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?


2 Answers 2


Any (2+1)-D TQFTs described by a unitary modular tensor category can be realized as the boundary of a Walker-Wang model. Moreover it is believed that any such Walker-Wang model is trivial in the bulk; hence there should exist a bulk disentangling unitary. Applying this unitary will give you a microscopic 2D lattice Hamiltonian. Of course turning this into a concrete construction is still challenging.


  • $\begingroup$ Is it clear how to truncate the disentangling unitary in the presence of a boundary when the UMTC is a chiral one? I thought this is the kind of argument that Fidkowski et al to show that the QCA for three fermion is nontrivial. $\endgroup$
    – Meng Cheng
    Oct 7, 2021 at 14:52
  • $\begingroup$ @MengCheng Any locally generated unitary can be truncated in the presence of a boundary. That should not be the issue, though constructing the local unitary in the first place may be difficult (it ought to exist, but it's hard to see it just from staring at the Walker-Wang model). $\endgroup$ Oct 7, 2021 at 17:04
  • $\begingroup$ @MengCheng Fidkowski et al are considering a unitary which maps the projectors of the Walker-Wang model to projectors of a product-state Hamiltonian, which is a stronger condition than what we need (it just needs to relate the ground states). $\endgroup$ Oct 7, 2021 at 17:05
  • $\begingroup$ Right, if the unitary is locally generated then there is no problem. $\endgroup$
    – Meng Cheng
    Oct 7, 2021 at 18:52

I'll offer an alternative route to engineer a Hamiltonian to realize a general 2+1 TQFT. Again working out details is very challenging and have only been done for some special cases.

The idea is to use bulk-boundary correspondence. We know chiral topological phases in 2+1 host gapless CFT edge modes. It is a conjecture that for any UMTC, you can find a chiral CFT that sits on the boundary of a 2+1 bulk described by the UMTC. If you take a thin strip of the bulk, with chiral and anti-chiral CFTs on the two edges, this should just be a purely 1+1 system without any anomaly. So now imagine first creating an array of such 1+1 gapless CFTs ("wires"). This will be the 2+1 bulk. Next turn on suitable interactions to gap out the chiral and anti-chiral modes from neighboring wires. Everything in the bulk will be gapped out, except the chiral CFT on the edge. Via bulk-boundary correspondence we know this bulk must be described by the desired UMTC.

This construction can be explicitly carried out for Abelian UMTCs, and I believe can also be done for any WZW CFTs (of course in these two cases one might argue that a 2+1 Chern-Simons theory would suffice).


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