Entanglement renormalization and topological order

TL;DR

The case of Kitaev's toric code is analyzed in detail and shown to possess a remarkably simple MERA description leading to distillation of the topological degrees of freedom at the top of the tensor network.

Abstract

The multi-scale entanglement renormalisation ansatz (MERA) is argued to provide a natural description for topological states of matter. The case of Kitaev's toric code is analyzed in detail and shown to possess a remarkably simple MERA description leading to distillation of the topological degrees of freedom at the top of the tensor network. Kitaev states on an infinite lattice are also shown to be a fixed point of the RG flow associated with entanglement renormalization. All these results generalize to arbitrary quantum double models.

Figures (7)

• Figure 1: RG transformation based on entanglement renormalization. In order to build an effective site from a block of four sites, we first apply disentanglers between sites of the block and surrounding sites. In this way part of the short-ranged entanglement between the block and its surroundings is removed. Then we coarse-grain the four sites into one by means of an isometry that selects the subspace ${\cal K'} \subseteq {\cal K}^{\otimes 4}$ to be kept. We show the case of a tilted square lattice in preparation for the toric code where, in addition, each site will contain four qubits.
• Figure 2: Elementary moves adding plaquettes and vertices to a toric code. Arrows stand for CNOT operations.
• Figure 3: (a) The square lattice $\Lambda$ for the toric code, with qubits (dots) on the links, is reorganized into a tilted square lattice ${\mathcal{L}}_0$ where each site is made of four qubits. The lattice constant is doubled (dotted lines dissapear) after the RG transformation, which produces a new four-qubit site for lattice ${\mathcal{L}}_1$ from every block of sixteen qubits (the twelve light qubits in the block are decoupled in known product states). (b) First step of the RG transformation: Disentanglers. Arrows stand for simultaneous CNOT operators from control to target qubits. Disentanglers act on sixteen-qubit domains overlapping with four blocks each (thick dashed line, cf. figure \ref{['figure:2DMERA']}.) Four qubits per block decouple in state $\lvert 0 \rangle$.
• Figure 4: (a)--(c) Second step of the RG transformation: Isometries. (a) Two qubits per block decouple in state $\lvert + \rangle$. (b) Two more qubits per block decouple in state $\lvert 0 \rangle$. (c) One qubit per edge, four per block, decouple in state $\lvert + \rangle$. The isometry also traces out the twelve decoupled qubits. (d) State of the system after the RG transformation.
• Figure 5: Local mapping between the toric code on a square lattice (a) and on a triangular lattice (b). The dual model in a honeycomb lattice (displayed for reference) is Levin and Wen's loop model.