A very good question. First of all, topological order strictly speaking is only defined for gapped states. But to some extent it can coexist with gapless degrees of freedom. A rather trivial example is just adding something gapless decoupled from the topological order (i.e. phonons). The example of the Kitaev model is quite different though, since the gapless part are fermionic spinons and the gapped part are visons ($Z_2$ gauge fields). The TEE says that the wavefunction of the $Z_2$ gauge field has non-local constraints(i.e. electric field lines must be closed, or only end on the spinons), which is also reflected on the mutual braiding statistics between spinons and visons. But on the other hand, the gapless spinons do have significant effect: for example, if one asks for the topological degeneracy (on top of the $1/L$ low-energy spinon excitations) on a torus, my thought is that the ground state degeneracy is reduced from $4$ to $1$, because spinons going around the large cycles can measure the vison fluxes and such a process has an amplitude $\sim 1/L$.
So indeed the distinction you draw is an important one. On the other hand, this gapless state is not robust: there is nothing preventing the spinons from opening a mass gap unless additional symmetries are imposed. So near the gapless phase, there are gapped A and B phases, both of which have the same value of TEE. So it can be considered as the "critical point" between the A and B phase.
I recommend a very insightful paper by Bonderson and Nayak, http://arxiv.org/abs/1212.6395, which discussed in great depth how one can define topological order in the presence of gapless degrees of freedom, and how ground state degeneracy and braiding statistics are affected.
For the second question, a gapped state should probably be defined as a state where all correlation functions (of physical, local operators) are short-ranged. It seems to me that if a state is gapless, then some correlation function should detect the gaplessness: for example, in your example of Kitaev honeycomb model, although the spin correlation functions are short-ranged, the bond energy correlations are algebraic. I have not seen a counterexample to this criteria, but I don't think it has been proved rigorously either. You can look at http://arxiv.org/abs/math-ph/0507008 for a proof of spectral gap implying short-ranged correlations (also need to be careful about whether one should use connected correlation functions, so many subtle details...).
This post imported from StackExchange Physics at 2015-03-15 11:30 (UTC), posted by SE-user Meng Cheng