Kitaev's Quantum Double Models Research Articles

Boundary algebras of the Kitaev quantum double model

The recent article by Jones et al. [arXiv:2307.12552 (2023)] gave local topological order (LTO) axioms for a quantum spin system, showed they held in Kitaev’s Toric Code and in Levin-Wen string net models, and gave a bulk boundary correspondence to describe bulk excitations in terms of the boundary net of algebras. In this article, we prove the LTO axioms for Kitaev’s Quantum Double model for a finite group G. We identify the boundary nets of algebras with fusion categorical nets associated to (Hilb(G),C[G]) or (Rep(G),CG) depending on whether the boundary cut is rough or smooth, respectively. This allows us to make connections to the work of Ogata [Ann. Henri Poincaré 25, 2353–2387 (2024)] on the type of the cone von Neumann algebras in the algebraic quantum field theory approach to topological superselection sectors. We show that the boundary algebras can also be calculated from a trivial G-symmetry protected topological phase (G-SPT), and that the gauging map preserves the boundary algebras. Finally, we compute the boundary algebras for the (3 + 1)D Quantum Double model associated to an Abelian group.

Algebraic aspects of boundaries in the Kitaev quantum double model

We provide a systematic treatment of boundaries based on subgroups K ⊆ G for the Kitaev quantum double D(G) model in the bulk. The boundary sites are representations of a *-subalgebra Ξ ⊆ D(G) and we explicate its structure as a quasi-Hopf *-algebra dependent on a choice of transversal R. We provide decomposition formulae for irreducible representations of D(G) pulled back to Ξ. As an application of our treatment, we study patches with boundaries based on K = G horizontally and K = {e} vertically and show how these could be used in a quantum computer using the technique of lattice surgery. More abstractly, we also provide explicitly the monoidal equivalence of the category of Ξ-modules and the category of G-graded K-bimodules and use this to prove that different choices of R are related by Drinfeld cochain twists. Examples include Sn−1 ⊂ Sn and an example related to the octonions where Ξ is also a Hopf quasigroup.

Quantum double models coupled to matter fields: A detailed review for a dualization procedure

In this paper, we investigate how it is possible to define a new class of lattice gauge models based on a dualization procedure of a previous generalization of the Kitaev Quantum Double Models. In the case of this previous generalization that will be used as a basis, it was defined by adding new qudits (which were denoted as matter fields in reference to some works) to the lattice vertices with the intention of, for instance, interpreting its models as Kitaev Quantum Double Models coupled with Potts ones. Now, with regard to the generalization that we investigate here, which we want to define as the dual of this previous one, these new qudits were added to the lattice faces. And as the coupling between gauge and matter qudits of the previous generalization was performed by a gauge group action, we show that the dual behavior of these two generalizations was achieved by coupling these same qudits in the second one through a gauge group co-action homomorphism. One of the most striking dual aspects of these two generalizations is that, in both, part of the quasiparticles that were inherited from the Kitaev Quantum Double Models become confined when these action and co-action are nontrivial. But the big news here is that, in addition to the group homomorphism (that defines this gauge group co-action) allows us to classify all the different models of this second generalization, this same group homomorphism also suggests that all these models can be interpreted as two-dimensional restrictions of the 2-lattice gauge theories.

Finite-group gauge theories on lattices as Hamiltonian systems with constraints

In this work, we present a brief but insightful overview of the gauge theories, which are defined on n-dimensional lattices by using finite gauge groups, in order to show how they can be interpreted as a Hamiltonian system with constraints, analogous to what happens with the classical (continuous) gauge (field) theories. As this interpretation is not usually explored in the literature that discusses/introduces the concept of lattice gauge theory, but some recent works have been exploring Hamiltonian models in order to support some kind of quantum computation, we use this interpretation to, for example, present a brief geometric view of one class of these models: the Kitaev Quantum Double Models.

