Topological order refers to long-range-entangled phases of matter that cannot be smoothly deformed into ordinary phases characterized by Landau’s symmetry breaking theory. A large number of topological orders are described and classified by topological quantum field theory and unitary modular tensor category theory [or unitary braided fusion category], the latter of which describes the rules governing the fusion and braiding process of topological excitations (anyons).
What is not clear to me is whether or not all of these phases are realizable in microscopic systems with local interactions, i.e. does there always exist a locally interacting Hamiltonian that have ground states and low energy excitations described by these macroscopic theories (TQFT and UMTC/UBFC)? The string-net models answer this question affirmatively for the case of "doubled topological order", but that is just a small subclass of topological order, and the answer for the general case is still missing.
[As a useful comparison with symmetry-breaking phases, UMTC or UBFC takes the role of group theory, while TQFT plays the role of some kind of effective field theory like Landau-Ginzberg. But to establish their existence, seems that we are still missing a microscopic Hamiltonian]
In particular, I want to ask: is there an example of an important topological order consistently described by TQFT or UMTC but not yet known to be microscopically realizable?