Continuous entanglement renormalization on the circle

TL;DR

This paper extends the continuous multi-scale entanglement renormalization ansatz (cMERA) from infinite systems to finite circular geometries, enabling better modeling of quantum fields on a circle and preserving ground state approximations.

Contribution

The authors generalize cMERA to finite circles by wrapping the entangler, and prove that good approximations on the line remain good on the circle for Gaussian states.

Findings

cMERA can be adapted to finite circular geometries.

Ground state approximations are preserved under the circle transformation.

Method of images helps prove the approximation quality on the circle.

Abstract

The continuous multi-scale entanglement renormalization ansatz (cMERA) is a variational class of states for quantum fields. As originally formulated, the cMERA applies to infinite systems only. In this paper we generalize the cMERA formalism to a finite circle, which we achieve by wrapping the action of the so-called entangler around the circle. This allows us to transform a cMERA on the line into a cMERA on the circle. In addition, in the case of a Gaussian cMERA for non-interacting quantum fields, the method of images allow us to prove the following result: if on the line a cMERA state is a good approximation to a ground state of a local QFT Hamiltonian, then (under mild assumptions on their correlation functions) the resulting cMERA on a circle is also a good approximation to the ground state of the same QFT Hamiltonian on the circle.