Geometric interpretation of the multi-scale entanglement renormalization ansatz

TL;DR

<3-5 sentence high-level summary>This work resolves the geometric interpretation of MERA by showing that MERA on the real line and on the circle implements a path integral for a two-dimensional CFT on a light sheet L2, rather than the hyperbolic plane H2 or de Sitter dS2. It then introduces euclidean and lorentzian generalizations of MERA (euclidean MERA and lorentzian MERA) that correspond to path integrals on H2^p and dS2^p, enabling discrete realizations of these geometries within tensor networks. The authors validate the line MERA picture by embedding MERA in AdS3^p, performing low-energy identifications between lattice spin chains and CFT states, and showing that the MERA layer W acts as the identity on low-energy subspaces, while W interleaved with euclideon/lorentzion layers yields the desired geometry-specific maps. These results provide a concrete, computationally accessible framework for studying CFT path integrals and quantum field theories in curved spacetimes using tensor networks. They also lay groundwork for extended holographic toy models and potential numerical simulations of QFTs in curved backgrounds.>

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.

Figures (22)

• Figure 1: (Left) Three candidate geometries for MERA on the real line, as embedded in the Poincare patch of AdS$_3$ with metric $dl^2(t,z,r)=(-dt^2+dz^2+dr^2)/(z/L)^2$, namely the hyperbolic plane H$_2^p$, light sheet L$_2^p$, and Poincare de Sitter dS$_2^p$ in Eqs. \ref{['eq:H2p']}-\ref{['eq:L2p']}Supplemental. (Right) Three candidate geometries for MERA on the circle, as embedded in Minkowski $\mathbb{R}^{1,2}$, namely the hyperbolic disk H$_2$, light cone L$_2$, and de Sitter dS$_2$, see Eq. \ref{['eq:embedding1']}-\ref{['eq:embedding3']}.
• Figure 2: (a) The MERA is made of disentanglers $u$ and isometries $w$ organized in infinite layers $\mathcal{W}$. (b) A finite, periodic layer $\mathcal{W}$ defines a liner map from the Hilbert space of a periodic spin chain with $N$ spins to that of a smaller periodic spin chain with $N/2$ spins.
• Figure 3: (a) The euclidean MERA is a tensor network made of layers $\mathcal{W}$ of optimized MERA interspersed with transfer matrices $\mathcal{T}$ made of euclideons $e$ and implementing $e^{-H}$. (b) The lorentzian MERA is made of layers $\mathcal{W}$ of optimized MERA insterspersed with trnasfer matrices $\mathcal{T}'$ made of lorentzions $l$ and implementing $e^{-iH}$.
• Figure 4: Graphical representation of several MERA tensor networks on the circle (only two concentric layers of each different MERA are displayed). Top: null MERA $\mathcal{M}$, with layers $\mathcal{W}$ made of disentanglers and isometries. Bottom left: euclidean MERA $\mathcal{M}_{+}$, with layers $\mathcal{W}_{+} = \mathcal{W}\mathcal{T}^{q}$ (for $q=1$), where $\mathcal{T}$ is an euclidean transfer matrix that implements euclidean time evolution. Bottom right: lorentzian MERA $\mathcal{M}_{-}$, with layers $\mathcal{W}_{-} = \mathcal{W}\mathcal{T}_{-}^{q}$ (for $q=1$), where $\mathcal{T}_{-}$ is a lorentzian transfer matrix that implements real time evolution.
• Figure 5: The three geometries H$_2$, dS$_2$, and L$_2$ of interest are shown embedded in three-dimensional Minkowski spacetime $\mathbb{R}^{1,2}$ with time coordinate $X_0$ and space coordinates $X_1$ and $X_2$, according to the restrictions \ref{['eqap:embedding1']}-\ref{['eqap:embedding3']}. Notice that both H$_2$ and dS$_2$ depend on a radius $R$, and that L$_2$ can be understood as the limit $R \rightarrow 0$ of either of these two geometries, see Fig. \ref{['fig:limit']}.