Given a spherical fusion category $\mathcal C$, the [Levin-Wen model](https://arxiv.org/pdf/cond-mat/0404617.pdf) constructs a lattice field theory: to each oriented surface with a triangulation, it assigns a state space $\mathcal H$ and a Hamiltonian $H$, whose space of ground states is independent of the choice of triangulation; and the [Turaev-Viro model](https://ncatlab.org/nlab/show/Turaev-Viro+model) uses $\mathcal C$ to construct a (2+1)-dimensional oriented TQFT $Z\_{\mathcal C}$.


[Kirrilov (2011)](https://arxiv.org/pdf/1106.6033.pdf) showed that for any spherical fusion category $\mathcal C$ and oriented surface $\Sigma$, the space of ground states of the Levin-Wen model for $\mathcal C$ on $\Sigma$ is canonically isomorphic to the state space $Z\_{\mathcal C}(\Sigma)$ in the Turaev-Viro TQFT.


Is there an analogous result in one dimension lower?


Presumably this construction would, given a semisimple commutative Frobenius algebra $A$, define a lattice model on the circle whose space of ground states can be canonically identified with $A$. Someone's probably worked on this, but I haven't found any references.



Is there a 1-dimensional analogue of the correspondence between the Levin-Wen and Turaev-Viro models?
Yes, there is an analogous 1d result, but it's not very interesting (which probably accounts for you not finding it in the literature).


In order for there to be a corresponding lattice model, you want the TQFT to be fully extended. This means you should start with a noncommutative Frobenius algebra (the Hilbert space of an interval), not a commutative Frobenius algebra (the Hilbert space of a circle). We can drop the "Frobenius" part, since we are not planning on constructing a path integral for 2-manifolds. Any semisimple algebra is Morita equivalent to a sum of trivial algebras. A trivial algebra corresponds to the trivial TQFT, and the associated lattice Hamiltonian is the trivial Hamiltonian.


If we consider a direct sum of trivial TQFTs, then the resulting lattice Hamiltonian is a ferromagnet (I think that's the right name). At each site on the 1d lattice we put a label (spin) indicating a summand of the algebra. The Hamiltonian projects to the subspace where adjacent labels are equal.


In conclusion, (1+ε)-dimensional TQFTs and 1d commuting projection Hamiltonians are both sort of boring, and these two facts are related.


(But if you add group actions to the picture, then things become more interesting. The lattice Hamiltonian corresponding to a trivial (1+ε)-dimensional TQFT equipped with a non-trivial finite group action is a 1d SPT phase. Both of these are indexed by group 2-cocycles.)


