condensed matter - Reconciling topological insulators and topological order

We make an important distinction between the topological insulators (which are essentially uncorrelated band insulators, "with a twist") and topological order (which covers a variety of exotic properties in certain quantum many-body ground states). The topological insulators are clearly "topological" in the sense of the connectedness of the single particle Hilbert space for one electron; however they are not "robust" in the same way as topologically ordered matter.

My question is this: Topological order is certainly the more general and intriguing situation, but the notion of "topology" seems actually less explicit than in the topological insulators. Is there an easy way to reconcile this?

Perhaps a starting point might be, can we imagine a "topological insulator in Fock space"? Would such a beast have "long range entanglement" and "topological order"?

Edit:

While this has received very nice answers, I should maybe clarify what I'm looking for a bit; I'm aware of the "standard definitions" of (symmetry protected) topological insulators and topological order and why they are very different phenomena.

However, if I'm talking to nonexperts, I can describe topological insulators as, more or less, "Berry phases can give rise to a nontrivial 'band geometry,' and analogous to Gauss-Bonnet there is a nice quantity calculable from this that characterizes instead the 'band topology' and this quantity is also physically measurable" and they seem quite happy with this.

On the other hand, while the connection to something like Gauss-Bonnet might be clear for topological order in "TQFTs" or in the ground state degeneracy, these seem a bit formal. I think my favorite answer is the adiabatic continuity (or lack thereof) that Everett pointed out, but now that I'm thinking about it perhaps what I should have asked for is -- What are the geometric properties of states with topological order from which we could deduce the topological order with some kind of Chern number (but without starting from a Chern-Simons field theory and putting in the right one by hand ;) ). Is there anything like this?

Answer

As you have mentioned, topological insulators (TI) are "topological" because they can not be smoothly connected to trivial band insulators without closing the band gap (and without breaking certain symmetry). Simply generalize this to the many-body case, we may say that the topologically ordered states are called "topological" because they can not be smoothly connected to the trivial product state without closing the many-body gap.

To gain a better understanding, one should realize that "topology" is a complement to "geometry". By geometry, we mean that there is a sense of measurement of the distance and angle etc., and the shape and the size of the object matters. While by topology, we mean that one can continuously deform the object, and the shape or the size does not matter. So the topological properties are those properties that can endure continuous deformation of the state (by continuity we mean without encountering a quantum phase transition). To protect the topological properties against deformation, a gap between the ground states and the excited states is always required. So topological property is only defined for gapped quantum matters (both TI and topological order are within this scope). On the other hand, gapless quantum matters do not have topological properties, and their properties are geometrical.

The topological/geometrical distinction is also reflected from the mathematical tools we used to study the physics. For gapped quantum matters, we use topological tools like homotopy, cohomology, K-theory, category theory etc. For gapless quantum matters, we use geometrical tools like gravity theory (AdS/CFT).