Real-space renormalization yields finite correlations

TL;DR

This paper demonstrates that, except in one dimension, MERA states derived from real-space renormalization are finitely correlated PEPS with system-size independent bond dimension, obeying the area law for entanglement entropy.

Contribution

It establishes that MERA states in higher dimensions are finitely correlated PEPS, revealing their local structure and efficient contractibility, and discusses schemes violating the area law.

Findings

MERA states are finitely correlated PEPS in dimensions higher than one.

MERA states obey the area law for entanglement entropy.

Existence of efficiently contractible schemes that violate the area law.

Abstract

Real-space renormalization approaches for quantum lattice systems generate certain hierarchical classes of states that are subsumed by the multi-scale entanglement renormalization ansatz (MERA). It is shown that, with the exception of one spatial dimension, MERA states are actually finitely correlated states, i.e., projected entangled pair states (PEPS) with a bond dimension independent of the system size. Hence, real-space renormalization generates states which can be encoded with local effective degrees of freedom, and MERA states form an efficiently contractible class of PEPS that obey the area law for the entanglement entropy. It is shown further that there exist other efficiently contractible schemes violating the area law.