Explicit tensor network representation for the ground states of string-net models

We provide a simple expression for the ground states of arbitrary string-net models in the form of local tensor networks. These tensor networks encode the data of the fusion category underlying a string-net model and thus represent all doubled topological phases of matter in the infrared limit according to Levin and Wen [Phys. Rev. B 71, 045110 (2005)]. Furthermore, our construction highlights the importance of the fat lattice equivalence between lattice and continuum descriptions of string-net models.

Figure 1

(Color online) Reducing the fat lattice. In (a) an isolated loop at face $p$ carries a label ${\alpha }_p$ and all edges $E\left(\Lambda \right)$ (gray) of the physical lattice are labeled by the vacuum 1. The overall quantum state is denoted by $|\left\{{\alpha }_p\right\}⟩$. (b) Three rounds of $F$ moves reduce to the fat lattice configuration which carries both the final physical labels $i_p$, $j_p$, and $k_p$ as well as the labels ${\alpha }_p$ at the vertex loops. This is denoted by $|\left\{{\alpha }_p,i_p,j_p,k_p\right\}⟩$. In (c) each vertex has been fully reduced to the physical lattice by a sequence of another two $F$ moves each. The corresponding physical quantum state is given by $|\left\{i_p,j_p,k_p\right\}⟩$.

Figure 2

(Color online) Hexagonal lattice $\Lambda$ and its dual ${\Lambda }^{\ast }$. Horizontal edges (blue, solid line) of ${\Lambda }^{\ast }$ belong to $E_1$, diagonal edges (red, dashed line) to $E_2$, and diagonal edges (green, dotted line) to $E_3$. The directed edges of $\Lambda$ are labeled by uniquely associating them to a face. Circled vertices of $\Lambda$ belong to the even sublattice ${\Lambda }_1$, while filled ones belong to the odd sublattice ${\Lambda }_2$.

Figure 3

(Color online) Tensor network describing the ground state of an arbitrary string-net model. Only virtual bonds (black, outer lines) are summed over while physical indices (red, inner lines) are left uncontracted.

Figure 4

(Color online) Alternative tensor network for ground states of arbitrary string-net models. Here all bonds are contracted.