@Corman2017TransmissionAtoms; @Jennewein2016CoherentCloud; @Keaveney2012MaximalNanolayer]. However, an underlying physical explanation is still missing, and our goal here is to understand better the mechanisms that might limit the index even when operating close to resonance.

Specifically, we investigate in detail the optical response of an ideal ensemble of identical, stationary atoms, as a function of density starting from the dilute limit, and well within the regime where the atoms do not interact chemically. In large scale numerics (involving up to $\sim 25000$ atoms, about an order of magnitude larger than comparable works [@Chomaz2012AbsorptionAnalysis; @Javanainen2014ShiftsSample; @Javanainen2016LightOptics; @Zhu2016LightMedia; @Schilder2016PolaritonicAtoms; @Schilder2017HomogenizationScatterers; @Schilder2020Near-ResonantAtoms; @Jennewein2016PropagationAtoms; @Jennewein2018CoherentTheory; @Corman2017TransmissionAtoms; @Jenkins2016CollectiveExperiment; @Dobbertin2020CollectiveNanocavities]), we find that the maximum index does not indefinitely grow with density, and saturates to a maximum value of $n\approx 1.7$, when the typical distance between atoms becomes smaller than the length scale associated with the resonant cross section, i.e. $d<\lambda_0$. Furthermore, we introduce an underlying theory based upon strong-disorder renormalization group (RG), which has been a very successful technique to deal with highly varying interaction strengths in a wide variety of condensed matter systems [@Levitov1990DelocalizationInteraction; @Fisher1994RandomChains; @Damle2000DynamicsChains; @Motrunich2000Infinite-randomnessPoints; @Refael2004EntanglementDimension; @Igloi2005StrongSystems; @Vosk2013Many-bodyPoint; @Refael2013StrongTransition]. In the context of our particular problem, the combination of strong near-field ($\sim 1/r^3$) optical interactions 