Holographic RG flow of thermoelectric transport with momentum dissipation

We construct the holographic renormalization group (RG) flow of thermoelectric conductivities when the translational symmetry is broken. The RG flow is probed by the intrinsic observers hovering on the sliding radial membranes. We obtain the RG flow by solving a matrix-form Riccati equation. The RG flow provides a high-efficient numerical method to calculate the thermoelectric conductivities of strongly coupled systems with momentum dissipation. As an illustration, we recover the AC thermoelectric conductivities in the Einstein-Maxwell-axion model. Moreover, in several homogeneous and isotropic holographic models which dissipate the momentum and have the finite density, it is found that the RG flow of a particular combination of DC thermoelectric conductivities does not run. As a result, the DC thermal conductivity on the boundary field theory can be derived analytically, without using the conserved thermal current.