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It is interesting to observe that all optical materials with a positiverefractive index have a value of index that is of order unity. Surprisingly,though, a deep understanding of the mechanisms that lead to this universalbehavior seems to be lacking. Moreover, this observation is difficult toreconcile with the fact that a single, isolated atom is known to have a giantoptical response, as characterized by a resonant scattering cross section thatfar exceeds its physical size. Here, we theoretically and numericallyinvestigate the evolution of the optical properties of an ensemble of idealatoms as a function of density, starting from the dilute gas limit, includingthe effects of multiple scattering and near-field interactions. Interestingly,despite the giant response of an isolated atom, we find that the maximum indexdoes not indefinitely grow with increasing density, but rather reaches alimiting value $n\approx 1.7$. We propose an explanation based uponstrong-disorder renormalization group theory, in which the near-fieldinteraction combined with random atomic positions results in an inhomogeneousbroadening of atomic resonance frequencies. This mechanism ensures thatregardless of the physical atomic density, light at any given frequency onlyinteracts with at most a few near-resonant atoms per cubic wavelength, thuslimiting the maximum index attainable. Our work is a promising first step tounderstand the limits of refractive index from a bottom-up, atomic physicsperspective, and also introduces renormalization group as a powerful tool tounderstand the generally complex problem of multiple scattering of lightoverall.

Maximum refractive index of an atomic medium