The complete set of infinite volume ground states for Kitaev's abelian quantum double models

Abstract

We study the set of infinite volume ground states of Kitaev's quantum double model on ${\mathbb{Z}}^2$ for an arbitrary finite abelian group $G$. It is known that these models have a unique frustration-free ground state. Here we drop the requirement of frustration freeness, and classify the full set of ground states. We show that the ground state space decomposes into $|G{|}^2$ different charged sectors, corresponding to the different types of abelian anyons (also known as superselection sectors). In particular, all pure ground states are equivalent to ground states that can be interpreted as describing a single excitation. Our proof proceeds by showing that each ground state can be obtained as the weak${}^{\ast }$ limit of finite volume ground states of the quantum double model with suitable boundary terms. The boundary terms allow for states which represent a pair of excitations, with one excitation in the bulk and one pinned to the boundary, to be included in the ground state space.

arXiv comment27 pages, 6 figures. v2: minor corrections, some simplifications and clarificactions added

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• Size-Driven Quantum Phase Transitions
• A derivation of braided $C^{\ast }$-tensor categories from gapped ground states satisfying the approximate Haag duality
• On the stability of charges in infinite quantum spin systems
• Local disorder, topological ground state degeneracy and entanglement entropy, and discrete anyons
• Quasi-Locality Bounds for Quantum Lattice Systems. Part II. Perturbations of Frustration-Free Spin Models with Gapped Ground States
• A dynamical Toric Code model with fusion and de-fusion
• Dynamical abelian anyons with bound states and scattering states
• Subfactors and quantum information theory
• Local topological order and boundary algebras
• A ${\mathbb{Z}}_2$-Topological Index for Quasi-Free Fermions
• Classification of the anyon sectors of Kitaev's quantum double model
• Topologically ordered states in infinite quantum spin systems
• Finite-group gauge theories on lattices as Hamiltonian systems with constraints
• Multidimensional Contours à la Fröhlich-Spencer and Boundary Conditions for Quantum Spin Systems