Quantum double model

The input data for Kitaev quantum double is a finite group ${\displaystyle G}$. Consider a directed lattice ${\displaystyle \Sigma }$, we put a Hilbert space ${\displaystyle \mathbb {C} [G]}$ spanned by group elements on each edge, there are four types of edge operators

${\displaystyle L_{+}^{g}|h\rangle =|gh\rangle ,L_{-}^{g}|h\rangle =|hg^{-1}\rangle ,}$

${\displaystyle T_{+}^{g}|h\rangle =\delta _{g,h}|h\rangle ,T_{-}^{g}|h\rangle =\delta _{g^{-1},h}|h\rangle .}$

For each vertex connecting to ${\displaystyle m}$ edges ${\displaystyle e_{1},\ldots ,e_{m}}$, there is a vertex operator

${\displaystyle A_{v}={\frac {1}{|G|}}\sum _{g\in G}L^{g}(e_{1})\otimes \ldots \otimes L^{g}(e_{m}).}$

Notice each edge has an orientation: when ${\displaystyle v}$ is the starting point of ${\displaystyle e_{k}}$, the operator is set as ${\displaystyle L_{-}}$, otherwise, it is set as ${\displaystyle L_{+}}$.

For each face surrounded by ${\displaystyle m}$ edges ${\displaystyle e_{1},\ldots ,e_{m}}$, there is a face operator

${\displaystyle B_{f}=\sum _{h_{1}\cdots h_{m}=1_{G}}\prod _{k=1}^{m}T^{h_{k}}(e_{k}).}$

Similar to the vertex operator, due to the orientation of the edge, when face ${\displaystyle f}$ is on the right-hand side when traversing the positive direction of ${\displaystyle e}$, we set ${\displaystyle T_{+}}$; otherwise, we set ${\displaystyle T_{-}}$ in the above expression. Also, note that the order of edges surrounding the face is assumed to be counterclockwise.

The lattice Hamiltonian of quantum double model is given by

${\displaystyle H=-\sum _{v}A_{v}-\sum _{f}B_{f}.}$

Both of ${\displaystyle A_{v}}$ and ${\displaystyle B_{f}}$ are Hermitian projectors, they are stabilizer when regard the model is a quantum error correcting code.

The topological excitations of the model is characterized by the representations of the quantum double of finite group ${\displaystyle G}$. The anyon types are given by irreducible representations. For the lattice model, the topological excitations are created by ribbon operators.^[1][3]

The gapped boundary theory of quantum double model can be constructed based on subgroups of ${\displaystyle G}$.^[4][5][6] There is a boundary-bulk duality for this model.

The topological excitation of the model is equivalent to that of the Levin-Wen string-net model with input given by the representation category of finite group ${\displaystyle G}$.