Geometric interpretation of the multi-scale entanglement renormalization ansatz

TL;DR

This paper provides a geometric interpretation of MERA as a light sheet or light cone, clarifying its relation to spacetime geometries and proposing generalizations to hyperbolic and de Sitter geometries.

Contribution

It rigorously interprets MERA as a specific 2D geometry and introduces Euclidean and Lorentzian generalizations corresponding to different spacetime geometries.

Findings

MERA on the real line is a light sheet geometry.

MERA on a finite circle is a light cone geometry.

Proposes Euclidean and Lorentzian MERA generalizations for hyperbolic and de Sitter geometries.

Abstract

The multi-scale entanglement renormalization ansatz (MERA) is a tensor network representation for ground states of critical quantum spin chains, with a network that extends in an additional dimension corresponding to scale. Over the years several authors have conjectured, both in the context of holography and cosmology, that MERA realizes a discrete version of some geometry. However, while one proposal argued that the tensor network should be interpreted as representing the hyperbolic plane, another proposal instead equated MERA to de Sitter spacetime. In this \letter we show, using the framework of path integral geometry [A. Milsted, G. Vidal, arXiv:1807.02501], that MERA on the real line (and finite circle) can be given a rigorous interpretation as a two-dimensional geometry, namely a light sheet (respectively, a light cone). Accordingly, MERA describes neither the hyperbolic plane nor de Sitter spacetime. However, we also propose euclidean and lorentzian generalizations of MERA that correspond to a path integral on these two geometries.