Ribbon operators in the generalized Kitaev quantum double model based on Hopf algebras

Kitaev’s quantum double model is a family of exactly solvable lattice models that realize two dimensional topological phases of matter. The model was originally based on finite groups, and was later generalized to semi-simple Hopf algebras. We rigorously define and study ribbon operators in the generalized quantum double model. These ribbon operators are important tools to understand quasi-particle excitations. It turns out that there are some subtleties in defining the operators in contrast to what one would naively think of. In particular, one has to distinguish two classes of ribbons which we call locally clockwise and locally counterclockwise ribbons. Moreover, we point out that the issue already exists in the original model based on finite non-abelian groups, but it seems to not have been noticed in the literature. We show how certain common properties would fail even in the original model if we were not to distinguish these two classes of ribbons. Perhaps not surprisingly, under the new definitions ribbon operators satisfy all properties that are expected. For instance, they create quasi-particle excitations only at the end of the ribbon, and the types of the quasi-particles correspond to irreducible representations of the Drinfeld double of the input Hopf algebra. However, the proofs of these properties are much more complicated than those in the case of finite groups. This is partly due to the complications in dealing with general Hopf algebras rather than group algebras.

Kitaev's quantum double model as an error correcting code

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.

On the Stability of Charges in Infinite Quantum Spin Systems

We consider a theory of superselection sectors for infinite quantum spin systems, describing charges that can be approximately localized in cone-like regions. The primary examples we have in mind are the anyons (or charges) in topologically ordered models such as Kitaev's quantum double models and perturbations of such models. In order to cover the case of perturbed quantum double models, the Doplicher-Haag-Roberts approach, in which strict localization is assumed, has to be amended. To this end we consider endomorphisms of the observable algebra that are almost localized in cones. Under natural conditions on the reference ground state (which plays a role analogous to the vacuum state in relativistic theories), we obtain a braided tensor $C^*$-category describing the sectors. We also introduce a superselection criterion selecting excitations with energy below a threshold. When the threshold energy falls in a gap of the spectrum of the ground state, we prove stability of the entire superselection structure under perturbations that do not close the gap. We apply our results to prove that all essential properties of the anyons in Kitaev's abelian quantum double models are stable against perturbations.

Higher lattices, discrete two-dimensional holonomy and topological phases in (3 + 1)D with higher gauge symmetry

Higher gauge theory is a higher order version of gauge theory that makes possible the definition of 2-dimensional holonomy along surfaces embedded in a manifold where a gauge 2-connection is present. In this paper, we study Hamiltonian models for discrete higher gauge theory on a lattice decomposition of a manifold. We show that a construction for higher lattice gauge theory is well-defined, including in particular a Hamiltonian for topological phases of matter in [Formula: see text] dimensions. Our construction builds upon the Kitaev quantum double model, replacing the finite gauge connection with a finite gauge 2-group 2-connection. Our Hamiltonian higher lattice gauge theory model is defined on spatial manifolds of arbitrary dimension presented by slightly combinatorialized CW-decompositions (2-lattice decompositions), whose 1-cells and 2-cells carry discrete 1-dimensional and 2-dimensional holonomy data. We prove that the ground-state degeneracy of Hamiltonian higher lattice gauge theory is a topological invariant of manifolds, coinciding with the number of homotopy classes of maps from the manifold to the classifying space of the underlying gauge 2-group. The operators of our Hamiltonian model are closely related to discrete 2-dimensional holonomy operators for discretized 2-connections on manifolds with a 2-lattice decomposition. We therefore address the definition of discrete 2-dimensional holonomy for surfaces embedded in 2-lattices. Several results concerning the well-definedness of discrete 2-dimensional holonomy, and its construction in a combinatorial and algebraic topological setting are presented.

Anyons are not energy eigenspaces of quantum double Hamiltonians

Kitaev's quantum double models, including the toric code, are canonical examples of quantum topological models on a 2D spin lattice. Their Hamiltonian defines the groundspace by imposing an energy penalty to any nontrivial flux or charge, but does not distinguish among those. We generalize this construction by introducing a novel family of Hamiltonians made of commuting four-body projectors that provide an intricate splitting of the Hilbert space by discriminating among non-trivial charges and fluxes. Our construction highlights that anyons are not in one-to-one correspondence with energy eigenspaces, a feature already present in Kitaev's construction. This discrepancy is due to the presence of local degrees of freedom in addition to topological ones on a lattice.

The Complete Set of Infinite Volume Ground States for Kitaev’s Abelian Quantum Double Models

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$^*$ 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.

Hamiltonian and Algebraic Theories of Gapped Boundaries in Topological Phases of Matter

We present an exactly solvable lattice Hamiltonian to realize gapped boundaries of Kitaev's quantum double models for Dijkgraaf-Witten theories. We classify the elementary excitations on the boundary, and systematically describe the bulk-to-boundary condensation procedure. We also present the parallel algebraic/categorical structure of gapped boundaries.

