RG FLOW OF TRANSPORT QUANTITIES

Abstract

The RG flow equation of various transport quantities are studied in arbitrary space–time dimensions, in the fixed as well as fluctuating background geometry both for the Maxwellian and DBI type of actions. The regularity condition on the flow equation of the conductivity at the horizon for the DBI action reproduces naturally the leading order result of Hartnoll et al. [J. High Energy Phys. 04, 120 (2010)]. Motivated by the result of van der Marel et al. [Science 425, 271 (2003], we studied, analytically, the conductivity versus frequency plane by dividing it into three distinct parts: ω < T, ω > T and ω ≫ T. In order to compare, we choose (3+1)-dimensional bulk space–time for the computation of the conductivity. In the ω < T range, the conductivity does not show up the Drude like form in any space–time dimensions. In the ω > T range and staying away from the horizon, for the DBI action with unit dynamical exponent, nonzero magnetic field and charge density, the conductivity goes as ω-2/3, whereas the phase of the conductivity, goes as, arctan ( Im σxx/ Re σxx) = π/6 and arctan ( Im σxy/ Re σxy) = -π/3. There exists a universal quantity at the horizon that is the phase angle of conductivity, which either vanishes or an integral multiple of π. Furthermore, we calculate the temperature dependence to the thermoelectric and the thermal conductivity at the horizon. The charge diffusion constant for the DBI action is studied.