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Non-local scaling operators with entanglement renormalization

The multi-scale entanglement renormalization ansatz (MERA) can be used, in its scale invariant version, to describe the ground state of a lattice system at a quantum critical point. From the scale invariant MERA one can determine the local scaling operators of the model. Here we show that, in the presence of a global symmetry $\mathcal{G}$, it is also possible to determine a class of non-local scaling operators. Each operator consist, for a given group element $g\in\mathcal{G}$, of a semi-infinite string $\tGamma_g$ with a local operator $ϕ$ attached to its open end. In the case of the quantum Ising model, $\mathcal{G}= \mathbb{Z}_2$, they correspond to the disorder operator $μ$, the fermionic operators $ψ$ and $\barψ$, and all their descendants. Together with the local scaling operators identity $\mathbb{I}$, spin $σ$ and energy $ε$, the fermionic and disorder scaling operators $ψ$, $\barψ$ and $μ$ are the complete list of primary fields of the Ising CFT. Thefore the scale invariant MERA allows us to characterize all the conformal towers of this CFT.