Generalized color codes supporting non-Abelian anyons

We propose a generalization of the color codes based on finite groups $G$. 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.

Generalized cluster states based on finite groups

We define generalized cluster states based on finite group algebras in analogy to the generalization of the toric code to the Kitaev quantum double models. We do this by showing a general correspondence between systems with CSS structure and finite group algebras, and applying this to the cluster states to derive their generalization. We then investigate properties of these states including their projected entangled pair state representations, global symmetries, and relationship to the Kitaev quantum double models. We also discuss possible applications of these states.

Anyon condensation and tensor categories

Instead of studying anyon condensation in various concrete models, we take a bootstrap approach by considering an abstract situation, in which an anyon condensation happens in a 2-d topological phase with anyonic excitations given by a modular tensor category C; and the anyons in the condensed phase are given by another modular tensor category D. By a bootstrap analysis, we derive a relation between anyons in D-phase and anyons in C-phase from natural physical requirements. It turns out that the vacuum (or the tensor unit) A in D-phase is necessary to be a connected commutative separable algebra in C, and the category D is equivalent to the category of local A-modules as modular tensor categories. This condensation also produces a gapped domain wall with wall excitations given by the category of A-modules in C. A more general situation is also studied in this paper. We will also show how to determine such algebra A from the initial and final data. Multi-condensations and 1-d condensations will also be briefly discussed. Examples will be given in the toric code model, Kitaev quantum double models, Levin–Wen types of lattice models and some chiral topological phases.

Entropic Barriers for Two-Dimensional Quantum Memories

Comprehensive no-go theorems show that information encoded over local two-dimensional topologically ordered systems cannot support macroscopic energy barriers, and hence will not maintain stable quantum information at finite temperatures for macroscopic time scales. However, it is still well motivated to study low-dimensional quantum memories due to their experimental amenability. Here we introduce a grid of defect lines to Kitaev's quantum double model where different anyonic excitations carry different masses. This setting produces a complex energy landscape which entropically suppresses the diffusion of excitations that cause logical errors. We show numerically that entropically suppressed errors give rise to superexponential inverse temperature scaling and polynomial system size scaling for small system sizes over a low-temperature regime. Curiously, these entropic effects are not present below a certain low temperature. We show that we can vary the system to modify this bound and potentially extend the described effects to zero temperature.

Twisted quantum double model of topological phases in two dimensions

We propose a new discrete model---the twisted quantum double model---of 2D topological phases based on a finite group $G$ and a 3-cocycle $\alpha$ over $G$. The detailed properties of the ground states are studied, and we find that the ground--state subspace can be characterized in terms of the twisted quantum double $D^{\alpha}(G)$ of $G$. When $\alpha$ is the trivial 3-cocycle, the model becomes Kitaev's quantum double model based on the finite group $G$, in which the elementary excitations are known to be classified by the quantum double $D(G)$ of $G$. Our model can be viewed as a Hamiltonian extension of the Dijkgraaf--Witten topological gauge theories to the discrete graph case with gauge group being a finite group. We also demonstrate a duality between a large class of Levin-Wen string-net models and certain twisted quantum double models, by mapping the string--net 6j symbols to the corresponding 3-cocycles. The paper is presented in a way such that it is accessible to a wide range of physicists.

A hierarchy of topological tensor network states

We present a hierarchy of quantum many-body states among which many examples of topological order can be identified by construction. We define these states in terms of a general, basis-independent framework of tensor networks based on the algebraic setting of finite-dimensional Hopf C*-algebras. At the top of the hierarchy we identify ground states of new topological lattice models extending Kitaev's quantum double models [Ann. Phys. 303, 2 (2003)10.1016/S0003-4916(02)00018-0]. For these states we exhibit the mechanism responsible for their non-zero topological entanglement entropy by constructing an entanglement renormalization flow. Furthermore, we argue that the hierarchy states are related to each other by the condensation of topological charges.

Toric codes and quantum doubles from two-body Hamiltonians

We present here a procedure to obtain the Hamiltonians of the toric code and Kitaev quantum double models as the low-energy limits of entirely two-body Hamiltonians. Our construction makes use of a new type of perturbation gadget based on error-detecting subsystem codes. The procedure is motivated by a projected entangled pair states (PEPS) description of the target models, and reproduces the target models' behavior using only couplings that are natural in terms of the original Hamiltonians. This allows our construction to capture the symmetries of the target models.