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

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 one 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 generalizing the map to broader classes of string nets is considered.

Figure 1

(Color online) Convention for the definition, Eq. (2), of operators $A_v^{\text{QD}}$ in quantum double models. A trivalent vertex is shown but the generalization to arbitrary vertices is straightforward. The operator is symmetric in the different edges.

Figure 2

(Color online) Convention for the definition, Eq. (3), of operators $B_p^{\text{QD}}$ in quantum double models. A hexagonal plaquette is shown but the generalization to arbitrary plaquettes is straightforward. The definition of $B_p^{\text{QD}}$ does not depend on the choice of $v_0$ or orientation of the loop.

Figure 3

(Color online) Convention for the definition, Eq. (5), of operators $A_v^{\text{SN}}$ in string-net models. The operator is symmetric in the different edges.

Figure 4

(Color online) Convention for the definition, Eq. (7), of operators $B_p^{\text{SN}}$ in string-net models. The definition of $B_p^{\text{SN}}$ does not depend on the choice of $v_0$ or orientation of the loop.

Figure 5

(Color online) Graphical representation of the projector $A_v^{\text{QD}}$ or more precisely its matrix element $⟨\left\{{\mu }_j,a_j,c_j\right\}|A_v^{\text{QD}}|\left\{{\mu }_i,a_i,b_i\right\}⟩$. Note that only the $b$ indices change (into the $c$’s), according to the corresponding $3j$ symbol $W_{\left\{c_i\right\},\left\{b_i\right\}}^{\left\{{\mu }_i\right\}}$; this is represented by the cylinder. The trivial propagation of the other indices is represented by the rectangular faces.

Figure 6

(Color online) Graphical representation of the projector $B_p^{\text{QD}}\left(\nu \right)$ or more precisely its matrix element $⟨\left\{{\mu }_j^{\prime },a_j^{\prime },b_j^{\prime }\right\}|B_p^{\text{QD}}\left(\nu \right)|\left\{{\mu }_j,a_j,b_j\right\}⟩$ as given in Eq. (17). 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).