Entanglement renormalization and topological order

L. P. Kadanoff, Physics 2

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X.-G. Wen, Quantum Field Theory of Many-Body Systems , Oxford University Press (2004)

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In a D D dimensional lattice the ground state typically obeys a boundary law S l ∼ l D − 1 S_{l}\sim l^{D-1} for the entanglement entropy of a block of l D l^{D} sites. In this case the dimension d τ d_{\tau} for a site of ℒ τ {\mathcal{L}}_{\tau} must at least scale doubly exponentially in τ \tau , d τ ∼ exp ⁡ ( exp ⁡ ( τ ) ) d_{\tau}\sim\exp(\exp(\tau)) . Indeed, one the one hand the dimension of an effective space for a block of l D l^{D} sites must be at least d ∼ exp ⁡ ( S l ) = exp ⁡ ( l D − 1 ) d\sim\exp(S_{l})=\exp(l^{D-1}) . On the other, l τ ∼ exp ⁡ ( τ ) l_{\tau}\sim\exp(\tau) after τ \tau iterations of the RG transformation, where n τ = l τ D n_{\tau}=l_{\tau}^{D} is the number of sites of ℒ 0 {\mathcal{L}}_{0} effectively described by a single site of ℒ τ {\mathcal{L}}_{\tau}

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G. Evenbly, G. Vidal, arXiv:0710.0692v2 [quant-ph]

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G. Vidal, arXiv:quant-ph/0610099

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Notice that the qubits being removed from the lattice during the RG transformation are exactly in a product state. In all previous examples [ 5 , 6 ] it was only possible to approximately disentangle the subsystems before their removal

Cited in the paper.

This points to a more general result, to be discussed elsewhere: under the present RG transformation based on entanglement renormalization, systems with topological order flow toward the string-net models of Levin and Wen [ 9 ] , substantiating ideas already advocated by these authors

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Z. Nussinov, G. Ortiz, arXiv:cond-mat/0702377v5 [cond-mat.str-el]

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M. A. Levin, X.-G. Wen, Phys. Rev. B71

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G. Vidal, Phys. Rev. Lett. 99

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Closest in time.