  It is interesting to observe that all optical materials with a positive
refractive index have a value of index that is of order unity. Surprisingly,
though, a deep understanding of the mechanisms that lead to this universal
behavior seems to be lacking. Moreover, this observation is difficult to
reconcile with the fact that a single, isolated atom is known to have a giant
optical response, as characterized by a resonant scattering cross section that
far exceeds its physical size. Here, we theoretically and numerically
investigate the evolution of the optical properties of an ensemble of ideal
atoms as a function of density, starting from the dilute gas limit, including
the effects of multiple scattering and near-field interactions. Interestingly,
despite the giant response of an isolated atom, we find that the maximum index
does not indefinitely grow with increasing density, but rather reaches a
limiting value $n\approx 1.7$. We propose an explanation based upon
strong-disorder renormalization group theory, in which the near-field
interaction combined with random atomic positions results in an inhomogeneous
broadening of atomic resonance frequencies. This mechanism ensures that
regardless of the physical atomic density, light at any given frequency only
interacts with at most a few near-resonant atoms per cubic wavelength, thus
limiting the maximum index attainable. Our work is a promising first step to
understand the limits of refractive index from a bottom-up, atomic physics
perspective, and also introduces renormalization group as a powerful tool to
understand the generally complex problem of multiple scattering of light
overall.
