Disordered Systems and the Replica Method in AdS/CFT

TL;DR

This work addresses quenched disorder in conformal field theories by formulating a holographic replica method in AdS/CFT, where randomness is captured by a double-trace deformation and disorder-averaged correlators are computed from bulk gravity. The authors derive explicit expressions for disorder-averaged two-point functions and demonstrate RG flows that interpolate between $\Delta_-$ and $\Delta_+$, with notable differences when the replica parameter is sent to zero. Field-theory analysis in the planar limit corroborates the holographic results and clarifies the beta functions and anomalous dimensions that govern the flows, including the interplay between the pure double-trace coupling $\tilde{\lambda}$ and the disorder coupling $\tilde{f}$. They illustrate the framework with concrete examples, such as random ${\cal N}=4$ SYM and random $O(N)$ magnets, showing that disorder can either be relevant or drive the system to infinite randomness, and they discuss the behavior of the effective central charge $c_{eff}$ under RG flow. Overall, the paper provides a controlled holographic approach to strongly disordered systems, linking replica techniques, AdS/CFT, and random critical phenomena, with implications for non-unitary fixed points and potential extensions to quantum disorder.

Abstract

We formulate a holographic description of effects of disorder in conformal field theories based on the replica method and the AdS/CFT correspondence. Starting with $n$ copies of conformal field theories, randomness with a gaussian distribution is described by a deformation of double trace operators. After computing physical quantities, we take the $n \to 0$ limit at the final step. We compute correlation functions in the disordered systems by using the holographic replica method as well as the formulation in the conformal field theory. We find examples where disorder changes drastically the scaling of two point functions. The renormalization group flow of the effective central charge in our disordered systems is also discussed.