Are there any materials for which the refractive index decreases with density? It seems that for most materials we would expect there to be a positive correlation between density and refractive index, but perhaps there are exceptions. I am particularly interested in any gases that might not obey this rule.
Since you want to modify the density of a specific material, you will need to change some of the parameters, like pressure or temperature.
For gases, if you increase the temperature, the refractive index will be lower as it is explained here. So in general, they would not follow the relation you want.
Silicon might be the material you're looking for. As this paper shows, the refractive index of silicon increases with temperature. And here you can see that silicon's density decreases with temperature. Then, you can conclude that silicon's refractive index decreases with density (not much, though).
For different materials, yes. For example, olive oil has a higher refractive index but has lower density.
This is a graph of the refractive index and density. As you can see, there is no clear correlation between index of refraction and density. The explanation is that light (electromagnetic waves) affects electric charge, not mass.
There is even a joke:
For more information, check http://dx.doi.org/10.1119/1.1934113