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arXiv:2509.12864

Higher Abelian Quantum Double Models

Flores, Jorge Acuña, Giuseppe De Nittis, J. Lorca Espiro

Published Sep 16, 2025• 0 citations
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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.

DOI
10.48550/arxiv.2509.12864

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This paper develops a rigorous C∗C^*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.
We first prove that the subalgebra C\mathcal{C}C generated by the vertex and face operators of an abelian Kitaev model is a C∗C^\astC∗-diagonal of the UHF algebra A\mathcal{A}A of quasilocal observables. This gives us access to the Weyl groupoid GC\mathcal{G}_\mathcal{C}GC​ associated with the C∗C^\astC∗-inclusion C↪A\mathcal{C} \hookrightarrow \mathcal{A}C↪A, which supplies a valuable presentation of A\mathcal{A}A as a groupoid C∗C^\astC∗-algebra where the dynamics of the model are generated by a groupoid 1-cocycle cHc_HcH​. Making appeal to the notion of (cH,β)(c_H,β)(cH​,β)-KMS measures for this groupoid, we identify the full set of KMS states of the model and prove its uniqueness for β∈[0,∞)β\in [0,\infty)β∈[0,∞). Furthermore, we show that its limit at β→∞β\rightarrow \inftyβ→∞ exists and coincides with the unique frustration-free ground state of the model.
In this paper, we determine the complete ground-state spaces for a broad class of generalized stabilizer quantum models defined over CW-decompositions of non-compact manifolds of arbitrary dimensions and topological type. We introduce a C∗C^*C∗-algebraic framework in which the quantum spin algebra $\A$ is realized as the twisted group C∗C^*C∗-algebra of an infinite discrete Pauli module PPP endowed with a symplectic structure. We establish a correspondence between stabilizer models and isotropic subgroups of PPP: Every Lagrangian extension of such a subgroup determines a C∗C^*C∗-diagonal of $\A$, whose Weyl groupoid encodes the dynamics through a continuous 1-cocycle. This realization yields an explicit family of extremal ground-states for every admissible Lagrangian model, together with their decomposition into sectors. When the sector space is T0T_0T0​, this gives a complete characterization of the ground-state spaces. We also prove that every Lagrangian stabilizer model admits a unique frustration-free ground state with a spectral gap. Our framework applies in arbitrary dimension, including higher abelian quantum doubles, homological codes, and a broad class of stabilizer models.
The goal of our work is to characterize the landscape of the frustration-free quantum spin models over the Cayley graph of a finitely generated group G. This is achieved by establishing GGG-equivariant morphisms from the partially ordered space of frustration-free models to the partially ordered spaces 1) of hereditary C*-algebras of the underlying UHF algebra of quasi-local observables, 2) of open projections in its double dual, and 3) of subsets of pure state space. Our main result consists of an intrinsic characterization of the images of these morphisms, which captures the essence of frustration-freeness and enables us to extend the concept to generic AF-C*-algebras. Additionally, using well established facts about AF-C*-algebras, we prove density theorems and provide intrinsic and optimal characterizations of frustration-free ground states.
We prove Haag duality for cone-like regions in the ground state representation corresponding to the translational invariant ground state of Kitaev’s quantum double model for finite abelian groups. This property says that if an observable commutes with all observables localized outside the cone region, it actually is an element of the von Neumann algebra generated by the local observables inside the cone. This strengthens locality, which says that observables localized in disjoint regions commute. As an application, we consider the superselection structure of the quantum double model for abelian groups on an infinite lattice in the spirit of the Doplicher–Haag–Roberts program in algebraic quantum field theory. We find that, as is the case for the toric code model on an infinite lattice, the superselection structure is given by the category of irreducible representations of the quantum double.
We develop and investigate several quantum many-body spin models of use for topological quantum information processing and storage. These models fall into two categories: those that are designed to be more realistic than alternative models with similar phenomenology, and those that are designed to have richer phenomenology than related models. In the first category, we present a procedure to obtain the Hamiltonians of the toric code and Kitaev quantum double models as the perturbative low-energy limits of entirely two-body Hamiltonians. This construction reproduces the target models' behavior using only couplings which are natural in terms of the original Hamiltonians. As an extension of this work, we construct parent Hamiltonians involving only local 2-body interactions for a broad class of Projected Entangled Pair States (PEPS). We define a perturbative Hamiltonian with a finite order low energy effective Hamiltonian that is a gapped, frustration-free parent Hamiltonian for an encoded version of a desired PEPS. For topologically ordered PEPS, the ground space of the low energy effective Hamiltonian is shown to be in the same phase as the desired state to all orders of perturbation theory. We then move on to define models that generalize the phenomenology of several well-known systems. We first define generalized cluster states based on finite group algebras, and investigate properties of these states including their PEPS representations, global symmetries, relationship to the Kitaev quantum double models, and possible applications. Finally, we propose a generalization of the color codes based on finite groups. For non-Abelian groups, the resulting model supports non-Abelian anyonic quasiparticles and topological order. We examine the properties of these models such as their relationship to Kitaev quantum double models, quasiparticle spectrum, and boundary structure.
We show that the local von Neumann algebras on convex areas of the frustration-free ground state of abelian quantum double models are of type II∞II_\inftyII∞​.