Mapping Kitaev's quantum double lattice models to Levin and Wen's string-net models

TL;DR

This work explicitly embeds Kitaev's quantum double models into Levin–Wen's string-net framework by enlarging local Hilbert spaces with auxiliary degrees of freedom and performing a Fourier basis transformation on edge states. The authors show that, on reduced states, vertex constraints induce fusion channels consistent with a string-net description, and plaquette operators map to a weighted sum over string-net plaquette actions, establishing an exact correspondence between the two formalisms. The main contributions include a concrete mapping that identifies the excitations with irreps of the quantum double D(G) and demonstrates Morita equivalence between QD and SN; the approach also extends to arbitrary planar lattices and provides a representation-theoretic interpretation of the excitations. This unifies two central topological lattice-model constructions, clarifies the structure of their anyonic excitations, and suggests avenues for generalizations to broader algebraic frameworks and tensor-network formulations.

Abstract

We exhibit a mapping identifying Kitaev's quantum double lattice models explicitly as a subclass of Levin and Wen's string net models via a completion of the local Hilbert spaces with auxiliary degrees of freedom. This identification allows to carry over to these string net models the representation-theoretic classification of the excitations in quantum double models, as well as define them in arbitrary lattices, and provides an illustration of the abstract notion of Morita equivalence. The possibility of generalising the map to broader classes of string nets is considered.

Figures (4)

• Figure 3: Convention for the definition (\ref{['stringnets:vertexsn']}) of operators $A_v^{ \mathrm{SN} }$ in string-net models. The operator is symmetric in the different edges.
• Figure 4: Convention for the definition (\ref{['stringnets:plaquettesnfull']}) of operators $B_p^{ \mathrm{SN} }$ in string-net models. The definition of $B_p^{ \mathrm{SN} }$ does not depend on the choice of $v_0$ or orientation of the loop.
• Figure 5: Graphical representation of the projector $A_v^{ \text{QD} }$, or more precisely its matrix element $\langle \{ \mu_j, \, a_j, \, c_j \} \rvert A_v^{ \text{QD} } \lvert \{ \mu_i, \, a_i, \, b_i \} \rangle$. Note that only the $b$ indices change (into the $c$'s), according to the corresponding $3j$ symbol $W_{ \{ c_i \}, \, \{ b_i \} }^{ \{ \mu_i \} }$; this is represented by the cylinder. The trivial propagation of the other indices is represented by the rectangular faces.
• Figure 6: Graphical representation of the projector $B_p^{ \text{QD} } ( \nu )$, or more precisely its matrix element $\langle \{ \mu'_j, a'_j, b'_j \} \rvert B_p^\text{QD} ( \nu ) \lvert \{ \mu_j, a_j, b_j \} \rangle$ as given in eq. (\ref{['qdouble:pprojector_d_full']}). The string-net picture of a loop associated with irrep $\nu$ is represented by a transversal face (hexagon delimited by bold lines) that interacts with the propagation (rectangular faces) of the plaquette via $3j$-symbols (bold lines).