  Increasing the refractive index available for optical and nanophotonic
systems opens new vistas for design: for applications ranging from broadband
metalenses to ultrathin photovoltaics to high-quality-factor resonators, higher
index directly leads to better devices with greater functionality. Although
standard transparent materials have been limited to refractive indices smaller
than 3 in the visible, recent metamaterials designs have achieved refractive
indices above 5, accompanied by high losses, and near the phase transition of a
ferroelectric perovskite a broadband index above 26 has been claimed. In this
work, we derive fundamental limits to the refractive index of any material,
given only the underlying electron density and either the maximum allowable
dispersion or the minimum bandwidth of interest. The Kramers--Kronig relations
provide a representation for any passive (and thereby causal) material, and a
well-known sum rule constrains the possible distribution of oscillator
strengths. In the realm of small to modest dispersion, our bounds are closely
approached and not surpassed by a wide range of natural materials, showing that
nature has already nearly reached a Pareto frontier for refractive index and
dispersion. Surprisingly, our bound shows a cube-root dependence on electron
density, meaning that a refractive index of 26 over all visible frequencies is
likely impossible. Conversely, for narrow-bandwidth applications, nature does
not provide the highly dispersive, high-index materials that our bounds suggest
should be possible. We use the theory of composites to identify metal-based
metamaterials that can exhibit small losses and sizeable increases in
refractive index over the current best materials.
