Kitaev models based on unitary quantum groupoids

TL;DR

The work extends Kitaev's exactly solvable models to inputs from unitary $C^*$-quantum groupoids $H_{\mathcal{C}}$ built from unitary fusion categories $\mathcal{C}$. It proves that the ground state space of the generalized Kitaev model is canonically isomorphic to the Levin-Wen ground space, thereby realizing the TV-TQFT target space $Z_{TV}(\Sigma)$ on closed surfaces. The construction hinges on defining frustration-free Hamiltonians via cocommutative elements $\Lambda$ and $\lambda$, and on establishing a precise correspondence between the representations of $H_{\mathcal{C}}$ and the category $\mathcal{C}$ (via the Kitaev–Kong framework). This provides a robust bridge between lattice Hamiltonians, unitary fusion categories, and TQFTs, with explicit examples such as Fibonacci theory illustrating the structure.

Abstract

We establish a generalization of Kitaev models based on unitary quantum groupoids. In particular, when inputting a Kitaev-Kong quantum groupoid $H_\mathcal{C}$, we show that the ground state manifold of the generalized model is canonical isomorphic to that of the Levin-Wen model based on a unitary fusion category $\mathcal{C}$. Therefore the generalized Kitaev models provide realizations of the target space of the Turaev-Viro TQFT based on $\mathcal{C}$.

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Theorems & Definitions (30)

• Definition 1
• Proposition 1
• Lemma 1
• proof
• Definition 2
• Proposition 2
• Proposition 3
• Lemma 2
• proof
• Proposition 4