Holographic RG flow of thermo-electric transports with momentum dissipation

We construct the holographic renormalization group (RG) flow of thermo-electric 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 thermo-electric conductivities of strongly coupled systems with momentum dissipation. As an illustration, we recover the AC thermo-electric 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 thermo-electric 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.