A Family of Non-Abelian Kitaev Models on a Lattice: Topological Confinement and Condensation

TL;DR

This work introduces a two-parameter family of non-Abelian lattice Hamiltonians H_G^{N,M} that extend Kitaev's model by adding single-qudit edge terms, effectively reducing the discrete gauge symmetry to G' = M/N. The authors develop a comprehensive ribbon-operator framework to classify ground states, excitations, domain walls, and charges, and show how condensation and confinement arise in this setting. In the large-μ limit, the low-energy sector maps to a Kitaev-like model for the reduced group M/N, enabling exact ground-state characterizations and controlled descriptions of topological phenomena. The paper also distinguishes two structural regimes (Case I and Case II) with distinct confinement and flux properties, offering a versatile platform to study symmetry breaking, topological order, and potential applications to topological quantum computation.

Abstract

We study a family of non-Abelian topological models in a lattice that arise by modifying the Kitaev model through the introduction of single-qudit terms. The effect of these terms amounts to a reduction of the discrete gauge symmetry with respect to the original systems, which corresponds to a generalized mechanism of explicit symmetry breaking. The topological order is either partially lost or completely destroyed throughout the various models. The new systems display condensation and confinement of the topological charges present in the standard non-Abelian Kitaev models, which we study in terms of ribbon operator algebras.

Figures (15)

• Figure 1: The two dimensional lattice that we consider are arbitrary in shape and have oriented edges. Thick lines display the support of a face operator (left) and a vertex operator (right).
• Figure 2: Thick lines correspond to the lattice and thin lines to the dual lattice. Arrows show the orientation of edges and dual edges. Note that dual edges are oriented in agreement with edges (see explanation in main text). The shaded area is a ribbon. All the sites that form the ribbon are displayed as dashed lines, thicker in the case of the two sites in the ends. The arrowed thick white line shows the orientation of the ribbon.
• Figure 3: Each figure represents a triangle $\tau$ (shaded area) that connects two sites (dashed lines): $\partial_0 \tau$ to the left and $\partial_1 \tau$ to the right. Thick lines correspond to the lattice and thin lines to the dual lattice. Arrows show the orientation of edges and dual edges. (a) A direct triangle with an edge which matches its direction. (b) A direct triangle with an edge which does not match its direction. (c) A dual triangle with a dual edge which matches its direction. (d) A dual triangle with a dual edge which does not match its direction.
• Figure 4: An example of a deformation of a ribbon. The endpoints are fixed, and the area in between the two ribbons does not contain any excited site, which are represented with dotted lines.
• Figure 5: Three examples of closed ribbons. $\sigma$ is a proper closed ribbon, containing both dual and direct triangles. It is also a boundary ribbon, as it encloses an area with the topology of a disc. $\alpha$ is a dual closed ribbon and thus encloses a single vertex. $\beta$ is a direct closed ribbon and thus encloses a single face.

Theorems & Definitions (19)

• Definition 1
• Proposition 2
• Definition 4
• Lemma 5
• Definition 6
• Proposition 7
• Definition 8
• Proposition 9
• Definition 10
• Proposition 11