RG flow of transport quantities

TL;DR

This paper investigates the renormalization group flow of transport properties in various spacetime dimensions using holographic models, revealing universal phase behavior and analyzing conductivity, thermoelectric, and thermal responses in different frequency regimes.

Contribution

It provides a detailed analytical study of the RG flow of transport quantities in arbitrary dimensions, including the universal phase of conductivity and the temperature dependence of thermoelectric and thermal conductivities.

Findings

Conductivity does not exhibit Drude behavior for  < T in any dimension.

At high frequencies, conductivity scales as ^{-2/3} with specific phase angles.

A universal phase angle of conductivity exists at the horizon, being either zero or a multiple of .

Abstract

The RG flow equation of various transport quantities are studied in arbitrary spacetime 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 {\it Hartnoll et al.}, [{\it JHEP}, {\bf 04}, 120 (2010)]. Motivated by the result of {\it van der Marel et al.}, [{\it science}, {\bf 425}, 271 (2003], we studied, analytically, the conductivity versus frequency plane by dividing it into three distinct parts: $\omega <T, \omega >T$ and $\omega >> T$. In order to compare, we choose 3+1 dimensional bulk spacetime for the computation of the conductivity. In the $\omega <T$ range, the conductivity does not show up the Drude like form in any spacetime dimensions. In the $\omega > T$ range and staying away from the horizon, for the DBI action with unit dynamical exponent, non-zero magnetic field and charge density, the conductivity goes as $\omega^{-2/3}$, whereas the phase of the conductivity, goes as, $ArcTan(Im\sigma^{xx}/Re\sigma^{xx})=\pi/6$ and $ArcTan(Im\sigma^{xy}/Re\sigma^{xy})=-\pi/3$. There exists a universal quantity at the horizon that is the phase angle of conductivity, which either vanishes or an integral multiple of $\pi$. 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.