Twisted Kitaev quantum double model as local topological order

TL;DR

The paper advances the understanding of twisted quantum double phases by embedding the Twisted Kitaev Quantum Double model into the Local Topological Order framework on arbitrary 2D lattices. By mapping general lattices to a canonical triangular lattice and exploiting monomial representations, it provides an explicit ground-state basis and shows the ground-state space forms a quantum error-correcting code. It reformulates LTO for arbitrary lattices and proves that the twisted model satisfies all four LTO axioms for both smooth and rough boundaries, highlighting boundary nets and operator algebras that realize the bulk-boundary correspondence. The results extend known LTO validations (previously for toric code and untwisted models) to the twisted setting, enabling robust topological encoding with a broad lattice flexibility and concrete cohomological data via $\alpha \in Z^3(G,U(1))$.

Abstract

We study the Twisted Kitaev Quantum Double model within the framework of Local Topological Order (LTO). We extend its definition to arbitrary 2D lattices, enabling an explicit characterization of the ground state space through the invariant spaces of monomial representations. We reformulate the LTO conditions to include general lattices and prove that the twisted model satisfies all four LTO axioms on any 2D lattice. As a corollary, we show that its ground state space is a quantum error-correcting code.

Figures (17)

• Figure 1: A portion of a triangular lattice (or $\Delta$-complex), where each edge is oriented according to the ordering of the vertices.
• Figure 2: Ordering of the vertices in a lattice on the torus, consistent with the identification of edges.
• Figure 3: The action of operator $B_f$.
• Figure 4: Example for the calculation of $\alpha_{v_3}^g$.
• Figure 5: Action of operator $A_{v_3}^g$, over a coloring $\varphi$.

Theorems & Definitions (50)

• Theorem 1.1
• Definition 2.1: Face Operator
• Definition 2.2
• Definition 2.3: Vertex Operator
• Example 2.4
• Example 3.1
• Proposition 3.2
• proof
• Proposition 3.3
• proof