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.