Abstract
Kitaev's quantum double models in 2D provide some of the most commonly studied examples of topological quantum order. In particular, the ground space is thought to yield a quantum error-correcting code. We offer an explicit proof that this is the case for arbitrary finite groups. Actually a stronger claim is shown: any two states with zero energy density in some contractible region must have the same reduced state in that region. Alternatively, the local properties of a gauge-invariant state are fully determined by specifying that its holonomies in the region are trivial. We contrast this result with the fact that local properties of gauge-invariant states are not generally determined by specifying all of their non-Abelian fluxes --- that is, the Wilson loops of lattice gauge theory do not form a complete commuting set of observables. We also note that the methods developed by P. Naaijkens (PhD thesis, 2012) under a different context can be adapted to provide another proof of the error correcting property of Kitaev's model. Finally, we compute the topological entanglement entropy in Kitaev's model, and show, contrary to previous claims in the literature, that it does not depend on whether the ``log dim R'' term is included in the definition of entanglement entropy.
Highlights
Topological phases of matter in two spatial dimensions are gapped quantum liquids that exhibit exotic properties such as stable ground state degeneracy, stable long-range entanglement, existence of quasi-particle excitations, non-Abelian exchange statistics, etc. These phases are characterized by a new type of order, topological quantum order (TQO), that is beyond the conventional Landau theory of spontaneous symmetry breaking and local order parameters
TQO-1 states that the ground state space is a quantum error correcting code with a macroscopic distance, and TQO-2 means that the local ground state space coincides with the global one
Contrary to intuitions one might have from typical gauge theory models, Wilson loop operators do not form a complete set of commuting observables
Summary
Topological phases of matter in two spatial dimensions are gapped quantum liquids that exhibit exotic properties such as stable ground state degeneracy, stable long-range entanglement, existence of quasi-particle excitations, (possibly) non-Abelian exchange statistics, etc. See Theorem 3.1 for a formal statement Another motivation for the current work is to construct quantum error correcting codes in lattice models. We have shown in this paper that all non-Abelian Kitaev’s models are quantum error correcting codes with macroscopic distance. The Levin-Wen models and the generalized Kitaev models are essentially equivalent It is an interesting question whether or not our current proof for the case of finite groups can be adapted to the case of Hopf algebras and/or to weak Hopf algebras. There are well-defined notions of local gauge transformations and holonomy which allow us to obtain an explicit characterization of the ground states, though this is not necessary for the proof of our main result.
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
