Are there experimental signatures unique to Mott insulators? Mott insulators are differentiated from band insulators because they are insulators due to their strong electron-electron interactions. 
Although Mott insulators are defined using a specific model,  I want to ask the following question:
Is there an experimental way to differentiate between an insulator due to strong Coulomb repulsion and a band insulator that does not rely on any calculations or assumptions about the Hamiltonian?
If we consider other phases such as ferromagnets, ferroelectrics, piezoelectrics, superconductors, etc. each have very specific experimental signatures that do not rely on doping, chemical changes,  calculations, models, fitting, etc. On the other hand, for Mott insulators I don't know of any such unique signatures without referring to external information. 
 A: Since Mott insulators and band insulators are both gapped, incompressible, insulating states, there aren't many signatures that would distinguish between them without some additional knowledge of the system. But here is a brief list of the tests that you might do, and the knowledge that each requires:


*

*Density measurements. Probably the most important and practical test is just measuring the density of the system. If it is an insulator, but it is at half-filling (i.e., there is exactly one particle per Wigner–Seitz cell), then it cannot be a band insulator. This does not require detailed knowledge of the microscopic Hamiltonian, just the crystal structure and carrier density.

*Excitation measurements. The fundamental excitations in a Mott insulator are particle-hole excitations, while in a band insulator they are band excitations. So, if one knows at least something about the relative hierarchy between the interaction scale and bandgap, by studying the excitations of the system it might also be possible to distinguish between the two states. This, again, requires some microscopic knowledge, but not necessary full information about the Hamiltonian.

*Finally, if it is possible to vary the interaction strength one can observe the metal-Mott insulator phase transition directly. As far as I know, this is the only test that doesn't require any microscopic knowledge. It isn't possible with electrons, but can be done in ultracold gases in optical lattices.
One final comment: in Hubbard's original paper (paywalled link, sorry), he develops an approximate picture for the Hubbard model under strong interactions as having two remormalized bands separated by the interaction energy. So, within this approximate picture, a Mott insulator is in a sense a type of band insulator, with a bandgap that is created by the interactions.
There are also various papers that tackle this question directly for a particular model (2).
