Higher Abelian Quantum Double Models

Abstract

This paper develops a rigorous $C^{\ast }$-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.

44 pages. v2: Added a new section (Section 4) on the (C^*)-diagonal structure and corrected minor typos. Keywords:Quantum Double Models, Frustration Free Ground States, Pure States Characterization, $C^{\ast }$-algebra, Cartan pair