On 1-loop diagrams in AdS space and the random disorder problem

TL;DR

This paper investigates 1-loop scalar field corrections in AdS space with boundary perturbations, revealing divergence issues and their implications for boundary conductivities in disordered systems via AdS/CFT correspondence.

Contribution

It introduces a method to handle divergences in 1-loop AdS calculations with boundary perturbations and applies it to analyze disorder effects on conductivity in dual CFTs.

Findings

IR-relevant disorder reduces conductivity.

Impurities' effects dominate low-frequency conductivity corrections.

Divergences in loop calculations are independent of boundary couplings.

Abstract

We study the complex scalar loop corrections to the boundary-boundary gauge two point function in pure AdS space in Poincare coordinates, in the presence of a boundary quadratic perturbation to the scalar. These perturbations correspond to double trace perturbations in the dual CFT and modify the boundary conditions of the bulk scalars in AdS. We find that, in addition to the usual UV divergences, the 1-loop calculation suffers from a divergence originating in the limit as the loop vertices approach the AdS horizon. We show that this type of divergence is independent of the boundary coupling, and making use of which we extract the finite relative variation of the imaginary part of the loop via Cutkosky rules as the boundary perturbation varies. Applying our methods to compute the effects of a time-dependent impurity to the conductivities using the replica trick in AdS/CFT, we find that generally an IR-relevant disorder reduces the conductivity and that in the extreme low frequency limit the correction due to the impurities overwhelms the planar CFT result even though it is supposedly $1/N^2$ suppressed. Comments on the effect of time-independent impurity in such a system are presented.