Abstract
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.
Highlights
“GR 1⁄4 renormalization group (RG)” [1]
The precise process of coarse graining is not clear, it is evident that the anti-de Sitter/conformal field theory (AdS=CFT) correspondence provides the geometrization of RG flow, in which the radial direction in the bulk can be identified with certain energy scale [5,6,7,8,9,10,11,12,13,14,15,16,17]
The essence of the RG flow is to reformulate the classical equations of motion (EOM) in terms of the transport coefficients measured by the intrinsic observers on the sliding membranes
Summary
“GR 1⁄4 RG” [1]. In the holographic theory, this short “equation” highlights that the renormalization group (RG), an iterative coarse-graining scheme to extract the relevant physics [2,3,4], is essential in generating the bulk gravity dual from the boundary field theory. Compared with the traditional method that solves the coupled second-order perturbation equations directly, the new method can greatly simplify the numerical calculation, for the AC transport or spatially inhomogeneous systems. The trivial RG flow interpolates the classical black hole membrane paradigm [25,26] and AdS=CFT smoothly Based on this universality argument, Blake and Tong identified a massless mode in the massive gravity and obtained the analytical expression of the DC electric conductivity [27]. Given the analytical expression of electric and thermoelectric conductivities that can be obtained from the conserved electric current, we can derive the thermal conductivity analytically by using the trivial RG flow of ZHC conductivity and the infrared boundary condition of the matrix Riccati equation. In two Appendices, we will present the thermodynamics on the membranes and a semianalytical proof for the trivial RG flow, respectively
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