Constructing holographic spacetimes using entanglement renormalization

TL;DR

The paper develops a framework tying entanglement renormalization (MERA) to holographic duality in the large-N limit, showing how a classical bulk geometry and area-law entanglement emerge from strongly coupled tensor networks. It analyzes operator dimensions, entropy, mutual information, and correlations within large-N MERA, and draws parallels with holography, including the Ryu–Takayanagi entropy formula and concepts of bulk locality. It also extends these ideas to continuous MERA (cMERA), explores Fermi-surface and black-hole analogues, and discusses pathways to covariant holography and deeper connections with quantum gravity. The work provides a structured bridge between quantum information approaches to many-body physics and gravitational holography, offering insights into how spacetime geometry and holographic duality may arise from entanglement patterns in strongly interacting quantum systems.

Abstract

We elaborate on our earlier proposal connecting entanglement renormalization and holographic duality in which we argued that a tensor network can be reinterpreted as a kind of skeleton for an emergent holographic space. Here we address the question of the large $N$ limit where on the holographic side the gravity theory becomes classical and a non-fluctuating smooth spacetime description emerges. We show how a number of features of holographic duality in the large $N$ limit emerge naturally from entanglement renormalization, including a classical spacetime generated by entanglement, a sparse spectrum of operator dimensions, and phase transitions in mutual information. We also address questions related to bulk locality below the AdS radius, holographic duals of weakly coupled large $N$ theories, Fermi surfaces in holography, and the holographic interpretation of branching MERA. Some of our considerations are inspired by the idea of quantum expanders which are generalized quantum transformations that add a definite amount of entropy to most states. Since we identify entanglement with geometry, we thus argue that classical spacetime may be built from quantum expanders (or something like them).

Figures (12)

• Figure 1: Basic setup of entanglement entropy calculations. The system is divided into two components, here called $A$ and $B$. The smaller of the two is $A$ which has linear size $R$. The entanglement entropy $S(A)$ is typically proportional to $|\partial A|$ which in $d=2$ dimensions would be $R$. This scaling of entanglement with boundary size is called the area law.
• Figure 2: Schematic structure of entanglement renormalization with $d=1$ and $k=2$. The blue squares represent unitaries, called disentanglers, that remove local entanglement. The red triangles represent isometries, called coarse-grainers, that thin out unentangled degrees of freedom.
• Figure 3: An operator (green box) is coarse-grained using one layer of the MERA. Most of the isometries and unitaries of the layer will give $1$ in this process (substitution in grey shaded region) because of identities like $U^\dagger U = 1$.
• Figure 4: The causal cone is the set of all unitaries, isometries, and sites of the network that can affect the state of a region in the UV lattice. The effective number of sites in a region, and hence the width of the causal cone, shrinks exponentially fast we descend in the network. This behavior persists until the causal cone width is of order a few lattice spacings where it will fluctuate depending on the details of the scheme.
• Figure 5: Entropy bounds in entanglement renormalization (schematic). The entropy of a region in the UV lattice (grey boxed in region) is bounded by the number of network bonds that must be cut to isolate it. The red curve is the corresponding minimal curve which is pierced by the minimal number of bonds. The length of this curve, suitably defined, or equivalently, the number of bonds cut bounds the entropy.