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All of the famous theories with topological order have some kind of gauge structure, for example the $Z_2$ lattice gauge theory or in QHE. So my question is that whether the existence of a kind of gauge structure is a necessary condition to have topological order in a system or not? If the answer is no, can you mention an example of such model which have no gauge structures but its low energy theory is described by a TQFT?

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First of all, it is dicey to say physically what it means for a system to be a gauge theory. Indeed, the gauge group itself is often just an artifact of how we describe the system. S-duality in SUSY Yang-Mills theories for instance change the gauge group (as do a number of finite gauge group dualities explained in my paper with Anton Kapustin.)

Probably the strongest question one can really ask along these lines is whether a system can be modelled with a local Hilbert space at low energies. That is, one which admits a factorization into a tensor product of local "atomic" Hilbert spaces. Note that deconfined gauge theories don't satisfy this condition because the Hilbert space is a set of charge zero states inside some local Hilbert space, and the charge operator is not at all local.

But note further that fermionic systems also don't satisfy this factorization property, since the fermionic creation operators anti-commute at arbitrarily large distances. And indeed, it has been shown that fermionic Hilbert spaces and operators can be exactly mapped to something like a higher $\mathbb{Z}_2$ gauge theory, so the difference between a theory of fermions and a gauge theory is even blurred.

Something with topological order intrinsically also lacks this factorization property due to the presence of nontrivial string and membrane operators, for basically the same reason as the fermionic systems.

One can maybe say a bit more. It is known that deconfined gauge theories come about from spontaneously broken higher symmetries (also explained in our paper above and in this paper) and various aspects of abelian Chern-Simons TQFTs can be explained from this perspective. This spontaneously broken higher symmetry idea precisely encodes what it means for a system to lack a model by local Hilbert spaces.

There's no reason one can't immediately extend this to the setting of arbitrary (extendable) TQFTs using the cobordism hypothesis. Indeed, Z(point) acts on Z(X) for all spaces X and the cobordism hypothesis roughly says that the TQFT behaves as the regular representation of Z(point), meaning this higher symmetry is completely broken in the TQFT ground states. Whether this is actually a useful perspective, probably it isn't but I think it's kind of cool.

For instance, a dimer model on a square lattice has "topological degeneracy" and doesn't look like a gauge theory but actually can be mapped to one.

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  • $\begingroup$ Very nice answer! I guess one could extend your notion to a system being $q$-factorizable, meaning that the system admits a factorization into a tensor product of Hilbert spaces which are defined/localized on a $q$-dimensional physical subspace. I.e. then your definition of gauge theory is 'not being $0$-factorizable'. I guess usual gauge theories are not factorizable for any $q < D$, with fracton order being a counter-example (i.e. for fracton order there is some $q_\textrm{frac}< D$ such that the system is $q$-factorizable iff $q \geq q_\textrm{frac}$ (?)). $\endgroup$ – Ruben Verresen Apr 17 '18 at 14:01
  • $\begingroup$ Hi! Is this the same as restricting the dimension of the logical operators somehow? $\endgroup$ – Ryan Thorngren Apr 17 '18 at 18:55

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