1. All the materials being studied as TIs [e.g. Bi2Se3] actually aren't TIs.
What!? A TI is by definition a bulk insulator with surface metallic states that are topologically protected. However, the actual materials turn out not to be bulk insulators. On a practical level this makes separating out bulk and surface contributions, particularly in transport measurements, tricky. But, also presents an ideological problem: one is not actually studying the phase of matter one wishes one was studying.
2. One could argue that TIs are "just a band structure effect", i.e., they do not involve any quantum many-body physics.
However, these objections are put to rest by a preprint
Discovery of the First True Three-Dimensional Topological Insulator: Samarium Hexaboride
Steven Wolgast, Cagliyan Kurdak, Kai Sun, J. W. Allen, Dae-Jeong Kim, Zachary Fisk
This is of particular interest for several reasons
a. The material really is a true topological insulator.
b. The material is a Kondo insulator. [Although strictly the material is in the mixed valence rather than the local moment regime.] The insulating state emerges from strong electronic correlations.
c. This resolves long standing puzzles about previous transport measurements on this material which did not show activated conductivity at low temperatures. This can now be explained as a sample dependent contribution from metallic surface states.
d. This material was predicted to be a topological Kondo insulator by Dzero, Sun, Coleman, and Galitski.
I also note a recent paper Actinide Topological Insulator Materials with Strong Interaction.
I thank Tony Wright for bringing the preprint to my attention.