Generalized string-net model for unitary fusion categories without tetrahedral symmetry

The Levin-Wen model of string-net condensation explains how topological phases emerge from the microscopic degrees of freedom of a physical system. However, the original construction is not applicable to all unitary fusion categories since some additional symmetries for the $F$-symbols are imposed. In particular, the so-called tetrahedral symmetry is not fulfilled by many interesting unitary fusion categories. In this paper, we present a generalized construction of the Levin-Wen model for arbitrary multiplicity-free unitary fusion categories that works without requiring these additional symmetries. We explicitly calculate the matrix elements of the Hamiltonian and, furthermore, show that it has the same properties as the original one.

Figure 1

Hamiltonian on the honeycomb lattice. A general Hamiltonian consists of operators that act on vertices $Q_{\mathbf{v}}$ and ones that act on plaquettes $B_{\mathbf{p}}$.

Figure 2

Projector. (a) We apply the operators $B_{\mathbf{p}}$ twice to the same plaquette, which insert two loops, one of type $s$ and one of type $t$ (keep in mind that we sum over $s$ and $t$ which is not depicted in the picture). (b) These two loops are then fused together. (c) The result is (a sum over) a single loop of type $\alpha$.

Figure 3

Commutativity. (a) In this scenario, the operators $B_{{\mathbf{p}}_{\mathbf{1}}}$ and $B_{{\mathbf{p}}_{\mathbf{2}}}$ are applied to neighboring plaquettes. (b) The operator $B_{{\mathbf{p}}_{\mathbf{1}}}^s$ puts a loop of type $s$ into plaquette ${\mathbf{p}}_{\mathbf{1}}$, and $B_{{\mathbf{p}}_{\mathbf{2}}}^t$ puts a loop of type $t$ into ${\mathbf{p}}_{\mathbf{2}}$ (although not depicted in the picture, we take the sum over $s$ and $t$). (c) After fusing in both loops, the result is a linear combination of string-net configurations which is independent of the order in which the loops have been fused in.

Figure 4

Closed string operator on the honeycomb lattice. A closed string operator $W^s\left(P\right)$ only acts nontrivially on spin states along the closed path $P$, depicted as thick red line. It creates a type-$s$ string and fuses it into each vertex ${\mathbf{v}}_1,{\mathbf{v}}_2,...{\mathbf{v}}_N$ along $P$. This action only changes the two sites $i_{k-1},i_k$ of ${\mathbf{v}}_k$, whereas the third site $e_k$ of the vertex is unaffected. In total, $W^s\left(P\right)$ transforms the initial spin state $i_1,i_2,...,i_N$ to the final spin state $i_1^{\prime },i_2^{\prime },...,i_N^{\prime }$. The labeling convention is demonstratively shown in the picture for $i_1,i_2,e_1,e_2$ and ${\mathbf{v}}_6$ of an example for a $W^s\left(P\right)$ operator.

Figure 5

Commuting diagrams. The fact that these two diagrams commute yields the equations required to prove the commutativity of the magnetic flux operator.