Entanglement Renormalization and Topological Order

The multiscale entanglement renormalization 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 renormalization group flow associated with entanglement renormalization. All of these results generalize to arbitrary quantum double models.

Figure 1

RG transformation based on entanglement renormalization. To build an effective site from a 4-site block, disentanglers are first applied between sites of the block and surrounding sites, removing part of the short-ranged entanglement between the block and its surroundings. Then the four sites are coarse-grained into one by an isometry selecting the subspace ${\mathcal{K}}^{\prime }\subseteq {\mathcal{K}}^{\otimes 4}$ to be kept. The case of a tilted square lattice is shown (in the toric code, each site will contain 4 qubits.)

Figure 2

Elementary moves adding plaquettes and vertices to a toric code. Arrows stand for CNOTs.

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 disappear) after the RG transformation, producing a new 4-qubit site for lattice ${\mathcal{L}}_1$ from every block of 16 qubits (the 12 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 16-qubit domains overlapping with four blocks each (thick dashed line, cf. Fig. 1.) 4 qubits per block decouple as $|0⟩$.

Figure 4

(a)–(c) Second step of the RG transformation: Isometries. (a) 2 qubits per block decouple as $|+⟩$. (b) 2 more qubits per block decouple as $|0⟩$. (c) One qubit per edge (4 per block), decouple as $|+⟩$. The isometry also traces out all 12 decoupled qubits. (d) State 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.