obtained can be different from the Ryu-Takayanagi proposal depending on the cohesion exponent.