Exact entanglement renormalization for string-net models

Abstract

We construct an explicit renormalization group (RG) 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 multi-scale 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.

Figures (3)

• Figure 1: An $F$-move reconnecting an edge $e$ of $\mathcal{G}$. Plaquettes of $\mathcal{G}$ and of $\mathcal{G}'$ 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 $\mathcal{G}\backslash\{e_1,e_2\}:\{e_1,e_2\}$ (\ref{['t:tadpoleremoval']}). 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.

Theorems & Definitions (9)

• Lemma 1
• Lemma 2
• Example 1
• Example 2
• Remark
• proof : Proof of \ref{['t:Fcommute']}
• Lemma 3
• proof
• proof : Proof of \ref{['t:tadpoleremoval']}