Exact entanglement renormalization for string-net models

We construct an explicit renormalization-group transformation for Levin and Wen’s string-net models on a hexagonal lattice. The transformation leaves invariant the ground-state “fixed-point” wave function of the string-net condensed phase. Our construction also produces an exact representation of the wave function in terms of the multiscale entanglement renormalization ansatz (MERA). This sets the stage for efficient numerical simulations of string-net models using MERA algorithms. It also provides an explicit quantum circuit to prepare the string-net ground-state wave function using a quantum computer.

Figure 1

An $F$-move reconnecting an edge $e$ of $\mathcal{G}$. Plaquettes of $\mathcal{G}$ and of ${\mathcal{G}}^{\prime }$ are in one-to-one correspondence.

Figure 2

When $\mathcal{G}$ contains a tadpole around plaquette $p$ (attached to vertex $v$) and the state is in the range of $B_p$, it is a product state with respect to the bipartition $\mathcal{G}\backslash \left\{e_1,e_2\right\}:\left\{e_1,e_2\right\}$ (Lemma 2). In this diagram, the $e_i$ are names for the directed edges and not string-net labels.

Figure 3

The RG transformation $\mathbf{R}$ coarse-grains lattice $\mathcal{L}$ into $\tilde{\mathcal{L}}$. Edges where $F$-moves are applied are marked by dots. Note that there are many alternative sequences of moves that work equally well.