Nonlocal scaling operators with entanglement renormalization

Abstract

The multiscale 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 nonlocal scaling operators. Each operator consists, for a given group element $g∊\mathcal{G}$, of a semi-infinite string ${\ensuremath{\Gamma}}_{g}^{\ensuremath{\vartriangleleft}}$ with a local operator $\ensuremath{\varphi}$ attached to its open end. In the case of the quantum Ising model, $\mathcal{G}={\mathbb{Z}}_{2}$, they correspond to the disorder operator $\ensuremath{\mu}$, the fermionic operators $\ensuremath{\psi}$ and $\overline{\ensuremath{\psi}}$, and all their descendants. Together with the local scaling operators identity $\mathbb{I}$, spin $\ensuremath{\sigma}$, and energy $ϵ$, the fermionic and disorder scaling operators $\ensuremath{\psi}$, $\overline{\ensuremath{\psi}}$, and $\ensuremath{\mu}$ are the complete list of primary fields of the Ising CFT. Therefore the scale invariant MERA allows us to characterize all the conformal towers of this CFT.