Paradoxical Crystal Baffles Physicists
wired.com
wired.com
Also there is an other type of non-conventional insulator which is considered to be well understood: the Mott-insulator[2]. It really doesn't fit in here as a candidate since one wouldn't expect cyclotron resonance from this model in the insulating regime. However it's a good learning example of strongly correlated electrons in materials.
[1]https://en.wikipedia.org/wiki/Electron_cyclotron_resonance#I... [2]https://en.wikipedia.org/wiki/Mott_insulator
I'm a total noob, so if that is correct it's a miracle.
"But this is no garden-variety insulator. Not only does its insulating behavior arise from strong correlations between its electrons, but in the past five years, mounting evidence has suggested that it is a “topological insulator” at low temperatures, a material that resists the flow of electricity through its three-dimensional bulk, while conducting electricity along its two-dimensional surfaces."
See https://en.wikipedia.org/wiki/Skin_effect and https://en.wikipedia.org/wiki/Litz_wire .
http://faculty.wwu.edu/vawter/physicsnet/Topics/Gauss/IdealC...
http://hyperphysics.phy-astr.gsu.edu/hbase/electric/gausur.h...
That doesn't apply to moving charge though (ie. an electric current)