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arXiv · 2509.12864

Higher Abelian Quantum Double Models

Jorge Acuña Flores·Giuseppe De Nittis·Javier Lorca Espiro
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Abstract

This paper develops a rigorous $C^*$-algebraic framework for higher abelian quantum double models, generalizing Kitaev's construction to simplicial complexes of arbitrary finite dimension and local regularity. We fully characterize the frustration-free ground state space through an associated algebra of logical operators: its state space is shown to be homeomorphic to the space of frustration-free ground states, and to obey generalized canonical commutation relations. When the relevant homology and cohomology groups are finite, the logical algebra decomposes into a commutative factor and a full matrix algebra, thereby separating the classical and quantum parts of the frustration-free ground state structure. We further prove that the vanishing of these groups is necessary and sufficient for the core algebra of the model to form a Cartan pair with the full algebra of observables, a property expected to pave the way toward classifying the model's equilibrium (KMS) states, extending recent results for the planar case.

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Jorge Acuña Flores, Giuseppe De Nittis, Javier Lorca Espiro. 2025-09-16. Higher Abelian Quantum Double Models. https://arxiv.org/abs/2509.12864

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