Non-local scaling operators with entanglement renormalization

TL;DR

This paper extends the multi-scale entanglement renormalization ansatz (MERA) to identify non-local scaling operators associated with global symmetries, enriching the understanding of quantum critical points and conformal field theories.

Contribution

It introduces a method to determine non-local scaling operators within the scale invariant MERA framework, including those related to global symmetries like $ ext{Z}_2$, and applies it to the quantum Ising model.

Findings

Identifies non-local operators such as disorder and fermionic operators in the Ising model.

Shows that MERA can characterize all conformal towers of the Ising CFT.

Provides a complete list of primary fields using the extended MERA approach.

Abstract

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 $\phi$ attached to its open end. In the case of the quantum Ising model, $\mathcal{G}= \mathbb{Z}_2$, they correspond to the disorder operator $\mu$, the fermionic operators $\psi$ and $\bar{\psi}$, and all their descendants. Together with the local scaling operators identity $\mathbb{I}$, spin $\sigma$ and energy $\epsilon$, the fermionic and disorder scaling operators $\psi$, $\bar{\psi}$ and $\mu$ 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.